A. Syrakos, S. Varchanis, Y. Dimakopoulos, A. Goulas, J. Tsamopoulos
Finite volume methods (FVMs) constitute a popular class of methods for the numerical simulation of fluid flows. Among the various components of these methods, the discretisation of the gradient operator has received less attention despite its fundamental importance with regards to the accuracy of the FVM. The most popular gradient schemes are the divergence theorem (DT) (or GreenGauss) scheme and the leastsquares (LS) scheme. Both are widely believed to be secondorder accurate, but the present study shows that in fact the common variant of the DT gradient is secondorder accurate only on structured meshes whereas it is zerothorder accurate on general unstructured meshes, and the LS gradient is secondorder and firstorder accurate, respectively. This is explained through a theoretical analysis and is confirmed by numerical tests. The schemes are then used within a FVM to solve a simple diffusion equation on unstructured grids generated by several methods; the results reveal that the zerothorder accuracy of the DT gradient is inherited by the FVM as a whole, and the discretisation error does not decrease with grid refinement. On the other hand, use of the LS gradient leads to secondorder accurate results, as does the use of alternative, consistent, DT gradient schemes, including a new iterative scheme that makes the common DT gradient consistent at almost no extra cost. The numerical tests are performed using both an inhouse code and the popular public domain partial differential equation solver OpenFOAM.
]]>S. Varchanis, Y. Dimakopoulos, J. Tsamopoulos
Film flow along an inclined, solid substrate featuring periodic rectangular trenches may either completely wet the trench floor (Wenzel state) orgetpinnedon the entrance and exit corners of the trench (Cassie state) or assume other configurationsin between these two extremes. Such intermediate configurations are examined in the present study. They are boundedby a second gasliquid interface inside the trench, which adheres to itswalls forming two threephase contact lines, and encloses a different amount of air under different physical conditions. The Galerkin finite element method is used to solve the NavierStokes equations in a physical domain, which is adaptively remeshed. Multiple steady solutions, connected by turning points and transcritical bifurcationsas well as isolated solution branches, are revealed by pseudo arclength continuation. Twopossible configurationsof a single air inclusion inside the trench are examined: the inclusion either surrounds the upstream convex corner or is attached to the upstream trench wall. The penetration of the liquid inside the trench is enhanced primarily by increasing either the wettability of the substrate orcapillaryover viscous forcesor by decreasing the flow rate. Flow hysteresis may occur when the liquid wetting of the upstream wall decreases abruptly leading to drastically different flow patternsfor the same parameter values. The interplay of inertia,viscous, gravity and capillary forces along with substrate wettability determines the volume of the air encapsulated in the trench and the extent of deformation of the outer free surface.
]]>E. Martino, G. Koilias, M. Athanasiou, A. Katsaounis, Y. Dimakopoulos, J. Tsamopoulos, C.G. Vayenas
The triode operation of humidified PEM fuel cells has been investigated both with pure H_{2} and with CO poisoned H_{2} feed over commercial Vulcan supported Pt(30%)Ru(15%) anodes. It was found that triode operation, which involves the use of a third, auxiliary, electrode, leads to up to 400% power output increase with the same CO poisoned H_{2} gas feed. At low current densities, the power increase is accompanied by an increase in overall thermodynamic efficiency. A mathematical model, based on Kirchhoff’s laws, has been developed which is in reasonably good agreement with the experimental results. In order to gain some additional insight into the mechanism of triode operation, the model has been also extended to describe the potential distribution inside the Nafion membrane via the numerical solution of the NernstPlanck equation. Both model and experiment have shown the critical role of minimizing the auxiliaryanode or auxiliarycathode resistance, and this has led to improved combshaped anode or cathode electrode geometries.
Nafion membrane, PEM fuel cell, Triode operation, CO poisoning, NernstPlanck equation
]]>D.Fraggedakis, J.Papaioannou, Y.Dimakopoulos, J.Tsamopoulos
A new boundaryfitted technique to describe free surface and moving boundary problems is presented. We have extended the 2D elliptic grid generator developed by Dimakopoulos and Tsamopoulos (2003) [19] and further advanced by Chatzidai et al. (2009) [18] to 3D geometries. The set of equations arises from the fulfillment of the variational principles established by Brackbill and Saltzman (1982) [21], and refined by Christodoulou and Scriven (1992) [22]. These account for both smoothness and orthogonality of the grid lines of tessellated physical domains. The ellipticgrid equations are accompanied by new boundary constraints and conditions which are based either on the equidistribution of the nodes on boundary surfaces or on the existing 2D quasielliptic grid methodologies. The capabilities of the proposed algorithm are first demonstrated in tests with analytically described complex surfaces. The sequence in which these tests are presented is chosen to help the reader build up experience on the best choice of the elliptic grid parameters. Subsequently, the mesh equations are coupled with the Navier–Stokes equations, in order to reveal the full potential of the proposed methodology in free surface flows. More specifically, the problem of gas assisted injection in ducts of circular and square crosssections is examined, where the fluid domain experiences extreme deformations. Finally, the flowmesh solver is used to calculate the equilibrium shapes of staticmenisci in capillary tubes.
Moving boundary problems, Mesh generation, Freesurface flows, Ellipticgrid generation, Moving contact line, Contact angle models
]]>A. Syrakos, S. Varchanis, Y. Dimakopoulos, A. Goulas, J. Tsamopoulos
The divergence theorem (or GreenGauss) gradient scheme is among the most popular methodsfor discretising the gradient operator in secondorder accurate finite volume methods, with a longhistory of successful application on structured grids. This together with the ease of applicationof the scheme on unstructured grids has led to its widespread use in unstructured finite volumemethods (FVMs). However, the present study shows both theoretically and through numericaltests that the common variant of this scheme is zerothorder accurate (it does not converge to theexact gradient) on grids of arbitrary skewness, such as typically produced by unstructured gridgeneration algorithms. Moreover, we use the scheme in the FVM solution of a diffusion (Poisson)equation problem, with both an inhouse code and the popular opensource solver OpenFOAM,and observe that the zerothorder accuracy of the gradient operator is inherited by the FVM solveras a whole. However, a simple iterative procedure that exploits the outer iterations of the FVMsolver is shown to effect firstorder accuracy to the gradient and secondorder accuracy to the FVMat almost no extra cost compared to the original scheme. Secondorder accurate results are alsoobtained if a leastsquares gradient operator is used instead.
Finite volume method; GreenGauss gradient; divergence theorem gradient; diffusionequation; OpenFOAM
]]>D.Pettas, G.Karapetsas, Y.Dimakopoulos, J.Tsamopoulos
Liquid film flow along an inclined plane featuring a slit, normal to the main directionof flow, creates a second gas–liquid interface connecting the two side walls of theslit. This inner interface forms two threephase contact lines and supports a widelyvarying amount of liquid under different physical and geometrical conditions. Theexact liquid configuration is determined by employing the Galerkin/finite elementmethod to solve the twodimensional Navier–Stokes equations at steady state. Theinterplay of inertia, viscous, gravity and capillary forces along with the substratewettability and orientation with respect to gravity and the width of the slit determinethe extent of liquid penetration and freesurface deformation. Finite wetting lengthsare predicted in hydrophilic and hydrophobic substrates for inclination angles moreor less than the vertical, respectively. Multiple steady solutions, connected by turningpoints forming a hysteresis loop, are revealed by pseudoarclength continuation. Underthese conditions, small changes in certain parameter values leads to an abrupt changein the wetting length and the deformation amplitude of the outer film surface. Inhydrophilic substrates the wetting lengths exhibit a local minimum for small valuesof the Reynolds number and a very small range of Bond numbers; when inertiaincreases, they exhibit the hysteresis loop with the second limit point in a very shortrange of Weber numbers. Simple force balances determine the proper rescaling ineach case, so that critical points in families of solutions for different liquids or contactangles collapse. The flow inside the slit is quite slow in general because of viscousdissipation and includes counterrotating vortices often resembling those reported byMoffatt (J. Fluid Mech., vol. 18, 1964, pp. 1–18). In hydrophobic substrates, thewetting lengths decrease monotonically until the first limit point of the hysteresisloop, which occurs in a limited range of Bond numbers when the Kapitza number isless than 300 and in a limited range of Weber numbers otherwise. Here additionalsolution families are possible as well, where one or both contact points (Cassie state)coincide with the slit corners.
coating, microfluidics, thin films
]]>Evan Mitsoulis, John Tsamopoulos
Viscoplasticity is characterized by a yield stress, below which the materials will not deform and above which they will deform and flow according to different constitutive relations. Viscoplastic models include the Bingham plastic, the HerschelBulkley model and the Casson model. All of these ideal models are discontinuous. Analytical solutions exist for such models in simple flows. For general flow fields, it is necessary to develop numerical techniques to track down yielded/unyielded regions. This can be avoided by introducing into the models a regularization parameter, which facilitates the solution process and produces virtually the same results as the ideal models by the right choice of its value. This work reviews several benchmark problems of viscoplastic flows, such as entry and exit flows from dies, flows around a sphere and a bubble and squeeze flows. Examples are also given for typical processing flows of viscoplastic materials, where the extent and shape of the yielded/unyielded regions are clearly shown. The abovementioned viscoplastic models leave undetermined the stress and elastic deformation in the solid region. Moreover, deviations have been reported between predictions with these models and experiments for flows around particles using Carbopol, one of the very often used and heretofore widely accepted as a simple “viscoplastic” fluid. These have been partially remedied in very recent studies using the elastoviscoplastic models proposed by Saramito.
Viscoplastic fluids, Bingham plastics, HerschelBulkley fluids, Viscoplastic models, Simulations, Yield stress, Yielded/unyielded regions, Elastoviscoplastic fluids
]]>G. Karapetsas, N. K. Lampropoulos, Y. Dimakopoulos, J. Tsamopoulos
We examine the transient film flow under the action of gravity over solid substrates with threedimensional topographical features. Our focus is placed on the coating of a periodic array of rectangular cuboid trenches. The Navier–Stokes equations are solved using the volumeoffluid method, fully taking into account the flow in both the liquid and gas phases. Using this scheme, we are able to determine the different wetting patterns that may arise depending on parameters such as the various geometrical characteristics of the trench, the lateral distance between them, the substrate wettability and the liquid viscosity. We present flow maps that describe the conditions under which the liquid film may successfully coat the patterned substrate, resulting in the socalled Wenzel state, or air may become entrapped inside the topography of the substrate. In the latter case, we describe in detail the position and shape of the air inclusions, how they are formed and the conditions under which coating can approach the ideal Cassie–Baxter state. We investigate in detail the effect of the sidewalls, typically ignored when considering the case of ideal 2D trenches (i.e., trenches extending to infinity in the lateral direction), through the enhancement of the viscous resistance inside the trench and the effect of capillarity in the case of narrow trenches. We also examine the coating behavior for a wide range of liquids and show that successful coating is favored for liquids with moderate viscosities. Finally, we perform simulations for the coating of two successive trenches in the flow direction and show that in the case of 3D trenches, the differences between the coating of the first and subsequent trenches are not significant.
Thinfilm flow, Flow over topography, Coating flows, Air entrapment, Cassie and Wenzel states
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ο Τμήμα Χημικών Μηχανικών του Αριστοτέλειου Πανεπιστημίου Θεσσαλονίκης έχει την τιμητική ευθύνη της διοργάνωσης του 11ου Πανελλήνιου Επιστημονικού Συνεδρίου Χημικής Μηχανικής (11 ΠΕΣΧΜ), το οποίο θα διεξαχθεί στη Θεσσαλονίκη, από τις 25 έως τις 27 Μαΐου 2017, στο Συνεδριακό Κέντρο του Μεγάρου Μουσικής Θεσσαλονίκης.
The 6th International Conference on Biomedical Engineering and Biotechnology (ICBEB 2017) will take place on 1720 October 2017 in Guangzhou, China
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23th Congress of the European Society of Biomechanics will take place on 25 July 2017 in Seville, Spain
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The current work investigates the effect of blood viscoelasticity on the stress, velocity and haematocrit fields as well as the formation of the celldepletionlayer for haemodynamics in microvessels lined with a glycocalyx layer. To this end, we advance the inhomogeneous constitutive model proposed by Moyers GonzalezOwensFang (2015), treating properly the shear induced RBC migration mechanism and incorporating the effect of plasma viscoelasticity. The enhanced model (enmodel) can accurately reproduce the experimental data by DamianoLongSmith (2004) and provide reasonable predictions for the RBC distribution and the shearstress along the luminal surface.
It is shown that blood viscoelasticity expressed in terms of the Deborah number is quite small (O(0.3)), but plays a crucial role in the formation of the celldepletionlayer. In particular, there is a critical value of Deborah number above which the formation of the celldepletionlayer occurs.
]]>This force is attractive, if the forcing frequency lies outside the range of eigenfrequencies for volume oscillation of the two bubbles. Here we study the nonlinear interaction of two deformable bubbles set in oscillation in water by a step change in the ambient pressure, by solving the Navier–Stokes equations numerically.
As in typical experiments, the bubble radii are in the range 1–1000 μm. We find that the smaller bubbles (῀5 μm) deform only slightly, especially when they are close to each other initially. Increasing the bubble size decreases the capillary force and increases bubble acceleration towards each other, leading to oblate or spherical cap or even globally deformed shapes.
These deformations may develop primarily in the rear side of the bubbles because of a combination of their translation and harmonic or subharmonic resonance between the breathing mode and the surface harmonics. Bubble deformation is also promoted when they are further apart or when the disturbance amplitude decreases.
The attractive force depends on the Ohnesorge number and the ambient pressure to capillary forces ratio, linearly on the radius of each bubble and inversely on the square of their separation. Additional damping either because of liquid compressibility or heat transfer in the bubble is also examined.
]]>The Laboratory of Fluid Mechanics and Rheology (The Fluids Lab) primary research areas are Bubble Dynamics, Multiphase / Biological Flows, Wetting Flows, Coating Process, Complex Flows & Interfacial Instabilities, Pressure Sensitive Adhesives.
Phones
Director: (+30) 2610997203
Lab: (+30) 2610969559
Fax: (+30) 2610996178
Email: tsamo [at] chemeng.upatras.gr
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Our Lab is equipped with 5 computer clusters with a total of 35 nodes and a total computational power of 214 cores. The total RAM of these clusters is 1.8 TB and the total hard disk capacity 16 TB. It also includes 11 single processor workstations having in total 49 cores, 620 GB RAM and 7 TB total hard disk capacity. 4 of these workstations are equipped with CUDAenabled graphic adapters.
Furthermore the Lab is equipped with a Network Accessible Storage (NAS) with a total capacity of 20TB used as backup server.
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Advanced grid generation algorithms have been implemented for the simulation of freesurface and multiphase flows that typically arise in chemical engineering processes.
Our research is focused on both structured and unstructured grid generation techniques that can optimally adjust the location of nodes according to the flow field.
Sets of elliptic or hyperbolic differential equations are solved for the description of mesh motion along curves, on surfaces, and in volumes.
Flow of highly viscous liquid (in red) through a low viscosity, lubricating liquid (in blue) in a constricted and periodic tube with increasing flow rates (from top to bottom).
]]>For diverse applications such as polymer devolatilization, fermentation, composites processing, plastic foam processing, gas absorption and others it is of interest to know the rise velocity (hence the residence time) and shape of bubbles in polymeric solutions. It is known that single bubbles rising in quiescent viscoelastic liquids may exhibit a sudden (discontinuous) 310 fold increase in their rise velocity, once their volume exceeds a critical value. The abrupt change of the bubble rise velocity is accompanied by a change in the bubble shape from a convex to a cusped or “teardrop” shape. (see below).
Predictions of the effect of fluid elasticity on the shape of a deformable air bubble in very good agreement with experiments.
The motion of air bubbles in complex fluids has drawn our attention because of its numerous applications. Characteristic examples of such applications include: a) bubble removal from structural materials (i.e. cement) through vibration, b) entrapment of air bubbles in many foods to improve their flavor and texture (i.e. chocolate, ketchup), c) prevention of large bubble formation in drilling mud, which may cause dangerous explosions, inhibit production and potentially inflict huge burden on the ecosystem and the finances of oildrilling companies.
]]>Topleft view: 163 Cavitating bubbles from the experiment by the French collaborating group. Bottomleft view: Cavitating bubbles from our simulations. Rightview: Detail of the cavitating bubbles
Cross sections of the mid plane of a cylindrical sample of PSA that contains an arbitrary number of deformable cavities, for different nominal elongational strains (ε) .
]]>During fabrication of microelectronic components, polymer solutions are used to produce thin films over surfaces with uneven topography. The patterning of the substrates may require using multilayer coatings, where one of the purposes of the bottom layer is to planarize the topography, i.e. to smooth out the uneven topography. A nonuniform coating leads to low quality products or to manufacturing failures.
The problem of liquid advancing over orthogonal 2D and 3D substrates with variable depth, width and length under the influence of the gravitational force resulting in different physical phenomena is addressed.
Firstly, we can achieve continuous coating by which a thin film of fluid is developed throughout the trench while a steady flow of fluid is established upstream and downstream the topography which is welcome by coating of microelectronic devices.
Secondly, we may result in poor coating, namely the entrainment of air between the trenches of the substrate and the coating liquid inducing less fluid friction, therefore desirable in producing superhydrophobic surfaces for microfluidic applications.
The software tool used is the application interFoam comprised in the OpenFOAM open source CFD software package for simulation of twophase flow of two immiscible fluids, namely air and a viscous liquid. All calculations were implemented on the laboratory HPC clusters using Message Passing Interface (MPI).
Snapshots of the the covering process (red color: liquid, blue color: water).
Sharkskin instability in an extruded polymer melt.
Spatial form of the eigenvector for a cylindrical die.
]]>APIVITA SA (Project Coordinator)
University of Patras (Scientific Coordinator)
University of Athens
Anna Lotan Ltd
Technion
Project Meetings Project Results
Mud is one of the most often used natural materials for preventive, healing and cosmetic reasons. Although mud has been used since the antiquity, little light has been shed on its physical, chemical and biological properties. Besides the lack of information concerning its chemical composition, mud demonstrates significant physicochemical stability issues. The main problem is the phase separation between aqueous and solid phase, which makes its use difficult a short while after storage. This fact limits the application of mud in the areas nearby its origin and its incorporation in high added value formulations. Furthermore, the biological activity of mud, although well accepted, has never been scientifically proven. The main goals of this coordinated research effort are:
a) Characterization in terms of chemical composition of widely used muds from different areas of Greece and Israel,
b) Stabilization of these muds over time and simultaneously theoretical examination of some of the open fundamental questions related to particle(s) motion in a viscoelastoplastic fluid,
c) Proof and promotion of their biological properties, and
d) Formulation of selected muds into face masks and scaling up the process.
In this way, mud will improve its potential to reach a wide range of people in need of its beneficial properties through its incorporation into cosmetic product forms.
Dead Sea mud from Israel is a betterknown and established material. On the contrary Greek mud, although it has been used since antiquity and is currently being used for mud baths at SPAs in different areas of Greece, it has remained unexplored scientifically and technologically. The unique advantage of Greek muds is their variable chemical composition, which is responsible for a variety of beneficiary activities. Therefore it is a challenge to investigate the composition of mud from different areas of Greece. The mutual interest of both companies in the project lies on the physicochemical instability of mud independent from its origin, a fact that makes its formulation and further exploitation rather difficult. The collaboration with the Chemical Engineering Department in the University of Patras is of utmost importance due to the unknown structure, physicochemical complexity and stability problems of the material, while the collaboration with the Medical school of the University of Athens is equally important because of the need to characterize its most relevant biological properties. Although muds from Israel and Greece are expected to differ in composition, the transfer of knowledge between all partners (Universities and Industry from Greece and Israel) is of crucial importance for the successful completion of the project.
APIVITA is a leading cosmetic company in Greece and demonstrates very significant sales worldwide in that area. As Greek mud is almost completely unexploited, it is certain that developing face masks with specific properties and incorporating mud as active ingredient will improve the competitiveness of APIVITA SA in the field. APIVITA face masks exhibited in 2012 an annual turnover of approximately 3 million euros, a fact that justifies the cost effectiveness of the investment even for the sustainability, if not the growth, of the company in the field. The market share of APIVITA combined with the fact of the rapid expansion of APIVITA in hotels & SPAs as well as the opening of new APIVITA standalone stores around the world strengthens the need for investment in new mud applications. It is certain that upon successful completion of the project, i.e. in early 2016, APIVITA will have at least 2 new face masks in the global market containing Greek mud.
Project Meetings Project Results
Contours of the velocity magnitude on the surface of an evolving air enclosure and on the solid substrate, via a slip condition.
Contours of the magnitude of the rate of strain tensor on the same surfaces.
]]>Two Dimensional FibreReinforced Polymer are characterized by a laminated structure in which the fibres are only aligned along the plane in xdirection and ydirection of the material. This means that no fibres are aligned in the through thickness or the zdirection, this lack of alignment in the through thickness can create a disadvantage in cost and processing. Costs and labour increase because conventional processing techniques used to fabricate composites, such as wet hand layup, autoclave and resin transfer moulding, require a high amount of skilled labor to cut, stack and consolidate into a preformed component.
When the raw material contains reinforcing fibres, a compression molded part qualifies as a fibrereinforced plastic. More typically the plastic preform used in compression molding does not contain reinforcing fibres. In compression molding, a “preform” or “charge”, of SMC, BMC is placed into mould cavity. The mould is closed and the material is formed & cured inside by pressure and heat. Compression moulding offers excellent detailing for geometric shapes ranging from pattern and relief detailing to complex curves and creative forms, to precision engineering all within a maximum curing time of 20 minutes.
The current effort deals with the numerical study of a polymer melt reinforced by a network of two dimensional fibers under compression. An advanced Lagrangian Finite Element Algorithm has been developed which assumes the Fibers as a stiff NeoHookean material and the melty matrix as a multimode viscoelastic fluid.
]]>Extrusion damper configuration
Viscous dampers dissipate mechanical energy into heat through the action of viscous stresses in a fluid. The damper response to external stimulus (force or displacement) depends on the properties of the enclosed fluid. NonNewtonian fluids have widely varying properties, enabling a range of possibilities for the damper response. We use numerical models to simulate the operation of dampers containing nonNewtonian fluids, such as viscoplastic fluids.
Flow field snapshot: Pressure contours, streamlines, and unyielded areas (white).
Rate of viscous dissipation of energy.
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Project Overview Project Results
Photos from the visit of Greekside Project Coordinator, Dr. Konstantinos Gardikis (APIVITA SA) to the premises of the Israeli partner Anna Lotan Ltd. at Ceseare on 2930 November 2015.
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Project Overview Project Results
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Project Overview Project Meetings
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]]>Professor John Tsamopoulos has been named a Fellow of the American Physical Society “For outstanding contributions, via insightful computations and analyses, to the fundamental understanding of flows of twophase materials and viscoplastic fluids” as stated in the official assessment. The nomination ceremony will be held on November 2016 during the 69th Annual Meeting of the APS Division of Fluid Dynamics and will be included in APS News magazine (December edition).
The American Physical Society (APS) is a nonprofit membership organization working to advance and diffuse the knowledge of physics through its outstanding research journals, scientific meetings, and education, outreach, advocacy, and international activities. APS represents over 51,000 members, including physicists in academia, national laboratories, and industry in the United States and throughout the world.
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The award is given biennially and recognises the contribution of members of the viscoplastic fluids community.
]]>Email: pel [at] uth.gr Professor, Dep. of Mechanical Engineering, Univ. of Thessaly, Greece Chen, M.F. Senior Engineer, TAIWAN Costas Aggelidis
Email costas.aggelidis [at] kavalaoil.gr Kavala oil, Greece Economou, K. Engineer, ENIA Co., Greece. Housiadas, K. Professor, Dep. of Mathematics, University of the Aegean, Greece. Kouris, Ch. Senior Engineer, Heracles Group of Companies, Greece. Smyrnaios, D. Engineer, ENIA Co., Greece. Chantzidai Nikoleta
Email: nchatzidai [at] gmail.com University of Pireaus, Greece. Klidis, G. Secondary Education, Greece. Foteinopoulou, K. Post Doctoral Fellow, Institute for Optoelectronics and Microsystems (ISOM), Universidad Politécnica de Madrid, Spain. Karakosta, N. Engineer, Unilever Co., Greece. Katsipou, I. Public sector, Greece. Tseropoulos George
Email: tgeorge88 [at] hotmail.com New York University, USA. Chatzikalymniou Alexandra Toronto University, Canada. Stylianos Theodoros Kondylis
Email: theodoros.stylianos.kondylis [at] gmail.com Forschungszentrum Jülich GmbH, Germany. Arindam Sen Email: sen [at] stce.rwthaachen.de RWTH Aachen University, Germany Iasonas HitzazisEmail: hitzazis [at] master.math.upatras.gr University of Patras, Greece Kelesides George
Email: gkelesidhs [at] gmail.com
ETH, Zurich, Switzerland.
Dr. Nguyen Phuc Khan
Email: nguyen1phuc1khanh [at] gmail.com
Managing Partner at Leonardo Group in Vietnam
Gkoutzioupa Victoria
Email: vgkoutzi [at] yahoo.com National Technical University of Athens (NTUA), Greece. Michalaki Eleftheria
Email: riamichalaki [at] gmail.com Stanford University, USA. Koilias George
Email: koiliasg21 [at] gmail.com University of Patras, Greece Delidakis George
Email: gdelidakis [at] gmail.com UT Austin, USA Kokolis Spyros
Email: kokolis.spy [at] hotmail.com Cranfield University, UK M.Eng. Tsouka Sophie
Email: sophiatsouka [at] hotmail.com EPFL, Switzerland Ph.D. Papaioannou John
Email: jpapaioan1982 [at] gmail.com University of Patras, Greece M.Eng. Vasilopoulos Yannis
Email: jean.vasil [at] gmail.com Private Sector, Greece Dr. Lampropoulos Nikolaos
Email: nikolaoslampropoulos [at] hotmail.com TEI of Athens, Greece Fraggedakis Dimitris
Email: dimfraged [at] gmail.com MIT, USA Tsolas Spyros
Email: cmng1993 [at] gmail.com
Texas A&NM, USA
Dr. Pavlidis Michael
Public sector, Greece
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Email: theamazingchemeng [at] outlook.com
]]>Place: Main Seminar Room, 1st Floor, Main Building of Chemical Engineering Department
Date: July 6, 2016 , Time: 12:00
Committee: J. Tsamopoulos (supervisor), V. Mavrantzas, Y. Dimakopoulos
Dr. Syrakos Alexandros
Email: alexandros.syrakos [at] mail.com 

Dr. Karapetsas George
Email: gkarapetsas [at] gmail.com 

Dr. Photeinos Dionysios
Email: photeinos [at] gmail.com 
M.Eng. Pettas Dionisis
Email: dspedi [at] gmail.com 

B.Eng. Varchanis Stelios
Email: varchanis [at] hotmail.com 

B.Sc. Giannokostas Konstantinos
Email: giannoko [at] hotmail.com 

B.Eng. Makrygiorgos George
Email: g.makrys [at] gmail.com 
Email: dimako [at] chemeng.upatras.gr
Download Business card (as PDF)
Diploma in Chemical Engineering, University of Patras (1997)
Master in the Simulation, Optimization and Control of Processes, University of Patras (2003)
Doctoral Diploma in Chemical Engineering, University of Patras (2003)
Dimakopoulos, Y., Karapetsas, G., Malamataris, N.A., Mitsoulis, E., “The Free (open) Boundary Condition at Inflow Boundaries”, J. NonNewtonian Fluid Mech., 187188, 1631 (2012) DOI: 10.1016/j.jnnfm.2012.09.001
Dimakopoulos, Y., Bogaerds, A., Anderson, P., Hulsen, M., Baaijens, F.P.T., “Direct numerical simulation of a 2D idealized aortic heart valve at physiological flow rates”, Comp. Meth. Biomech. and Biomed. Engrg, 15 (11), 11571179 (2012) DOI: 10.1080/10255842.2011.581238
Dimakopoulos, Y., “An efficient parallel fully implicit algorithm for the simulation of transient free surface flows of multimode viscoelastic liquids”, J. NonNewtonian Fluid Mech., 165 (78), 409424 (2010) DOI: 10.1016/j.jnnfm.2010.01.017
Dimakopoulos, Y., and Tsamopoulos, J., “A quasielliptic transformation for moving boundary problems with large anisotropic deformations”, J. Comp. Physics, 192, 494522 (2003) DOI: 10.1016/j.jcp.2003.07.027
Tseropoulos, G., Dimakopoulos, Y., Tsamopoulos, J., and Lyberatos, G., “On the flow characteristics of the conical Minoan pipes used in water supply systems, via Computational Fluid Dynamics”, J. Archaeological Sc. 40 (4), 20572068 (2013) DOI: 10.1016/j.jas.2012.11.025
Pavlidis, M., Dimakopoulos, Y., and Tsamopoulos, J., “Analytical and numerical solutions for the flow of an undeformed viscoelastic film down a vertical cylinder”, Rheologica Acta, 48 (9), 10311048(2009)
Dimakopoulos, Y., Bogaerds, A., Anderson, P., and Baaijens, F., “Application of the ficticious domain/Lagrange multiplier method on the simulation of the aortic valve”, GRACM 2011, JuneJuly 2011, Athens, GREECE.
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Email: tsamo [at] chemeng.upatras.gr
Download Business card (as PDF)
Diploma, Chemical Engineering, National Technical University of Athens, 1979.
M.S. Chemical Engineering, Mass. Inst. of Technology, 1981.
Ph.D.,Chemical Engineering, Mass. Inst. of Technology, 1985.
Tsamopoulos, J., Dimakopoulos, Y Chatzidai N., Karapetsas, G. and Pavlidis M., “Steady bubble rise and deformation in Newtonian and viscoplastic fluids and conditions for bubble entrapment” J. Fluid Mech., 601, 123–164 (2008), DOI: 10.1017/S0022112008000517
Chatzidai, N. Giannousakis, A. Dimakopoulos, Y. and Tsamopoulos, J. “On the elliptic mesh generation in domains containing multiple inclusions and undergoing large deformations”, J. Comp. Phys. 228 1980–2011 (2009), DOI: 10.1017/S002211200800051710.1016/j.jcp.2008.11.020
Papaioannou, J., Karapetsas, G., Dimakopoulos Y. and Tsamopoulos, J. “Injection of a viscoplastic material inside a tube or between parallel disks: conditions for wall detachment of the advancing front J. Rheol. 53(5), 11551191 (2009), DOI: 10.1122/1.3191779
Pavlidis, M., Dimakopoulos, Y. and Tsamopoulos, J. ‘Steady viscoelastic film flow over 2D topography: I. The effect of viscoelastic properties under creeping flow”, J. Non Newt. Fluid Mech., 165, 576591 (2010), DOI: 10.1016/j.jnnfm.2010.02.017
Chatzidai, A. Dimakopoulos, Y. and Tsamopoulos, J., “Viscous effects on two interacting and deformable bubbles under a step change in pressure” J. Fluid Mech., 673, 513547 (2011), DOI: 10.1017/S0022112010006361
Dimakopoulos, Y., Pavlidis, M. and Tsamopoulos, J. “Steady bubble rise in HerschelBulkley fluids and comparison of predictions via the Augmented Lagrangian Method with those via the Papanastasiou model” J. Non Newt. Fluid Mech., 200, 3451 (2013), DOI: 10.1016/j.jnnfm.2012.10.012
Karapetsas, G. and Tsamopoulos, J. “On the stickslip flow from slit and cylindrical dies of a PhanTien and Tanner fluid model: II. Linear stability analysis to two and threedimensional disturbances”, Phys. Fluids, 25, 093105 (2013), DOI: 10.1063/1.4821805
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The Annual European Rheology Conference is coorganized with the 26th Nordic Rheology Conference on 3 – 6 April 2017, in Copenhagen, Denmark.
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Place: Main Seminar Room, 1st Floor, Main Building of Chemical Engineering Department
Date: February 25, 2016 , Time: 12:00
Committee: J. Tsamopoulos (supervisor), Ko Van der Weele, Y. Dimakopoulos
Global Engage’s 2nd Microfluidics Congress will be held on 20 – 21 October 2016 in London UK.
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The American Physical Society – Division of Fluid Dynamics (APSDFD) Meeting will be held in Portland, Oregon 20 – 22 November, 2016. The Meeting is hosted by Portland State University, University of Washington, and Oregon State University.
The Meeting will be held at the Oregon Convention Center. The scientific program will include award lectures, invited talks and minisymposia, many parallel sessions with contributed papers, poster presentations, a poster competition, and the Gallery of Fluid Motion.
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Place: Main Seminar Room, 1st Floor, Main Building of Chemical Engineering Department
Date: December 21, 2015 , Time: 11:00
Committee: J. Tsamopoulos (supervisor), J. Ekaterinaris, Y. Dimakopoulos, V. Mavratzas, S. Pandis, Ch. Paraskeva, N.Pelekasis
Place: Main Seminar Room, 1st Floor, Main Building of Chemical Engineering Department
Date: June 8, 2015 , Time: 9:00
Committee: J. Tsamopoulos (supervisor), V. Mavrantzas, Y. Dimakopoulos
Place: Main Seminar Room, 1st Floor, Main Building of Chemical Engineering Department
Date: June 4, 2015 , Time: 17:00
Committee: J. Tsamopoulos (supervisor), V. Mavrantzas, Y. Dimakopoulos
Fraggedakis D., Dimakopoulos Y., Tsamopoulos J.
The sedimentation of a single particle in materials that exhibit simultaneously elastic, viscous and plastic behavior is examined in an effort to explain phenomena that contradict the nature of purely yieldstress materials. Such phenomena include the loss of the foreandaft symmetry with respect to an isolated settling particle under creeping flow conditions and the appearance of the “negative wake” behind it. Despite the fact that similar observations have been reported in studies involving viscoelastic fluids, researchers conjectured that thixotropy is responsible for these phenomena, as the aging of yieldstress materials is another common feature. By means of transient calculations, we study the effect of elasticity on both the fluidized and the solid phase. The latter is considered to behave as an ideal Hookean solid. The material properties of the model are determined under the isotropic kinematic hardening framework via Large Amplitude Oscillatory Shear (Laos) measurements. In this way, we are able to predict accurately the unusual phenomena observed in experiments with simple yieldstress materials, irrespective of the appearance of slip on the particle surface. Viscoelasticity favors the formation of intense shear and extensional stresses downstream of the particle, significantly changing the entrapment mechanism in comparison to that observed in viscoplastic fluids. Therefore, the critical conditions under which the entrapment of the particle occurs deviate from the wellknown criterion established theoretically by Beris et al. (1985) and verified experimentally by Tabuteau et al. (2007) for similar materials under conditions that elastic effects are negligible. Our predictions are in quantitative agreement with published experimental results by Holenberg et al. (2012) on the loss of the foreaft symmetry and the formation of the negative wake in Carbopol with wellcharacterized rheology. Additionally, we propose simple expressions for the Stokes drag coefficient, as a function of the gravity number, Yg (related to the Bingham number), for different levels of elasticity and for its critical value, under which entrapment of particles occurs. These criteria are in agreement with the results found in the recent work by Ahonguio et al. (2014). Finally, we propose a method to quantify experimentally the elastic effects in viscoplastic particulate systems. © 2016 The Royal Society of Chemistry.
Fraggedakis D., Pavlidis M., Dimakopoulos Y., Tsamopoulos J.
We examine the abrupt increase in the rise velocity of an isolated bubble in a viscoelastic fluid occurring at a critical value of its volume, under creeping flow conditions. This ‘velocity discontinuity’, in most experiments involving shearthinning fluids, has been somehow associated with the change of the shape of the bubble to an inverted teardrop with a tip at its pole and/or the formation of the ‘negative wake’ structure behind it. The interconnection of these phenomena is not fully understood yet, making the mechanism of the ‘velocity jump’ unclear. By means of steadystate analysis, we study the impact of the increase of bubble volume on its steady rise velocity and, with the aid of pseudo arclength continuation, we are able to predict the stationary solutions, even lying in the discontinuous area in the diagrams of velocity versus bubble volume. The critical area of missing experimental results is attributed to a hysteresis loop. The use of a boundaryfitted finite element mesh and the openboundary condition are essential for, respectively, the correct prediction of the sharply deformed bubble shapes caused by the large extensional stresses at the rear pole of the bubble and the accurate application of boundary conditions far from the bubble. The change of shape of the rear pole into a tip favours the formation of an intense shear layer, which facilitates the bubble translation. At a critical volume, the shear strain developed at the front region of the bubble sharply decreases the shear viscosity. This change results in a decrease of the resistance to fluid displacement, allowing the developed shear stresses to act more effectively on bubble motion. These coupled effects are the reason for the abrupt increase of the rise velocity. The flow field for stationary solutions after the velocity jump changes drastically and intense recirculation downstream of the bubble is developed. Our predictions are in quantitative agreement with published experimental results by Pilz & Brenn (J. NonNewtonian Fluid Mech., vol. 145, 2007, pp. 124–138) on the velocity jump in fluids with wellcharacterized rheology. Additionally, we predict shapes of larger bubbles when both inertia and elasticity are present and obtain qualitative agreement with experiments by Astarita & Apuzzo (AIChE J., vol. 11, 1965, pp. 815–820). © 2016 Cambridge University Press
drops and bubbles, interfacial flows (free surface), nonNewtonian flows
]]>Tsouka S., Dimakopoulos Y., Tsamopoulos J.
We consider the two dimensional, steady flow of a dilute polymer solution over a solid substrate with periodic topography under the action of a body force. We examine how the distribution of polymer is affected by flow conditions, physical properties and substrate geometry and how it affects the dynamics of the freesurface. The MavrantzasBeris twofluid Hamiltonian model is used in order to account for polymer migration due to stress gradients. The model is solved via the mixed finite element method combined with a quasielliptic grid generation technique. Results for polymer concentration, stress and velocity fields are presented as a function of the nondimensional parameters of the problem. The basic phenomena that appear in the case of polymer migration in undulating channels (Tsouka et al., 2014), also appear in free surface flows. Increasing the elastic stresses, increases migration of macromolecules towards the free surface, developing a polymerdepleted layer especially over the substrate maxima, which finally gives rise to “apparent” slip. Increasing the cavity steepness also enhances migration and the thickness of the depletion layer and induces strong variation in the stresses away from the substrate wall, especially in low polymer concentrations. All these phenomena cannot be captured by the homogenous OldroydB model. The evaluation of the Stokes number shows that flow resistance decreases as elasticity increases, increases monotonically up to an asymptote with the capillary number, and exhibits a nonmonotonic dependence on the Reynolds number. Moderate inertial effects can give rise to large deformations of the free surface, developing waves with complex shapes. If we use a similarly modified ePTT model, we observe that its inherent shearthinning is enhanced by the stress induced migration and tends to reduce the gradients on both polymeric stresses and polymer concentration along the film.
Film flow over topography, Finite element solution, Stressinduced migration, Viscoelastic fluids
]]>Lampropoulos N.K., Dimakopoulos Y., Tsamopoulos J.
We study the transient, twodimensional film flow over solid substrates with variable topography, a flow that has practical applications in microelectronics and microfluidics. The problem we address here is the advancing of a thin liquid film over squareshaped trenches with different depths and widths, under the influence of the gravitational force. We use the volumeoffluid method to obtain completely different wetting patterns depending on the dimensions of the topography, the capillary and Reynolds numbers, and the contact angle. On one hand, we predict continuous coating, i.e., the formation of the Wenzel state, in which a thin liquid film covers the entire trench, while steady flow is established upstream and downstream this topographical feature. This is the desirable pattern, when perfect coating is sought, as in the manufacturing of microelectronic devices. Under different conditions, we predict that the film almost completely bypasses the trench, entrapping air inside it, i.e., forming the Cassie state. The coating quality is clearly poor in this case, but this pattern reduces the drag on the film, and therefore, it is desirable in the operation of superhydrophobic surfaces for microfluidic applications. Between these two extreme configurations, we uncover a large variety of patterns, in which the film partially wets the trench forming an air inclusion all along its bottom or its upstream or downstream inner corners or the film may break up periodically. We produce comprehensive flow maps covering a wide range of relevant parameter values.
Air entrapment, Cassie and Wenzel states, Coating flows, Flow over topography, Thinfilm flow
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Syrakos A., Dimakopoulos Y., Georgiou G.C., Tsamopoulos J.
Numerical simulations of the flow in an extrusion damper are performed using a finite volume method. The damper is assumed to consist of a shaft, with or without a spherical bulge, oscillating axially in a containing cylinder filled with a viscoplastic material of Bingham type. The response of the damper to a forced sinusoidal displacement is studied. In the bulgeless case the configuration is the annular analogue of the wellknown liddriven cavity problem, but with a sinusoidal rather than constant lid velocity. Navier slip is applied to the shaft surface in order to bound the reaction force to finite values. Starting from a base case, several problem parameters are varied in turn in order to study the effects of viscoplasticity, slip, damper geometry and oscillation frequency to the damper response. The results show that, compared to Newtonian flow, viscoplasticity causes the damper force to be less sensitive to the shaft velocity; this is often a desirable damper property. The bulge increases the required force on the damper mainly by generating a pressure difference across itself; the latter is larger the smaller the gap between the bulge and the casing is. At high yield stresses or slip coefficients the amount of energy dissipation that occurs due to sliding friction at the shaftfluid interface is seen to increase significantly. At low frequencies the flow is in quasi steady state, dominated by viscoplastic forces, while at higher frequencies the fluid kinetic energy storage and release also come into the energy balance, introducing hysteresis effects.
Annular cavity, Bingham flow, Finite volume method, Slip, Viscous damper, Viscous dissipation
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Pavlidis M., Karapetsas G., Dimakopoulos Y., Tsamopoulos J.
We examine and compare five constitutive models, which have been recently proposed to describe the behavior of elastoviscoplastic fluids. The comparison is performed in simple rheometric flows, i.e. simpleshear, uniaxial elongation and large amplitude oscillatory tests and in the complex flow generated by a falling spherical particle. The first set of three models do not explicitly include shearthinning. These are the ones proposed by Saramito [21], Park & Liu [27] and Belblidia et al. [28]. The first one has been derived under a thermodynamic framework, while the other two have been based on viscosity regularization methods. When spatial and temporal inhomogeneity are not present in the flow field, the models generally produce acceptable predictions, except for: (a) the BWW in predicting the primary normal stress under small shear rate and under small strain in the LAOStrain test, (b) all models in predicting different parts of the spectra of G′ and G″, although the predictions of the SRM can be corrected when kinematic hardening is accounted for and (c) the P&L model in LAOStress because of nonexistence or multiplicity of solutions. In the complex flow, the predictions of each model are compared with the experimental data of Holenberg et al. [34] under non shearthinning conditions and the predictions of the SRM model are clearly superior. The second set of models have been proposed by Saramito [22] to explicitly account for shear thinning either by extending the Herschel–Bulkley model to include elastic effects, SRMHB, or by introducing a PTTtype term in the constitutive model, SRMPTT. Both these models provide acceptable results in the rheometric tests. In the falling sphere test, their predictions are compared with the experimental results by Putz et al. [15] for the settling of a particle under conditions that the Carbopol solution exhibits shearthinning. Here the SRMHB is found to be superior.
Elastoviscoplastic fluids, Falling sphere in EVP fluids, LAOS tests, Viscometric tests
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Fraggedakis D., Dimakopoulos Y., Tsamopoulos J.
We examine and compare five constitutive models, which have been recently proposed to describe the behavior of elastoviscoplastic fluids. The comparison is performed in simple rheometric flows, i.e. simpleshear, uniaxial elongation and large amplitude oscillatory tests and in the complex flow generated by a falling spherical particle. The first set of three models do not explicitly include shearthinning. These are the ones proposed by Saramito [21], Park & Liu [27] and Belblidia et al. [28]. The first one has been derived under a thermodynamic framework, while the other two have been based on viscosity regularization methods. When spatial and temporal inhomogeneity are not present in the flow field, the models generally produce acceptable predictions, except for: (a) the BWW in predicting the primary normal stress under small shear rate and under small strain in the LAOStrain test, (b) all models in predicting different parts of the spectra of G′ and G″, although the predictions of the SRM can be corrected when kinematic hardening is accounted for and (c) the P&L model in LAOStress because of nonexistence or multiplicity of solutions. In the complex flow, the predictions of each model are compared with the experimental data of Holenberg et al. [34] under non shearthinning conditions and the predictions of the SRM model are clearly superior. The second set of models have been proposed by Saramito [22] to explicitly account for shear thinning either by extending the Herschel–Bulkley model to include elastic effects, SRMHB, or by introducing a PTTtype term in the constitutive model, SRMPTT. Both these models provide acceptable results in the rheometric tests. In the falling sphere test, their predictions are compared with the experimental results by Putz et al. [15] for the settling of a particle under conditions that the Carbopol solution exhibits shearthinning. Here the SRMHB is found to be superior.
Elastoviscoplastic fluids, Falling sphere in EVP fluids, LAOS tests, Viscometric tests
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Dimakopoulos Y., Kelesidis G., Tsouka S., Georgiou G.C., Tsamopoulos J.
BACKGROUND: In microcirculation, the nonNewtonian behavior of blood and the complexity of the microvessel network are responsible for the high flow resistance and the large reduction of the blood pressure. Red blood cell aggregation along with inward radial migration are two significant mechanisms determining the former. Yet, their impact on hemodynamics in nonstraight vessels is not well understood. OBJECTIVE: In this study, the steady state blood flow in stenotic rigid vessels is examined, employing a sophisticated nonhomogeneous constitutive law. The effect of red blood cells migration on the hydrodynamics is quantified and the constitutive model’s accuracy is evaluated. METHODS: A numerical algorithm based on the twodimensional mixed finite element method and the EVSS/SUPG technique for a stable discretization of the mass and momentum conservation equations in addition to the constitutive model is employed. RESULTS: The numerical simulations show that a celldepleted layer develops along the vessel wall with an almost constant thickness for slow flow conditions. This causes the reduction of the drag force and the increase of the pressure gradient as the constriction ratio decreases. CONCLUSIONS: Viscoelastic effects in blood flow were found to be responsible for steeper decreases of tube and discharge hematocrits as decreasing function of constriction ratio. © 2015 – IOS Press and the authors.
blood viscoelasticity, celldepleted layer, FahraeusLindqvist effect, RBC migration, shearinduced migration, stenotic microvessels
]]>Fraggedakis D., Kouris C., Dimakopoulos Y., Tsamopoulos J.
We study the flow of two immiscible, Newtonian fluids in a periodically constricted tube driven by a constant pressure gradient. Our volumeoffluid algorithm is used to solve the governing equations. First, the code is validated by comparing its predictions to previously reported results for stratified and pulsing flow. Then, it is used to capture accurately all the significant topological changes that take place. Initially, the fluids have a coreannular arrangement, which is found to either remain the same or change to a different arrangement depending on the fluid properties, the pressure driving the flow, or the flow geometry. The flowpatterns that appear are the coreannular, segmented, churn, spray, and segregated flow. The predicted scalings near pinching of the core fluid concur with similarity predictions and earlier numerical results [I. Cohen et al., “Two fluid drop snapoff problem: Experiments and theory,” Phys. Rev. Lett. 83, 11471150 (1999)]. Flowpattern maps are constructed in terms of the Reynolds and Weber numbers. Our result provides deeper insights into the mechanism of the pattern transitions and is in agreement with previous studies on coreannular flow [Ch. Kouris and J. Tsamopoulos, “Coreannular flow in a periodically constricted circular tube, I. Steady state, linear stability and energy analysis,” J. Fluid Mech. 432, 3168 (2001) and Ch. Kouris et al., “Comparison of spectral and finite element methods applied to the study of interfacial instabilities of the coreannular flow in an undulating tube,” Int. J. Numer. Methods Fluids 39(1), 4173 (2002)], segmented flow [E. Lac and J. D. Sherwood, “Motion of a drop along the centreline of a capillary in a pressuredriven flow,” J. Fluid Mech. 640, 2754 (2009)], and churn flow [R. Y. Bai et al., “Lubricated pipeliningStability of core annularflow. 5. Experiments and comparison with theory,” J. Fluid Mech. 240, 97132 (1992)]. © 2015 AIP Publishing LLC.
Pettas D., Karapetsas G., Dimakopoulos Y., Tsamopoulos J.
It is wellknown that by increasing the flow rate in polymer extrusion, the flow becomes unstable and the smooth extrudate surface becomes wavy and disordered to an increasing degree. In order to investigate the mechanisms responsible for these instabilities we perform a linear stability analysis of the steady extrusion of a viscoelastic fluid flowing through a planar die under creeping flow conditions. We consider the PhanThienTanner (PTT) model to account for the viscoelasticity of the material. We employ the mixed finite element method combined with an elliptic grid generator to account for the deformable shape of the interface. The generalized eigenvalue problem is solved using Arnoldi’s algorithm. We perform a thorough parametric study in order to determine the effects of all material properties and rheological parameters. We investigate in detail the effect of the interfacial tension and the presence of a deformable interface. It is found that the presence of a finite surface tension destabilizes the flow as compared to the case of the stickslip flow. We recognize two modes, which become unstable beyond a critical value of the Weissenberg number and perform an energy analysis to examine the mechanisms responsible for the destabilization of the flow and compare against the mechanisms that have been suggested in the literature. © 2015 Elsevier B.V.
Extrudate swell, Extrusion instabilities, Polymer melts, Sharkskin, Stability analysis, Surface tension
]]>Papaioannou J., Giannousakis A., Dimakopoulos Y., Tsamopoulos J.
Motivated by the probe experiment for characterizing the adhesion strength of polymeric materials, we studied the axisymmetric extensional flow of a viscoelastic liquid filament that contains one or three initially spherical bubbles along its axis of symmetry. The filament is confined between two parallel and coaxial disks of the same radius and its initially cylindrical outer surface is surrounded by air. The flow is induced by the axial extension of the upper disk under constant velocity, while the lower disk remains stationary. The rheology of the liquid is described by the exponential PhanThien and Tanner viscoelastic model. A quasielliptic transformation combined with domain decomposition and local mesh refinement is employed to discretize the domain. The mixed finite element Galerkin method for the mass and momentum balances combined with the EVSSG method and SUPG weighting for the constitutive equation are used to obtain their weak forms. All these are necessary for the successful simulation of much larger filament extensions and of liquids with higher elasticity than those reported earlier. The evolution of the filament and the bubble(s)free surfaces depends on the interplay of the viscous, elastic, and capillary forces. It was found that, when the ratio of the elastic and the capillary forces is small compared to the viscous forces, the bubble attains very large deformations. The onset of cusps was observed at the bubble poles for intermediate values of capillarity and elasticity. The force varies with the filament extension (or time) in a way reported in experiments measuring the strength of adhesive materials: Its initial increase up to a maximum is followed by a plateau, and finally it drops to zero when the adhesive tends to fail. Increasing the filament elasticity delays the development of stresses (and the applied force). © 2014 American Chemical Society.
Tseropoulos G., Dimakopoulos Y., Tsamopoulos J., Lyberatos G.
The Minoan Terracotta pipes with their conical shape were widely used in the water distribution system in the ancient Minoan civilization. They remain one of the brightest achievements of the Minoan tribe in water supply technology and raise admiration as well as many questions about the technological advancements of antiquity, that are yet to be understood. The present work aims at answering the following questions: a) what inspired the Minoans to manufacture pipes with such a peculiar shape, that differs greatly not only from later pipe designs of antiquity, but also from contemporary cylindrical pipes and b) why was the design of those pipes abandoned after the fall of the Minoan civilization? It tries to address these questions by investigating the flow physics and dynamics that take place in such pipes, adopting advanced numerical and computational methods. The timeaveraged NavierStokes equations along with the k – e{open} turbulence model are solved for a variety of geometrical parameters, pipe orientations and flow rates, in order to produce a comparative picture of the hydraulic efficiency of the conical Minoan pipes. The flow field is visualized and critical flow parameters, such as the head loss, the velocity magnitude and turbulence intensity, are calculated. These calculations show clearly that the conical Minoan pipes exhibit significantly higher pressure drops along their length compared to an equivalent straight pipe. In their widest part an extended recirculation appears, which could wash out impurities that may be present in the water, which at the same time cannot be deposited on the pipe wall. This evidence proves that the Minoan pipes are energetically expensive to operate and consequently their replacing by cylindrical pipes was inevitable. Therefore, it seems that the main advantage and purpose of the particular geometry was that they could be easily connected on site, forming long straight or slowly bending lines without having to add straight or many different fittings in between. © 2012 Elsevier Ltd.
Computational fluid mechanics, Minoan civilization, Minoan hydraulic pipes, Turbulence modeling
]]>Karapetsas G., Tsamopoulos J.
During extrusion of viscoelastic fluids various flow instabilities may arise resulting in a distorted free surface. In order to investigate the factors generating these instabilities we performed a linear stability analysis at zero Reynolds number around the steady solution of the cylindrical or planar stickslip flow for a viscoelastic fluid following the affine exponential PhanThien Tanner (PTT) model. Stickslip flow is an important special case of the extrudate swell problem, since the latter reduces to it in the limit of infinite surface tension but avoids the complications of a freeboundary flow. The linear stability analysis is performed for various values of the rheological parameters of the PTT model in order to determine the effects of all material properties. It is found that the flow becomes unstable as the Weissenberg number increases above a critical value, due to a Hopf bifurcation suggesting that the flow will become periodic in time. Both the critical value of the Weissenberg number and the frequency of the instability depend strongly on the rheological parameters of the viscoelastic model. The corresponding eigenvectors indicate that the perturbed flow field has a spatially periodic structure, initiated at the rim of the die, extending for up to 57 die gaps downstream, but confined close to the surface of the extrudate, in qualitative agreement with existing experiments. This suggests that instability is generated by the combination of the singularity in the velocity and stress fields at the die lip and the strong extension that the extruded polymer undergoes near its surface. The elasticity alone can be responsible for the appearance of instabilities in the extrusion process of viscoelastic fluids and the often used assumptions of wall slip or compressibility, although they might be present, are not required. Finally, the mechanisms that produce these instabilities are examined through energy analysis of the disturbance flow. © 2013 AIP Publishing LLC.
Dimakopoulos Y., Pavlidis M., Tsamopoulos J.
The steady, buoyancydriven rise of a bubble in a HerschelBulkley fluid is examined assuming axial symmetry. The variation of the rateofstrain tensor around a rising bubble necessitates the coexistence of fluid and solid regions in this fluid. In general, a viscoplastic fluid will not be deforming beyond a finite region around the bubble and, under certain conditions, it will not be deforming either just behind it or around its equatorial plane. The accurate determination of these regions is achieved by introducing a Lagrange multiplier and a quadratic term in the corresponding variational inequality, resulting in the socalled Augmented Lagrangian Method (ALM). Additionally here, the augmentation parameters are determined following a nonlinear conjugate gradient procedure. The new predictions are compared against those obtained by the much simpler Papanastasiou model, which uses a continuous constitutive equation throughout the material, irrespective of its state, but does not determine the boundary between solid and liquid along with the flow field. The flow equations are solved numerically using the mixed finiteelement/Galerkin method on a mesh generated by solving a set of quasielliptic differential equations. The accuracy of solutions is ascertained by mesh refinement and comparison with our earlier and new predictions for a bubble rising in a Newtonian and a Bingham fluid. We determine the bubble shape and velocity and the shape of the yield surfaces for a wide range of material properties, expressed in terms of the Bingham, Bn, Bond, and Archimedes numbers. As Bn increases, the bubble decelerates, the yield surfaces at its equatorial plane and away from it approach each other and eventually merge immobilizing the bubble. For small and moderate Bingham numbers, the predictions using the Papanastasiou model satisfactorily approximate those of the discontinuous HerschelBulkley model for sufficiently large values of the normalization exponent (≥104). On the contrary, as Bn increases and the rateofstrain approaches zero almost throughout the fluidlike region, much larger values of the exponent are required to accurately compute the yield surfaces. Bubble entrapment does not depend on the power law index, i.e. a bubble in a HerschelBulkley fluid is entrapped under the same conditions as in a Bingham fluid. © 2012 Elsevier B.V.
Augmented Lagrangian Method, Free surface flow, HerschelBuckley model, Papanastasiou model, Rising bubble, Viscoplastic material
]]>Papaioannou J.G., Dimakopoulos Y., Tsamopoulos J.A.
Because of the three dimensional nature of many physical problems encountered in industrial processes and applications, we develop a novel 3D elliptic grid generator for the construction and the deformation of structured grids. The specific application we consider is related to the large deformations of an air bubble in a liquid between two parallel plates. The new technique is based on the inverted Laplace equations along with internal constraints, which are considered essential in order to adjust grid spacing. This set of equations is accompanied with appropriate set of boundary conditions that is the imposition of the 1D differential arclength at each of the two coordinates of the boundary surfaces in the computational domain. The new 3D grid generator provides the smoothness of the coordinate lines not only in the bulk, but also on the boundary surfaces. The kinematic condition is invoked for the motion of the 3D surface of the bubble and we also consider the free motion of the 2D contact line (points where the air in the bubble and the surrounding liquid meet the solid boundary). In order to resolve better the flow field and the shape of the bubble free surface we have also developed a local mesh refinement technique. In this way the demand for computational memory is overly reduced. Our method in 3D grid generation follows the Arbitrary Lagrangian Eulerian (A.L.E.) formulation, which when combined with the finite element/Galerkin method provides a flexible and robust tool for simulating free surface flows in three dimensions.
3D simulations, Continuum mechanics, Elliptic mesh generator, Free surface flows
Chatzidai N., Dimakopoulos Y., Tsamopoulos J.
According to linear theory and assuming the liquids to be inviscid and the bubbles to remain spherical, bubbles set in oscillation attract or repel each other with a force that is proportional to the product of their amplitude of volume pulsations and inversely proportional to the square of their distance apart. This force is attractive, if the forcing frequency lies outside the range of eigenfrequencies for volume oscillation of the two bubbles. Here we study the nonlinear interaction of two deformable bubbles set in oscillation in water by a step change in the ambient pressure, by solving the NavierStokes equations numerically. As in typical experiments, the bubble radii are in the range 11000 m. We find that the smaller bubbles (~5 m) deform only slightly, especially when they are close to each other initially. Increasing the bubble size decreases the capillary force and increases bubble acceleration towards each other, leading to oblate or spherical cap or even globally deformed shapes. These deformations may develop primarily in the rear side of the bubbles because of a combination of their translation and harmonic or subharmonic resonance between the breathing mode and the surface harmonics. Bubble deformation is also promoted when they are further apart or when the disturbance amplitude decreases. The attractive force depends on the Ohnesorge number and the ambient pressure to capillary forces ratio, linearly on the radius of each bubble and inversely on the square of their separation. Additional damping either because of liquid compressibility or heat transfer in the bubble is also examined. Copyright © Cambridge University Press 2011.
bubble dynamics, interfacial flows (free surface)
]]>Pavlidis M., Dimakopoulos Y., Tsamopoulos J.
Twodimensional, steady flow of a viscoelastic film over a periodic topography under the action of a body force is studied. The exponential PhanThien and Tanner (ePTT) constitutive model is used. The conservation equations are solved via the usual mixed finite element method combined with a quasielliptic grid generation scheme in order to capture the large deformations of the free surface. The constitutive equation is weighted using the SUPG method and solved via the polymeric stress splitting EVSSG technique. First, the code is validated by verifying that in isolated topographies the periodicity conditions result in fully developed viscoelastic film flow at the inflow/outflow boundaries and that its predictions for Newtonian fluids over 2D topography under creeping flow conditions coincide with those of previous works. Since the lubrication approximation is not invoked here, the topographical features can have wall segments that form any angle with the main flow, but only slight smoothing of the convex corners assists in reducing the stress singularity there. Thus, steadystate solutions are computed accurately up to high Deborah numbers, resulting in large deformations of the free surface. The magnitude of the capillary ridge in the film before the entrance to a step down of the substrate and of the capillary depression before a step up is increased as De increases up to ∼0.7 due to increased fluid elasticity. Above this value they decrease, because increasing De increases also the shear and elongational thinning, which eventually affect them more. Increasing the ratio of solvent to polymer viscosities, β, the elongational parameter, e{open} and the molecular slip parameter, ξ, monotonically increases their magnitudes and especially that of the capillary ridge, but the mechanisms leading to these changes are different as explained in the text. © 2010 Elsevier B.V.
Elliptic mesh generation, Flow over topography, Viscoelastic film flow
]]>Chatzidai N., Giannousakis A., Dimakopoulos Y., Tsamopoulos J.
We present an improved method to generate a sequence of structured meshes even when the physical domain contains deforming inclusions. This method belongs to the class of Arbitrary LagrangianEulerian (ALE) methods for solving moving boundary problems. Its tools are either (a) separate mappings of the domain boundaries and enforcing the node distribution on lines emanating from singular points or (b) domain decomposition and separate mappings of each subdomain using suitable coordinate systems. The latter is shown to be more versatile and general. In both cases a set of elliptic equations is used to generate the grid extending in this way the method advanced by Dimakopoulos and Tsamopoulos [Y. Dimakopoulos, J.A. Tsamopoulos, A quasielliptic transformation for moving boundary problems with large anisotropic deformations, J. Comput. Phys. 192 (2003) 494522]. We shall present examples where this earlier method and all other mesh generating methods which are based on a conformal mapping or solving a quasielliptic set of PDEs fail to produce an acceptable mesh and accurate solutions in such geometries. Furthermore, in contrast to other methods, appropriate boundary conditions and constraints such as, orthogonality of specific mesh lines and prespecified node distributions on them, can be easily implemented along a specific part of the domain or its boundary. Hence, no attractive terms at specific corners or singular points are needed. To increase the mesh resolution around the moving interfaces while keeping low the memory requirements and the computational time, a local mesh refinement technique has been incorporated as well. The method is demonstrated in two challenging examples where no remeshing is required in spite of the large domain deformations. In the first one, the transient growth of two bubbles embedded in a viscoelastic filament undergoing stretching in the axial direction is examined, while in the second one the linear and nonlinear dynamics of two bubbles in a viscous medium are determined in an acoustic field. The large elasticity of the filament in the first case or the large inertia in the second case coupled with the externally induced large deformations of the liquid domain requires the accurate calculation which is achieved by the method we propose herein. The governing equations are solved using the finite element/Galerkin method with appropriate modifications to solve the hyperbolic constitutive equation of a viscoelastic fluid. These are coupled with an implicit Euler method for time integration or with Arnoldi’s algorithm for normal mode analysis. © 2008 Elsevier Inc. All rights reserved.
Elliptic mesh generation, Moving boundary problems, Structured meshes
]]>Dimakopoulos Y., Tsamopoulos J.
We examine the pressuredriven coating of a straight, long tube with a finite amount of viscoelastic liquid. This combines the formation and elongation of an open bubble pushing the liquid downstream with the motion of the advancing liquid front and the deposition of a film along the tube wall. The mixed finite element method is combined with the discontinuous Galerkin method for computing the polymeric contribution to the stress tensor, while a quasielliptic grid generation scheme is used for tessellating the highly deforming domain. Global remeshing is adopted to overcome local distortion and coarsening of the mesh which arises from the attachment of advancing front nodes on the wall. The accuracy of the solution is achieved by selectively refining the mesh along areas where sharp boundary layers in stress arise and in particular at the bubble front. An extensive parametric analysis is performed using the affine, singlemode PhanTien and Tanner constitutive model. The thickness of the remaining film increases as the solvent contribution to viscosity decreases, because of the sharply increased normal viscoelastic stresses along the bubble front. The effect of decreasing the extensional parameter εPTT is similar because this decreases the shear and extensional thinning in the liquid. The transient nature of the process delays the development of the viscoelastic stresses as the Deborah number increases. The streamlines relative to the liquid tip follow the usual ‘fountain flow’ pattern as long as the bubble and the liquid front do not interact. The nonaffine, multimode PTT model with constants obtained from the literature for a PIB solution in C14 predicts a slightly thinner deposited film and a flatter bubble front because of its stronger shear thinning. Moreover, it predicts a higher dimensionless bubble velocity because of the very small viscosity ratio of the solvent and a Vshape in the radial distribution of the first normal stress difference and the birefringence just behind the contact line in qualitative agreement with experiments and theory for planar fountain flow. © 2009 Elsevier B.V. All rights reserved.
Coating flows, Discontinuous Galerkin, Elliptic mesh generation, Exponential PTT, Fountain flow, Injection molding
]]>Pavlidis M., Dimakopoulos Y., Tsamopoulos J.
The onedimensional, gravitydriven film flow of a linear (l) or exponential (e) PhanThien and Tanner (PTT) liquid, flowing either on the outer or on the inner surface of a vertical cylinder or over a planar wall, is analyzed. Numerical solution of the governing equations is generally possible. Analytical solutions are derived only for: (1) lPTT model in cylindrical and planar geometries in the absence of solvent, β ≡ η̃s(η̃s + η̃p) = 0, where η̃p and η̃s are the zeroshear polymer and solvent viscosities, respectively, and the affinity parameter set at ξ = 0; (2) lPTT or ePTT model in a planar geometry when β = 0 and ξ ≠ 0; (3) ePTT model in planar geometry when β = 0 and ξ = 0. The effect of fluid properties, cylinder radius, R̃, and flow rate on the velocity profile, the stress components, and the film thickness, H̃, is determined. On the other hand, the relevant dimensionless numbers, which are the Deborah, De = λ̃Ũ/H̃, and Stokes, St = ρ̃g̃H̃2/ (η̃p + η̃s)Ũ, numbers, depend on H̃ and the average film velocity, Ũ. This makes necessary a trial and error procedure to obtain H̃ a posteriori. We find that increasing De, ξ, or the extensibility parameter ε increases shear thinning resulting in a smaller St. The Stokes number decreases as R̃/H̃ decreases down to zero for a film on the outer cylindrical surface, while it asymptotes to very large values when R̃/H̃ decreases down to unity for a film on the inner surface. When ξ ≠ 0, an upper limit in De exists above which a solution cannot be computed. This critical value increases with ε and decreases with ξ. © SpringerVerlag 2009.
Gravitydriven flow, PTT fluid model, Viscoelastic film flow
]]>Papaioannou J., Karapetsas G., Dimakopoulos Y., Tsamopoulos J.
The injection of a viscoplastic material, driven by a constant pressure drop, inside a pipe or between two parallel coaxial disks under creeping flow conditions is examined. The transient nature of both flow arrangements requires solving a timedependent problem and fully accounting for the advancing liquid/air interface. Material viscoplasticity is described by the Papanastasiou constitutive equation. A quasielliptic grid generation scheme is employed for the construction of the mesh, combined with local mesh refinement near the material front and, periodically, full mesh reconstruction. All equations are solved using the mixed finite element/Galerkin formulation coupled with the implicit Euler method. For a viscoplastic fluid, the flow field changes qualitatively from that of a Newtonian fluid because the material gets detached from the walls. For small Bingham numbers, the contact line moves in the flow direction, so that initially the flow resembles that of a Newtonian fluid, but even in that case detachment eventually occurs. The distance covered by the contact line, before detachment takes place, decreases as the Bingham number increases. For large enough Bingham numbers, the fluid may even detach from the wall without advancing appreciably. In pipe flow, when detachment occurs, unyielded material arises at the front and the flow changes into one under constant flow rate with pressure distribution that does not vary with time. In the flow between disks, it remains decelerating and the material keeps rearranging at its front because of the increased cross section through which it advances. The wall detachment we predict has been observed experimentally by Bates and Bridgwater [Chem. Eng. Sci. 55, 30033012 (2000)] in radial flow of pastes between two disks. © 2009 The Society of Rheology.
Karapetsas G., Tsamopoulos J.
The steady planar and cylindrical stickslip flows for a viscoelastic fluid are computed using the PhanThien and Tanner (PTT) constitutive model. The mixed finite element method is used in combination with the elasticviscous stresssplitting technique and the streamline upwind PetrovGalerkin discretization for the constitutive equation. This combination of methods when applied to the PTT constitutive model allows us to compute steady state solutions up to high Weissenberg numbers; practically without an upper limit. Equally important, the global Jacobian matrix is generated in order to be able to perform a linear stability analysis of the computed steady state. The dependence of the steady solutions on all the problem parameters is examined. In the limit of a Newtonian fluid, the expansion coefficients near the singularity are computed with comparable accuracy to those from previous analytical and numerical studies, which include the singular finite element method. In the case of a viscoelastic liquid, it is shown that the computed solutions converge quadratically with mesh refinement even at the exit plane of the die and also locally very close to the singularity. The form of the converged solution near the singularity is examined as well as its dependence on various rheological parameters. It is shown that the singularity at the die exit is a logarithmic one and always integrable. Under such conditions our calculations can be extended to determine the linear stability of the herein computed steady states. © 2009 American Institute of Physics.
Tsamopoulos J., Dimakopoulos Y., Chatzidai N., Karapetsas G., Pavlidis M.
We examine the buoyancydriven rise of a bubble in a Newtonian or a viscoplastic fluid assuming axial symmetry and steady flow. Bubble pressure and rise velocity are determined, respectively, by requiring that its volume remains constant and its centre of mass remains fixed at the centre of the coordinate system. The continuous constitutive model suggested by Papanastasiou is used to describe the viscoplastic behaviour of the material. The flow equations are solved numerically using the mixed finiteelement/Galerkin method. The nodal points of the computational mesh are determined by solving a set of elliptic differential equations to follow the often large deformations of the bubble surface. The accuracy of solutions is ascertained by mesh refinement and predictions are in very good agreement with previous experimental and theoretical results for Newtonian fluids. We determine the bubble shape and velocity and the shape of the yield surfaces for a wide range of material properties, expressed in terms of the Bingham Bn = tauy*/rho*g*Rb* Bond Bo = rho*g*Rb* 2/γ* and Archimedes Ar = ρ*2g* Rb *3/μo*2 numbers, where ρ* is the density, μ*o the viscosity, γ* the surface tension and τ*y the yield stress of the material, g* the gravitational acceleration and R*b the radius of a spherical bubble of the same volume. If the fluid is viscoplastic, the material will not be deforming outside a finite region around the bubble and, under certain conditions, it will not be deforming either behind it or around its equatorial plane in contact with the bubble. As Bn increases, the yield surfaces at the bubble equatorial plane and away from the bubble merge and the bubble becomes entrapped. When Bo is small and the bubble cannot deform from the spherical shape the critical Bn is 0.143, i.e. it is a factor of 3/2 higher than the critical Bn for the entrapment of a solid sphere in a Bingham fluid, in direct correspondence with the 3/2 higher terminal velocity of a bubble over that of a sphere under the same buoyancy force in Stokes flow. As Bo increases allowing the bubble to squeeze through the material more easily, the critical Bingham number increases as well, but eventually it reaches an asymptotic value. Ar affects the critical Bn value much less. © 2008 Cambridge University Press.
Housiadas K., Tsamopoulos J.
The effect of the air jet, which is used to increase the process stability and cool the polymeric film which is produced by the widelyused Film Blowing Process (FBP) is presented. The air jet is supplied circumferentially and tangentially on the outside surface of the tubular film that is pulled in the same upward direction. The two steady flow fields interact through the outside surface of the film which must be determined as part of the solution. Viscous, elastic, inertial, gravitational and interfacial forces are taken into account for the film, while only viscous and inertial forces determine the airflow. The film viscoelasticity is described using a variety of constitutive equations including the UCM and the affine exponential PPT models. The governing equations are simplified under the thin film approximation and a boundary layer formulation for the air. A similarity solution is obtained for the air jet and the air drag on the film is evaluated in terms of the bubble radius and introduced in the equations governing the film flow which are solved numerically by finite differences. It is found that the aerodynamic force on the film is very important in determining its shape, keeping it much closer to the vertical position for a longer distance from its extrusion level and, thus, stabilizing the process. © 2008 American Institute of Physics.
Film blowing, Free surface flows, Laminar boundary layer, Tangential wall jet, Thin film approximation
]]>Dimakopoulos Y., Pavlidis M., Tsamopoulos J.
We examine the pressuredriven coating of a straight tube with a viscoelastic liquid using an advanced and stable finite element algorithm, which is based on an elliptic grid generator along with local refinement and reconstruction of the mesh wherever this is needed. In particular, we investigate the case where a finite amount of viscoelastic material is allowed to deform inside a very long tube. A complete parametric analysis is performed in order to examine the effects of the elastic and inertia forces, and the various rheological parameters including the solvent to polymer viscosity ratio. Results using the exponential PTT constitutive model show that the thickness of the remaining film increases as the solvent viscosity decreases, because of the development of sharp boundary layers in the normal components of the viscoelastic stress tensor along the bubble front. In all cases the streamlines follow a ‘fountain flow’ pattern when the origin of the coordinate system is located at the liquid tip, due to the small effect of capillary forces. The other two constitutive models that have been examined are the FENECR and the Giesekus model. © 2008 American Institute of Physics.
Coating process, Free surface flow, Liquid displacement, Numerical solution via finite elements
]]>Karapetsas G., Tsamopoulos J.
The steady extrusion of viscoelastic materials from a straight, annular die is studied theoretically. The viscoelastic behavior is modelled using the affine PhanThien and Tanner (PTT) constitutive equation of the exponential form. For the numerical solution of the governing equations the mixed finite element method is combined with a quasielliptic mesh generation scheme in order to capture the large deformations of the two free surfaces of the extrudate. The elasticviscous stress splitting technique (EVSSG) is used to separate the elastic and viscous contributions to the polymeric part of the stress tensor together with a streamline upwind PetrovGalerkin (SUPG) weighting for the discretization of the constitutive equation. This combination of solution methods and constitutive model allows us (i) to compute accurate steadystate solutions up to very high Weissenberg numbers resulting in very high deformations of the free surfaces (ii) construct and store the Jacobian matrix, which is necessary to conduct linear stability analysis for this flow. First, results for the fully developed flow of a PTT liquid inside an annular die are presented. They reveal a complex interplay between material elasticity, shear thinning and solvent viscosity. Next, a complete parametric analysis of annular extrusion is performed. Such a complete study using the PTT model has not been reported before, even at much lower Wi numbers. It is found that swelling of the material increases sharply up to moderate Weissenberg numbers, whereas its rate of increase is reduced for higher values of Wi, as shear thinning becomes increasingly important. The latter generally plays a crucial role, in addition to elasticity, on the swelling of the extrudate. Moreover, as the contribution of the solvent viscosity increases, the contribution of elastic stresses decreases causing a decrease in the swelling of the material which approaches the Newtonian limit. The predicted swelling ratios, which characterize the geometry of the extrudate, are in satisfactory agreement with earlier experimental and theoretical data for three particular HDPE resins. © 2008 Elsevier B.V. All rights reserved.
Annular flow, Free surface flows, Polymer extrusion, Viscoelastic fluids
]]>Housiadas K.D., Klidis G., Tsamopoulos J.
We examine the film blowing process (FBP), which is widely used for manufacturing biaxially stretched films of polymeric materials. The viscoelastic property of the material is taken into account by employing the Upper Convected Maxwell, the OldroydB or the PhanThien and Tanner constitutive model. In contrast to all previous theoretical works, which followed the now classical method developed by Pearson and Petrie [J.R.A. Pearson, C.J.S. Petrie, The flow of a tubular film. Part 1. Formal mathematical representation, J. Fluid Mech. 40 (1) (1970) 119; J.R.A. Pearson, C.J.S. Petrie, The flow of a tubular film. Part 2. Interpretation of the model and discussion of solutions, J. Fluid Mech. 42 (3) (1970) 609625], we analyze the process by starting with the general threedimensional mass and momentum balances and by formally and systematically applying the thinfilm approximation. This procedure results in twodimensional dynamic balances in both the axial and azimuthal directions. Although these balances are highly nonlinear and more complicated than the original momentum balance, they are reduced by one spatial dimension and, more importantly, they are more general than the classical ones, whereas they are developed in a rigorous and straightforward manner. When we assume axial symmetry and steady state, we recover the earlier model equations. However, this new methodology allows us to examine not only axisymmetric, but also nonaxisymmetric disturbances to this base flow and to retain the time derivatives in all the governing equations. This procedure is an extension of our earlier one used to study transient annular extrusion [K. Housiadas, J. Tsamopoulos, Unsteady flow of an axisymmetric annular film under gravity, Phys. Fluids 10 (10) (1998) 25002516; K. Housiadas, J. Tsamopoulos, Unsteady extrusion of a viscoelastic annular film: I. General model and its numerical solution, J. NonNewton. Fluid Mech. 88 (3) (2000) 229259], which also involved the thinfilm approximation and three moving interfaces, but under the assumption of axial symmetry. Viscous, elastic, inertial, gravitational and capillary forces are included in our model. The base state is computed using finite differences to simultaneously predict bubble shape, film thickness, velocity, pressure and polymer extrastress profiles. Subsequently, its linear stability is examined to two and threedimensional disturbances by solving the full eigenvalue problem to determine the stability regions of the process. It is shown that under typical operating conditions the bubble becomes unstable first to nonaxisymmetric disturbances, although twodimensional instability is also predicted by our model, in agreement with recent experiments. © 2006 Elsevier B.V. All rights reserved.
Annular films, Film blowing, Linear stability analysis, OldroydB, Perturbation method, PTT constitutive models, UCM
]]>Dimakopoulos Y., Tsamopoulos J.
We study the transient displacement of Newtonian and viscoplastic liquids by highly pressurized air in cylindrical tubes of finite length with an expansion followed by a contraction in their cross section. Papanastasiou’s formula is employed to regularize the discontinuous Bingham model. For both fluid models considered, the distribution of the remaining film on the inner tube wall is nonuniform and only partly follows the tube geometry: it is thinner in the expanding section of the tube, thicker in the contracting one, and as thin as observed in relevant experiments for straight tube segments, if these are long enough. The effect of changes in the diameter of the narrow introductory tube on the width of the remaining film depends on liquid inertia and yield stress. It is confined near the expansion corner for small Reynolds numbers, but causes extended distortions on the free surface for larger ones due to the development of ‘lip’ or ‘corner’ vortices on the expanding side of the tube. The increased viscosity of viscoplastic materials, especially where velocity gradients are smaller, reduces the extent of flow and interface fluctuations. Unyielded regions appear along the film remaining on the tube wall, the core area of the main and exit tubes and around the concave corners of the tube. These cause the thinning of the remaining material inside the entrance tube, around the centerline of the main section and the flattening of the bubble front. The size of the solidlike areas increases at higher Reynolds numbers. The tip velocity for both viscous and viscoplastic materials increases along the introductory tube until it attains a maximum, then decreases down to a plateau while it moves inside the main tube, and finally increases again as it approaches the contraction corner. Its values decrease as the Bingham number increases. © 2006 Elsevier B.V. All rights reserved.
Elliptic grid generation, Free boundary problems, Liquid displacement by air, Viscoplastic fluids
]]>Zacharioudaki M., Kouris C., Dimakopoulos Y., Tsamopoulos J.
A Volume Tracking (VT) and a Front Tracking (FT) algorithm are implemented and compared for locating the interface between two immiscible, incompressible, Newtonian fluids in a tube with a periodically varying, circular crosssection. Initially, the fluids are stationary and stratified in an axisymmetric arrangement so that one is around the axis of the tube (core fluid) and the other one surrounds it (annular fluid). A constant pressure gradient sets them in motion. With both VT and FT, a boundaryfitted coordinate transformation is applied and appropriate modifications are made to adopt either method in this geometry. The surface tension force is approximated using the continuous surface force method. All terms appearing in the continuity and momentum equations are approximated using centered finite differences in space and onesided forward finite differences in time. In each time step, the incompressibility condition is enforced by a transformed Poisson equation, which is linear in pressure. This equation is solved by either direct LU decomposition or a Multigrid iterative solver. When the two fluids have the same density, the former method is about 3.5 times faster, but when they do not, the Multigrid solver is as much as 10 times faster than the LU decomposition. When the interface does not break and the Reynolds number remains small, the accuracy and rates of convergence of VT and FT are comparable. The wellknown failure of centered finite differences arises as the Reynolds number increases and leads to nonphysical oscillations in the interface and failure of both methods to converge with mesh refinement. These problems are resolved and computations with Reynolds as large as 500 converged by approximating the convective terms in the momentum equations by thirdorder upwind differences using Lagrangian Polynomials. When the volume of the core fluid or the Weber number decrease, increasing the importance of interfacial tension and leading to breakup of the interface forming a drop of core fluid, the FT method converges faster with mesh refinement than the VT method and upwinding may be required. Finally, examining the generation of spurious currents around a stationary “bubble” in the tube for Ohnesorge numbers between 0.1 and 10 it is found that the maximum velocity remains approximately the same in spite mesh refinements when VT is applied, whereas it is of the same order of magnitude for the coarsest mesh and monotonically decreases with mesh refinement when FT is applied. © 2007 Elsevier Inc. All rights reserved.
Front tracking, Multigrid, Multiphase flow, Upwinding, Volume tracking
]]>Lac E., BarthèsBiesel D., Pelekasis N.A., Tsamopoulos J.
The dynamic response of an initially spherical capsule subject to different externally imposed flows is examined. The neoHookean and Skalak et al. (Biophys. J., vol. 13 (1973), pp. 245264) constitutive laws are used for the description of the membrane mechanics, assuming negligible bending resistance. The viscosity ratio between the interior and exterior fluids of the capsule is taken to be unity and creepingflow conditions are assumed to prevail. The capillary number ε is the basic dimensionless number of the problem, which measures the relative importance of viscous and elastic forces. The boundaryelement method is used with bicubic Bsplines as basis functions in order to discretize the capsule surface by a structured mesh. This guarantees continuity of second derivatives with respect to the position of the Lagrangian particles used for tracking the location of the interface at each time step and improves the accuracy of the method. For simple shear flow and hyperbolic flow, an interval in ε is identified within which stable equilibrium shapes are obtained. For smaller values of ε, steady shapes are briefly captured, but they soon become unstable owing to the development of compressive tensions in the membrane near the equator that cause the capsule to buckle. The postbuckling state of the capsule is conjectured to exhibit small folds around the equator similar to those reported by Walter et al. Colloid Polymer Sci. Vol. 278 (2001), pp. 123132 for polysiloxane microcapsules. For large values of ε, beyond the interval of stability, the membrane has two tips along the direction of elongation where the deformation is most severe, and no equilibrium shapes could be identified. For both regions outside the interval of stability, the membrane model is not appropriate and bending resistance is essential to obtain realistic capsule shapes. This pattern persists for the two constitutive laws that were used, with the Skalak et al. law producing a wider stability interval than the neoHookean law owing to its strain hardening nature. © 2004 Cambridge University Press.
Dimakopoulos Y., Tsamopoulos J.
We examine the displacement by pressurized air of a liquid, which only partially occupies straight or complex tubes, according to the GasAssisted Injection Molding (GAIM) process. The process involves the formation and continuous elongation of a gaseous finger, which sets in motion the liquid, which, in turn, forms a second moving interface with gas downstream, the advancing front, and is simultaneously deposited on the tube walls. The motion of the advancing front is simulated using a Naviertype slip condition. A complete parametric analysis is performed in order to examine the effects of the initial amount of liquid, inertia, liquid compressibility, and the slip coefficient. Simulations under creeping flow conditions and relatively large initial amounts of liquid show that the thickness of the deposited film on the inner tube wall is uniform for the most part, except near the tube entrance and where the still moving portion of the liquid is nearly depleted. Increasing inertia causes flattening of the liquid front, nonuniform film distribution along the wall and eventually a tipsplitting instability. Liquid compressibility influences the phenomenon only slightly. The difference between the two interfacial velocities increases as the noslip condition is approached and eventually leads to their collision. Finally, coating of an expanding tube of finite length and either closed or open downstream is examined for various amounts of liquid initially placed in it. © 2005 Society of Plastics Engineers.
Karapetsas G., Tsamopoulos J.
The transient, axisymmetric squeezing of viscoplastic materials under creeping flow conditions is examined. The flow of the material even outside the disks is followed. Both cases of the disks moving with constant velocity or under constant force are studied. This timedependent simulation of squeeze flow is performed for such materials in order to determine very accurately the evolution of the force or the velocity, respectively, and the distinct differences between these two experiments, the highly deforming shape and position of all the interfaces, the effect of possible slip on the disk surface, especially when the slip coefficient is not constant, and the effect of gravity. All these are impossible under the quasisteady state condition used up to now. The exponential constitutive model, suggested by Papanastasiou, is employed. The governing equations are solved numerically by coupling the mixed finite element method with a quasielliptic mesh generation scheme in order to follow the large deformations of the free surface of the fluid. As the Bingham number increases, large departures from the corresponding Newtonian solution are found. When the disks are moving with constant velocity, unyielded material arises only around the two centers of the disks verifying previous works in which quasisteady state conditions were assumed. The size of the unyielded region increases with the Bingham number, but decreases as time passes and the two disks approach each other. Their size also decreases as the slip velocity or the slip length along the disk wall increase. The force that must be applied on the disks in order to maintain their constant velocity increases significantly with the Bingham number and time and provides a first method to calculate the yield stress. On the other hand, when a constant force is applied on the disks, they slow down until they finally stop, because all the material between them becomes unyielded. The final location of the disk and the time when it stops provide another, probably easier, method to deduce the yield stress of the fluid. © 2005 Elsevier B.V. All rights reserved.
Free surface flows, Transient squeeze flow, Viscoplastic fluids
]]>Foteinopoulou K., Mavrantzas V.G., Dimakopoulos Y., Tsamopoulos J.
Our recent finite elementbased study of the deformation of a single bubble in a Newtonian or viscoelastic filament undergoing stretching is extended here to the case of multiple bubbles simultaneously growing in the stretched medium. The filament, having initially the shape of a cylinder with uniform radius, is confined between two disks and is continuously stretched by pulling the upper disk along the filament axis with a constant velocity; the lower disk is assumed stationary. All bubbles are taken to lie along the axis of symmetry of the filament and undergo deformation and/or growth with the medium being stretched. The governing equations are solved by a finite element/Galerkin method coupled with an implicit Euler method for the time integration, using an adaptive time step. The problem of the multiple bubbleliquid interfaces is addressed by a robust meshgeneration scheme that solves a set of elliptic differential equations for the locations of the nodal points. The resulting numerical scheme is accurate and extremely stable, independently of the number of bubbles assumed in the filament. It has allowed us to address multiple bubble growth in the filament, and investigate how their interaction affects the response of the system to the applied deformation. Numerical results are presented quantifying the dependence of bubble dynamics on bubbleliquid surface tension, filament aspect ratio (especially as this is decreased to very low values), relative bubble size, and bubblebubble separation. The tensile force on the upper plate is also calculated and reported as a function of time and number of bubbles present in the film. Overall, our numerical calculations demonstrate the dominant role of the pressure field in the elongating filament: for the case of Newtonian fluids considered here, the force needed to maintain the flow comes solely from the pressure field, while for geometries with small aspect ratio, pressure attains large negative values near the centerline, in accord with the predictions of simple arguments from lubrication theory to the extent this applies to this problem. © 2006 American Institute of Physics.
Dimakopoulos Y., Tsamopoulos J.
The displacement of viscous liquids by pressurized gas from harmonically undulated tubes of finite length is examined. This unsteady process gives rise to a long open bubble of varying radius, increasing length and surrounded by the liquid. In general, the thickness of the liquid film that remains on the tube wall is nonuniform. Under creeping flow conditions, it varies periodically, but with a phase difference from the tube radius. The liquid fraction remaining in each periodic segment of the tube increases as the ratio between the minimum and maximum of the tube radius S decreases, whereas it tends to the wellknown asymptotic value for straight tubes as S → 1, or as the wavelength of the tube undulation increases, although here the flow is accelerating. At highvalues of the Reynolds number, the film thickness increases with the axial distance, and the periodicity of the flow field ahead of the bubble tip, which exists under creeping flow conditions, is broken. At even higher Reynolds numbers, recirculating vortices develop inside each tube expansion and when S also decreases significantly, nearly isolated bubbles are formed in each tube segment. The location of the bubble tip can be monitored by examining the time variation of the pressure at the tube wall. © 2006 American Institute of Chemical Engineers.
Elliptic mesh generation, Flow in undulated tubes, Gasassisted injection molding, Liquid displacement by gas, Moving boundary problems, Oil recovery
]]>Pelekasis N.A., Gaki A., Doinikov A., Tsamopoulos J.A.
The translational velocities of two spherical gas bubbles oscillating in water, which is irradiated by a highintensity acoustic wave field, are calculated. The two bubbles are assumed to be located far enough apart so that shape oscillations can be neglected. Viscous effects are included owing to the small size of the bubbles. An asymptotic solution is obtained that accounts for the viscous drag on each bubble, for large Re based on the radial part of the motion, in a form similar to the leadingorder prediction by Levich (1962), CD = 48/ReT; ReT → ∞ based on the translational velocity. In this context the translational velocity of each bubble, which is a direct measure of the secondary Bjerknes force between the two bubbles, is evaluated asymptotically and calculated numerically for sound intensities as large as the Blake threshold. Two cases are examined. First, two bubbles of unequal size with radii on the order of 100 μm are subjected to a sound wave with amplitude PA < 1.0 bar and forcing frequency ωf = 0.51ω10, so that the second harmonic falls within the range defined by the eigenfrequencies of the two bubbles, ω10 < 2ωf < ω20. It is shown that their translational velocity changes sign, becoming repulsive as PA increases from 0.05 to 0.1 bar due to the growing second harmonic, 2ωf, of the forcing frequency. However, as the amplitude of sound further increases, PA ≈ 0.5 bar, the two bubbles attract each other due to the growth of even higher harmonics that fall outside the range defined by the eigenfrequencies of the two bubbles. Second, the case of much smaller bubbles is examined, radii on the order of 10 μm, driven well below resonance, ωf/2π = 20 kHz, at very large sound intensities, PA ≈ 1 bar. Numerical simulations show that the forces between the two bubbles tend to be attractive, except for a narrow region of bubble size corresponding to a nonlinear resonance related to the Blake threshold. As the distance between them decreases, the region of repulsion is shifted, indicating sign inversion of their mutual force. Extensive numerical simulations indicate the formation of bubble pairs with constant average interbubble distance, consisting of bubbles with equilibrium radii determined by the primary and secondary resonance frequencies for small and moderate sound amplitudes or by the Blake threshold for large sound amplitudes. It is conjectured that in experiments where ‘acoustic streamers’ are observed, which are filamentary structures consisting of bubbles that are aligned and move rapidly in a cavitating fluid at nearly constant distances from each other, bubbles with size determined by the Blake threshold are predominant because those with size determined by linear resonance are larger and therefore become unstable due to shape oscillations. © 2004 Cambridge University Press.
Talaslidis D.G., Manolis G.D., Paraskevopoulos E., Panagiotopoulos C., Pelekasis N., Tsamopoulos J.A.
A modular analysis package is assembled for assessing risk in typical industrial structural units such as steel storage tanks, due to extreme transient loads that are produced either as a result of chemical explosions in the form of atmospheric blasts or because of seismic activity in the form of ground motions. The main components of the methodology developed for this purpose are as follows: (i) description of blast overpressure and ground seismicity, (ii) transient nonlinear finite element analysis of the industrial structure, (iii) development of 3Dequivalent, continuous beam multidegreeoffreedom structural models, (iv) introduction of soilstructure interaction effects, (v) probabilistic description of the loading process and the stiffness/mass characteristics of the structure, and (vi) generation of fragility curves for estimation of structural damage levels by using the Latin hypercube statistical sampling method. These fragility curves can then be used within the context of the engineering analysisdesign cycle, so as to minimize structural failure probability under both manmade hazards such as blasts and natural hazards such as earthquakeinduced transient loads. © 2004 Elsevier Ltd. All rights reserved.
Chemical explosions, Finite elements, Fragility curves, Industrial structures, Latin hypercube method, Risk analysis, Seismic loads, Structural dynamics
]]>Dimakopoulos Y., Tsamopoulos J.
We examine the transient displacement of viscoelastic fluids by a gas in straight cylindrical tubes of finite length. For the simulation of the processes, the mixed finite element method is combined with a quasielliptic grid generation scheme for discretizing the highly deforming domain of the liquid and the discontinuous Galerkin (DG) method for calculating the polymeric stresses. In addition the effectiveness of other formulations is examined: streamline upwind/Petrov (SUPG), DG/discrete elastic viscous stress splitting (DEVSS), SUPG/DEVSS and the latter two also with the DEVSSG approximation with either the PTT or Giesekus constitutive models. A parametric analysis is made in order to examine the effects of elastic and inertia forces and of the Newtonian viscosity on the process. Results using the PTT constitutive model show that the thickness of the remaining film increases as the Deborah number increases and that remaining fluid fractions greater than 0.60 arise, the Newtonian limit, in agreement with experiments. However, the thickness of the remaining film is not as large as with Boger fluids due to the shear thinning nature of the PTT fluid model. Moderate and high values of the Deborah number cause the development of extremely sharp stress boundary layers, which affect eventually the stability of the applied numerical scheme. Increasing the contribution of the Newtonian viscosity decreases the effects of the viscoelastic model. In particular, the bubble motion is decelerated, the thickness of the remaining film tends to the Newtonian limit and the polymeric stresses decrease in magnitude. In the fully developed region, wellahead of the penetrating gas, the predictions for the distribution of vz, τzz, τ rz are in qualitative agreement with the semianalytical work of [J. Fluid Mech. 387 (1999) 271]. © 2004 Elsevier B.V. All rights reserved.
Finite elements, Gasassisted injection molding, Viscoelasticity
]]>Foteinopoulou K., Mavrantzas V.G., Tsamopoulos J.
Numerical results are presented concerning bubble growth in Newtonian and viscoelastic filaments undergoing stretching. In practice, such bubbles or cavities develop in materials (either in their bulk or at their interface with a substrate) such as the pressure sensitive adhesives. The problem we address here is how such an initially spherical bubble deforms inside a filament (which at the beginning has the shape of a cylinder with uniform radius) undergoing stretching with a constant pulling velocity. The lower end of the liquid filament is fixed at the substrate; stretching along the filament axis is achieved by means of an imposed force on the upper end. The governing equations consist of the momentum, continuity and constitutive equations and the free surface boundary conditions at the two interfaces (the bubbleliquid and the liquidair interfaces). These are solved by a finite element/Galerkin method coupled with a firstorder implicit Euler scheme for time integration. In the case of a viscoelastic medium (here the PhanThien/Tanner constitutive model is chosen), the elastic viscous stress splittingG (EVSSG) technique is used to separate the elastic and viscous contributions to the stress tensor together with a streamline upwind PetrovGalerkin (SUPG) discretization of the constitutive equation. Our numerical calculations provide information on the effect of a number of important parameters on bubble and filament growth rate and deformation. These include mainly the capillary and Deborah numbers. The effect of other variables such as the relative size of the bubble, the proximity of the bubble to and liquid slippage on the substrate are also studied and discussed in detail. © 2004 Elsevier B.V. All rights reserved.
Bubble growth, Cavitation, EVSSG, Filament stretching, Finite elements, Moving boundary problems, PhanThien/Tanner constitutive equation, SUPG, Surface tension, Viscoelasticity
]]>Kouris C., Tsamopoulos J.
The nonlinear dynamics of the concentric, twophase flow of two immiscible fluids in a circular tube is studied when the viscosity ratio of the fluid in the annulus to that in the core of the tube, μ, is larger than or equal to unity. For these values of the viscosity ratio the perfect coreannular flow (CAF) is linearly unstable and it is necessary to keep the ratio of the thickness of the annulus to the radius of the tube small so that the solutions remain uniformly bounded. The simulations are based on a pseudospectral numerical method while special care has been taken in order to minimize as far as possible the effect of the boundary conditions imposed in the axial direction allowing for multiple waves of different lengths to develop and interact. The time integration originates with the analytical solution for the pressure driven, perfect CAF or the perfect CAF seeded with either the most unstable mode or random disturbances. Quite regular wave patterns are predicted in the first two cases, whereas multiple unstable modes grow and remain even after saturation of the instability in the last case. The resulting waves generally travel in the same direction and faster than the undisturbed interface, except for the case with μ=1 for which they are stationary with respect to it. Depending on parameter values, waves move with the same velosity or interact with each other exchanging their amplitudes or merge and split giving rise to either chaotic or organized solutions. For fluids of equal viscosities and densities (μ=ρ1) and for a Reynolds number, on the properties of the inner fluid, the tube radius, R̂2, and the average flow velocity, Ŵ0, small amplitude waves are predicted. The increase of μ by almost two orders of magnitude does not affect their amplitudes, but increases their temporal period linearly. Varying W by more than three orders of magnitude increases their amplitudes proportionately, while their period increases with the logarithm of W. Similar to that is the effect of increasing Re. The present analysis confirms and extends results based on long wave expansions, which lead to the KuramotoSivashinsky equation and modifications of it. © 2002 American Institute of Physics.
Dimakopoulos Y., Tsamopoulos J.
We examine the transient displacement of a viscoplastic material from straight or suddenly constricted cylindrical tubes of finite length. Our general goal is to develop accurate and efficient numerical methods for the fundamental study of processes in which a gas is displacing a liquid from prototype geometries under various operating conditions. Such processes can be part of the Gas Assisted Injection Molding (GAIM) or enhanced oil recovery. To this end, we use the mixed finite element method coupled with a quasielliptic mesh generation scheme in order to follow the very large deformations of the fluid volume. The displacing fluid is gas at high pressure, which forms a bubble of increasing length and a shape that depends on the fluid properties, the flow conditions, and the tube geometry. The crosssection of the bubble is always smaller than that of the tube due to adherence of fluid on the tube walls. The thickness of the remaining film depends on the same parameters and for most of its length it behaves as unyielded material. Unyielded material also arises in front of the bubble, around the axis of symmetry of the tube(s) and in the case of a constricted tube near the recirculation corner, but not around the entrance of the secondary tube. The rate of growth of the ‘tip splitting’ instability, that arises at relatively large values of the Reynolds number for Newtonian fluids in straight tubes, decreases as the Bingham number increases and, eventually, the instability disappears. The resistance provided by the constricted tube downstream makes the bubble move at a nearly constant velocity only when the Bingham number is not large. When the bubble approaches the constriction it becomes more pointed, but after entering it, the bubble reassumes its welldeveloped profile. Depending on parameter values, the bubble in the secondary tube may periodically split, thus forming a train of smaller bubbles directed towards the exit of the tube, a phenomenon for which experimental evidence exists. © 2003 Elsevier Science B.V. All rights reserved.
Finite element methods, Free surface flows, Gas displacing liquid from a tube, Viscoplastic fluids
]]>Dimakopoulos Y., Tsamopoulos J.
We study the displacement of a viscous fluid by highly pressurized air in a straight or a suddenly constricted cylindrical tube of finite length. In contrast to previous efforts, the transient situation is examined. A long, narrower than the tube and roundended bubble is created during the process. This is sometimes called “fingering instability” and is often encountered in several applications, but we will focus on process parameters that are relevant to the gasassisted injection molding. For our numerical simulations we have combined the mixed finite element method with an appropriate system of elliptic partial differential equations and boundary conditions, capable of generating a boundaryfitted finite element mesh. The bubble front and the thickness of the deposited film on the tube wall are affected by the properties of the fluid being displaced and the flow conditions. Specifically, in straight tubes, the bubble keeps accelerating due to the decreasing fluid mass ahead of it. Increasing the Reynolds number decreases the film thickness and makes the bubble front steeper. When inertia becomes significant a tipsplitting instability arises. For sufficiently low Reynolds numbers, but still large applied pressures, a steady bubble shape is attained even in a straight tube and the fraction of the liquid deposited on the wall of the tube reaches the asymptotic value of 0.60, as observed by Taylor [J. Fluid Mech. 10, 161 (1961)] and Cox [J. Fluid Mech. 14, 81 (1962)]. In a constricted tube, the bubble temporarily attains a nearly constant velocity. When its front approaches the tube constriction it becomes pointed, due to the extensional flow that prevails there, but it reassumes its fingerlike profile, after it goes through the constriction. © 2003 American Institute of Physics.
Kouris C., Dimakopoulos Y., Georgiou G., Tsamopoulos J.
A Galerkin/finite element and a pseudospectral method, in conjunction with the primitive (velocitypressure) and streamfunctionvorticity formulations, are tested for solving the twophase flow in a tube, which has a periodically varying, circular cross section. Two immiscible, incompressible, Newtonian fluids are arranged so that one of them is around the axis of the tube (core fluid) and the other one surrounds it (annular fluid). The physical and flow parameters are such that the interface between the two fluids remains continous and singlevalued. This arrangement is usually referred to as CoreAnnular flow. A nonorthogonal mapping is used to transform the uneven tube shape and the unknown, time dependent interface to fixed, cylindrical surfaces. With both methods and formulations, steady states are calculated first using the NewtonRaphson method. The most dangerous eigenvalues of the related linear stability problem are calculated using the Arnoldi method, and dynamic simulations are carried out using the implicit Euler method. It is shown that with a smooth tube shape the pseudospectral method exhibits exponential convergence, whereas the finite element method exhibits algebraic convergence, albeit of higher order than expected from the relevant theory. Thus the former method, especially when coupled with the streamfunctionvorticity formulation, is much more efficient. The finite element method becomes more advantageous when the tube shape contains a cusp, in which case the convergence rate of the pseudospectral method deteriorates exhibiting algebraic convergence with the number of the axial spectral modes, whereas the convergence rate of the finite element method remains unaffected. Copyright © 2002 John Wiley & Sons, Ltd.
Coreannular flow, Finite element method, Pseudospectral method, Twophase flow, Undulating tube
]]>Dimakopoulos Y., Tsamopoulos J.
We have developed a quasielliptic set of equations for generating a discretization mesh that optimally conforms to an entire domain that undergoes large deformations in primarily one direction. We have applied this method to the axisymmetric problem of the transient displacement of a viscous liquid by a highpressure gas. The liquid initially fills completely a tube the diameter of which may be constant or change smoothly, suddenly or periodically along its finite length. Key ingredients for the success of the proposed transformation are limiting the orthogonality requirements on the mesh and employing an improved node distribution function along the deforming boundary. The retained orthogonal term along with the penalty method for the imposition of the boundary conditions overcome the inherent restrictions of a conformal transformation, producing meshes of high quality. This term also eliminates the discontinuous slopes of the coordinate lines that are normal to the free surface. These usually arise due to the harmonic transformation around highly deforming surfaces. The mathematical anisotropy in the mesh generating equations directly corresponds to the physical one, when the initially straight and normal to the axis of symmetry air/fluid interface, at the tube entrance, develops mostly along the axial direction, generating a long, open bubble. Moreover, the generalized node distribution imposes an optimal discretization of the bubble surface itself. Combining this scheme with the mixed finite element method produces a powerful tool, with extended robustness and accuracy. In order to reduce the computational cost, the resulting set of equations is solved with a nonlinear, GaussSeidel technique and a variable time step. Specific applications of the proposed scheme are presented. © 2003 Elsevier B.V. All rights reserved.
Adaptive timestepping, Finite element methods, Liquid displacement by gas, Mesh generation, Moving boundary problems
]]>Smyrnaios D.N., Pelekasis N.A., Tsamopoulos J.A.
The steady twodimensional laminar flow of a stream of saturated vapor flowing over a tube that is kept at a uniform temperature, below the saturation temperature, is examined. Owing to the temperature difference between the vapor stream and the solid surface a film of condensate is generated that flows along the surface due to shear, pressuredrop, and gravity. In the limit as the boundary layer and film thickness remain smaller than the radius of curvature of the surface a simplified lubricationtype formulation describes the temperature and flow fields in the film, whereas the usual boundary layer formulation is applied in the vapor boundary layer. The case of flow past a horizontal cylinder of radius R is investigated numerically with the oncoming stream aligned with gravity. The parameters that control momentum and heat transfer in this problem are the viscosity ratio, μcs/μss, the density ratio ρcs/ρss, the Prandtl number, Pr=Cpcsμcs/Kcs, the Froude number, Fr=U2∞/(gπR), and finally the thickness ratio between the condensate and the vapor boundary layer, ε, which is also a measure of the temperature difference between the vapor stream and the tube wall. Then, the Nusselt number and the skin friction coefficient, averaged over the upper half of the cylinder, are calculated for a wide parameter range. When Fr is very small and ε relatively large the flow remains attached until the trailing stagnation point of the cylinder. As the effect of adverse pressure drop becomes more pronounced (Fr increases or ε decreases) it is shown that the solution exhibits two different types of singularity in the rear part of the cylinder. The first one is a typical Goldstein singularity because it appears at the tube wall and it is associated with vanishing skin friction (wall shear) and rapidly increasing film thickness. The second one takes place near the interface between the vapor stream and the film of condensate in a region where very small velocities prevail in conjunction with vanishing shear rate. The latter has not been reported so far and it is expected to affect the flow locally, as opposed to the Goldstein singularity which is known to lead to massive separation in the case of a cylindrical surface. Upon proper rescaling of Fr and ε, Fr′= Fr ρss/ρcs, © 2002 American Institute of Physics.
Kouris C., Tsamopoulos J.
Nonlinear dynamics of the concentric, twophase flow of two immiscible fluids in a circular tube of variable crosssection is studied for parameter values where the steady coreannular flow (CAF) is linearly unstable. The simulations are based on a pseudospectral numerical method. They are carried out assuming axial symmetry, that the total flow rate remains constant and that all dependent variables are periodic in the axial direction, which includes the minimum necessary number of repeated units so that the obtained solution is independent of this number. The time integration originates with the numerically computed steady CAF or the steady CAF seeded with either the most unstable mode or random small disturbances. Only a limited number of the most interesting cases are presented. For the most part, the values of the majority of the dimensionless parameters are such that oil flows in the centre of the tube driven by an applied pressure gradient against gravity, whereas water is flowing in the annulus. It is shown that, whereas the steady (unstable) solution may indicate that the heavier water flows countercurrently with respect to the oil, the time periodic (observable) solution may indicate the same, albeit at a much smaller core flow rate or that concurrent flow occurs. This is due to the water being trapped between the largeamplitude interfacial waves that are generated and being convected by the oil. It is also shown that increasing the inverse Weber number increases the wave amplitude to the point that the flow of the core fluid may become discontinous with a mechanism that depends on the viscosity ratio between the two fluids. Increasing the amplitude of the sinusoidal variation of the tube leads to a combination of travelling and standing waves, which interact to produce a time periodic solution with a long period associated with the time it takes the travelling wave to travel through the computational domain and a second much shorter period that is related to their interaction time. Qualitative agreement has been obtained upon comparing our numerical simulations with limited experimental reports, even though the experimental conditions were not identical to those in our model.