Two- and three-dimensional instabilities in the film blowing process


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 Oldroyd-B or the Phan-Thien 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) 1-19; 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) 609-625], we analyze the process by starting with the general three-dimensional mass and momentum balances and by formally and systematically applying the thin-film approximation. This procedure results in two-dimensional dynamic balances in both the axial and azimuthal directions. Although these balances are highly non-linear 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 non-axisymmetric 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) 2500-2516; K. Housiadas, J. Tsamopoulos, Unsteady extrusion of a viscoelastic annular film: I. General model and its numerical solution, J. Non-Newton. Fluid Mech. 88 (3) (2000) 229-259], which also involved the thin-film 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 extra-stress profiles. Subsequently, its linear stability is examined to two- and three-dimensional 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 non-axisymmetric disturbances, although two-dimensional 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, Oldroyd-B, Perturbation method, PTT constitutive models, UCM

DOI: 10.1016/j.jnnfm.2006.09.006