Kouris C., Tsamopoulos J.
The concentric, two-phase flow of two immiscible fluids in a tube of sinusoidally varying cross section is studied. Assuming that the tube radius is much smaller than the period of the constriction, the Navier-Stokes equations in each phase are simplified accordingly. This geometry is used often as a model to study the onset of different flow regimes in packed beds. The relevant Reynolds number is not assumed to be a priori small, since inertia in the axial momentum balance is known to be important in generating different flow regimes. The curvature of the fluid/fluid interface is not approximated according to the lubrication approximation in order to create a well-defined set of equations in the sense of Hadamard. The model depends on six dimensionless parameters: the Reynolds, Froude and Weber numbers and the ratios of density, viscosity and volume of the two fluids. Two more dimensionless numbers describe the shape of the solid wall: the constriction ratio and the ratio of its maximum radius to its period. The equations are solved using the pseudo-spectral methodology and the Arnoldi algorithm for eigenvalue calculations. Stationary solutions are obtained for a wide parameter range and may exhibit flow recirculation in the wider part of the tube. Extensive calculations for the dependence of the neutral stability boundaries on the various parameters are performed. In all cases that the steady solution is found to become unstable it does so through a Hopf bifurcation. Finally, the time evolution of the nonlinear equations for parameter values well into the unstable region is performed and it shows that a perturbed steady solution eventually leads to an oscillatory flow of constant amplitude.