Transient two-layer thin-film flow inside a channel

The flow of two superimposed newtonian layers in a channel is investigated numerically in this study. The two-layer film flows inside a long channel due to a pressure gradient. The scaled conservation equations for two-layer incompressible newtonian film flow are first introduced. The weighted residual approach first proposed by Amaouche et al. [Phys. Fluids 19, 084106 (2007)] is used for finding the suitable weight functions before depth averaging. Subsequently, a linear stability analysis of thin-film equations for two-layer Poiseuille flow is carried out. The formulas which give the asymptotic stability with respect to long-wave perturbations obtained with Navier-Stokes equations [J. Non-Newtonian Fluid Mech. 71, 1 (1997)] are recovered with our averaging equations. In order to mimic the disturbance effect on the coextrusion flow, the steady flow, which is simply the uniform flow of two layers of fluid inside a channel, is then perturbed at inception. Following a finite difference based scheme, two types of boundary conditions are considered at the channel inception, namely a Dirac-type pulse and periodic forcing. The perturbation takes the form of a wave packet which may or may not be amplified as it moves downstream, depending on the values of the parameters involved in the problem. Furthermore, Gaster's relation is used to calculate the spatial growth rate of perturbation. The values obtained from this relation are in good quantitative agreement with those coming from the numerical simulations of thin-film equations. Then, our averaging equations also describe the nonlinear behavior of the interfacial instabilities occurring for the Poiseuille flow of two thin layers.