Direct numerical simulations of two-layer viscosity-stratified flow

Two-dimensional simulations of flow instability at the interface of a two-layer, density-matched, viscosity-stratified Poiseuille flow are performed using a front-tracking/finite difference method. We present results for the small-amplitude (linear) growth rate of the instability at small to medium Reynolds number for varying thickness ratio n, viscosity ratio m, and wavenumber. We also present results for large-amplitude non-linear evolution of the interface for varying viscosity ratio and interfacial tension. For the linear case, the interfacial mode is neutrally stable for n = m as predicted by analysis. The growth rate is proportional to Reynolds number for small Re, and increases with viscosity ratio. The growth rate also increases when the thickness of the more viscous layer is reduced. Strong non-linear behavior is observed for relatively large initial perturbation amplitude. The higher viscosity fluid is drawn out as a finger that penetrates into the lower viscosity layer. The simulated interface shape compares well with previously reported experiments. Increasing interfacial tension retards the growth rate of the interface as expected, whereas increasing the viscosity ratio enhances it. Drop formation at the small Reynolds number considered in this study is precluded by the two-dimensional nature of the calculations.