Instability modes of a two-layer Newtonian plane Couette flow past a porous medium

Abstract

We explore the salient features of the different instability modes of a pressure-driven two-layer plane Couette flow confined between a moving wall and a Darcy-Brinkman porous layer. A linear stability analysis of the conservation laws leads to an Orr-Sommerfeld system, which is solved numerically with appropriate boundary conditions to identify the time and length scales of the instability modes. The study reveals that the movement of the confining wall together with the slippage at the porous-liquid interface originating from the flow inside the porous layer can stimulate a pair of finite-wave-number shear modes in addition to the long-wave interfacial mode of instability. The shear modes dominate the interfacial mode, especially when the frictional influence at the liquid layers is smaller due to the movement of the confining plate or due to the larger slippage at the porous-liquid interface. The shear modes are found to be present under all combinations of the viscosity μ(r) and thickness h(r) ratios of the liquid layers. This is in stark contrast to the two-layer flow confined between nonporous plates where the interfacial (shear) mode is observed only when μ(r)>h(r)(2) (μ(r)<h(r)(2)). Interestingly, the strength of one of the shear modes is found to increase with the velocity of the bounding moving plate, whereas the other shear mode gains strength in the presence of highly porous, permeable, and thick porous layers. The results reported can be of significant importance in the microscale two-phase flows where the presence of a bounding porous layer or moving wall can expedite the intermixing of layers to improve the multiphase mixing, heat and mass transfer, and emulsification characteristics.