Role of wall deformability on interfacial instabilities in gravity-driven two-layer flow with a free surface

Abstract

The linear stability of gravity-driven flow of two superposed Newtonian liquid layers down a deformable, inclined, wall is analyzed in order to examine the effect of wall deformability on the interfacial instabilities in the system. There are three distinct interfacial modes in this composite system, viz., gas-liquid (GL), liquid-liquid (LL), and liquid-solid (LS) modes. For a rigid-wall, the GL interface becomes unstable above a critical Reynolds number, while the stability of the LL interface depends on the relative placement of the liquid layers. When the more viscous liquid is adjacent to rigid surface, the LL mode becomes unstable beyond a critical Reynolds number (Re), while it becomes unstable even at Re=0 when the less viscous liquid is next to rigid-wall. Our asymptotic results show that solid deformability has a stabilizing effect on both GL and LL modes in the low-wavenumber limit when the more viscous liquid layer is near the deformable wall. Numerical results reveal that both the GL and LL interfacial instabilities can be suppressed for all wavenumbers when the solid layer becomes sufficiently deformable. With further increase in solid deformability, all three interfacial modes become unstable. However, the parameters characterizing the solid (shear modulus, thickness, and solid viscosity) can be chosen such that the GL and LL interfaces remain stable (which are otherwise unstable in flow down a rigid incline) at all wavenumbers without the destabilization of LS interface. When the thickness of the top (less viscous) liquid layer is greater, it is more difficult to obtain stable flow configuration by manipulating the solid parameters. When the less viscous liquid is adjacent to the deformable surface, solid deformability always has a destabilizing effect on LL interfacial mode, and it is not possible to simultaneously stabilize both GL and LL interfaces for this configuration.