Investigating the influence of surface shear stress on the stability of gravity‐driven multilayer liquid flow

Abstract

We conducted a linear stability analysis on the behaviour of two gravity‐driven Newtonian liquid layers flowing down an inclined rigid plane, subjected to external shear stress at the free surface, in both the direction of and opposite to gravity. A long wave asymptotic analysis was conducted to examine the influence of several parameters, including liquid film thickness (β), viscosity ratio (μr), and external shear stress (τ), on the interfacial modes of both gas–liquid (GL) and liquid–liquid (LL). For viscosity ratios (μr) < 1 and Reynolds number (Re) = 0, the observations indicate that the LL mode exhibits instability attributed to O(k) instability. This phenomenon is manifest when an external shear stress is applied at the GL interface, specifically within a defined range of negative shear stress values. When and , the investigation reveals the presence of two complex conjugate roots when external shear stress opposes the direction of gravity. One of these roots, referred to as the positive root, is associated with the leading (or zero) order instability that renders the GL mode unstable. The other root corresponds to the leading (or zero) order stability, contributing to the stability of the LL mode. In contrast, when external shear stress acts along the gravity direction, we observe that the GL mode is stable while the LL mode becomes unstable due to O(k) instability for various values of β. This finding highlights the importance of the direction of external shear stress in determining the interfacial mode's stability behaviour.