Stability of two-layer viscoelastic plane Couette flow past a deformable solid layer

Abstract

The stability of two-layer plane Couette flow of upper-convected Maxwell (UCM) fluids of thicknesses (1− β) R and βR, with matched viscosities η, and relaxation times τ 1 and τ 2 past a soft, deformable solid layer (modeled here as a linear viscoelastic solid fixed to a rigid plate) of thickness HR, shear modulus G and viscosity η w is determined using a temporal linear stability analysis in the creeping-flow regime where the inertia of the two fluids and the solid layer is negligible. Interfaces with and without the interfacial tension between the two fluids are considered, while the interfacial tension between the UCM fluid and the deformable solid is neglected. It is well known that the two-layer Couette flow of UCM fluids with different relaxation times undergoes a purely elastic interfacial instability (referred here as mode 1) in rigid-walled channels, even in the creeping flow limit. It was recently shown [J. Non-Newtonian Fluid Mech 116 (2004) 371] that the plane Couette flow of a single UCM fluid past a deformable solid layer also undergoes an instability (referred here as mode 2) in the creeping flow limit when the nondimensional solid elasticity parameter Γ= Vη/( GR) exceeds a certain critical value, for a given Weissenberg number W= τV/ R. In this study, the respective effects of solid layer deformability and fluid elasticity stratification on mode 1 and mode 2, and the concomitant interaction of these two qualitatively different interfacial modes are analyzed in detail. It is shown that the deformability of the solid layer has a dramatic effect on the purely elastic (mode 1) interfacial instability between the two UCM fluids: When the more elastic UCM fluid is present in between the less elastic UCM fluid and the solid layer, the layer deformability has a stabilizing effect. In this configuration, if the thickness of the more elastic fluid is smaller than the less elastic fluid, it is shown that it is possible to stabilize the mode 1 purely elastic instability (except for very long and very short waves, the latter being stabilized usually by the nonzero interfacial tension between the two fluids) by making the solid layer sufficiently deformable (i.e., Γ exceeding a critical value), while such a configuration is unstable in the case of two-layer flow in rigid channels (i.e., in the absence of the deformable solid layer). Increase in the solid layer deformability, on the other hand, destabilizes the interfacial mode between the UCM fluid and the deformable solid (mode 2). It is demonstrated that it is possible to choose the solid elasticity parameter Γ and viscosity ratio η w/ η such that both mode 1 and mode 2 are completely stable in finite experimental geometries. In marked contrast, when the less elastic fluid is present in between the more elastic fluid and the solid layer, it is shown that solid layer deformability has a destabilizing effect on mode 1. For this configuration, if the thickness of the less elastic fluid is smaller, it is shown that mode 1 can be rendered unstable by increasing solid layer deformability, while it is stable in rigid channels (in the absence of the solid layer). In both of the above configurations, the nondimensional solid layer elasticity required respectively to stabilize or destabilize mode 1 varies as Γ∝( k * R) −1 for ( k * R)≪1, where k * is the wavenumber of perturbations. However, for finite experimental geometries the minimum allowed k * is dictated by the system length, and in such cases, it is argued that complete stabilization or destabilization can be achieved for realistic values of Γ. In both the configurations, our results show that deformable solid layer coatings, by design or default, can completely suppress or induce purely elastic interfacial instabilities in two-layer flow of viscoelastic fluids past a deformable solid surface.