Stability of two-layer viscoelastic plane Couette flow past a deformable solid layer: implications of fluid viscosity stratification

The linear stability of two-layer plane Couette flow of upper-convected Maxwell (UCM) fluids of thicknesses (1−β)R and βR, with viscosities μa and μb, and relaxation times τa and τb past a linear viscoelastic solid layer (of thickness HR, shear modulus G, and viscosity ηw) is determined using a combination of low wavenumber asymptotic analysis and a numerical method. The asymptotic analysis is used to determine the effect of the deformable solid layer on the two-fluid interfacial instability due to elasticity and viscosity stratification in the low wavenumber limit. The asymptotic results show that as the solid layer is made more deformable, the two-fluid interfacial instability could be completely stabilized in the long wave limit, when the non-dimensional parameter Γ=Vμb/(GR)∼O(1) exceeds a certain critical value. Here, V is the dimensional velocity of the moving top plate. In marked contrast with the previous results obtained for matched fluid viscosities [V. Shankar, J. Non-Newtonian Fluid Mech., 117 (2004) 163], the present asymptotic results show that the stabilizing or destabilizing nature of the solid layer on the two-fluid interfacial instability is controlled only by the fluid viscosity ratio and is independent of the relaxation times of the two UCM fluids. In general, it is found that the solid layer has a stabilizing effect for β<0.5,μr<1 and β>0.5,μr>1, while it has both stabilizing and destabilizing effects (depending on its thickness H, and the fluid thickness ratio β) for β>0.5,μr<1 and β<0.5,μr>1. In the absence of the solid layer, the two-fluid interfacial mode is unstable or stable depending on the ratio of relaxation times between the two fluids and the thickness ratio β. It is thus possible, under appropriate combinations of relaxation times, viscosities, and thicknesses of the two fluids, to stabilize (destabilize) the two-fluid interfacial mode by the deformable solid layer while it is unstable (stable) in the absence of the solid layer. Another important result from the present low wavenumber (denoted by k) analysis is that the non-dimensional solid elasticity parameter Γ required to stabilize or destabilize the two-fluid interfacial mode is an O(1) quantity in the low-k-limit when μa≠μb, with numerical values of Γ significantly smaller than 1 for H∼O(1). Whereas when μa=μb our asymptotic analysis (as well as the earlier study) shows that Γ∝k−1 for k≪1. The results from the low-k-asymptotic analysis are continued numerically to finite values of k and Reynolds number, and the numerical results confirm that the stabilization of the two-fluid interfacial mode by the solid layer extends to finite values of k. However, for short wavelength fluctuations with k≫1, the fluid velocity perturbations are localized near the two-fluid interface, so the solid layer has no effect on these fluctuations. These short wavelength unstable modes can be stabilized only by the presence of a sufficiently strong interfacial tension between the two fluids. Thus, the present study shows that the viscosity mismatch between the two fluids profoundly changes the stabilizing or destabilizing effect of the deformable solid layer on the two-fluid interfacial mode in viscoelastic fluids when compared to the conclusions reached using matched fluid viscosities.