Stability of the viscous flow of a polymeric fluid past a flexible surface

The instability in plane Couette flow of a viscoelastic fluid past a deformable surface is examined using the temporal linear stability theory in the zero Reynolds number limit. The polymeric fluid is described using the Oldroyd-B model and the flexible wall is modeled as a linear viscoelastic solid surface. The analysis shows that the wall flexibility tends to reduce the decay rate of the stable discrete modes for the polymeric flow past a rigid wall, and one of the discrete modes becomes unstable when the wall deformability parameter $\Gamma=V \eta /(GR)$ exceeds a certain critical value $\Gamma_c$. Here, V is the top-plate velocity, \eta is the zero shear viscosity of the polymeric fluid, G is the shear modulus of the wall, and R is the width of the fluid layer. The analysis reveals the presence of two classes of modes, the first of which becomes unstable for perturbations with wavelength comparable to the channel width (finite wavelength modes), and the second becomes unstable for perturbations with wavelength small compared to the channel width (short wave modes). The latter class of modes are found to be absent for the highly concentrated polymer solutions with \beta\leq 0.23, where \beta is the ratio of solvent-to-solution viscosity. We have mapped out the regions in the parameter space \bar{W}-H) where the finite wavelength and short wave modes are unstable, where $\bar{W}=(\lambda G/ \eta)$, and lambda is the relaxation time of the viscoelastic fluid. Fluid elasticity is found to have a stabilizing influence on the unstable mode, such that when the shortwave instability is absent for \beta \leq 0.23, the flow becomes stable for any Weissenberg number $\bar{W}>\bar{W}_ {max}$. Here, $\bar{W}_{max}$ increases proportional to H for H\gg1. However, when the shortwave instability is present, the instability persists for \bar{W}\gg 1. The behavior of both classes of modes with respect to the parameters, like \bar{W}, H, \beta, and the ratio of solid-to-fluid viscosity $\eta_o$ , is examined