Flow Induced Instability of the Interface between a Fluid and a Gel

The stability of the flow of a fluid adjacent to a polymer gel is studied using a linear stability analysis. The system consists of a Newtonian fluid of density rho, viscosity eta and thickness R flowing adjacent to a polymer gel of density rho, modulus of elasticity E, viscosity eta(g) and thickness HR. The base flow in the fluid is a plane Couette flow. The Navier-Stokes equations for the fluid and the elasticity equations for the gel are solved numerically, and the characteristic equation is obtained using the boundary conditions at the interface. The characteristic equation for the growth rate is a non-linear equation, so analytical solutions cannot be obtained in general. Numerical solutions are obtained by analytic continuation using the exact solutions at zero Reynolds number as the starting guess. The growth rate depends on the parameter Sigma = (rho ER2/eta(2)), the ratio of thickness H, the ratio of viscosities eta(r) = (eta(g)/eta) and wave number k. For eta(r) = 0, it is found that the perturbations become unstable when the Reynolds number is increased beyond a transition value Re-t for all Sigma and k. The critical Reynolds number Re-c, which is the minimum of the Re-t - K curve, increases proportional to Sigma for Sigma much less than 1, and it shows a scaling behavior Re-c proportional to Sigma(beta) for Sigma much greater than 1, where 0.75 less than or equal to beta less than or equal to 0.8. Re-c decreases with increase in the ratio of thickness of gel to fluid H, but the scaling behavior remains unchanged. A variation in the ratio of viscosities eta(r) qualitatively changes the stability characteristics. For relatively low values of 1 less than or equal to Sigma less than or equal to 10(3), it is found that the transition Reynolds number decreases as eta(r) is increased, indicating that an increase in the gel viscosity has a destabilizing effect. For relatively higher values of 10(4) less than or equal to Sigma less than or equal to 10(5), the transition Reynolds number increases as eta(r) is increased and goes through a turning point. In this case, perturbations are unstable only when eta(r) is less than a maximum value eta(max), and there is no instability for eta(r) > eta(max).