J. Phys. II France
Volume 7, Numéro 6, June 1997
Page(s) 947 - 963
DOI: 10.1051/jp2:1997164
J. Phys. II France 7 (1997) 947-963

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

L. Srivatsan and V. Kumaran

Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India

(Received 25 September 1996, received in final form 24 February 1997, accepted 28 February 1997)

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 $\rho_{\rm g}$, and thickness H R. 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 $\Sigma=(\rho ER^2/\eta^2)$, the ratio of thickness H, the ratio of viscosities $\eta_{\rm r}=(\eta_{\rm g}/\eta)$ and wave number k. For $\eta_{\rm r}=0$, it is found that the perturbations become unstable when the Reynolds number is increased beyond a transition value ${\rm Re_t}$ for all $\Sigma$ and k. The critical Reynolds number ${\rm Re_c}$, which is the minimum of the ${\rm Re_t}-k$ curve, increases proportional to $\Sigma$ for $\Sigma \ll 1$, and it shows a scaling behavior ${\rm Re_c} \varpropto \Sigma^\beta$ for $\Sigma \gg 1$, where $0.75 \le \beta \le 0.8.$ ${\rm Re_c}$, 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 $\eta_{\rm r}$ qualitatively changes the stability characteristics. For relatively low values of $1\le \Sigma \le 10^3$, it is found that the transition Reynolds number decreases as $\eta_{\rm r}$ is increased, indicating that an increase in the gel viscosity has a destabilizing effect. For relatively higher values of $10^4 \le \Sigma \le 10^5$, the transition Reynolds number increases as $\eta_{\rm r}$ is increased and goes through a turning point. In this case, perturbations are unstable only when $\eta_{\rm r}$ is less than a maximum value $\eta_{\max}$, and there is no instability for $\eta_{\rm r}>\eta_{\max}$.

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