J. Phys. II France
Volume 4, Numéro 6, June 1994
Page(s) 893 - 911
DOI: 10.1051/jp2:1994173
J. Phys. II France 4 (1994) 893-911

Flow induced instability of the interface between a fluid and a gel at low Reynolds number

V. Kumaran, G. H. Fredrickson and P. Pincus

Department of Chemical and Nuclear Engineering and Materials Department, University of California, Santa Barbara, CA 93106, U.S.A.

(Received 10 October 1993, received in final form 28 December 1993, accepted 21 February 1994)

The stability of the interface between a gel of thickness HR and a Newtonian fluid of thickness R subjected to a linear shear flow is studied in the limit where inertial effects are negligible. The shear stress for the gel contains an elastic part that depends on the local displacement field and a viscous component that depends on the velocity field. The shear flow at the surface tends to destabilize the surface fluctuations, and the critical strain rate $\gamma_{\rm c}$, which is the minimum strain rate required for unstable fluctuations, is determined as a function of the dimensionless quantities H, $\eta_{\rm r} = (\eta_{\rm g}/\eta_{\rm f})$, and $T = (\Gamma/ER)$. Here $\eta_{\rm g}$ and $\eta_{\rm f}$ are the gel and fluid viscosities, E is the gel elasticity, $\Gamma$ is the surface tension of the gel - fluid interface and the strain rate $\gamma$ is scaled by ( $E/\eta_{\rm f}$). In the limit $H \to \infty$, we find that $\gamma_{\rm c}$ decreases proportional to H-1 independent of $\eta_{\rm r}$ and T. But at finite H, $\gamma_{\rm c}$ is strongly dependent on $\eta_{\rm r}$ and T. For $\eta_{\rm r} \geqslant 1$, the interface is stable for all values of the strain rate for $H < \sqrt{\eta_{\rm r}}$, while there are unstable traveling waves for $H > \sqrt{\eta_{\rm r}}$. For $\eta_{\rm r} = 1$ and $H \to 1$, we find that $\gamma_{\rm c} \propto(H - 1)^{1/2}$ for T=0 and $\gamma_{\rm c} \propto(H - 1)^{3/4}T^{1/4}$ for $T \ne 0$. For $\eta_{\rm r} > 1$, the analysis indicates that $\gamma_{\rm c} \propto(H-\sqrt{\eta_{\rm r}})^{-1}$ independent of T for $H \to \sqrt{\eta_{\rm r}}$. For $\eta_{\rm r} < 1$, the onset of instability depends strongly on the parameter T. For T = 0, the critical strain rate is finite in the limit $H \to 0$, while for $T \ne 0$ the critical strain rate diverges at a finite value of H. This minimum H decreases proportional to $\eta_{\rm r}$ for large T. The instability is caused by the energy transfer from the mean flow to the fluctuations due to the work done by the mean flow at the interface.

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