The stability of inertialess gravity-driven free-surface flow of a Newtonian fluid down an inclined plane covered with a deformable elastic solid is examined in this work. Although the inertialess flow of a Newtonian fluid down a rigid inclined plane cannot become unstable, the present system can become unstable if the applied strain, which is proportional to the ratio of gravity forces to elastic forces, becomes larger than a critical value. For linear elastic solids, the critical strain and corresponding wave number asymptote to constant values as the ratio of the solid thickness to the fluid thickness increases. In contrast, the critical strain and wave number for neo-Hookean solids continue to decrease as the thickness ratio increases. Examination of the eigenfunctions for the neo-Hookean solid reveals that the difference in the critical conditions is due to a coupling between base state and perturbation quantities in the neo-Hookean model that remains important even for large values of the thickness ratio. The instability arises due to deformability of the solid-fluid interface, and flow past this interface is found to be more stable than when the free surface is absent. The results of this work reinforce the importance of accounting for large displacement gradients in the solid when modeling elastohydrodynamic instabilities, and suggest that results obtained with linear constitutive models may be misleading when multiple interfaces are present.
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Acknowledgment is made to the Donors of the American Chemical Society Petroleum Research Fund for partial support of this research. S.K. also thanks the Shell Oil Company Foundation for support through its Faculty Career Initiation Funds program, and 3M for a Nontenured Faculty Award. We are grateful for resources from the University of Minnesota Supercomputing Institute.