Instability of gravity-driven free-surface flow past a deformable elastic solid

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 ...