Instability of creeping flow past a deformable wall: the role of depth-dependent modulus

Linear stability analysis is carried out to examine the effect of a depth-dependent modulus on the stability of creeping flow of a Newtonian fluid past an incompressible and impermeable linear elastic solid. Two different systems are considered: (i) Couette flow past a solid with a continuously varying modulus, and (ii) Couette flow past two adjacent solids with different thicknesses and moduli. For the first system, we find that between two configurations having the same average modulus, the more stable configuration is the one that has the higher modulus at the interface. In the case of two different configurations having the same interfacial modulus and the same average modulus, the more stable configuration is the one that has the higher modulus right below the interface. For the second system, we find that stability depends in a non-monotonic way on the modulus ratio (top modulus to bottom modulus) of the two solids. If the thickness of the top solid is less than a critical value, then increasing the modulus ratio initially causes the system to be less stable. Since this critical thickness decreases as the modulus ratio increases, increasing the modulus ratio beyond a certain point causes the system to be more stable. An analysis of the solid-solid interfacial boundary conditions suggests that the relationship between the stiffness of the top solid and the stability of the system is due to a jump in the base-state displacement gradient at the interface which creates a net perturbation displacement.