Stability of plane Poiseuille flow and plane Couette flow

The stability of plane Poiseuille flow of an incompressible viscous fluid between a rigid plate and a layer of incompressible isotropic elastic solid was studied. Two distinct modes of instability were found. The interfacial mode occurred at small Reynolds number when the solid was soft. For this mode, the rigidity of the solid was a stabilizing factor, whereas the viscosity of the fluid was destabilizing. There was a critical elastic parameter, beyond which the flow became unstable regardless of the viscosity of the fluid. The shear wave mode occurred at large Reynolds number. The stability of plane Couette flow of an incompressible viscous fluid between a moving plate and a layer of incompressible isotropic elastic solid was also investigated. It was found that the instability of interfacial mode was essentially similar to that of plane Poiseuille flow. It required, however, softer solid or higher critical speed to induce the instability than does the plane Poiseuille flow.