The instability of sheared liquid layers

A prescribed shear stress applied to the free surface of a thin liquid layer sets up a steady shear flow. When the shear flow has a linear velocity, profile, Miles, using asymptotic analysis, finds critical values R c of the Reynolds number above which unstable travelling waves exist. However, Miles omits a term in the normal-stress boundary condition. We correct this omission and solve the appropriate Orr-Sommer-feld system numerically to obtain the critical conditions. For the case of a zero-surface-tension interface, we find that R c = 34.2, as compared with Miles’ value of R c = 203. As surface tension increases, R c asymptotes to the inviscid limit developed by Miles. The critical Reynolds number, critical wavenumber and critical phase speed are presented as functions of a non-dimensional surface tension. We investigate the mechanism of the instability through an examination of the disturbance-energy equation. When the shear flow has a parabolic velocity profile, we find a long-wave instability at small values of the Reynolds number. Numerical methods are used to extend these results to larger values of the wavenumber. Examination is made of the relation between this long-wave instability and profile curvature.