The instability of sheared liquid layers

Abstract

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.