The instability of sheared liquid layers

The critical values of the Reynolds number (Rc) above which unstable travelling waves exist in a steady shear flow are calculated with the inclusion of a term in the normal stress boundary condition omitted by Miles (1960). Results show a value of Rc = 34.2, compared to Miles' value of 203, for the case of a zero-surface tension interface. The critical wavenumber, critical phase speed, and Rc are calculated as functions of a nondimensional surface tension, and the mechanism of the instability is investigated by examining the disturbance energy equation. A long wave instability is found at small values of the Reynolds number when the shear flow has a parabolic velocity profile. In addition, these results are extended to larger values of the wavenumber by numerical methods, and the relation between this long wave instability and the profile curvature is examined.