A short note on Squire's theorem for interfacial instabilities in a stratified flow of two superposed fluids

Abstract

In a recent paper Schaflinger (1994) investigated interfacial instabilities in a stratified flow of two superposed fluids. The paper focused on a viscous resuspension flow (Schaflinger et al., 1990) in which a sediment of non-buoyant, solid particles is resuspended and driven by a clear fluid flow. At the suggestion of one reviewer, Schaflinger (1994) mentions that "[a]ccording to Squire (1933) three-dimensional disturbances in the classical, inviscid problem can be transformed into an equivalent two-dimensional problem, i.e. to each unstable three-dimensional disturbance there corresponds a more unstable two-dimensional one (Drazin and Reid, 1981, p. 129). Even though it seems to be imaginative to carry over Squire's theorem to the current study, the foundation supporting Squire's statement may not hold in the present problem. No studies dealing with this specific issue have been found in the literature. Therefore, this matter is subject to further, thorough investigations." Here we show that Squire's transformation indeed reduces the equations and the boundary conditions for a stratified flow to an equivalent two-dimensional problem.