Plane Channel Flows

Abstract

The problem of stability of plane Couette flow of two fluids, treated here (figure 1.1), is the simplest of all the shearing flows of two fluids we might consider. This is a good problem in its own right and is well suited for teaching the physical concepts and analytical methods which are appropriate in the study of shearing flows of two fluids. Squire’s transformation for layered flows, discussed in section IV.3, is well known in the problem of stability of shearing flows of one fluid, where it guarantees that the smallest critical Reynolds number will occur when the disturbances are two-dimensional in the plane of the flow. The transformation leading to the theorem requires that planes parallel to the bottom and top of the channel should be unbounded in all directions. In practice, channels are nearly always bounded by side walls so that the application of Squire’s Theorem to practical situations is still open to questions.