On the hydrodynamic stability of two viscous incompressible fluids in parallel uniform shearing motion

Abstract

A new problem in hydrodynamic stability is investigated. Given two contiguous viscous incompressible fluids, the fluid on one side of the plane interface being bounded by a solid wall and that on the other side being unbounded, the problem is to determine the hydrodynamic stability when the fluids are in steady unidirectional motion, parallel to the interface, with uniform rate of shear in each fluid. The mathematical analysis, based on small disturbance theory, leads to a characteristic value problem in a system of two linear ordinary differential equations. The essential dimensionless parameters that appear in the present problem are the viscosity ratiom, the density ratior, the Froude numberF, and the Weber numberW, as well as the parameters α,R(which is proportional here to the flow rate of the inner fluid) andc, that occur in the study of hydrodynamic stability of a single fluid. The results obtained are presented graphically for most fluid combinations of possible interest. The neutral stability curve in the (α,R)-plane is single-looped, as in the boundary layer case. The calculated critical Reynolds numbers are higher than the values observed in liquid film cooling experiments. (In these experiments, the outer fluid is usually a turbulent gas, in which the thickness of the laminar sublayer is of the same order of magnitude as the liquid film thickness.) General agreement between the theoretical and experimental values exists for all critical quantities except the Reynolds number. Gravity and surface tension are found here to have a destabilizing effect on the flow, in agreement with experimental evidence. Semi-infinite plane Couette flow is a special case of the present problem and the known stability of this flow is recovered. The linear velocity profile of two adjacent fluids with the same viscosity, but different densities, is shown to be unstable for high enough Reynolds numbers. The Reynolds stress distribution for a neutral oscillation in the general case is discussed qualitatively.