Stability of a laminar boundary layer in an incompressible fluid with variable physical properties

Abstract

The stability of plane-parallel flows of an incompressible fluid with variable kinematic viscosity in the presence of solid walls has been discussed in [1–5]. The stability of Couette flow is considered in [1]. The method of solution, which is the same as that used in [2], differs from the Tollmien-Schlichting method, since the expansion of the solutions in powers of αR which is used assumes the smallness of this quantity. A general formulation of the problem of stability of a nonuniform fluid is presented in [3]. Di Prima and Dunn [4] used the Galerkin method to study the stability of the boundary layer relative to vortex-like disturbances in the case of variable kinematic viscosity. Since the development of this sort of disturbance depends only weakly on the form of the velocity profile in the boundary layer, a marked change of the viscosity had little effect on the critical Reynolds number. This same problem is considered in [5], The present author was not able to find in the literature any references to study of the stability of the laminar boundary layer in an incompressible fluid with variable kinematic viscosity relative to disturbances of the Tollmien-Schlichting type, with the exception of mention in [4] of an unpublished work of MacIntosh which showed the essential dependence of the critical Reynolds number on the viscosity gradient.

In Section 1 an analysis is made of the effect of variability of the kinematic viscosity on the stability of the boundary layer relative to Tollmien-Schlichting waves under the condition of constant fluid density.

Two approaches are possible to the study of the development of disturbances in a heterogeneous fluid. On the one hand, we can assume that the displacement of the fluid particles does not cause changes in the distribution of p(y) and υ_*(y), i.e., the velocity pulsations are not accompanied by pulsations of ρ and υ_*. This will be the case if a particle which is characterized by the quantities ρ₁,υ₁ entering a layer with the different values ρ₂υ₂, instantaneously alters its properties so that its density becomes equal to ρ₂, and its kinematic viscosity becomes equal to υ₂. On the other hand, we can consider that a fluid particle moving from layer 1 into layer 2 fully retains the properties which it had in layer 1. In this case the velocity pulsations naturally lead to pulsations of the quantities ρ and υ_*.

In actuality, the phenomenon develops along some intermediate scheme, since the particle alters its properties as it moves in a heterogeneous fluid. The degree of approximation of the process to the first or second scheme depends on the rate of these changes. The analyses below are based on the first scheme.