On hydrodynamic stability in unlimited fields of viscous flow

Abstract

This paper considers the hydrodynamic stability of flows in which there are no solid boundaries in the field of flow. The method used is an extension of that initiated by McKoen (1957), in which the fourth derivative, 0 iv , is assumed to be significant only near to the singular layer, but otherwise the complete fourth-order Orr—Sommerfeld equation is considered. An alternative derivation is given for McKoen’s integral form of the boundary condition for an antisymmetrical perturbation. In this integral it is necessary to approximate for (j) but not for any of its derivatives. It is shown that the present method will always lead to a neutral stability curve of wave number against Reynolds number, having two branches as R ->oo and hence a least critical R . The case of the plane laminar jet is considered, and a critical Reynolds number of 4 is obtained, which does not compare unreasonably with experiment in which unsteadiness is first detected at a Reynolds number of about 10. The lower branch of the neutral curve is found to be almost coincident with the R -axis.