On the Stability Problem in Hydrodynamics

Abstract The paper deals with the problem of the effect of two-dimensional first order disturbances on the linear and parabolic flows of a viscous fluid. If the laminar flow is in the direction x and is bounded by the planes y = ± h , and if the stream function of the disturbance is assumed to be of the form , it is found that, for small Reynolds numbers R (= Uh/v ), C can be developed in the power series The first term in (1) corresponds to a disturbance in still water and is known from Rayleigh's investigation of that problem; the succeeding C 's are real and can be obtained by an application of Schrödinger's perturbation theory of wave mechanics. In the case of the parabolic flow, expressions for C 0 and C 1 were thus derived for both the symmetrical and anti-symmetrical types of disturbance. The imaginary part of C , as represented by in this approximation, is found to be positive. When α is of the order of unity, the value of R which makes the second term of the first is about 50 for the symmetrical disturbance and 100 for the antisymmetrical disturbance. C 0 , the first term in the expression for the phase velocity, is less than 1 and increases with α, which implies a group velocity greater than the phase velocity. In the case of the linear flow it is found that the C's with even subscripts vanish, implying a vanishing phase velocity. This verifies the assumption to that effect made by Southwell and Chitty, and it also suggests that the region of convergence of ( A ) is limited to the range of values of R covered by these authors. Expressions are derived for C 1 and C 3 and are evaluated for α=1. The formulae thus obtained for the damping constant give an initial decrease with R , which is not shown in the corresponding curve of Southwell and Chitty. An alternative method of determining C is outlined in the last section. Use is made of the orthogonal system of the characteristic functions for the vorticity of a disturbance in still water. The equation determining C is reduced to an infinite determinant in which C occurs only in the diagonal elements. When R →0, the determinant can be evaluated and the roots C are found to coincide with the first two terms in ( A ). Finally it is shown that the two-dimensional laminar flow is stable for values of R less than the smaller of the quantities 2α and α 3 .