On the spatio-temporal development of small perturbations of Jeffery-Hamel flows

McAlpine, Alan and Drazin, P.G. (1998) On the spatio-temporal development of small perturbations of Jeffery-Hamel flows. Fluid Dynamics Research, 22 (3), 123-138. (doi:10.1016/S0169-5983(97)00049-X).

Record type: Article

Abstract

This paper presents a new linear theory of small two-dimensional perturbations of a Jeffery-Hamel flow of a viscous incompressible fluid, in order to understand better the stability of the steady flow driven between inclined plane walls by a line source at the intersection of the walls. Because the variables of space and time are not all separable, a modified form of normal modes is used in solving the linearized equations of motion. The modes only satisfy the equations asymptotically far downstream. They are proportional to an exponential function of the ratio of time to the square of the radial distance, rather than of time alone. An eigenvalue problem to determine the modes is derived, a problem which reduces to the Orr-Sommerfeld problem in the special case when the walls are parallel, that is when the primary Jeffery-Hamel flow is plane Poiseuille flow. The results indicate that a small divergence of the walls is an astonishingly strong destabilizing influence on plane Poiseuille flow, and a small convergence a strong stabilizing influence. The relationship of the modes to the stability of the flow is discussed critically.

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