The time-dependent Stokes paradox
When a uniform stream starts to flow impulsively at time t = 0 past a two-dimensional body, the solution of the Stokes equation indicates that the velocity grows without bound as / -too. It is seen that this is a natural extension of the Stokes paradox to unsteady flows. Two resolutions of this result are presented: firstly, through an analysis of the Oseen equation on the basis of singular perturbation theory and, secondly, through considering the two-dimensional body as the limit of a threedimensional body when the length increases without bound. When Stokes [1] tried to describe the uniform streaming flow past a circular cylinder, he found it was impossible to find a solution of the governing two-dimensional equations for slow viscous flow which gave finite velocities at large distances—a conclusion now known as the "Stokes paradox." Although Oseen showed that the velocity is bounded when the convection terms are included for 0 < Re «: 1 , the full resolution of the paradox required the perspective of singular perturbation theory as developed by Kaplun and Lagerstrom [2], and Proudman and Pearson [3]. They showed that the Stokes equation is only valid in the inner domain where the radial distance r = 0{ 1), whereas the Oseen equation is appropriate in the outer domain where Re r = 0{ 1); the solutions in the separate regions can then be matched across the boundary where r —> oo , Re r —> 0 . In his classical paper, Stokes writes (in paragraph 46) We have evidently a right to conceive a sphere or infinite cylinder to exist at rest in an infinite mass of fluid also at rest, to suppose the sphere or cylinder to be then moved with a uniform velocity V , and to propose for determination the motion of the fluid at the end of the time t. But we have no right to assume that the motion approaches a permanent state as t increases indefinitely. We may follow either of two courses. We may proceed to solve the general problem in which the sphere or cylinder is supposed to move from rest, and then examine what results we obtain by supposing t to increase indefinitely, or else we may assume for trial that the motion is steady, and proceed to inquire whether we can satisfy all the conditions of the problem on Received June 13, 1989. ©1991 Brown University 427