On the stability for three-dimensional disturbances of viscous fluid flow between parallel walls

Abstract

The turbulence problem is still unsolved, through a number of valuable papers have been published on it comparatively recently. But, since Hopf and von Mises proved that uniform shearing motion between two parallel planes was stable for infinitesimal disturbances but unstable for disturbances of a finite size has become more and more widely held. Von mises suggested that the reoughness of the walls might be the determining factor, but the experiments of Schiller have shown that the degree of roughness of the walls is of negligible influence on the critical value of Reynold's number. He concluded that the breakdown of laminar flow depended primarily on the size of the initial disturbance, in agreement eith Osborne Reynold's view. Important papers have been published by Noether and Tollmien, whose conclusions are in contradiction to one another. On the one hand, Noether, by a formal investigation of the asymptotic solutions of the equation governing the two-dimensional disturbances of flow between parallel walls, claims to have proved that all velocity profiles are stable for all values of Reynolds' number. On the other hand, Tollmien has determined a critical value of Reynolds' number for the flow past a flat plate placed edgeways to the stream. This value is in good agreement with the experimental results. There are, however, certain points in his analysis which are not clear and it would be useful to know if the method gave results in agreement with those derived more strictly.