Non-linear Stability of Asymptotic Suction

Abstract

A flow over a plane y = 0 in R3 given by U(x,y,z) = (1 - e-y, -1/R, 0) is called an asymptotic suction velocity profile [12]. R>0 is the Reynolds number. U satisfies the Navier-Stokes equation ðv / ðt + (v · ) v = - p0 + 1/R v div v = 0 with p0 = 0. In the present paper it is proved that the stability of U for small perturbations which initially decay exponentially in the y direction and are periodic in the x and z direction is governed by the eigenvalues of the classical Orr-Sommerfeld equation [1, 8, 12]. For precise statements see Theorems 4, 5, 9, and 15.