On the stability of parallel flows with respect to periodic disturbances

Abstract

The stability of a two-dimensional parallel flow has been investigated by the energy method. It was found that the flow is stable with respect to spatially periodic disturbances of the type u′=Re[f(y, z, t) exp(iγx)] where x is the distance along the direction of the basic flow. This result holds true for both finite and infinitesimal disturbances. It also holds for both two-dimensional and three-dimensional disturbances. The analysis follows essentially the procedure used by Thomas. A comparison of the present analysis with that used by Synge is given. The reason for the different results obtained by the method of Thomas and that of Synge is pointed out. The Poiseuille flow in a cylindrical tube of arbitrary section is also stable with respect to disturbance of the type ( A).