Nonlinear Stability of the Plane Couette and the Hagen-Poiseuilie Flows

Abstract

The stability problems of the plane Couette and the Hagen-Poiseuille flows are studied. The nonlinear integro-differential equations which govern the mean and disturbance velocities are solved by making use of orthogonal function expansion and the Galerkin method. The stability characteristics are expressed as the equilibrium surface in a three-dimentional space of the Reynolds number, the wave number, and the disturbance energy. The critical Reynolds number is found to be 1.8105×10⁵ (based on the relative velocity of boundary walls and the distance of walls) for the Couette flow and 1212.9 (based on the mean velocity and the diameter of the pipe) for the Hagen-Poiseuille flow.