Nonlinear instability in plane Poiseuille flow: a quantitative comparison between the methods of amplitude expansions and the method of multiple scales

Abstract

This paper shows that two different expansion procedures for hydrodynamical stability problems are equivalent. The method of multiple scales of Stewartson & Stuart (1971) is extended to calculate the stream function up to order ε2. Watson’s (1960) rigorous amplitude expansion of the solution of the Navier-Stokes equations is also used to calculate the stream function up to the same order of magnitude, and a complete equi­valence between the two results is found. An analysis of the Eckhaus model equations has been made and the results are equivalent.