On asymptotic methods for bifurcation and stability: confined Marangoni convection

Abstract

In 1979 Rosenblat developed a spectral method for studying bifurcation and stability problems. Drawing on an example from ordinary differential equations, he showed quite elegantly that, although the method bore striking similarities to the Lyapunov-Schmidt procedure, the range of validity of his method was significantly greater. In the early eighties Rosenblat, Homsy and Davis developed these pioneering ideas and extended them to partial differential equations, using Marangoni flow as an example. Several workers have subsequently employed the method to analyse other important problems. In this paper we caution against following their precise implementation and suggest a modified procedure. We reconsider their Marangoni problem and show that the approximations they used to represent temperature and velocity are inadequate. In particular, the approximation provides a poor representation of the vertical component of the velocity as the number of members in the basis is increased. We remedy this by constructing a new set of basis functions which we show represents the solution well and, unlike the previous work, provides results in agreement with weakly nonlinear theory. The shortcomings lead to quantitative, rather than qualitative, changes in the results for the Marangoni problem considered here.