Nonlinear waves in two-fluid flows

Abstract

Typically, the interaction of two miscible fluids leads to the formation of segregation patterns in the form of plane concentration waves and two- or three-dimensional voids or clusters. The development of small-amplitude plane travelling waves and the subsequent onset of higher-dimensional instabilities can be elucidated analytically; in addition, numerical methods allow the computation of higher-amplitude solutions. As both direct approaches become costly, although at different amplitude levels, one is tempted to look for a simpler model problem. Therefore, a nonlinear wave equation has been derived for two-dimensional dispersed two-phase flow in fluidized beds near the stability boundary using the Froude number as small parameter. This equation is a two-dimensional perturbation of the Korteweg-de Vries equation containing all low-order terms. It allows to identify the transition point between voids and clusters, and to investigate these two regions for solitary and periodic waves. A Ginzburg-Landau equation is derived to describe transverse perturbations of one-dimensional wavetrains bifurcating at the stability boundary.