Evolution equation for bidirectional surface waves in a convecting fluid

Abstract

Surface waves in a heated viscous fluid exhibit a long wave oscillatory instability. The nonlinear evolution of unidirectional waves is known to be described by a modified Korteweg-deVries-Kuramoto-Sivashinsky equation. In the present work we eliminate the restriction of unidirectional waves and find that the evolution of the wave is governed by a modified Boussinesq system . A perturbed Boussinesq equation of the form $y_{tt}-y_{xx}-{\epsilon }^2(y_{xxtt}+(y^2{)}_{xx})+{\epsilon }^3(y_{xxt}+y_{xxxxt}+(y^2{)}_{xxt})=0$ which includes instability and dissipation is derived from this system.

8 pages, no figures