Pulse evolution for Marangoni-Benard convection

Marangoni-Benard convection is the process by which oscillatory waves are g enerated on an interface due to a change in surface tension. This process, which can be mass or temperature driven, is described by a perturbed Kortew eg-de Vries (KdV) equation. Far a certain parameter range, this perturbed K dV equation has a solitary wave solution with an unique steady-state amplit ude for which the excitation and friction terms in the perturbed KdV equati on are in balance. The evolution of an initial sech(2) pulse to the steady- state solitary wave governed by the perturbed KdV equation of Marangoni-Ben ard convection is examined. Approximate equations, derived from mass conser vation, and momentum evolution for the perturbed KdV equation, are used to describe the evolution of the initial pulse into steady-state solitary wave (s) plus dispersive radiation. Initial conditions which result in one or tw o solitary waves are considered. A phase plane analysis shows that the puls e evolves on two timescales, initially to a solution of the KdV equation, b efore evolving to the unique steady solitary wave of the perturbed KdV equa tion. The steady-state solitary wave is shown to be stable. A parameter reg ime for which the steady-state solitary wave is never reached, with the pul se amplitude increasing without bound, is also examined. The results obtain ed from the approximate conservation equations are found to be in good agre ement with full numerical solutions of the perturbed KdV equation governing Marangoni-Benard convection. (C) 1998 Elsevier Science Ltd. All rights res erved.