Abstract

MarangoniBenard convection is the process by which oscillatory waves are generated on an interface due to a change in surface tension. This process, which can be mass or temperature driven, is described by a perturbed Kortewegde Vries (KdV) equation. For a certain parameter range, this perturbed KdV equation has a solitary wave solution with an unique steadystate amplitude for which the excitation and friction terms in the perturbed KdV equation are in balance. The evolution of an initial sech2 pulse to the steadystate solitary wave governed by the perturbed KdV equation of MarangoniBenard convection is examined. Approximate equations, derived from mass conservation, and momentum evolution for the perturbed KdV equation, are used to describe the evolution of the initial pulse into steadystate solitary wave(s) plus dispersive radiation. Initial conditions which result in one or two solitary waves are considered. A phase plane analysis shows that the pulse evolves on two timescales, initially to a solution of the KdV equation, before evolving to the unique steady solitary wave of the perturbed KdV equation. The steadystate solitary wave is shown to be stable. A parameter regime for which the steadystate solitary wave is never reached, with the pulse amplitude increasing without bound, is also examined. The results obtained from the approximate conservation equations are found to be in good agreement with full numerical solutions of the perturbed KdV equation governing MarangoniBenard convection.