Nonlinear Surface Waves Driven by the Marangoni Instability in a Heat Transfer System

Nonlinear surface waves of long wavelength driven by the Marangoni instability are investigated theoretically for a heat transfer system, in which a temperature of a thin liquid layer causes variations of the surface tension coefficient. For two examples considered here, the surface waves are governed, respectively, by a nonlinear evolution equation of diffusion type and the Kuramoto-Sivashinsky equation. It is shown that steady solutions for the first example, which are expressed by the cnoidal function, can be realized if a condition prescribed by values of parameters involved is satisfied at initial instants. It is also shown that the surface waves for the second example include two types of shock wave solutions.