The nonlinear stability of dynamic thermocapillary liquid layers

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Abstract
When a temperature gradient is imposed on the free surface of a thin liquid layer, fluid motion can develop due to thermocapillarity. Previous work using linear theory has shown that the layer can become unstable to a pair of obliquely travelling hydrothermal waves. Here, we shall study the nonlinear behaviour of this system to determine possible equilibrium waveforms for the instability when the critical point from the linear theory is slightly exceeded. We find that for all Prandtl numbers and small Biot numbers, possible waveforms are composed of only one of the unstable linear waves. For small Prandtl number and larger Biot numbers, a combination of the two linear waves is a possible waveform. Further analysis of these equilibrium states shows that both exhibit the Eckhaus and Benjamin-Feir sideband instability and a corresponding phase instability. Thus, they become modulated on long length-and timescales as the system develops.