Thermocapillary Flow in a Thin Cavity

The final example of a shallow-cavity problem that we consider is the flow driven by surface-tension gradients at the upper surface, which is again assumed to be an interface as sketched in Fig. 7(c).13 The surface-tension gradient is produced when a fixed temperature differential is maintained between the two walls at x = 0 and L. 

Above the interface at z = h (x ), we assume that there is a gas in which the temperature varies linearly with position from the cold to the hot end of the cell, i.e., 

The thermal boundary condition at the interface is continuity of the heat flux  

Here k is the thermal conductivity of the liquid in the shallow cavity and kg is a heat transfer coefficient for the gas. Finally, we assume, for simplicity, that the interfacial tension o( T) depends linearly on the interface temperature, 

by definition, o0 is o(T) at T = T. The temperature distribution within the liquid in the thin film is determined by means of the thermal energy equation  

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