H Marangoni Instability

In this section, we consider the classic problem of a fluid layer of depth d, with an upper surface that is an interface with air that is maintained at an ambient temperature 7o. The fluid layer is heated from below, and we shall assume that the lower fluid boundary is isothermal with temperature T ( To). This problem sounds exactly like the Rayleigh-Benard problem with a free upper surface. However, we consider the fluid layer to be very thin (i.e., d small) so that the Rayleigh number, which depends on d3, is less than the critical value for this configuration. Nevertheless, as previously suggested, the fluid layer may still undergo a convective motion that is due to Marangoni instability. 

The governing equations for the linear stability theory are the same as for the Rayleigh-Benard problem, namely (12-215), except that it is customary to drop the buoyancy terms because these are of secondary importance for very thin fluid layers where Marangoni instabilities are present but Ra C 1. Furthermore, we state without proof (see Problems section) that the exchange of stabilities is valid. Thus a = 0 at the neutral state. Assuming that 

The boundary conditions at the lower solid boundary are the same as for the Rayleigh Benard problem, 

Apart from the assumption that the normal velocity at the upper interface is zero, 

As already noted we assume that the gas above the fluid layer is held at a constant temperature To. Assuming local thermal equilibrium, we have seen [Eqs. (12-214)] that the thermal boundary condition for the disturbance temperature at the upper surface can be expressed in the nondimensional form 

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