Steady thermocapillary-buoyant flow in an unbounded liquid layer heated nonuniformly from above

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Thermocapillary and buoyant flows induced by nonuniform heating of the free surface of a horizontally unbounded liquid layer over a cold solid bottom are studied numerically and by order of magnitude analyses for large Marangoni and Rayleigh numbers. The Prandtl number of the liquid is assumed to be of order unity or large, which are the cases of most interest in combustion. The asymptotic structures of plane and axisymmetric stationary flows are described qualitatively, showing that they consist of several horizontally spaced regions. Heat conduction and viscous forces are confined to thin boundary layers in a region around the heat source, while viscous forces extend to the whole liquid layer in a longer region where the flow is driven by the momentum imparted to the liquid by thermocapillary stresses around the source, in the case of plane thermocapillary flow; by this momentum and remaining thermocapillary stresses, in the case of axisymmetric thermocapillary flow; and by the horizontal gradient of a hydrostatic pressure distribution, in the case of buoyant flows. For large values of the Prandtl number, this region is followed by a region of viscosity-dominated flow which may be responsible for a large fraction of the heat loss to the bottom. A linear stability analysis of the surface boundary layer in the vicinity of the heat source gives values of the critical Marangoni number for the transition to oscillatory flow that are comparable to known experimental results.