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The {\it unsteady} process for thermocapillary droplet migration at largeReynolds and Marangoni numbers has been previously reported by identifying anonconservative integral thermal flux across the surface in the {\it steady}thermocapillary droplet migration, [Wu and Hu, J. Math. Phys. {\bf 54} 023102,(2013)]. Here we add a thermal source in the droplet to preserve the integralthermal flux across the surface as conservative, so that thermocapillarydroplet migration at large Reynolds and Marangoni numbers can reach a {\itquasi-steady} process. Under assumptions of {\it quasi-steady} state andnon-deformation of the droplet, we make an analytical result for the {\itsteady} thermocapillary migration of droplet with the thermal source at largeReynolds and Marangoni numbers. The result shows that the thermocapillarydroplet migration speed slowly increases with the increase of Marangoni number.

Thermocapillary migration of a droplet with a thermal source at large Reynolds and Marangoni numbers