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

The {\it unsteady} process for thermocapillary droplet migration at large Reynolds and Marangoni numbers has been previously reported by identifying a nonconservative 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 integral thermal flux across the surface as conservative, so that thermocapillary droplet migration at large Reynolds and Marangoni numbers can reach a {\it quasi-steady} process. Under assumptions of {\it quasi-steady} state and non-deformation of the droplet, we make an analytical result for the {\it steady} thermocapillary migration of droplet with the thermal source at large Reynolds and Marangoni numbers. The result shows that the thermocapillary droplet migration speed slowly increases with the increase of Marangoni number.