Continuous surface force based lattice Boltzmann equation method for   simulating thermocapillary flow
  In this paper, we extend a lattice Boltzmann equation (LBE) with continuous surface fore (CSF) to simulate thermocapillary flows. The model is designed on our previous CSF LBE for athermal two phase flow, in which the interfacial tension forces and the Marangoni stresses as the results of the interface interactions between different phases are described by a conception of CSF. In this model, the sharp interfaces between different phases are separated by a narrow transition layers, and the kinetics and morphology evolution of phase separation would be characterized by an order parameter visa Cahn-Hilliard equation which is solved in the frame work of LBE. The scalar convection-diffusion equation for temperature field is also solved by thermal LBE. The models are validated by thermal two layered Poiseuille flow, and a two superimposed planar fluids at negligibly small Reynolds and Marangoni numbers for the thermocapillary driven convection, which have analytical solutions for the velocity and temperature. Then thermocapillary migration of two dimensional deformable droplet are simulated. Numerical results show that the predictions of present LBE agreed with the analytical solution/other numerical results. 
Lin Zheng, Song Zheng, and Qinglan Zhai
physics.comp-ph physics.flu-dyn