  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.
