  A mesoscopic or coarse-grained approach is presented to study
thermo-capillary induced flows. An order parameter representation of a
two-phase binary fluid is used in which the interfacial region separating the
phases naturally occupies a transition zone of small width. The order parameter
satisfies the Cahn-Hilliard equation with advective transport. A modified
Navier-Stokes equation that incorporates an explicit coupling to the order
parameter field governs fluid flow. It reduces, in the limit of an infinitely
thin interface, to the Navier-Stokes equation within the bulk phases and to two
interfacial forces: a normal capillary force proportional to the surface
tension and the mean curvature of the surface, and a tangential force
proportional to the tangential derivative of the surface tension. The method is
illustrated in two cases: thermo-capillary migration of drops and phase
separation via spinodal decomposition, both in an externally imposed
temperature gradient.
