Coarse-grained Description of Thermo-capillary Ow Typeset Using Revt E X 1

A mesoscopic or coarse-grained approach is presented to study thermo-capillary induced ows. An order parameter representation of a two-phase binary uid is used in which the interfacial region separating the phases naturally occupies a transition zone of small width. The order parameter satisses the Cahn-Hilliard equation with advective transport. A modiied Navier-Stokes equation that incorporates an explicit coupling to the order parameter eld governs uid ow. It reduces, in the limit of an innnitely 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.