Coarse‐grained description of thermo‐capillary flow

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.