Unsteady Stokes flow near an oscillating, heated contact line

Abstract

A contact line on a heated oscillating plate is investigated. The interface is a non-deformable plane and the contact angle is π/2. The amplitude of the oscillation and the temperature deviation of the plate from the ambient temperature of the fluid are assumed to be much smaller than the viscous velocity scale. This flow is then governed by the unsteady Stokes equations coupled to the heat equation in a frame of reference moving with the contact line. Evaporation is assumed to be neglible, but the effects of heat transfer across the interface and unsteadiness are assumed to be significant. For a stationary heated plate, there are two distinct regions of flow that is induced by Marangoni stresses. An outer stagnation-point-type flow is seen, which separates from the plate for non-zero Biot numbers. For an oscillatory, isothermal plate, vortices are generated at the plate during plate reversal and are propagated along the interface. Dissipation of these vortices occurs on the Stokes layer scale. The order-Péclet-number correction in the thermal field is also found, and the presence of the flow field leads to a heated region in the steady case along the separating streamline. For the unsteady case, a localized cooled region propagates into the bulk with a trajectory determined by the relative scale of the thermal diffusive scale and the rate of heat transfer across the interface.