Tangential stress and Marangoni effects at a fluid-fluid interface in a Hele-Shaw cell

Although the depth-averaged equations describing fluid flow in a Hele-Shaw cell resemble those of potential flow, the appropriate tangential stress boundary condition at a fluid-fluid interface is different from that for potential flow. The precise form of this boundary condition is derived herein by solving the complete microscale problem near the interface followed by depth-averaging of the results. The shear stress exerted on the interface by each phase is found to be proportional to its viscosity, the tangential velocity of the interface relative to that in the bulk phase, and the reciprocal of a 'slip layer' thickness whith depends only on the gap width in the Hele-Shaw cell. The results are applied to the problem of translation of a circular drop or bubble in a Hele-Shaw cell, in the presence of Marangoni effects. For instance, the thermocapillary migration velocity of a circular bubble in a constant temperature gradient is found to be inversely proportional to its radius, in contrast with the case of a spherical bubble migrating in an infinite liquid, for which the velocity is directly proportional to the radius.