The effect of surface contamination on thermocapillary flow in a two-dimensional slot. Part 2. Partially contaminated interfaces

We consider the flow driven by a thermally induced surface stress in a fluid held in a shallow two-dimensional slot, and show that, for low Maragoni number, the extent of surface stagnation due to the presence of a non-diffusing surfactant depends on a single parameter E, the elasticity number defined in Part 1 of this analysis (Homsy & Meiburg 1984). For situations in which the adsorbed species are insufficient to result in a fully covered surface, we find that the interface is either clean and subject to constant stress, or contaminated and no-slip. There is a region in which one type of surface is replaced by the other. The Wiener–Hopf technique is used to obtain an analytic expression for the stream function in the vicinity of the leading edge of the stagnant surface in the limit of creeping flow. This result shows that the flow dies off under the stagnant surface at a distance of the order of the depth of the fluid, in a series of vortices of exponentially decreasing magnitude.