The effect of surface contamination on thermocapillary flow in a two-dimensional slot

Abstract

We consider the effect of insoluble surfactants on the steady thermocapillary flow in a differentially heated slot treated previously by Sen & Davis (1982). The equation of state for interfacial tension is taken to be linear in both temperature and surfactant concentration. We treat the problem in the limit of shallow slots and low thermal Marangoni numbers so that the effect of surfactants is described by only two parameters: a surface Péclet number Pe and an elasticity parameter denoted by E, the ratio of the compositional elasticity to the tension difference due to the imposed temperature difference. Using lubrication theory and matched asymptotic expansions, we reduce the problem to a single nonlinear integral–algebraic equation (for the outer core variables), which we solve both numerically and in various asymptotic limits by perturbation theory. It is shown that the general effect of surfactants is to retard the strength of the motion, but that this retardation is not necessarily uniform in space. Surprisingly, there are only extreme cases in which condensed surfactant layers will form, these being E [Lt ] 1, Pe [Gt ] 1. Sharp gradients in surfactant concentrations will not form in the general case of E = O(1). This behaviour is due to the strong coupling between the flow and the interfacial stress, and is contrasted with certain well-known forced-convection problems.