Convective mass transfer from a submerged drop in a thin falling film

Abstract

We study the fluid mechanics of removing a passive tracer contained in small, viscous drops attached to a flat inclined substrate using thin gravity-driven film flows. A convective mass transfer establishes across the drop-film interface and the tracer in the drop diffuses into the film flow. The Peclet number for the tracer in the film is large. The Peclet number Pe_d in the drop varies from 0.01 to 1. The characteristic transport time in the drop is much larger than in the film. We model the mass transfer of the tracer from the drop bulk into the film using an empirical model based on Newton's law of cooling. This model is supported by a theoretical model solving the quasi-steady 2D advection-diffusion equation in the film coupled with a time-dependent 1D diffusion equation in the drop. We find excellent agreement between our experimental data and the 2 models, which predict an exponential decrease in time of the tracer concentration in the drop. The results are valid for all drop and film Peclet numbers studied. The transport characteristic time is related to the drop diffusion time scale, as diffusion within the drop is the limiting process. Our theoretical model predicts the well-known relationship between the Sherwood and Reynolds numbers in the case of a well-mixed drop Sh~Re_L^{1/3}=\gamma L^2/\nu_f, based on the drop length L, film shear rate \gamma and film kinematic viscosity \nu_f. We show that this relationship is mathematically equivalent to a more physically intuitive relationship Sh~Re_\delta, based on the diffusive boundary layer thickness \delta. The model also predicts a correction in the case of a non-uniform drop concentration, which depends on Re_\delta, the Schmidt number, the drop aspect ratio and the diffusivity ratio. This prediction is in agreement with experiments at low Pe_d. It also agrees as Pe_d approaches 1, although the influence of Re_\delta increases.