Surface-tension- and injection-driven spreading of a thin viscous film

We consider the spreading of a thin viscous droplet, injected through a finite region of a substrate, under the influence of surface tension. We neglect gravity and assume that there is a precursor layer covering the whole substrate and that the rate of injection is constant. We analyse the evolution of the film profile for early and late time, and obtain power-law dependencies for the maximum film thickness at the centre of the injection region and the position of an apparent contact line, which compare well with numerical solutions of the full problem. We relax the conditions on the injection rate to consider more general time-dependent and spatially varying forms. In the case of power-law injection of the form t k , we observe a switch in the behaviour of the evolution of the film thickness for late time from increasing to decreasing at a critical value of k. We show that point-source injection can be treated as a limiting case of a finite-injection slot and the solutions exhibit identical behaviours for late time. Finally, we formulate the problem with thickness-dependent injection rate, discuss the behaviour of the maximum film thickness and the position of the apparent contact line and give power-law dependencies for these.