On a model for a sliding droplet:Well-posedness and stability of translating circular solutions

Abstract

In this paper the model for a highly viscous droplet sliding down an inclined plane is analyzed. It is shown that, provided the slope is not too steep, the corresponding moving boundary problem possesses classical solutions. Well-posedness is lost when the relevant linearization ceases to be parabolic. This occurs above a critical incline which depends on the shape of the initial wetted region as well as on the liquid's mass. It is also shown that translating circular solutions are asymptotically stable if the kinematic boundary condition is given by an affine functionand provided the incline is small.