Energy dissipation due to viscosity during deformation of a capillary surface subject to contact angle hysteresis

A capillary surface is the boundary between two immiscible fluids. When the two fluids are in contact with a solid surface, there is a contact line. The physical phenomena that cause dissipation of energy during a motion of the contact line are hysteresis in the contact angle dynamics, and viscosity of the fluids involved. In this paper, we consider a simplified problem where a liquid and a gas are bounded between two parallel plane surfaces with a capillary surface between the liquid-gas interface. The liquid-plane interface is considered to be non-ideal, which implies that the contact angle of the capillary surface at the interface is set-valued, and change in the contact angle exhibits hysteresis. We analyze a two-point boundary value problem for the fluid flow described by the Navier-Stokes and continuity equations, wherein a capillary surface with one contact angle is deformed to another with a different contact angle. The main contribution of this paper is that we show the existence of non-unique classical solutions to this problem, and numerically compute the dissipation.