On a Two-point Boundary Value Problem for the 2-D Navier-Stokes Equations arising from Capillary Effect

In this article, we consider the motion of a liquid surface between<br>two parallel surfaces. Both surfaces are non-ideal, and hence, subject to <br>contact angle hysteresis effect. Due to this effect, the angle of contact between a capillary <br>surface and a solid surface takes values in a closed interval. Furthermore, the evolution of the contact angle as a function of the contact area exhibits hysteresis. We study the two point boundary value problem in time whereby a liquid surface with one contact<br>angle at t=0 is deformed to another with a different contact angle at t = infinity while the volume remains constant, with the goal of determining the energy loss due to viscosity. The fluid flow is modeled by the Navier-Stokes equations, while the Young-Laplace equation models the initial and final capillary surfaces. It is well-known even for ordinary differential equations that two-point boundary value problems may not have solutions.<br>We show existence of classical <br>solut