An analytical study on the dynamical contact angle of a drop in steady-state motion

In this paper we study the dynamical contact angle at three phase (fluid-fluid-solid) contact points for the 2 dimensional steady-state moving liquid drop. We present a result giving the speeds at the contact points as a function of the dynamical contact angle, in the form of a simple algebraic expression. The only physical constants involved are the viscosities of the fluids and the surface tension. Our approach is based only on the hypotheses that the fluids obey Navier-Stokes equation and that on the fluid-fluid interface the free stress equation holds. So, here the dynamical contact angle is defined as the angle in the region where the Navier-Stokes model is valid. Under these hypotheses it follows that at the contact points the local speeds are related through simple equations and the stress is singular, but well-defined in the Cauchy sense. The leading order singularity at the contact point is identified using a blow-up technique which leads to a reduced problem that contains the most singular behavior. This problem is a Stokes one in a sector plane which has a simple solution. The result we obtain is based on a dynamical force balance equation which leads to a simple contact point speed versus contact angle relation. The originality of this work is that it is based on simple hypotheses and it deals with the singularity, without additional assumption on the stress.