Theoretical model of a finite force at the moving contact line

Abstract: In theoretical analyses of the moving contact line, an infinite force along the solid wall has been reported based off the non-integrable stress along a single interface. In this investigation we demonstrate that the stress singularity is integrable and results in a finite force at the moving contact line if the contact line is treated as a one-dimensional manifold and all three interfaces that make up the moving contact line are taken into consideration. This is due to the dipole nature of the vorticity and pressure distribution around the moving contact line. Mathematically, this finite force is determined by summing all the forces that act over an infinitesimally small cylindrical control volume that encloses the entire moving contact line. With this finite force, we propose a new dynamic Young's equation for microscopic dynamic contact angle that is a function of known parameters only, specifically the interface velocity, surface tension, and fluid viscosity. We combine our model with Cox's model for apparent dynamic contact angle and find good agreement with published dynamic contact angle measurements.