Integrable stress and viscous dissipation singularities of the Moving Contact Line

The classical hydrodynamic solution to the moving contact line (MCL) reported by Huh & Scriven (1971) is often associated with a singular force and viscous dissipation due to a diverging stress and viscous dissipation per unit volume. In our recent analysis, we find that these singular fields arise from the application of the divergence theorem to a volume cut by interfacial and line discontinuities. Furthermore, we find that the previously reported singular force and total viscous dissipation are a consequence of integral relations derived for interfacial surfaces. Unlike interfacial surfaces, contact lines are one-dimensional manifolds that have their own set of integral relations for quantities like force. In order to determine the total force and rate of work at the MCL, we integrate the stress and viscous dissipation per unit volume over an infinitely small cylindrical control volume that encloses the contact line only. Using the classical hydrodynamic solution, we find that the total integrated force and total rate of work is finite as this cylindrical volume captures the physical effects of the fluid-fluid interface, in addition to the fluid-solid interfaces.