Inertial effects on the flow near a moving contact line

The wetting or dewetting of a solid substrate by a liquid involves the motion of the contact line between the two phases. One of the parameters that govern the dynamics of the flow near a moving contact line is the local Reynolds number, rho. At sufficient proximity to the moving contact line, where rho << 1, the flow is dominated by viscous forces over inertia. However, further away from the contact line, or at higher speeds of motion, inertia is also expected to be influential. In such cases, the current contact line models, which assume Stokes flow and neglect inertia entirely, would be inaccurate in describing the hydrodynamic flow fields. Hence, to account for inertia, here we perform a regular perturbation expansion in rho, of the streamfunction near the Stokes solution. We, however, find that the leading-order inertial correction thus obtained is singular at a critical contact angle of 0.715 pi. We resolve this spurious mathematical singularity by incorporating the eigenfunction terms, which physically represent flows due to disturbances originating far from the contact line. In particular, we propose a stick slip on the solid boundary - arising from local surface heterogeneities - as the mechanism that generates these disturbance flows. The resulting singularity-free, inertia-corrected streamfunction shows significant deviation from the Stokes solution in the visco-inertial regime (rho similar to 1). Furthermore, we quantify the effect of inertia by analysing its contribution to the velocity at the liquid interface. We also provide the leading-order inertial correction to the dynamic contact angles predicted by the classical Cox-Voinov model; while inertia has considerable effect on the hydrodynamic flow fields, we find that it has little to no influence on the dynamic contact angles.