Transition in a numerical model of contact line dynamics and forced dewetting

摘要

We investigate the transition to a Landau-Levich-Derjaguin film in forceddewetting using a quadtree adaptive solution to the Navier-Stokes equationswith surface tension. A discretization of the capillary forces near thereceding contact line is used that yields an equilibrium for a specifiedcontact angle $\theta_\Delta$ called the numerical contact angle. Despite thewell-known contact line singularity, dynamic simulations can proceed withoutany explicit additional numerical procedure, yielding an implicitly dynamiccontact angle model. We investigate angles from 15 to 110 degrees and capillarynumbers from 0.001 to 0.1. The observed dynamics at small capillary number arein agreement with the Cox-Voinov theory, and yield a dynamic contact anglemeasurable at scales larger than the grid size $\Delta$ that dependslogarithmically on the distance to the contact line. The fit to the logarithmicdependence involves a "microscopic" distance $r_m$ that characterizes thenumerics. This distance $r_m$ is proportional to $\Delta/\phi$, where $\phi$ isa scaling factor or gauge function. This scaling factor is shown to depend onlyon the equilibrium angle $\theta_\Delta$ and the viscosity ratio. We rework theprediction of Eggers [Phys. Rev. Lett., vol. 93, pp 094502, 2004] of thecritical capillary number for the forced dewetting transition to include finiteangles $\theta_\Delta$ and the gauge function $\phi$. The numerical results arein agreement with this theory. An analogy can be drawn between the numericalcontact angle condition and a regularization of the Navier-Stokes equation by apartial Navier slip model. The analogy leads, at small angles, to a value forthe numerical length scale proportional to the slip length. The knowledge ofthis microscopic length scale and the associated gauge function can be used torealize grid-independent simulations of dynamic contact lines problems.