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

Abstract

We investigate the transition to a Landau-Levich-Derjaguin film in forced dewetting using a quadtree adaptive solution to the Navier-Stokes equations with surface tension. We use a discretization of the capillary forces near the receding contact line that yields an equilibrium for a specified contact angle ${\theta }_{\Delta }$ called the numerical contact angle. Despite the well-known contact line singularity, dynamic simulations can proceed without any explicit additional numerical procedure. We investigate angles from $15^{\circ }$ to $110^{\circ }$ and capillary numbers from $0.00085$ to $0.2$ where the mesh size $\Delta$ is varied in the range of $0.0035$ to $0.06$ of the capillary length $l_c$. To interpret the results, we use Cox's theory which involves a microscopic distance $r_m$ and a microscopic angle ${\theta }_e$. In the numerical case, the equivalent of ${\theta }_e$ is the angle ${\theta }_{\Delta }$ and we find that Cox's theory also applies. We introduce the scaling factor or gauge function $\varphi$ so that $r_m=\Delta /\varphi$ and estimate this gauge function by comparing our numerics to Cox's theory. The comparison provides a direct assessment of the agreement of the numerics with Cox's theory and reveals a critical feature of the numerical treatment of contact line dynamics: agreement is poor at small angles while it is better at large angles. This scaling factor is shown to depend only on ${\theta }_{\Delta }$ and the viscosity ratio $q$. In the case of small ${\theta }_e$, we use the prediction by Eggers [Phys. Rev. Lett., vol. 93, pp 094502, 2004] of the critical capillary number for the Landau-Levich-Derjaguin forced dewetting transition. We generalize this prediction to large ${\theta }_e$ and arbitrary $q$ and express the critical capillary number as a function of ${\theta }_e$ and $r_m$. An analogy can be drawn between $r_m$ and the numerical slip length.

arXiv commentThis version of the paper includes the corrections indicated in Ref. [1]

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