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 θ Δ , 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 ∘ to 110 ∘ and capillary numbers from 0.00085 to 0.2 where the mesh size Δ 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 θ e . In the numerical case, the equivalent of θ e is the angle θ Δ and we find that Cox's theory also applies. We introduce the scaling factor or gauge function ϕ so that r m = Δ / ϕ 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 θ Δ and the viscosity ratio q. In the case of small θ e , we use the prediction by Eggers [Phys. Rev. Lett. 93 (2004) 094502] of the critical capillary number for the Landau–Levich–Derjaguin forced dewetting transition. We generalize this prediction to large θ e and arbitrary q and express the critical capillary number as a function of θ e and r m . This implies also a prediction of the critical capillary number for the numerical case as a function of θ Δ and ϕ. The theory involves a logarithmically small parameter ϵ = 1 / ln ⁡ ( l c / r m ) and is thus of moderate accuracy. The numerical results are however in approximate agreement in the general case, while good agreement is reached in the small θ Δ and q case. An analogy can be drawn between the numerical contact angle condition and a regularization of the Navier–Stokes equation by a partial Navier-slip model. The analogy leads to a value for the numerical length scale r m proportional to the slip length. Thus the microscopic length found in the simulations is a kind of numerical slip length in the vicinity of the contact line. The knowledge of this microscopic length scale and the associated gauge function can be used to realize grid-independent simulations that could be matched to microscopic physics in the region of validity of Cox's theory.