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

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 θΔ, 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 lc. To interpret the results, we use Cox's theory which involves a microscopic distance rm and a microscopic angle θe. In the numerical case, the equivalent of θe is the angle θΔ and we find that Cox's theory also applies. We introduce the scaling factor or gauge function ϕ so that rm=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 θΔ and the viscosity ratio q. In the case of small θe, 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 θe and arbitrary q and express the critical capillary number as a function of θe and rm. This implies also a prediction of the critical capillary number for the numerical case as a function of θΔ and ϕ. The theory involves a logarithmically small parameter ϵ=1/ln⁡(lc/rm) 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 θΔ 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 rm 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