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 ∆ 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 ∆ 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 ∆ and we find that Cox's theory also applies. We introduce the scaling factor or gauge function φ so that r m = ∆/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 ∆ and the viscosity ratio q. In the case of small θ 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 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 ∆ and φ. The theory involves a logarithmically small parameter = 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 ∆ 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. This version of the paper includes the corrections indicated in [1].

Simulations of microlayer formation in nucleate boiling

“…However, one could replace it with a specific mobility law that would describe the behavior of the microscopic contact angle in the conditions of interest. Mesh dependence of moving contact lines simulations with grid size as previously shown by Moriarty & Schwartz (1992), Weinstein & Pismen (2008), and Afkhami et al (2009) is reported elsewhere, see Afkhami et al (2017).…”

Two mechanisms of droplet splashing on a solid substrate

“…Physically, the flow is regularized by small-scale effects, that can be accounted for through an effective boundary slip condition at the solid surface, with a slip length of the order of a few tens of nanometers (Bonn et al 2009). To obtain numerical convergence for these dynamics, complex boundary conditions at the moving contact line involving the mesh size have to be introduced (Afkhami et al 2009;Legendre & Maglio 2015;Afkhami et al 2017). Keeping a fixed maximum refinement level avoids the question of the numerical convergence of the moving contact line dynamics, since it corresponds then to a fixed slip length boundary condition at the solid substrate.…”

“…It also indicates that the details of the moving contact line does not play a significant role in the impact dynamics, because of the rapid spreading, although it could be crucial in the receding dynamics of the drop as observed experimentally (Bartolo et al 2005). Notice however, that the full account of the moving contact line dynamics can be implemented in numerical simulations using specific boundary conditions (Afkhami et al 2009;Legendre & Maglio 2015;Mahadi et al 2015;Mahady et al 2016;Afkhami et al 2017). Experiments are typically achieved by releasing a pendant liquid droplet from a height H D, that then falls under gravity,…”

Breakup of finite-size liquid filaments: Transition from no-breakup to breakup including substrate effects⋆

“…We however note that a prior work in the literature has considered such effect [10] and the interested reader is referred to this study for further discussions. Additionally, we refer to the work carried out by two of the co-authors on the issues related to the computations of moving contact line problem [17]. Nevertheless, our study provides a most complete picture, that has not been available in the literature so far, of the breakup of finite-size filaments, when also considering the substrate effect.…”

“…where λ is the slip length and (u, w) are the components of the velocity field tangential to the solid boundary. For issues regarding the regularization of the viscous stress singularity at the contact line, the reader is referred to [17]. When the filament is placed on a solid surface, we impose a contact angle of 90 • ; this is due to the limitation of the code for having an arbitrary contact angle and the simplicity in imposing a 90 • contact angle.…”