Scale Dependence of Contact Line Computations

Abstract

Abstract. The shape and velocity of a sliding droplet are computed by solving the NavierStokes equation with free interface boundary conditions. The Galerkin finite element method is implemented in a 2D computation domain discretized using an unstructured mesh with triangular elements. The mesh is refined recursively at the corners (contact points). The stationary sliding velocity is found to be strongly dependent on grid refinement, which is a consequence of the contact line singularity resolved through the effective slip across the finite elements adjacent to the contact point. For small droplets, this dependence is well approximated by a theoretical estimates obtained using multiscale expansion and matching technique in lubrication approximation, where the corner element size is used as a microscale parameter. For larger droplets, the shape is also dependent on grid refinement. This questions the validity of numerous computations of flows with moving contact line where grids are invariably much more coarse than molecular scales on which the singularity is resolved. It is suggested that extrapolation to molecular scales should be used to obtain realistic results.

A mesh-dependent model for applying dynamic contact angles to VOF simulations

“…In particular, one cannot expect the results of simulations to converge when the only slip is the ''effective slip" associated with the particular implementation of an algorithm [22,24,25]. Here we presented further evidence of this mesh dependence, when applying both no-slip and Navier-slip boundary conditions.…”

“…This implies that the methodology includes an ''implicit" (or ''effective") slip length at no-slip boundaries. However, relying on this implicit slip length leads to a convergence breakdown, because the stress singularity at the contact line is resolved by a slip length that is proportional to mesh spacing [22,24,25].…”

Viscous flow in domains with corners: Numerical artifacts, their origin and removal

“…The bulk finite element equations are found by substituting equations (30) into the Navier-Stokes equations, see (18), weighting incompressibility by w i and the momentum equations by / i , then integrating over the entire domain X and requiring that the resultant expression is zero. We arrive at Z X w i r Á u dX ¼ 0 ð32Þ and, after using integration by parts and the divergence theorem, Z X / i Re @u @t ðu À _ xÞ Á ru þ Stĝ þ P : r/ i dX þ I @X / i P Á nds ¼ 0;…”

“…The first of these difficulties, caused by the region where the noslip boundary condition is relaxed being extremely small, requires one to ensure that the computational mesh near the corner is sufficiently refined for the slip region to be well resolved. Otherwise, the global effect of the under-resolved corner region could be quite dramatic, as has been emphasized in several recent works [16][17][18]: this error generation has been quantified in Sprittles and Shikhmurzaev [19] and lies outside the scope of this article in which sufficiently well resolved meshes are always employed. At the same time, away from the contact line, where only bulk length scales need to be resolved, the meshing should not be excessive, for the computations to still be feasible.…”

Shear driven droplet shedding and coalescence on a superhydrophobic surface

“…This implementation makes our model unlike the one reported in the literature, to not rely systematically on experiments for imposing the contact angle. It is worth noting that the contact angle model such as Kistler model, 31 may be quite sensitive to the mesh resolution as recently pointed out by Weinstein and Pismen, 32 Afkhami et al, 33 and Sui and Spelt. 34 Although the improvement by Afkhami et al 33 using VOF seems to achieve solution convergence with mesh refinement, it relies on adjusting parameters such as the slip length and another parameter K, accounting for the effect of the outer length scale, which depends on the droplet spreading configurations.…”