On the Navier-Slip Boundary Condition for Computations of Impinging Droplets

Abstract

Abstract-A mesh-dependent relation for the slip number in the Navier-slip with friction boundary condition for computations of impinging droplets with sharp interface methods is proposed. The relation is obtained as a function of Reynolds number, Weber number and the mesh size. The proposed relation is validated for several test cases by comparing the numerically obtained wetting diameter with the experimental results. Further, the computationally obtained maximum wetting diameter using the proposed slip relation is verified with the theoretical predictions. The relative error between the computationally obtained maximum wetting diameter and the theoretical predictions is less than 10% for impinging droplet on a hydrophilic surface, and the error increases in the case of hydrophobic surface.

A particle finite element-based model for droplet spreading analysis

“…where f Γ S and β Γ s are the dissipative force and the slip coefficient applied at the solid-liquid interface, respectively, and v is the slip velocity of the fluid on the solid-liquid interface. A variety of models have been proposed in the literature for the slip coefficient, β Γ s , at the solid-liquid interface such as, Navier-slip condition (β s ) [14][15][16]44 , prescribed slip profile condition 15 , and a constant slip coefficient that depends on the grid size 22 . The Navier-slip model is considered in this work, as it accounts for the shear rates and viscous dissipation along the solid-liquid interface during droplets deformation [14][15][16]21 .…”

“…In these applications, characterized by dominant capillary forces, the liquid phase is found in contact with solid substrates that can be hydrophobic, hydrophilic, or chemically heterogeneous 11 . Droplet dynamics models typically encounter several challenges when studying these phenomena: a) tracking of the free liquid surface [12][13][14] , b) identifying the interaction forces between liquids and substrates [15][16][17][18] , and hence tracking the transition between inertial and viscoelastic regimes 19,20 , and c) obtaining mesh-dependent solutions [21][22][23][24] .…”

“…A dynamic contact angle condition is needed to account for this phenomenon 9,34 . The most basic dynamic contact line condition imposes a slip boundary condition, i.e., contact line velocity normal to wall surface is zero, when a critical con-tact angle is reached 16 . This condition implies that no energy is dissipated as the contact line moves on the solid substrate, and that the velocity of the contact line is not restricted, which contradicts experimental observations 35 .…”

“…In addition to the dissipative force condition, Ren and E 15 observed that the normal stress inside the solid-liquid interface exhibits a large jump across the contact line, which varies linearly with its velocity, and hence should be balanced and considered as an additional boundary condition. Moreover, Venkatesan et al 16 observed that the tangential stress at the contact line is proportional to its tangential velocity, which was included as an additional term in their numerical formulation.…”

“…They concluded that increasing the dissipative force term leads to a less mesh-dependent solution. In addition, Venkatesan et al 16 used an Arbitrary Lagrangian-Eulerian (ALE) finite element formulation and introduced a slip coefficient in the Navierslip term that is a function of the mesh size, Weber number, and Reynolds number. They managed to alleviate the spurious mesh dependency.…”