Numerical simulation of static and sliding drop with contact angle hysteresis

Abstract

A numerical “macroscopic-scale” method for static (including hysteresis) and moving contact lines for partially wetting liquids is presented. The numerical method is based on the implementation of a “sub-grid” description of the contact line that consists in imposing the apparent angle for static and moving contact lines. The numerical simulations are validated against several well controlled bi-dimensional situations: the equilibrium shape of a drop released on a hydrophobic or hydrophilic wall, the axisymmetric spreading of a drop for a partially wetting liquid, the migration of a drop placed on a inclined wall and submitted to a Couette or Poiseuille flow.

Introduction

Despite its apparent simplicity, the behavior of a drop spreading on a wall is a very difficult problem. Indeed the wetting of a solid by a liquid concerns scales from the capillary length to the Van der Waals forces and remains only partially understood [4], [17], [24]. The description of the contact line (where the fluid–fluid interface intersects the solid surface) is complicated by the fact that the Navier–Stokes equations with standard no-slip conditions produce an infinite viscous dissipation [28]. Consequently a microscopic description has to be introduced to cut of this singularity. From the numerical point of view, this particular complexity of the physics of the contact line makes the simulations very delicate. For instance, the numerical description of the deformation and the motion of a drop on an inclined wall or submitted to a shear flow requires a numerical model able to simulate the static deformation of the drop, the transition from the static to the motion and then the motion of the receding and advancing contact lines. Only recently methods have been developed to account for the numerical implementation of the hysteresis and the dynamics of the contact angle, the angle that the fluid–fluid interface makes with the solid surface at the contact line. Few numerical studies presented in the literature are able to describe the complete behavior of the contact line (see [55] for a recent review on the subject). The numerical studies differ: (i) by the numerical strategy used to describe the interface displacement and deformation: Boundary Integral Method [14], [27], [52], Adaptative grid methods [21], [53], Level Set Method [38], [55], Volume of Fluid Method [1], [47] or Front-Tracking Method [31]; (ii) by the way that the moving contact angle is imposed: constant angle with no-slip conditions [1], [31], [47], static angle with a “slip length” [2], [38], [55], dynamic model for the apparent angle [27], Diffuse Interface Method [15], [29]. Moving contact lines have also been modeled with contact angle hysteresis in two-dimensional situations [38], [55] and three-dimensional situations [16]. Most of these approaches do not describe the microscopic (Van der Waals) interactions between the fluids and the solid wall but solve the flow on a macroscopic-scale to access to large scale of the interface. The limitation of such numerical approach is that it is not possible to perform direct numerical simulations of the flow up to the molecular scale responsible for the wetting as Molecular Dynamics simulations can do [4]. For macroscopic-scale simulations, adapted models have to be selected and implemented to describe the contact angle at a sub-grid scale. This is the objective of the numerical approach presented in this paper. We present a numerical method to describe static (including hysteresis) and moving contact lines for partially wetting liquids. The numerical code JADIM used for this study and the numerical modeling introduced to describe the physics of the contact line are presented in Sections 2 Numerical method, 3 Numerical modeling of the contact angle, respectively. Section 4 presents numerical tests performed to characterize the spurious currents for isolated and wetting drops. In Section 5, the numerical modeling of the contact line is validated by considering the equilibrium shape of a drop deposited on hydrophobic or hydrophilic walls. The implementation of the dynamic angle is tested in Section 6 by computing the axisymmetric spreading of a drop for a partially wetting liquid. In Section 7 we consider the two-dimensional problem of a drop initially placed on a horizontal wall which is slowly inclined until the motion of the drop. Finally, we consider in Section 8 the deformation and the migration of two-dimensional drops submitted to a shear flow.