Numerical simulation of static and sliding drop with contact angle hysteresis
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.
Section snippets
Numerical method
The numerical code used for this study is the JADIM code developed to perform local analyses of dispersed two-phase flows [34], [35], [41], [43]. The objective of this work is to introduce the modeling of the contact line in the Volume of Fluid (VoF) modulus of JADIM [3], [7], [8]. The implemented VoF method consists in an Eulerian description of each phase on a fixed grid, the interface between the two-phases being calculated using the transport equation of the local volume fraction of one
Numerical modeling of the contact angle
The main challenge for numerical simulations of drops spreading on surfaces concerns the accurate description of the surface tension effects controlled by the hysteresis and the motion of contact lines on walls. The numerical model for the contact angle developed in this study is presented in the following. We would like to stress here that the interface is not reconstructed in our approach so that it is not possible to impose directly the slope of the interface (or its normal) in the cell
Spurious currents characterization
The surface tension contribution in the momentum equation is calculated using the CSF method of [9]. The CSF method has the property to generate artificial flows called “spurious currents”. The aim of this section is to characterize these unphysical flows for isolated and wetting drops. For this purpose we consider situations of drops at equilibrium corresponding to exact solutions characterized by a zero velocity field at any time. The interface has also a constant curvature so that the
Equilibrium shape of a drop released on a wall
In this section, we consider some validations of the implementation of the static angle in bidimensional (plane) configurations. For the simulations reported in this section the contact angle is imposed to be constant and equal to the static angle . We consider the shape at equilibrium of a drop released on a horizontal wall. The drop is initially a semicircle and the initial contact angle with the wall is equal to (Fig. 6(a)). If the contact angle is different from the initial angle,
Partial wetting dynamics
The objective of this section is to test the method for moving contact lines. For this purpose we consider the axisymmetric spreading of a droplet for a partially wetting liquid. Some preliminary tests have been performed in order to investigate the convergence of the results with the spatial and time refinement. The simulations are compared with the experiments of [33].
Drop on an inclined wall
We consider here the two-dimensional deformation of a drop placed on a horizontal wall which is slowly inclined. We denote by the angle between the wall and the initial horizontal position. An hemi-circular drop is initially placed on the wall (see Fig. 6) and its shape evolves to satisfy the equilibrium between gravity, surface tension and the condition imposed by the contact angle at the wall. After the stabilization of the drop, the wall is inclined slowly with a characteristic time larger
Drop subjected to a shear flow
We consider in this section the problem of a drop subjected to a shear flow. A Couette flow and a Poiseuille flow are considered (Fig. 25). We first compare and validate our simulations with previous numerical works under creeping flow conditions. In the second part we present results concerning the drop deformation and migration in a Poiseuille flow and a criteria is proposed for the drop migration.
Conclusion
In this paper, a numerical”macroscopic-scale” method to describe static (including hysteresis) and moving contact lines for partially wetting liquids has been presented and tested in several well controlled situations. The numerical method is based on the implementation of the apparent angle observed at intermediate length scales. A“sub-grid” description of the contact line consists in imposing the apparent angle for the hysteresis as well as for the dynamic description of a moving line. For
Acknowledgments
We are grateful to Peter Spelt for several interesting discussions on moving contact line modeling and for a careful reading of the manuscript. We would like to acknowledge RENAULT (Technocentre, Guyancourt, France) and ADEME for the financial support of this work.
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