Curvature boundary condition for a moving contact line
Introduction
Wetting or dewetting of a solid surface occurs in a number of applications, such as coating, lamination, inkjet printing and spray painting [44], [33], [35]. A phenomenon common to these applications is the moving contact line, where one fluid displaces the other. A fundamental difficulty in the numerical treatment of the contact line arises with the conventional no-slip boundary condition, as it leads to inconsistent, grid-dependent results [10], [34].
Various approaches have been proposed to alleviate this problem, such as the slip model [39], [37], the precursor film model [28], the diffusive interface model [45], and the multi-scale model combining molecular dynamics and diffusive interface [22]. Convergence of results is controlled by prescribed parameters in these models, such as the slip length in the slip model, the film thickness in the precursor film model, and the interface thickness in the diffusive interface model. With the slip model, a slip length is introduced to replace the no-slip boundary condition. For grid convergence, the mesh size needs to be smaller than the introduced slip length [41]. However, experimental studies suggest that the physical slip length is on the order of the intermolecular distance [8], which is far beyond what can be achieved with acceptable grid resolution [7]. An efficient approach for circumventing extremely small grid sizes is to model the microscopic region near the contact line based on hydrodynamic theories and to resolve only the macroscopic region away from the contact line. Schönfeld and Hardt [29] obtained almost grid-independent results by a combined model with a macroscopic length and a near-wall body force. Similar results are obtained in [1] of Afkhami and Zaleski by modeling the dynamic contact angle based on an analog to Cox's theory. Sui and Spelt [36] revisited and extended this method by considering higher-order terms in Cox's theory.
One difficulty of such methods based on Cox's theory is to solve the complex, implicit integral function relating the microscopic angle to the macroscopic angle, which is either replaced by a simplified form [1], [7] or obtained numerically [36]. Another issue associated with the boundary condition is how to compute curvature at the contact line. In numerical simulations, imposing a contact angle explicitly at the wall is a common way to define the interface in the vicinity of the contact line [2], [25], [17], [20], [42]. Afkhami and Zaleski reconstructed a height function at the contact line based on the contact angle for their Volume-of-Fluid method [1]. Also, in Sui and Spelt's approach, an iterative procedure is used to impose the contact angle for their level-set method [36]. Note that, since in these cases the contact angle is defined on the numerical grid, undesirable artifacts can occur due to a mismatch with the exact contact angle which is defined on a length scale much smaller than the grid size. For example, imposing the contact angle explicitly can spuriously displace the zero level-set and the location of contact line, and thus induce large mass-conservation errors [40], [26].
In this paper, we rather impose a curvature boundary condition for predicting moving contact lines. An effective curvature is formulated explicitly based on the asymptotic theories of Cox [4], which avoids solving complex, implicit integral function. For simulating incompressible multi-phase flow, we adopt the conservative sharp-interface method, where surface tension and viscosity jump are treated without numerical smearing [18]. Since the present method does not impose the contact angle explicitly on the level-set function representing the interface, it avoids a mismatch between the exact and numerically imposed contact angles and hence spurious level-set displacement and the corresponding mass-conservation errors. Several multi-phase flow problems, including drop spreading on the wall, steady Couette and Poiseuille flows, are considered to demonstrate the ability of the present method. The numerical results suggest grid-convergence and good agreement with previous theoretical, numerical and experimental results.
Section snippets
Weakly compressible model
We use a weakly compressible model for incompressible multiphase flows [30], [3], [19]. The mass and momentum conservation for a weakly compressible flow can be described as where U is the density of mass and momentum and F represents the convective fluxes. On the right-hand side for the density component of U, and for the momentum components. Here, are the viscous fluxes. The surface-tension force is given by where σ, κ, N are surface tension, curvature
Curvature boundary condition
The boundary condition for a moving contact line is presented in the following. As in Ref. [1] the no-slip boundary condition is employed for fluid fields by defining velocities of the so-called “ghost cells” within the wall with where v is the tangential velocity in the cell adjacent to the wall and is the velocity of the wall. Within a level-set framework the interface curvature in a cut-cell sufficient far away from the moving contact line can be obtained directly from the
Numerical validation and examples
In the following numerical examples are considered to illustrate the capability of the present method for handling moving contact lines. Simulations have been performed in 2D (plane or axisymmetric) configurations where analytical solutions, previous numerical results or experimental data are available. We first consider the equilibrium shape of a water drop on a wall to demonstrate that the presented method recovers the static contact angle, when the curvature correction in Eq. (17) vanishes.
Concluding remarks
A curvature boundary condition is presented for simulating multi-phase flow problems with moving contact line. The boundary condition, which is derived from the theory of Cox, is explicitly defined and valid for variable viscosity ratios. Furthermore, since the present method does not prescribe the value of the contact angle explicitly, the contact-line evolution is determined by the flow field directly, and artifacts by mass-conservation errors are avoided. The numerical model has been tested
Acknowledgements
The first author has partially been financed by China Scholarship Council (No. 2010629044).
References (46)
- et al.
A mesh-dependent model for applying dynamic contact angles to VOF simulations
J. Comput. Phys.
(2009) - et al.
A continuum method for modeling surface tension
J. Comput. Phys.
(1992) On imposing dynamic contact-angle boundary conditions for wall-bounded liquid–gas flows
Comput. Methods Appl. Mech. Eng.
(November 2012)- et al.
Numerical simulation of static and sliding drop with contact angle hysteresis
J. Comput. Phys.
(2010) - et al.
A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method)
J. Comput. Phys.
(1999) - et al.
Adaptive multi-resolution method for compressible multi-phase flows with sharp interface model and pyramid data structure
J. Comput. Phys.
(April 2014) - et al.
A conservative interface method for compressible flows
J. Comput. Phys.
(2006) - et al.
An interface interaction method for compressible multifluids
J. Comput. Phys.
(2004) - et al.
Efficient implementation of weighted ENO schemes
J. Comput. Phys.
(1996) - et al.
Sharp interface Cartesian grid method II: a technique for simulating droplet interactions with surfaces of arbitrary shape
J. Comput. Phys.
(2005)