Moving Contact Line Problem Research Articles

Enhanced conservative phase field method for moving contact line problems

AbstractThe traditional Allen–Cahn phase field model doesn't conserve mass and is mostly used in solidification microstructure formation. However, a recently modified Allen–Cahn phase field model has riveted the attention of the academic community. It was obtained by subtracting the curvature‐driven flow term from the advective Allen–Cahn phase field model, and thus improves the boundedness of the phase field. More recently, the model has been further refined with the recovered signed distance function to compute interface normal vectors. This paper develops a three dimensional phase field model, based on the abovementioned Allen–Cahn phase field model. The model was discretized using a finite difference method on a half‐staggered grid. More important, interfacial tension was expressed in a potential form. The model was tested against a number of cases and was applied to impacts in various conditions. Besides, the model was parallelized using the shared memory parallelism, OpenMP, to facilitate computation.

A pressure Poisson equation-based second-order method for solving two-dimensional moving contact line problems with topological changes

We develop a second-order Cartesian grid based numerical method to solve two-dimensional moving contact line problems, which are modeled by the incompressible Navier–Stokes equations with the Navier-slip condition and the contact angle condition (CAC). The solutions of the flow field and the interface motion are decoupled in an alternating way. For a given interface, the velocity field is solved via a pressure Poisson equation formulation of the incompressible Navier–Stokes equations, which is numerically discretized by the immersed interface method. Once the velocity field is obtained, the interfacial kinematics together with the CAC is reformulated into a variational system, which is solved by the parametric finite element method (FEM). With this hybrid method, we detect topological changes in the interface by the inconsistency of neighboring normal vectors, which are directly computed through the parametric FEM. Second-order accuracy of the proposed method in both the interface and the contact line positions before and after topological changes has been numerically validated. Moreover, with the help of the numerical method, the merging and collision dynamics of droplets on the substrates are easily investigated.

Implementing contact angle boundary conditions for second-order Phase-Field models of wall-bounded multiphase flows

In the present work, a general formulation is proposed to implement the contact angle boundary conditions for the second-order Phase-Field models, which is applicable to N-phase (N⩾2) moving contact line problems. To remedy the issue of mass change due to the contact angle boundary condition, a source term or Lagrange multiplier is added to the original second-order Phase-Field models, which is determined by the consistent and conservative volume distribution algorithm so that the summation of the order parameters and the consistency of reduction are not influenced. To physically couple the proposed formulation to the hydrodynamics, especially for large-density-ratio problems, the consistent formulation is employed. The reduction-consistent conservative Allen-Cahn models are chosen as examples to illustrate the application of the proposed formulation. The numerical scheme that preserves the consistency and conservation of the proposed formulation is employed to demonstrate its effectiveness. Results produced by the proposed formulation are in good agreement with the exact and/or asymptotic solutions. The proposed method captures complex dynamics of moving contact line problems having large density ratios.

Effective boundary conditions for dynamic contact angle hysteresis on chemically inhomogeneous surfaces

Recent experiments (Guan et al., Phys. Rev. Lett., vol. 116, issue 6, 2016a, p. 066102; Guan et al., Phys. Rev. E, vol. 94, issue 4, 2016b, p. 042802) showed many interesting phenomena on dynamic contact angle hysteresis while there is still a lack of complete theoretical interpretation. In this work, we study the time averaging of the apparent advancing and receding contact angles on surfaces with periodic chemical patterns. We first derive a new Cox-type boundary condition for the apparent dynamic contact angle on homogeneous surfaces using the Onsager variational principle. Based on this condition, we propose a reduced model for some typical moving contact line problems on chemically inhomogeneous surfaces in two dimensions. Multiscale expansion and averaging techniques are employed to approximate the model for asymptotically small chemical patterns. We obtain a quantitative formula for the averaged dynamic contact angles. It describes explicitly how the advancing and receding contact angles depend on the velocity and the chemical inhomogeneity of the substrate. The formula is a coarse-graining version of the Cox-type boundary condition on inhomogeneous surfaces. Numerical simulations are presented to validate the analytical results. The numerical results also show that the formula characterizes well the complicated behaviour of dynamic contact angle hysteresis observed in the experiments.

Physical and Mechanical Analysis of Moving Contact Line

As a typical moving contact line problem, the industrial coating and coating problem has a very important application. The purpose of this paper is to study the physico-mechanical analysis of moving contact lines. The research background and significance of the moving contact line problem are given, the research progress of the plate pull-up coating is reviewed, and the research status of the moving contact line problem on the chemically uneven surface is given. The tilt-pulling problem of chemically inhomogeneous surfaces with horizontal fringe distribution is studied. Numerical simulations investigate the effects of different strip widths on the dynamic behavior of the contact line and the evolution of the liquid film morphology. The effects of the plate width and the angle between the plate edge and the pulling direction on the critical state of the wetting transition, the dynamic process of the inclined contact line and the evolution of the liquid film morphology were investigated. When the widths of the hydrophilic and hydrophobic strips are sufficiently small, both the critical velocity of the wetting transition and the dynamic evolution process of the contact line and the liquid film are the same as the uniform surface results with the equilibrium contact angle being the predicted value of Cassie theory.

A second order accuracy preserving method for moving contact lines with Stokes flow

The immersed interface method (IIM) has been widely used in simulations of multiphase flows with closed interfaces. We generalize IIM to simulate the moving contact line problems, which are modeled by the Stokes equation with the Navier-slip boundary condition and the contact angle condition. With the help of variational formulation, the contact angle condition can be combined with the interfacial kinematics in a weak form. A parametric finite element method (parametric FEM) is applied to solve for the interface motion as well as the curvature, which are in turn used to update the correction terms for the irregular points in IIM. The hybrid IIM-parametric FEM method is Cartesian grid based, and achieves second order accuracy not only in the velocity field but also in the interface and the contact line motion. This is validated by numerical results. Moreover, we generalize the method to account for discontinuous viscosity. Various numerical experiments are presented in the study of droplet motion and contact angle hysteresis.

The extended discontinuous Galerkin method adapted for moving contact line problems via the generalized Navier boundary condition

AbstractIn this work, an extended discontinuous Galerkin (extended DG/XDG also called unfitted DG) solver for two‐dimensional flow problems exhibiting moving contact lines is presented. The generalized Navier boundary condition is employed within the XDG discretization for the handling of the moving contact lines. The spatial discretization is based on a symmetric interior penalty method and the numerical treatment of the surface tension force is done via the Laplace–Beltrami formulation. The XDG method adapts the approximation space conformal to the position of the interface and allows a sub‐cell accurate representation within the sharp interface formulation. The interface is described as the zero set of a signed‐distance level‐set function and discretized by a standard DG method. No adaption of the level‐set evolution algorithm is needed for the extension to moving contact line problems. The developed solver is validated against typical two‐dimensional contact line driven flow phenomena including droplet simulations on a wall and the two‐phase Couette flow.

Thermodynamically consistent phase-field modelling of contact angle hysteresis

Abstract

A Multiple Scalar Auxiliary Variables Approach to the Energy Stable Scheme of the Moving Contact Line Problem

A Multiple Scalar Auxiliary Variables Approach to the Energy Stable Scheme of the Moving Contact Line Problem

Comparison of energy stable simulation of moving contact line problems using a thermodynamically consistent Cahn–Hilliard Navier–Stokes model

Liquid droplets sliding along solid surfaces are a frequently observed phenomenon in nature, e.g., raindrops on a leaf, and in everyday situations, e.g., drops of water in a drinking glass. To model this situation, we use a phase field approach. The bulk model is given by the thermodynamically consistent Cahn–Hilliard Navier–Stokes model from Abels et al. (2012) [8]. To model the contact line dynamics we apply the generalized Navier boundary condition for the fluid and the dynamically advected boundary contact angle condition for the phase field as derived in Qian et al. (2006) [7]. In recent years several schemes were proposed to solve this model numerically. While they widely differ in terms of complexity, they all fulfill certain basic properties when it comes to thermodynamic consistency. However, an accurate comparison of the influence of the schemes on the moving contact line is rarely found. Therefore, we thoughtfully compare the quality of the numerical results obtained with three different schemes and two different bulk energy potentials. Especially, we discuss the influence of the different schemes on the apparent contact angles of a sliding droplet.

A phase-field-based lattice Boltzmann method for moving contact line problems on curved stationary boundaries in two dimensions

In this work, we propose a phase-field-based lattice Boltzmann method to simulate moving contact line (MCL) problems on curved boundaries. The key point of this method is to implement the boundary conditions on curved solid boundaries. Specifically, we use our recently proposed single-node scheme for the no-slip boundary condition and a new scheme is constructed to deal with the wetting boundary conditions (WBCs). In particular, three kinds of WBCs are implemented: two wetting conditions derived from the wall free energy and a characteristic MCL model based on geometry considerations. The method is validated with several MCL problems and numerical results show that the proposed method has utility for all the three WBCs on both straight and curved boundaries.

Simulation of micron-scale drop impact

The numerical simulation of droplet impact is of interest for a vast variety of industrial processes, where practical experiments are costly and time-consuming. In these simulations, the dynamic contact angle is a key parameter, but the modeling of its behavior is poorly understood so far. One of the few models which considers the overall physical context of the involved ‘moving contact line problem’ is Shikhmurzaev’s interface formation model (Shikhmurzaev, 2008). In addition to keeping the problem well-posed, all surface and bulk parameters, such as the contact angle, are determined as part of the solution rather than being prescribed functions of contact line speed. In this article, we couple an asymptotic version of the interface formation model with our three-dimensional incompressible two-phase Navier–Stokes solver. Additionally, we employ a simple constant contact angle approach. We compare the results of these two numerical models with those from laboratory experiments for the micron-scale droplet impact on substrates with a variety of wetting characteristics. With our sophisticated asymptotic interface formation model, the droplet shapes, heights and diameters compare very well with practical experiments.

A kinematic evolution equation for the dynamic contact angle and some consequences

We investigate the moving contact line problem for two-phase incompressible flows with a kinematic approach. The key idea is to derive an evolution equation for the contact angle in terms of the transporting velocity field. It turns out that the resulting equation has a simple structure and expresses the time derivative of the contact angle in terms of the velocity gradient at the solid wall. Together with the additionally imposed boundary conditions for the velocity, it yields a more specific form of the contact angle evolution. Thus, the kinematic evolution equation is a tool to analyze the evolution of the contact angle. Since the transporting velocity field is required only on the moving interface, the kinematic evolution equation also applies when the interface moves with its own velocity independent of the fluid velocity. We apply the developed tool to a class of moving contact line models which employ the Navier slip boundary condition. We derive an explicit form of the contact angle evolution for sufficiently regular solutions, showing that such solutions are unphysical. Within the simplest model, this rigorously shows that the contact angle can only relax to equilibrium if some kind of singularity is present at the contact line. Moreover, we analyze more general models including surface tension gradients at the contact line, slip at the fluid–fluid interface and mass transfer across the fluid–fluid interface.

Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation.

Energetic Variational Approach is used to derive a novel thermodynamically consistent three-phase model of a mixture of Newtonian and visco-elastic fluids. The model which automatically satisfies the energy dissipation law and is Galilean invariant, consists of coupled Navier-Stokes and Cahn-Hilliard equations. Modified General Navier Boundary Condition with fluid elasticity taken into account is also introduced for using the model to study moving contact line problems. Energy stable numerical scheme is developed to solve system of model equations efficiently. Convergence of the numerical scheme is verified by simulating a droplet sliding on an inclined plane under gravity. The model can be applied for studying various biological or biophysical problems. Predictive abilities of the model are demonstrated by simulating deformation of venous blood clots with different visco-elastic properties and experimentally observed internal structures under different biologically relevant shear blood flow conditions.

Thin wedge evaporation/condensation controlled by the vapor dynamics in the atmosphere.

Evaporation or condensation in the vicinity of the immobile (pinned) contact line in an atmosphere of some inert (noncondensable) gas is considered here in a partial wetting configuration. Such a problem is relevant to many situations, in particular to a drop or a liquid film drying in open air. The thermal effects are not important and the mass exchange rate is controlled by the vapor dynamics in the gas. By following previous works, we account for the weak coupling between the diffusion in the gas and flow in the liquid through the Kelvin effect. Such a problem is nonlocal because of the diffusion in the gas. For generality, we consider a geometry of a liquid wedge posed on a flat and homogeneous substrate surrounded by a gas phase with a diffusion boundary layer of uniform thickness [Formula: see text]. Similarly to the moving contact line problem, the phase change leads to the hydrodynamic contact line singularity. The asymptotic analysis of this problem is carried out for the liquid wedge of the length [Formula: see text]. Three asymptotic regions are identified: the microscopic one (in which the singularity is relaxed, in the present case with the Kelvin effect) and two intermediate regions. The Kelvin effect alone turns to be sufficient to relax the singularity. The scaling laws for the interface slope and mass evaporation/condensation flux in each region are discussed. It is found that the difference of the apparent contact angle (i.e., interface slope in the second intermediate region) and the equilibrium contact angle is inversely proportional to the square root of [Formula: see text] and square root of the microscopic length, whatever is the singularity relaxation mechanism.

On the Kinematics of Contact Line Motion

AbstractWe investigate the moving contact line problem for two‐phase incompressible flows by a kinematic approach. The key idea is to derive an evolution equation for the contact angle assuming the transporting velocity field to be given. It turns out that the resulting equation expresses the time derivative of the contact angle in terms of the velocity gradient at the solid wall. Together with the additionally imposed boundary conditions for the velocity, it yields a more specific form of the contact angle evolution. In this paper we consider the Navier slip boundary condition, which is frequently used for the modeling of moving contact lines. Exploiting furthermore the interfacial transmission condition for the viscous stress, we derive an explicit form of the contact angle evolution for sufficiently regular solutions. In the absence of phase change it only involves the contact line velocity and the slip length from the Navier condition. From this equation we can read off the qualitative behavior of the contact angle evolution for sufficiently regular solutions to these models, which turns out to be unphysical.

A lattice Boltzmann model for multiphase flows with moving contact line and variable density

In this paper, we develop an efficient lattice Boltzmann model for the two-phase moving contact line problem with variable density. The Navier–Stokes and Cahn–Hilliard equations are recovered from the lattice Boltzmann model proposed by Fakhari and Rahimian [5]. To improve numerical stability, we present a semi-implicit lattice Boltzmann method together with a mixed finite difference scheme. In order to describe the behavior of the contact line motion on the boundary, we incorporate the generalized Navier boundary condition [25] by the nonequilibrium extrapolation method [8]. The proposed method is easy to implement and retains the advantage of the standard lattice Boltzmann method. Numerical tests are carried out to verify the proposed method. Our numerical results show that the present approach is able to model two-phase flows with variable density and moving contact line.

A Parallel Finite Element Method for 3D Two-Phase Moving Contact Line Problems in Complex Domains

Moving contact line problem plays an important role in fluid-fluid interface motion on solid surfaces. The problem can be described by a phase-field model consisting of the coupled Cahn–Hilliard and Navier–Stokes equations with the generalized Navier boundary condition (GNBC). Accurate simulation of the interface and contact line motion requires very fine meshes, and the computation in 3D is even more challenging. Thus, the use of high performance computers and scalable parallel algorithms are indispensable. In this paper, we generalize the GNBC to surfaces with complex geometry and introduce a finite element method on unstructured 3D meshes with a semi-implicit time integration scheme. A highly parallel solution strategy using different solvers for different components of the discretization is presented. More precisely, we apply a restricted additive Schwarz preconditioned GMRES method to solve the systems arising from implicit discretization of the Cahn–Hilliard equation and the velocity equation, and an algebraic multigrid preconditioned CG method to solve the pressure Poisson system. Numerical experiments show that the strategy is efficient and scalable for 3D problems with complex geometry and on a supercomputer with a large number of processors.

Bridging length and time scales in moving contact line problems

Physical, mechanical, chemical, and biological processes for various problems span length and time scales of many orders of magnitude. As a typical dynamic process, moving contact line (MCL) problem or droplet spreading consists of 7-8 length and time scales from the atomistic to the continuum, as shown in Figure 1. A grand challenge in MCL problems is to link these vastly different length and time scales.

An efficient and energy stable scheme for a phase‐field model for the moving contact line problem

SummaryIn this paper, we propose for the first time a linearly coupled, energy stable scheme for the Navier–Stokes–Cahn–Hilliard system with generalized Navier boundary condition. We rigorously prove the unconditional energy stability for the proposed time discretization as well as for a fully discrete finite element scheme. Using numerical tests, we verify the accuracy, confirm the decreasing property of the discrete energy, and demonstrate the effectiveness of our method through numerical simulations in both 2‐D and 3‐D. Copyright © 2015 John Wiley & Sons, Ltd.