Prescribed Contact Angle Research Articles

On a minimizing movement scheme for mean curvature flow with prescribed contact angle in a curved domain and its computation

We introduce a capillary Chambolle-type scheme for mean curvature flow with prescribed contact angle. Our scheme includes a capillary functional instead of just the total variation. We show that the scheme is well-defined and has consistency with the energy minimizing scheme of Almgren–Taylor–Wang type. Moreover, for a planar motion in a strip, we give several examples of numerical computation of this scheme based on the split Bregman method instead of a duality method.

Numerical Stability and Accuracy of Contact Angle Schemes in Pseudopotential Lattice Boltzmann Model for Simulating Static Wetting and Dynamic Wetting

There are five most widely used contact angle schemes in the pseudopotential lattice Boltzmann (LB) model for simulating the wetting phenomenon: The pseudopotential-based scheme (PB scheme), the improved virtual-density scheme (IVD scheme), the modified pseudopotential-based scheme with a ghost fluid layer constructed by using the fluid layer density above the wall (MPB-C scheme), the modified pseudopotential-based scheme with a ghost fluid layer constructed by using the weighted average density of surrounding fluid nodes (MPB-W scheme) and the geometric formulation scheme (GF scheme). But the numerical stability and accuracy of the schemes for wetting simulation remain unclear in the past. In this paper, the numerical stability and accuracy of these schemes are clarified for the first time, by applying the five widely used contact angle schemes to simulate a two-dimensional (2D) sessile droplet on wall and capillary imbibition in a 2D channel as the examples of static wetting and dynamic wetting simulations respectively. (i) It is shown that the simulated contact angles by the GF scheme are consistent at different density ratios for the same prescribed contact angle, but the simulated contact angles by the PB scheme, IVD scheme, MPB-C scheme and MPB-W scheme change with density ratios for the same fluid-solid interaction strength. The PB scheme is found to be the most unstable scheme for simulating static wetting at increased density ratios. (ii) Although the spurious velocity increases with the increased liquid/vapor density ratio for all the contact angle schemes, the magnitude of the spurious velocity in the PB scheme, IVD scheme and GF scheme are smaller than that in the MPB-C scheme and MPB-W scheme. (iii) The fluid density variation near the wall in the PB scheme is the most significant, and the variation can be diminished in the IVD scheme, MPB-C scheme and MPB-W scheme. The variation totally disappeared in the GF scheme. (iv) For the simulation of capillary imbibition, the MPB-C scheme, MPB-W scheme and GF scheme simulate the dynamics of the liquid-vapor interface well, with the GF scheme being the most accurate. The accuracy of the IVD scheme is low at a small contact angle (44 degrees) but gets high at a large contact angle (60 degrees). However, the PB scheme is the most inaccurate in simulating the dynamics of the liquid-vapor interface. As a whole, it is most suggested to apply the GF scheme to simulate static wetting or dynamic wetting, while it is the least suggested to use the PB scheme to simulate static wetting or dynamic wetting.

Gradient estimates for the solutions of higher order curvature equations with prescribed contact angle

<abstract><p>In this paper, we use the maximum principle and moving frame technique to prove the global gradient estimates for the higher-order curvature equations with prescribed contact angle problems.</p></abstract>

Gradient estimate of the solutions to Hessian equations with oblique boundary value

Abstract In this paper, we study Hessian equations with the prescribed contact angle boundary value or oblique derivative boundary value and finally derive the a priori global gradient estimate for the admissible solutions.

Dynamic and static stability of a drop attached to an inhomogeneous plane wall

This article concerns the stability of a drop on a wall for which the contact angle, \(\theta _\mathrm{{{w}}}\), varies from place to place. Such a wall may allow unstable equilibria of the drop, i.e. ones for which small perturbations to equilibrium grow, making the equilibrium unrealisable in practice. This will be referred to as dynamic instability and is one of the two versions of instability considered. The other arises from consideration of potential energy, which is the sum of surface (liquid/gas, liquid/solid and solid/gas) components and the gravitational potential energy. Equilibria are extrema of the potential energy with respect to variations of drop geometry which preserve its volume. An equilibrium is said to be statically stable if it is a local minimum of the potential energy for volume-preserving perturbations of the drop. The relationship between static and dynamic stability is the main subject of this paper. The liquid flow is governed by the incompressible Navier–Stokes equations. To allow for the moving contact line, a Navier slip condition with slip length \(\lambda \) is used at the wall, as is a prescribed contact angle, \(\theta _\mathrm{{{w}}} =\theta _\mathrm{{{w}}} ({x,y})\), at the contact line, where x, y are Cartesian coordinates on the wall. The perturbation is assumed small, allowing linearisation of the governing equations and, in the usual manner of stability analysis, complex modes having the time dependency \(e^\mathrm{{{st}}}\) are introduced. This leads to an eigenvalue problem with eigenvalue s, the sign of whose real part determines dynamic stability/instability. A quite different eigenvalue problem, which describes static stability/instability is also derived. It is shown that, despite this difference, the conditions for dynamic and static instability are in fact the same. This conclusion is far from evident a priori but should be good news for interested numerical analysts because determination of static stability is much less numerically costly than a dynamic stability study, whereas it is the latter which gives a true determination of stability.

Non-Parametric Mean Curvature Flow with Prescribed Contact Angle in Riemannian Products

Abstract Assuming that there exists a translating soliton u ∞ with speed C in a domain Ω and with prescribed contact angle on ∂Ω, we prove that a graphical solution to the mean curvature flow with the same prescribed contact angle converges to u ∞ + Ct as t →∞. We also generalize the recent existence result of Gao, Ma, Wang and Weng to non-Euclidean settings under suitable bounds on convexity of Ω and Ricci curvature in Ω.

Phase-field simulations of precursor film in microcapillary imbibition for liquid-liquid systems

• Parameters' effects on the accuracy of phase-field simulations are discussed for Liquid-Liquid microcapillary imbibition. • The impacts of contact angle and viscosity ratio on precursor film are discussed. • Viscosity ratios are considered in the Cox-Voinov model as the contact angle between 0 o and 30 o . • Precursor films cause the dynamic contact angle smaller than predictions of the Cox-Voinov model. • Precursor films reduce 10% to 20% of the theoretical total imbibition time. Precursor film flow has been extensively studied on flat surfaces in Liquid-Gas systems. However, its effects in capillary imbibition processes, especially in Liquid-Liquid systems, are lack investigation. Using the phase-field method, this work simulates microcapillary imbibition processes in strong imbibition regimes with the viscosity ratio magnitude from 10 −1 to 10. The numerical model is first tuned with a prescribed contact angle of 30 o under which the precursor film is less likely occurs. The model validation results indicate that simulation accuracy of microcapillary imbibition is more sensitive to the Cn number and the model length than the mobility tuning parameter and viscosity ratio. After that, we discussed the effects of contact angle and viscosity ratio on imbibition processes considering precursor film. 25 o is the critical contact angle for occurring the precursor film in the capillary filling. Compared with the contact angle, the effects of the viscosity ratio on film flow is limited. Though the viscosity ratio effects are considered in the Cox-Voinov model by solving the power-law approximations of the integrand, the simulated dynamic contact angle is still observed lower than the theoretical values because of the precursor film. Moreover, it also found that the simulated dynamic contact angle could reach a value lower than the static contact angle in tested unfavorable viscosity ratios. The significant effects of precursor film make the simulated total imbibition time roughly 10% to 20% less than the theoretical imbibition time without considering precursor film flow.

The response of a 2D droplet on a wall executing small sinusoidal vibrations

This study concerns a two-dimensional liquid drop surrounded by gas and attached to a sinusoidally vibrating wall. Gravity is neglected and the moving contact lines are modelled using a Navier-type boundary condition at the wall and a prescribed contact angle, θ¯, which can take any value in the range 0<θ¯<π. The vibration amplitude, and hence the departure from equilibrium of the drop, is assumed sufficiently small that the problem can be linearized. Wall vibration can have components both normal and tangential to the wall. The solution of the linear problem can be expressed as the sum of two decoupled components corresponding to the response to purely normal and purely tangential vibration, which are respectively symmetric and antisymmetric with respect to reflection in the symmetry plane of the equilibrium drop. Asymptotic analysis of the drop oscillations for small Ohnesorge number, Oh, brings out two distinct damping mechanisms, both of which are accounted for. One, arising from viscous dissipation in the regions near the contact lines, is characterized by a parameter β. The other comes from the boundary layer at the wall and is of order Oh1/2. The small-Oh problem has been implemented numerically. As expected, lightly damped normal modes are found to have resonant response close to their inviscid oscillation frequencies. Damping coefficients for each of the two damping mechanisms and lightly damped modes are determined as a function of contact angle. The relative importance of boundary-layer and contact-line damping is quantified and found to depend both on the contact angle and on β/Oh1/2. Cases can be found in which the two damping mechanisms have comparable effects, as well as others for which one or other of the mechanisms is dominant. Comparison with DNS, which allows for nonlinear effects and has the same contact-line model, shows agreement for a particular case having small Oh and small wall displacement amplitude.

A diffuse domain method for two-phase flows with large density ratio in complex geometries

Abstract

Additive Eigenvalue Problems of the Laplace Operator with the Prescribed Contact Angle Boundary Condition

Additive eigenvalue problem appears in ergodic optimal control or the homogenization of Hamilton–Jacobi equations. It has wide applications in several fields including computer science and then attracts the attention. In this paper, we consider the Poisson equations with the prescribed contact angle boundary condition and finally derive the existence and the uniqueness of the solution to the additive problem of the Laplace operator with the prescribed contact angle boundary condition.

Short time existence for the curve diffusion flow with a contact angle

We show short-time existence for curves driven by curve diffusion flow with a prescribed contact angle α∈(0,π): The evolving curve has free boundary points, which are supported on a line and it satisfies a no-flux condition. The initial data are suitable curves of class W2γ with γ∈(32,2]. For the proof the evolving curve is represented by a height function over a reference curve: The local well-posedness of the resulting quasilinear, parabolic, fourth-order PDE for the height function is proven with the help of contraction mapping principle. Difficulties arise due to the low regularity of the initial curve. To this end, we have to establish suitable product estimates in time weighted anisotropic L2-Sobolev spaces of low regularity for proving that the non-linearities are well-defined and contractive for small times.

Maximum curvature for curves in manifolds of sectional curvature at most zero or one

We prove a sharp lower bound for the maximum curvature of a closed curve in a complete, simply connected Riemannian manifold of sectional curvature at most zero or one. When the bound is attained, we get the rigidity result. The proof utilizes the maximum principle for a suitable two-point function. In the same spirit, we also obtain a lower bound for the maximum curvature of a curve in the same ambient manifolds which has the same endpoints with a fixed geodesic segment and has a prescribed contact angle. As a corollary, the latter result applies to a curve with free boundary in geodesic balls of Euclidean space and hemisphere.

Minimizing movements for mean curvature flow of droplets with prescribed contact angle

We study the mean curvature motion of a droplet flowing by mean curvature on a horizontal hyperplane with a possibly nonconstant prescribed contact angle. Using the solutions constructed as a limit of an approximation algorithm of Almgren–Taylor–Wang and Luckhaus–Sturzenhecker, we show the existence of a weak evolution, and its compatibility with a distributional solution. We also prove various comparison results.

An energy-based equilibrium contact angle boundary condition on jagged surfaces for phase-field methods

We consider an energy-based boundary condition to impose an equilibrium wetting angle for the Cahn–Hilliard–Navier–Stokes phase-field model on voxel-set-type computational domains. These domains typically stem from μCT (micro computed tomography) imaging of porous rock and approximate a (on μm scale) smooth domain with a certain resolution. Planar surfaces that are perpendicular to the main axes are naturally approximated by a layer of voxels. However, planar surfaces in any other directions and curved surfaces yield a jagged/topologically rough surface approximation by voxels. For the standard Cahn–Hilliard formulation, where the contact angle between the diffuse interface and the domain boundary (fluid–solid interface/wall) is 90°, jagged surfaces have no impact on the contact angle. However, a prescribed contact angle smaller or larger than 90° on jagged voxel surfaces is amplified. As a remedy, we propose the introduction of surface energy correction factors for each fluid–solid voxel face that counterbalance the difference of the voxel-set surface area with the underlying smooth one. The discretization of the model equations is performed with the discontinuous Galerkin method. However, the presented semi-analytical approach of correcting the surface energy is equally applicable to other direct numerical methods such as finite elements, finite volumes, or finite differences, since the correction factors appear in the strong formulation of the model.

Boundary gradient estimate of the solution to mean curvature equations with Neumann boundary or prescribed contact angle boundary

In this paper, we use the moving frame on the surface and the maximum principle to estimate the gradient of the solutions to the mean curvature equations with Neumann boundary value and the prescribed contact angle boundary value, which simplifies the proof of the gradient estimate of the solutions to the mean curvature equations.

Gradient estimates for semi-linear elliptic equations with prescribed contact angle problem

In this paper, we find a new auxiliary function and use the maximum principle to give a new proof for the gradient of a solution for semi-linear elliptic equations with capillarity-type problem.

Nonparametric Mean Curvature Type Flows of Graphs with Contact Angle Conditions

In this paper we study nonparametric mean curvature type flows in $M\times\mathbb{R}$ which are represented as graphs $(x,u(x,t))$ over a domain in a Riemannian manifold $M$ with prescribed contact angle. The speed of $u$ is the mean curvature speed minus an admissible function $\psi(x,u,Du)$. Long time existence and uniformly convergence are established if $\psi(x,u, Du)\equiv 0$ with vertical contact angle and $\psi(x,u,Du)=h(x,u)\omega$ with $h_u(x,u)\geq h_0>0$ and $\omega=\sqrt{1+|Du|^2}$. Their applications include mean curvature type equations with prescribed contact angle boundary condition and the asymptotic behavior of nonparametric mean curvature flows of graphs over a convex domain in $M^2$ which is a surface with nonnegative Ricci curvature.

On the uniform convergence of piecewise linear solutions of an equilibrium capillary surface equation

We consider piecewise-linear solutions of the equilibrium equation of a capillary surface over a given triangulation of a multifaceted (polygonal) region. It is shown that, under certain conditions, the gradients of these functions remain bounded in absolute value when the partition is refined; i.e., the maximum diameter of the triangulation triangles tends to zero. This property is fulfilled if the piecewise-linear functions approximate the energy integral of a smooth function with the required accuracy. Some consequence of the obtained properties is the uniform convergence of the piecewise-linear solutions to the exact solution of the equilibrium capillary surface equation with the boundary condition in the form of a prescribed contact angle.

Mean curvature flow by the Allen–Cahn equation

In this paper, we investigate motion by mean curvature using the Allen–Cahn (AC) equation in two and three space dimensions. We use an unconditionally stable hybrid numerical scheme to solve the equation. Numerical experiments demonstrate that we can use the AC equation for applications to motion by mean curvature. We also study the curve-shortening flow with a prescribed contact angle condition.

Darcy’s Flow with Prescribed Contact Angle: Well-Posedness and Lubrication Approximation

We consider the spreading of a thin two-dimensional droplet on a solid substrate. We use a model for viscous fluids where the evolution is governed by Darcy’s law. At the contact point where air and liquid meet the solid substrate, a constant, non-zero contact angle (partial wetting) is assumed. We show local and global well-posedness of this free boundary problem in the presence of the moving contact point. Our estimates are uniform in the contact angle assumed by the liquid at the contact point. In the so-called lubrication approximation (long-wave limit) we show that the solutions converge to the solution of a one-dimensional degenerate parabolic fourth order equation which belongs to a family of thin-film equations. The main technical difficulty is to describe the evolution of the non-smooth domain and to identify suitable spaces that capture the transition to the asymptotic model uniformly in the small parameter $${\varepsilon}$$ .