Generalized Navier Boundary Condition Research Articles

An improved phase-field algorithm for simulating the impact of a drop on a substrate in the presence of surfactants

This paper is devoted to the numerical study of droplet impact on solid substrates in the presence of surfactants. We formulate the problem in an energetically variational framework and introduce an incompressible Cahn-Hilliard-Navier-Stokes system for the phase-field modeling of two-phase flows. Flory-Huggins potential and generalized Navier boundary condition are used to account for soluble surfactants and moving contact lines. Based on the convex splitting and pressure stabilization technique, we develop unconditionally energy stable schemes for this model. The discrete energy dissipation law for the original energy is rigorously proved for the first-order scheme. The numerical methods are implemented using finite difference method in three-dimensional cylindrical coordinates with axisymmetry. Using the proposed methods for this model, we systematically study the impact dynamics of both clean and contaminated droplets (with surfactants) in a series of numerical experiments. In general, the dissipation in the impact dynamics of a contaminated drop is smaller than that in the clean case, and topological changes are more likely to occur for contaminated drops. Adding surfactants can significantly influence the impact phenomena, leading to the enhancement of adherence effect on hydrophilic surfaces and splashing on hydrophobic surfaces. Some qualitative agreements with experiments are also obtained.

Stream function solutions for some contact line boundary conditions: Navier slip, super slip and the generalized Navier boundary condition

The stream function solution for the inner region Stokes flow, for a locally plane moving fluid interface near the triple point, is derived considering three different boundary conditions: the Navier slip boundary condition (NBC), the super-slip boundary condition and the generalized Navier boundary condition (GNBC). The NBC, incorporating a slip length parameter λ , is a well-known method for regularization in the context of the three-phase dynamic contact line problem. It is demonstrated that the velocity field solution under this boundary condition maintains a C 0 continuity at the contact line, resulting in a logarithmic divergence of the pressure at the contact line. By contrast, the super-slip boundary condition establishes a proportional relationship between the wall velocity and the normal derivative of the shear stress, leading to a C 1 velocity field. Furthermore, the GNBC, which introduces an uncompensated Young stress to drive the contact line, yields a C 2 velocity field. The dominant terms are explicitly derived, and the analytical approach presented here can be extended to other bi-harmonic problems as well.

Energy stable finite element strategy for simulating spreading, sliding and rolling flow dynamics of viscoelastic droplets

Moving mesh finite element method (FEM) for simulating dynamics of viscoelastic droplets on inclined surfaces is proposed. Viscoelasticity is incorporated into the governing system employing the Oldroyd–B constitutive model. The supporting (inclined) surface is allowed to have non–homogeneous properties incorporated into the mathematical model through the generalized Navier boundary conditions (GNBC). The droplet motion (sliding and/or rolling) is handled by employing arbitrary Lagrangian Eulerian (ALE) framework. Energy balance is derived from the governing system and it is a starting point in the derivation of the numerical scheme. The overall numerical strategy is designed in such a way that a counterpart of the (continuous) energy balance holds on the discrete level. This ensures that no spurious energy is introduced into the discrete system which, in turn, guarantees the stability of the scheme in the energy norm. The framework proposed is very general and encapsulates both two and three dimensional scenarios. Numerical studies in this work focus primarily on the three dimensional scenarios since such are significantly more challenging and seem to be much scarcer in the literature. The newly proposed numerical strategy is validated on several examples to confirm the theoretical predictions. The role of viscoelasticity in the overall droplet dynamics is briefly investigated and the behaviors of Newtonian and non–Newtonian droplets are compared.

Front-tracking simulation of the wetting behavior of an impinging droplet using a relaxed impermeability condition and a generalized Navier boundary condition

In this paper, we present a new method for simulating the wetting behavior of a droplet impinging on a solid wall. It works continuously from the high-speed regime (capillary number ∼ 1) to the low-speed regime (capillary number < 10−2). We combined front-tracking with a generalized Navier boundary condition (GNBC)—to represent low-speed wetting that takes into account slip at the molecular scale—with a rolling-belt model to represent high-speed wetting due to the rolling motions of fluid elements. Usually, a rolling-belt model requires both a threshold distance and compulsory switching from the high-speed regime. To overcome this defect, we utilized the modified relaxed impermeability condition originally proposed by Lukyanov and Pryer (Langmuir, Vol. 33, (2017), pp. 8582–8590.) for the rolling-belt model. By applying this condition, interface markers very near the wall can have a velocity normal to the wall as they approach it in reproducing the rolling motion. There is consequently no need of either an adjustable threshold distance or a compulsory switch-off of the rolling-belt model. We confirmed the results obtained using the proposed simulation method by comparing them with published experimental data. The validity and applicability of the present method are confirmed by the very good agreement of the simulations with the experimental data for a variety of conditions.

Study of the trapping mechanism of merging drops moving under thermocapillary migration on a surface with wettability contrast

Trapping of drops due to wettability contrast in porous materials emerges in a variety of subsurface and manufacturing applications. In carbon capture and storage, carbon dioxide must be trapped in order to avoid its release into the atmosphere, while trapped oil must be displaced. Both carbon dioxide and oil can exist as drops in porous media. The migration of a single drop attached to a wall can be hindered if the wall surface has a different wettability. However, the trapping condition becomes more complicated in the presence of two or more drops. In this work, we aim to study the trapping mechanism during thermocapillary migration of a merging drop on a heterogeneous surface. To do so, numerical simulations have been performed using the Front-Tracking/Finite Volume Method, where the Navier-Stokes equations are coupled with the conservation equations. The generalized Navier boundary condition (GNBC) has been used as the slip model to remove the viscous singularity. The combined finite-volume and tracking method is able to deal with different types of discontinuities in compressible or incompressible fluid flows, as e.g., interfaces. The material properties of the drop and the ambient fluid are different, and surface tension depends on the temperature. The results have shown that there exist three regimes characterizing the motion of merging drops, including the passage of the leading drop, the trapping of the merging drop, and the partial trapping, which correspond to the trapping of the leading drop but the passage of the merging drop. We show that there is a critical wettability contrast at which the merging drop gets trapped. The effects of Marangoni number (Ma) and viscosity ratio are investigated. The critical wettability contrast decreases with the increase of Ma number and the regime shift as a function of the viscosity and Marangoni number. These findings have implications for the design of geological carbon dioxide storage, improvement of oil recovery and microfluidic device development.

Role of surfactant-induced Marangoni effects in droplet dynamics on a solid surface in shear flow

We address the dynamics of a surfactant-laden droplet on a solid surface in simple shear flow numerically. Our analysis uses the front-tracking method to take surfactant transport into account. The interfacial tension and the slip coefficient, both of which depend heavily on the surfactant concentration, are fully integrated into the generalized Navier boundary condition to model the moving contact lines. Accurate prediction of droplet motion indicates that the surfactant can change droplet behavior drastically. Surfactant-induced effects, such as interfacial tension reduction, the Marangoni stress, and wettability alternation, are investigated for various capillary numbers, surface wettabilities, elasticity numbers, and surface Péclet numbers. Deformation and motion of a sliding droplet are enhanced by the Marangoni effect, which is associated with an interfacial tension gradient. When the capillary number reaches a critical value, the sliding-to-detachment and detachment-to-pinch-off transitions occur. Both transitions can be triggered and accelerated by a surfactant, especially when convection is dominant. As a result, the critical capillary number decreases, but exhibits a non-monotonic relationship with the elasticity number and Péclet number. The mechanisms that underlie the effect of Marangoni stress are discussed by analyzing the distributions of the surfactant concentration and the hydrodynamic forces exerted on the droplet. Accumulation of surfactants near the receding contact line reverses the local concentration gradient, attempts to change its direction along the interface, and delays droplet detachment. Furthermore, the strong surfactant dilution reduces both the surfactant concentration and the interfacial tension gradient, and thereby increasing the critical value for droplet pinch-off.

Energy stable arbitrary Lagrangian Eulerian finite element scheme for simulating flow dynamics of droplets on non–homogeneous surfaces

An energy stable finite element scheme within arbitrary Lagrangian Eulerian (ALE) framework is derived for simulating the dynamics of millimetric droplets in contact with solid surfaces. Supporting surfaces considered may exhibit non–homogeneous properties which are incorporated into the governing system through generalized Navier boundary conditions (GNBC). Numerical scheme is constructed such that the counterpart of (continuous) energy balance holds on the discrete level. This ensures that no spurious energy is introduced into the discrete system, i.e. the discrete formulation is stable in the energy norm. The newly proposed scheme is numerically validated to confirm the theoretical predictions. Of a particular interest is the case of droplet on a non–homogeneous inclined surface. This case shows the capabilities of the scheme to capture the complex droplet dynamics (sliding and rolling) while maintaining stability during the long time simulation.

Energy-stable numerical method for compressible flow with generalized Navier boundary condition

In this paper, we derive a dimensionless model for pure-component two-phase compressible flows with Van der Waals equation of state (EoS) and generalized Navier boundary condition (GNBC). We propose three energy-stable numerical schemes. One of them is based on the scalar auxiliary variable (SAV) approach for Helmholtz free energy for bulk and surface free energy, which leads to a modified energy and is proved to be unconditional stable. Another numerical scheme is based on the Lagrange multiplier approach for Helmholtz free energy for bulk and surface free energy, which leads to the original energy and is proved to be unconditional stable. Numerical results are presented to verify the effectiveness of the proposed methods.

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.

The effect of surface pattern property on the advancing motion of three-dimensional droplets

<p style='text-indent:20px;'>We investigate numerically the advancing motion of 3D droplets spreading on physically flat chemically heterogeneous surfaces with periodic structures. We use the Navier-Stokes-Cahn-Hilliard equations with the generalized Navier boundary conditions to model the motion of droplets. Based on a convex splitting scheme, we have done numerical simulations and compared the results between different surface patterns quantitatively. We study the effect of pattern property on the advancing motion of three phase contact lines, the critical volume at the contact line jump and the effective advancing angles. By increasing the volume of droplet slowly on heterogeneous surfaces with different pattern property, we find that the advancing contact line approaches an equiangular octagon for the patterned surface with periodic squares separated by channels and approaches a regular hexagon for the patterned surface with periodic circles in regular hexagonal arrays. The shape of three-phase contact line is much more determined by the macro structure of the pattern than the micro structure of the pattern in each period.

Slip transition in dynamic wetting for a generalized Navier boundary condition

HypothesisComputer fluid dynamics simulations of dynamic wetting are often performed using a slip model on the substrate. In previous studies, the generalized Navier boundary condition (GNBC) has shown promising results and could help clear the gap between molecular and continuum scales, but lacks quantitative comparisons to experiments. We seek to investigate the dependence between the contact-line velocity and the slip length in a GNBC, by confronting numerical simulations to experimental data. ExperimentsThe physical properties of a molten polymer (polyethylene glycol) were assessed thoroughly. Its dynamic contact angle on a cellulosic substrate was measured carefully using the Wilhelmy method. The experiment was reproduced in a finite elements model using a GNBC. It was repeated for capillary numbers between 10−6 and 10−1, and slip lengths ranging from 1 μm to 1 mm. FindingsA realistic value of the slip length was selected by matching the dynamic contact angles issued from numerical simulations and their experimental counterparts. The slip length behavior as a function of contact line velocity displayed a clear transition. The model also reproduced a dynamic wetting transition between frictional and viscous dissipations, which seems to be linked to an increasing difference between microscopic and macroscopic contact angles.

Fully discrete energy stable scheme for a phase-field moving contact line model with variable densities and viscosities

In this study, we propose a fully discrete energy stable scheme for the phase-field moving contact line model with variable densities and viscosities. The mathematical model comprises a Cahn–Hilliard equation, Navier–Stokes equation, and the generalized Navier boundary condition for the moving contact line. A scalar auxiliary variable is employed to transform the governing system into an equivalent form, thereby allowing the double well potential to be treated semi-explicitly. A stabilization term is added to balance the explicit nonlinear term originating from the surface energy at the fluid–solid interface. A pressure stabilization method is used to decouple the velocity and pressure computations. Some subtle implicit–explicit treatments are employed to deal with convention and stress terms. We establish a rigorous proof of the energy stability for the proposed time-marching scheme. A finite difference method based on staggered grids is then used to spatially discretize the constructed time-marching scheme. We also prove that the fully discrete scheme satisfies the discrete energy dissipation law. Our numerical results demonstrate the accuracy and energy stability of the proposed scheme. Using our numerical scheme, we analyze the contact line dynamics based on a shear flow-driven droplet sliding case. Three-dimensional droplet spreading is also investigated based on a chemically patterned surface. Our numerical simulation accurately predicts the expected energy evolution and it successfully reproduces the expected phenomena where an oil droplet contracts inward on a hydrophobic zone and then spreads outward rapidly on a hydrophilic zone.

A phase-field moving contact line model with soluble surfactants

A phase-field moving contact line model is presented for a two-phase system with soluble surfactants. With the introduction of some scalar auxiliary variables, the original free energy functional is transformed into an equivalent form, and then a new governing system is obtained. The resulting model consists of two Cahn–Hilliard-type equations and incompressible Navier–Stokes equation with variable densities, together with the generalized Navier boundary condition for the moving contact line. We prove that the proposed model satisfies the total energy dissipation with time. To numerically solve such a complex system, we develop a nonlinearly coupled scheme with unconditional energy stability. A splitting method based on pressure stabilization is used to solve the Navier–Stokes equation. Some subtle implicit–explicit treatments are adopted to discretize convection and stress terms. A stabilization term is artificially added to balance the explicit nonlinear term associated with the surface energy at the fluid–solid interface. We rigorously prove that the proposed scheme can preserve the discrete energy dissipation. An efficient finite difference method on staggered grids is used for the spatial discretization. Numerical results in both two and three dimensions demonstrate the accuracy and energy stability of the proposed scheme. Using our model and numerical scheme, we investigate the wetting behavior of droplets on a solid wall. Numerical results indicate that surfactants can affect the wetting properties of droplet by altering the value of contact angles.

Thermodynamically consistent modelling of two-phase flows with moving contact line and soluble surfactants

Droplet dynamics on a solid substrate is significantly influenced by surfactants. It remains a challenging task to model and simulate the moving contact line dynamics with soluble surfactants. In this work, we present a derivation of the phase-field moving contact line model with soluble surfactants through the first law of thermodynamics, associated thermodynamic relations and the Onsager variational principle. The derived thermodynamically consistent model consists of two Cahn–Hilliard type of equations governing the evolution of interface and surfactant concentration, the incompressible Navier–Stokes equations and the generalized Navier boundary condition for the moving contact line. With chemical potentials derived from the free energy functional, we analytically obtain certain equilibrium properties of surfactant adsorption, including equilibrium profiles for phase-field variables, the Langmuir isotherm and the equilibrium equation of state. A classical droplet spread case is used to numerically validate the moving contact line model and equilibrium properties of surfactant adsorption. The influence of surfactants on the contact line dynamics observed in our simulations is consistent with the results obtained using sharp interface models. Using the proposed model, we investigate the droplet dynamics with soluble surfactants on a chemically patterned surface. It is observed that droplets will form three typical flow states as a result of different surfactant bulk concentrations and defect strengths, specifically the coalescence mode, the non-coalescence mode and the detachment mode. In addition, a phase diagram for the three flow states is presented. Finally, we study the unbalanced Young stress acting on triple-phase contact points. The unbalanced Young stress could be a driving or resistance force, which is determined by the critical defect strength.

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.

Numerical simulation of self-propelled non-equal sized droplets

In general, external energy is needed to remove a liquid from a solid wall during cooling by dropwise condensation. However, experiments have shown that in some cases, droplets can propel themselves from the wall, without any external energy, due to the coalescence-induced conversion of surface energy to kinetic energy. Several authors have reported scaling analysis combined with an energy balance of kinetic energy, surface energy, and viscous dissipation to estimate whether the droplets can be self-propelled or not. Here, we use numerical simulation to describe the coalescence and self-propelling for nonequal sized droplets based on a finite-volume/front-tracking method and the generalized Navier boundary condition to model the moving contact lines. We find that a slightly smaller contact angle (165°) will give a larger out-of-plane jumping velocity than a superhydrophobic surface (with a contact angle of 180°). Further decreasing the contact angles results in “immobile coalescence.” The speed of the moving contact line does not influence the spontaneous removal process as long as it is large enough to let the contact areas detach. Nonequal sized drops are much more difficult to be spontaneously removed from a wall compared to equal-sized ones. Two spherical drops with a diameter ratio of 2.0 have 60% total usable energy compared to equal-sized ones, and only 0.5% of the total released energy can be effectively used for out-of-plane jumping.

Global Weak Solutions to a Diffuse Interface Model for Incompressible Two-Phase Flows with Moving Contact Lines and Different Densities

In this paper, we analyze a general diffuse interface model for incompressible two-phase flows with unmatched densities in a smooth bounded domain $$\Omega \subset {\mathbb {R}}^d$$ ( $$d=2,3$$ ). This model describes the evolution of free interfaces in contact with the solid boundary, namely the moving contact lines. The corresponding evolution system consists of a nonhomogeneous Navier–Stokes equation for the (volume) averaged fluid velocity $${\mathbf {v}}$$ that is nonlinearly coupled with a convective Cahn–Hilliard equation for the order parameter $$\varphi $$ . Due to the nontrivial boundary dynamics, the fluid velocity satisfies a generalized Navier boundary condition that accounts for the velocity slippage and uncompensated Young stresses at the solid boundary, while the order parameter fulfils a dynamic boundary condition with surface convection. We prove the existence of a global weak solution for arbitrary initial data in both two and three dimensions. The proof relies on a combination of suitable approximations and regularizations of the original system together with a novel time-implicit discretization scheme based on the energy dissipation law.

Numerical simulation of dynamic behavior of compound droplets on solid surface in shear flow by front-tracing method

The dynamic behavior of compound droplets on the wall of a rectangle channel by the action of an imposed shear flow is simulated using our developed three-dimensional front-tracking method combined with generalized Navier boundary condition. The validity of the present method was confirmed by comparing results of the compound droplet spreading under gravity force with analytical solutions. To determine the physical condition required for detaching/pinching-off the compound droplet, we have performed a large number of simulations with varying capillary numbers of two interfaces and obtained a phase diagram of compound droplets on solid surface in shear flow. The deformation and motion of the compound droplet including its contact line motion are investigated. It is found that the behavior of the compound droplet is controlled by two dimensionless parameters, the capillary numbers of the outer interface and the inner interface. Moreover, we also analyze the deformation and migration of the inner droplet and discuss its effect on the compound droplet. The simulation demonstrate that the lateral migration of the small inner droplet could accelerates the pinch-off process and the large inner droplet could promote the detachment for a moderate capillary number of the outer interface.

GNBC-based front-tracking method for the three-dimensional simulation of droplet motion on a solid surface

Previous front-tracking (FT) method-based models to simulate droplet motion on a solid surface with a moving contact line (MCL) are limited to two-dimensional models in which the Navier boundary condition (NBC) is employed for the MCL. In this paper, we develop a three-dimensional FT method that integrates the generalized Navier boundary condition (GNBC) to model the MCL. This GNBC-based FT method addresses several key issues, such as the integration of GNBC for the dynamic description of the MCL and its coupling with the surrounding flow, the accurate updating of the density and viscosity of the two-phase fluid near the contact line, and the restructuring of the Lagrangian mesh for tracking the drop surface, especially near the contact line. The stability and accuracy of the present numerical method are validated by several tests: (1) numerical performance tests, (2) simulation of the transient and steady-state shapes of droplets under flow with a fixed contact line, and (3) simulation of a droplet spreading under gravity and moving under a shear flow with MCLs. Excellent agreement is achieved between the results obtained by our model and the data obtained by other theoretical and numerical approaches.

Direct simulation of multiphase flows with modeling of dynamic interface contact angle

We describe a modeling technique for dynamic contact angle between a phase interface and a solid wall using a generalized Navier boundary condition in the context of a front-tracking-based multiphase method. The contact line motion is determined by the generalized Navier slip boundary condition in order to eliminate the infinite shear stress at the contact line. Applying this slip boundary condition only to the interface movement with various slip ratios shows good agreement with experimental results compared to allowing full fluid slip along the solid surface. The interface slip model performs well on grid convergence tests using both the slip ratio and slip length models. A detailed energy analysis was performed to identify changes in kinetic, surface, and potential energies as well as viscous and contact line dissipation with time. A friction coefficient for contact line dissipation was obtained based on the other computed energy terms. Each energy term and the friction coefficient were compared for different grid resolutions. The effect of varying the slip ratio as well as the contact angle distribution versus contact line speed was analyzed. The behavior of drop impact on a solid wall with different advancing and receding angles was investigated. Finally, the proposed dynamic contact model was extended to three dimensions for large-scale parallel calculations. The impact of a droplet on a solid cylinder was simulated to demonstrate the capabilities of the proposing formulation on general solid structures. Widely different contact angles were tested and showed distinctive characteristic behavior clearly.