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The ``no-slip'' boundary condition, i.e., zero fluid velocity relative to the solid at the fluid-solid interface, has been very successful in describing many macroscopic flows. A problem of principle arises when the no-slip boundary condition is used to model the hydrodynamics of immiscible-fluid displacement in the vicinity of the moving contact l...
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... consider a displacement of fluid 2 by fluid 1 over distance L (see Fig. 6). According to the Young's equation for the equilibrium contact angle, the force πr 2 ∆p 0 equals 2πrγ 12 cos θ 0 = 2πr(γ S1 − γ S2 ), where γ S1 and γ S2 are the interfacial tensions for the S/1 and S/2 interfaces, respectively. Thus the change of the interfacial free energy at the fluid-solid interface is given by πr 2 ∆p 0 L = ...
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Citations
... [13][14][15] The situation is more complex at the dynamic CL (DCL)-appearing typically during droplet spreading or moving on a substrate, where the advancing and receding CAs are different. To model the CA difference, a number of theoretical, [16][17][18][19] computational [20][21][22][23][24][25] and experimental [26][27][28] studies about the DCL have been carried out and have indicated that this dynamic effect is induced by the viscosity and friction in the vicinity of the DCL; however, the governing principle of the DCL motion still remains unclear mainly due to the lack of detailed information on the nanoscale thermal and flow fields around the DCL, and it is considered to be one of the long-standing unsolved problems of fluid dynamics. ...
When a contact line (CL)—where a liquid–vapor interface meets a substrate—is put into motion, it is well known that the contact angle differs between advancing and receding CLs. Using non-equilibrium molecular dynamics simulations, we reveal another intriguing distinction between advancing and receding CLs: while temperature increases at an advancing CL—as expected from viscous dissipation, we show that temperature can drop at a receding CL. Detailed quantitative analysis based on the macroscopic energy balance around the dynamic CL showed that the internal energy change of the fluid due to the change of the potential field along the pathline out of the solid–liquid interface induced a remarkable temperature drop around the receding CL, in a manner similar to latent heat upon phase changes. This result provides new insights for modeling the dynamic CL, and the framework for heat transport analysis introduced here can be applied to a wide range of nanofluidic systems.
... It is the driving force in continuum equations, 40 is necessary for the understanding of nonequilibrium molecular dynamics (NEMD), 41,42 and is essential for calculating the viscosity in Newtonian fluids using both equilibrium 43,44 and nonequilibrium methods. 45 Pressure/stress can provide insight into tribology, 46 reveal the slip length, 47,48 and capture many aspects of multiphase flows, including the moving contact line for the wetting behavior [see Fig. 2(d)] 49,50 and dynamics of active liquid interfaces. 51 These molecular details can be included in continuum-based engineering simulation, such as computational fluid dynamics (CFD), through coupled simulation 52,53 where stress coupling is useful in both fluid simulation 54 and for solid mechanics. ...
The pressure tensor (equivalent to the negative stress tensor) in both microscopic and macroscopic levels is fundamental to many aspects of engineering and science, including fluid dynamics, solid mechanics, biophysics, and thermodynamics. In this perspective paper, we review methods to calculate the microscopic pressure tensor. Connections between different pressure forms for equilibrium and non-equilibrium systems are established. We also point out several challenges in the field, including the historical controversies over the definition of the microscopic pressure tensor; the difficulties with many-body and long-range potentials; the insufficiency of software and computational tools; and the lack of experimental routes to the probe of the pressure tensor at the nanoscale. Possible future directions are suggested.
... On the fluid-solid interface, the classical no-slip boundary conditions are inadequate to capture the dynamics of the moving triple contact line. In [10][11][12], the generalized Navier boundary conditions (GNBC) have been developed within the continuum mechanics framework from the molecular dynamics (MD) theory for the moving contact line problems. In there, they resorted to an intermediate phase-field model in the process. ...
... These properties of Γ may be controlled by alternating its microstructure or by applying hydrophobic coatings (see [4,5] and references therein). (3) In its dimensional form, the Navier slip coefficient is given by ς = µ/l s , where l s is the slip length (see [10,12,17,25]) and it depends on the properties of the underlying surface. Typically of order 10 −9 m, l s can be greatly altered, for example, by applying artificial hydrophobic coatings (reported in [4]). ...
Finite element method (FEM) moving mesh strategy for simulating dynamics of droplets on inclined surfaces is designed within arbitrary Lagrangian Eulerian (ALE) framework. Depending on the physical and chemical properties of the fluid and the supporting surface, resulting droplet dynamics can exhibit sliding and/or rolling regimes. Exploiting the full potential of the ALE framework, moving mesh strategy designed in this work is capable of tracking the droplet evolution regardless of the flow regime and without the need for frequent mesh adaptation. Additional attention is invested into the discrete energy balance: possible sources of spurious energy due to the mesh motion are identified and investigated. The overall strategy exhibits a good tradeoff between the stability and efficiency, and demonstrates the ability to perform long-time simulations.
The capabilities of the proposed strategy are demonstrated on a couple of fairly complex (3D) scenarios motivated by the industrial applications. In particular, droplet dynamics is simulated for the case of inclined and heterogeneous supporting surfaces, which are designed with the aim to manipulate the droplet motion. Complex dynamics, including sliding, rolling and change in movement direction and wetting area, is successfully captured by the numerical simulation.
... However, adopting the linear surface-energy formulation will also result in the appearance of unphysical film [26,31,32]. To overcome it, several different forms of the surface-energy formulation are developed such as the sine form surface energy [33] and the cubic form surface energy [32,34,35]. Huang et al. [36] compared the performance of different surface-energy formulation in the phase-field LB model based on the Cahn-Hilliard (C-H) equation [37]. ...
In this work, wetting boundary schemes in a modified phase-field lattice Boltzmann (LB) method are developed to simulate the contact-line motion and contact angle hysteresis of binary fluids with large density ratios. In the modified phase-field LB method, two multiple-relaxation-time (MRT) LB equations are utilized, one with a nondiagonal relaxation matrix used to solve the conserved Allen-Cahn (A-C) equation and the other with a modified discrete source term used to solve the incompressible Navier-Stokes (N-S) equations. Based on the idea of geometric formulation, two wetting boundary schemes corresponding to the modified phase-field LB model are proposed. In the first scheme, a layer of virtual nodes is utilized to realize the geometric formulation. The values of the order parameter at the virtual nodes are determined by the central difference method. In the second scheme, the geometric formulation is realized through a local scheme for the gradient terms of the order parameter where the nonequilibrium parts of the distribution function are utilized. Compared with the first wetting boundary scheme and some previous models, the second wetting boundary scheme does not require the virtual nodes and simplifies the numerical implementation. The developed two schemes are validated through a series of benchmark simulations, including droplet spreading on idea substrate, capillary intrusion and droplet shearing on nonideal substrate. It is found that satisfactory accuracy can be achieved by both two schemes, and the second scheme can obtain better stability while the first scheme shows better performance in simulating contact angle hysteresis.
... Remark 2.1. Here we borrow the idea introduced in moving contact line models [43,44]: ...
A thermodynamically consistent phase-field model is introduced for simulating motion and shape transformation of vesicles under flow conditions. In particular, a general slip boundary condition is used to describe the interaction between vesicles and the wall of the fluid domain in the absence of cell-wall adhesion introduced by ligand-receptor binding. A second-order accurate in both space and time $C^0$ finite element method is proposed to solve the model governing equations. Various numerical tests confirm the convergence, energy stability, and conservation of mass and surface area of cells of the proposed scheme. Vesicles with different mechanical properties are also used to explain the pathological risk for patients with sickle cell disease.
... It has been observed that the contact line is able to move despite the apparent no-slip of the fluid at the wall. So called generalized Navier boundary conditions (GNBC) derived from the molecular dynamics (MD) theory for the Navier-Stokes equations have been proven to capture the dynamics of the triple contact line adequately (see [16,17,18,19]). GNBC have already been successfully applied for simulating free surface flows with ALE FEM for various multiphysics phenomena such as free surface flows in [13], sliding/rolling droplets kinematics in [20,21], droplets kinematics on non-isothermal surfaces in [22], chemotaxis phenomena in [23,24], to name a few. The main goal of this paper is the derivation of the energy stable numerical scheme based on the mathematical model incorporating GNBC and not the investigation of GNBC themselves. ...
... Regarding the Navier slip coefficient, in its dimensional form it is given by ς = µ/l s , where l s is the slip length (see [16,17,20,37]). l s is typically of order 10 −9 m (i.e. order of nanometers) but it can be greatly altered on manufactured surfaces by, e.g, applying hydrophobic coatings ( [28]). ...
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 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.
... In this paper, we show a calculation method of the MoP-based local stress tensor appli-cable to NEMD systems with a proper definition of the density and macroscopic velocity consistent with the mass and momentum conservation. We provide the formulation for systems consisting of single-component mono-atomic fluid molecules for simplicity, while the present framework is also applicable to systems of multi-component or poly-atomic fluid molecules, in which the intra-molecular interaction term (for details, see Appendix B) gives rise to a difficulty for the stress definition and resulting flux 33 ; not the kinetic term in Eq. (2) discussed in the present study. Related to this point, the non-uniqueness problem of the local stress/pressure tensor due to the inhomogeniety of liquids typically at the interface [34][35][36][37][38][39] is also an issue regarding the interaction term. ...
In this work, we developed a calculation method of local stress tensor applicable to non-equilibrium molecular dynamics (NEMD) systems, which evaluates the macroscopic momentum advection and the kinetic term of the stress in the framework of the Method of Plane (MoP), in a consistent way to guarantee the mass and momentum conservation. From the relation between the macroscopic velocity distribution function and the microscopic molecular passage across a fixed control plane, we derived a method to calculate the basic properties of the macroscopic momentum conservation law including the density, the velocity, the momentum flux, the interaction and kinetic terms of the stress tensor defined on a surface with a finite area. Any component of the streaming velocity can be obtained on a control surface, which enables the separation of the kinetic momentum flux into the advection and stress terms in the framework of MoP, and this enebles strict satisfaction of the the mass and momentum conservation for an arbitrary closed control volume (CV) set in NEMD systems. We validated the present method through the extraction of the density, velocity and stress distributions in a quasi-1D steady-state Couette flow system and in a quasi-2D steady-state NEMD system with a moving contact line. We showed that with the present MoP, in contrast to the volume average method (VA), the conservation law was satisfied even for a CV set around the moving contact line, which was located in a strongly inhomogeneous region.
... Lattice-Boltzmann methods have also treated the contact angle with a wall energy contribution [39,40]. Molecular dynamics (MD) simulations [41][42][43][44] have also been considered, where the interaction between fluid and solid particles is described by the Lennard-Jones potential, albeit mainly for simulating nanoscale systems. ...
Wetting is fundamental to many technological applications that involve the motion of the fluid-fluid interface on a solid. While static wetting is well understood in the context of thermodynamic equilibrium, dynamic wetting is more complicated in that liquid interactions with a solid phase, possibly on molecular scales, can strongly influence the macroscopic scale dynamics. The problem with continuum models of wetting phenomena is then that they ought to be augmented with microscopic models to describe the molecular neighborhood of the moving contact line. In this review, widely used models for the computation of wetting flows are summarized first, followed by an overview of direct numerical simulations based on the Volume-of-Fluid approach. Recent developments in the Volume-of-Fluid simulations of the wetting are then reviewed, with a particular attention paid on combining macro-scale simulations with the hydrodynamic theory near the moving contact line, as well as including a microscopic description by coupling with the van der Waals interface model. Finally, the extension to modeling the contact line motion on non-flat surfaces is surveyed, followed by hot topics in nucleate boiling.
... It has been recognized that there exists a nearly complete slip at the CL region, where the slip velocity is significantly larger than that of a small partial slip in the inner liquid-solid interface. 51,52 Since this distinct slip is only found to occur within a length scale of nanometers around the contact line, a single Navier slip model can provide overall good results if the characteristic scale of a simulation is much larger than this width of CL region. However, if the scale of a simulated system is below sub-micrometer, the particular slip for the CL region becomes more dominant, at which the single Navier slip model would fail on quantifying the speed of dynamic wetting unless an artificial large slip length is used. ...
Molecular dynamics (MD) and volume of fluid (VOF) are powerful methods for the simulation of dynamic wetting at the nanoscale and macroscale, respectively, but the massive computational cost of MD and the sensitivity and uncertainty of boundary conditions in VOF limit their applications to other scales. In this work, we propose a multiscale simulation strategy by enhancing VOF simulations using self-consistent boundary conditions derived from MD. Specifically, the boundary conditions include a particular slip model based on the molecular kinetic theory for the three-phase contact line to account for the interfacial molecular physics, the classical Navier slip model for the remaining part of the liquid–solid interface, and a new source term supplemented to the momentum equation in VOF to replace the convectional dynamic contact angle model. Each slip model has been calibrated by the MD simulations. The simulation results demonstrate that with these new boundary conditions, the enhanced VOF simulations can provide consistent predictions with full MD simulations for the dynamic wetting of nanodroplets on both smooth and pillared surfaces, and its performance is better than those with other VOF models, especially for the pinning–depinning phenomenon. This multiscale simulation strategy is also proved to be capable of simulating dynamic wetting above the nanoscale, where the pure MD simulations are inaccessible due to the computational cost.
... We employ a diffuse interface method, based on a phase-field model, to describe the interfacial dynamics. One of the prime advantages of such models over other approaches is their natural ability to handle contact line motion thanks to intrinsic diffusive processes [19,20,21,22,23]. When the phase-field model is combined with classical conservation equations for fluid mass and momentum, and complemented by particles' equations of motion, a full description of the system dynamics can be achieved. ...
... motion without any stress singularity or ad hoc fluid slip models [19,20,21,22,23]. The details of the coupled phase-field/Navier-Stokes equations have already been described in our previous work [24] and by a number of authors [23,25,26,27]. ...
We use dynamic numerical simulations to investigate the role of particle rotation in pairwise capillary interactions of particles trapped at a fluid interface. The fluid interface is modeled with a phase-field method which is coupled to the Navier-Stokes equations to solve for the flow dynamics. Numerical solutions are found using a finite element scheme in a bounded two-dimensional geometry. The interfacial deformations are caused by the buoyant weight of the particles, which are allowed to both translate and rotate due to the capillary and viscous forces and torques at play. The results show that the capillary attraction is faster between freely rotating particles than if particle rotation is inhibited, and the higher the viscosity mismatch, the greater the effect. To explain this result, we analyze the drag force exerted on the particles and find that the translational drag force on a rotating particle is always less than its non-rotating counterpart due to attenuated velocity gradients in the vicinity of the particle. We also find that the influence of interfacial deformations on particle rotation is minute.
