Cahn-Hilliard Navier-Stokes Research Articles

Thermodynamically consistent Cahn–Hilliard–Navier–Stokes equations using the metriplectic dynamics formalism

Cahn–Hilliard–Navier–Stokes (CHNS) systems describe flows with two-phases, e.g., a liquid with bubbles. Obtaining constitutive relations for general dissipative processes for such systems, which are thermodynamically consistent, can be a challenge. We show how the metriplectic 4-bracket formalism (Morrison and Updike, 2024) achieves this in a straightforward, in fact algorithmic, manner. First, from the noncanonical Hamiltonian formulation for the ideal part of a CHNS system we obtain an appropriate Casimir to serve as the entropy in the metriplectic formalism that describes the dissipation (e.g. viscosity, heat conductivity and diffusion effects). General thermodynamics with the concentration variable and its thermodynamics conjugate, the chemical potential, are included. Having expressions for the Hamiltonian (energy), entropy, and Poisson bracket, we describe a procedure for obtaining a metriplectic 4-bracket that describes thermodynamically consistent dissipative effects. The 4-bracket formalism leads naturally to a general CHNS system that allows for anisotropic surface energy effects. This general CHNS system reduces to cases in the literature, to which we can compare.

Convergence analysis of a second order numerical scheme for the Flory–Huggins–Cahn–Hilliard–Navier–Stokes system

We present an optimal rate convergence analysis for a second order accurate in time, fully discrete finite difference scheme for the Cahn–Hilliard–Navier–Stokes (CHNS) system, combined with logarithmic Flory–Huggins energy potential. The numerical scheme has been recently proposed, and the positivity-preserving property of the logarithmic arguments, as well as the total energy stability, have been theoretically justified. In this paper, we rigorously prove second order convergence of the proposed numerical scheme, in both time and space. Since the CHNS is a coupled system, the standard ℓ∞(0,T;ℓ2)∩ℓ2(0,T;Hh2) error estimate could not be easily derived, due to the lack of regularity to control the numerical error associated with the coupled terms. Instead, the ℓ∞(0,T;Hh1)∩ℓ2(0,T;Hh3) error analysis for the phase variable and the ℓ∞(0,T;ℓ2) analysis for the velocity vector, which shares the same regularity as the energy estimate, is more suitable to pass through the nonlinear analysis for the error terms associated with the coupled physical process. Furthermore, the highly nonlinear and singular nature of the logarithmic error terms makes the convergence analysis even more challenging, since a uniform distance between the numerical solution and the singular limit values of is needed for the associated error estimate. Many highly non-standard estimates, such as a higher order asymptotic expansion of the numerical solution (up to the third order accuracy in time and fourth order in space), combined with a rough error estimate (to establish the maximum norm bound for the phase variable), as well as a refined error estimate, have to be carried out to conclude the desired convergence result. To our knowledge, it will be the first work to establish an optimal rate convergence estimate for the Cahn–Hilliard–Navier–Stokes system with a singular energy potential.

Convergence analysis of a decoupled pressure-correction SAV-FEM for the Cahn–Hilliard–Navier–Stokes model

In this paper, a linear, decoupled and unconditional energy-stable numerical scheme of Cahn–Hilliard–Navier–Stokes (CHNS) model is constructed and analyzed. We reformulate the CHNS model into an equivalent system based on the scalar auxiliary variable approach. The Euler implicit/explicit scheme is used for time discretization, that is, linear terms are implicitly processed, nonlinear terms are explicitly processed, and pressure-correction method of the Navier–Stokes equations is also adopted. In this way, we achieve the decoupling of phase field, velocity and pressure, and only need to solve a series of linear equations with constant coefficients at each time step, thereby improving computational efficiency. Then the finite element method is used for space discretization, and the finite element spaces of phase field, velocity and pressure are taken as Pl−Pl−Pl−1 respectively. We also verify that the proposed scheme satisfies the discrete energy dissipation law. The optimal error estimates of the fully discrete scheme is proved, which the phase field, velocity and pressure satisfy the first-order accuracy in time and the (ℓ+1,ℓ+1,ℓ)th order accuracy in space, respectively. Finally, numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

Novel turbulence and coarsening arrest in active-scalar fluids.

We uncover a new type of turbulence - activity-induced homogeneous and isotropic turbulence - in a model that has been employed to investigate motility-induced phase separation (MIPS) in a system of microswimmers. The active Cahn-Hilliard-Navier-Stokes (CHNS) equations, also called active model H, provide a natural theoretical framework for our study. In this CHNS model, a single scalar order parameter ϕ, positive (negative) in regions of high (low) microswimmer density, is coupled with the velocity field u. The activity of the microswimmers is governed by an activity parameter ζ that is positive for extensile swimmers and negative for contractile swimmers. With extensile swimmers, this system undergoes complete phase separation, which is similar to that in binary-fluid mixtures. By carrying out pseudospectral direct numerical simulations (DNSs), we show, for the first time, that (a) this model develops an emergent nonequilibrium, but statistically steady, state (NESS) of active turbulence, for the case of contractile swimmers, if ζ is sufficiently large and negative, and (b) this turbulence arrests the phase separation. We quantify this suppression by showing how the coarsening-arrest length scale does not grow indefinitely, with time t, but saturates at a finite value at large times. We characterise the statistical properties of this active-scalar turbulence by employing energy spectra and fluxes and the spectrum of ϕ. For sufficiently high Reynolds numbers, the energy spectrum (k) displays an inertial range, with a power-law dependence on the wavenumber k. We demonstrate that, in this range, the flux Π(k) assumes a nearly constant, negative value, which indicates that the system shows an inverse cascade of energy, even though energy injection occurs over a wide range of wavenumbers in our active-CHNS model.

Evaluating the forces involved in bubble management in DMEK surgery: mathematical and computational model with clinical implications.

To model postoperative forces involved in Descemet membrane endothelial keratoplasty (DMEK) tissue adherence and bubble management, including the impact of surface tension on graft support, with a view towards clinical applications. Tennent Institute of Ophthalmology, Glasgow, and James Watt School of Engineering, University of Glasgow, Glasgow, United Kingdom. Mathematical modelling and computer simulation. Theoretical modelling of biphasic flow and interaction of gas, liquid and tissue within the anterior chamber for static horizontal scenario A (adherent DMEK with mobile bubble) and dynamic vertical scenario B (release of bubble due to pupil block following DMEK). The model assumed incompressibility for both fluids within realistically achievable pressure ranges. Cahn-Hilliard Navier-Stokes equations were discretised through the application of the Finite Element Method. Mathematical modelling and computer simulation showed bubble size, corneal curvature and force intensity influences surface tension support for DMEK tissue in scenario A. Scenario B demonstrated complex, uneven distribution of surface pressure on the DMEK graft during uncontrolled bubble release. Uneven pressure concentration can cause local tissue warping, with air/fluid displacement via capillary waves generated on the fluid-air interface adversely impacting DMEK support. We have quantitatively and qualitatively modelled the forces involved in DMEK adherence in normal circumstances. We have shown releasing air/gas can abruptly reduce DMEK tissue support via generation of large pressure gradients at the liquid/bubble/graft interfaces, creating negative local forces. Surgeons should consider these principles to reduce DMEK graft dislocation rates via optimised bubble size to graft size, longer acting bubble support and avoiding rapid decompression where possible.

Exponential stability and stabilization of a stochastic 2D nonlocal Cahn-Hilliard-Navier-Stokes equations with multiplicative noise

In this article, we study the asymptotic stability of the unique strong solution to a stochastic version of a diffuse interface model which describes the motion of an incompressible isothermal mixture of two immiscible fluids. The model consists of the 2D Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard equation. We establish the existence, uniqueness and regularity results for the stationary solution of the stochastic 2D nonlocal Cahn-Hilliard Navier-Stokes equations. Furthermore, we prove that under some conditions on the forcing terms, the strong solution converges exponentially in the mean square and almost surely exponentially to the stationary solution. Finally, we also prove a result related to the stabilization of the stochastic 2D nonlocal Cahn-Hilliard-Navier-Stokes equations.

Optimal error estimates of a SAV–FEM for the Cahn–Hilliard–Navier–Stokes model

We carry out the optimal error estimates of a scalar auxiliary variable (SAV) based the Euler semi-implicit finite element method for the Cahn–Hilliard–Navier–Stokes (CHNS) system in this paper. A SAV is introduced to reformulate the CHNS system into an equivalent system. We apply the Euler semi-implicit method and finite element method for the temporal and spatial discretizations to obtain a linear and unconditional energy stable scheme. The stability of numerical solutions under different norms and the optimal error estimates for our proposed numerical schemes are also presented. Finally, numerical examples are presented to demonstrate the accuracy of the proposed method, and the discrete energy is dissipated in numerical simulations.

Modified diffuse interface fluid model and its consistent energy-stable computation in arbitrary domains

Binary fluid flows in irregular domains are common phenomena in natural and industrial fields. To describe these physical processes, we herein present a novel fluid flow-coupled two-material model reflecting different wetting conditions on an arbitrary fluid-solid interface. A modified ternary Cahn–Hilliard (CH) diffuse interface model is adopted to capture the fluid-fluid interface. One of three components is fixed for all time to represent the profile of complex domains. By considering the contact angle condition on solid-fluid interface and equilibrium interface assumption, an extra wetting term is derived. To treat the fluid flow in domains with solid obstacles, we consider the entire system as a porous medium with variable permeability and add a large penalty term into the incompressible Navier–Stokes (NS) equations. It can be proven that the proposed modified Cahn–Hilliard–Navier–Stokes (CHNS) system satisfies the energy dissipation law (energy stability). Based on the scalar auxiliary variable (SAV) approach with appropriate correction techniques, the linear and consistent energy-stable scheme is developed. We introduce the numerical implementation and estimate the discrete version of energy stability in detail. The proposed method can be implemented on regular Cartesian grids with the absence of explicit boundary treatment. Moreover, the calculations are totally decoupled in each time step. Extensive numerical experiments not only indicate the expected accuracy and stability but also show the superior performance in arbitrary domains with different wetting conditions.

An energy-stable Smoothed Particle Hydrodynamics discretization of the Navier-Stokes-Cahn-Hilliard model for incompressible two-phase flows

Varieties of energy-stable numerical methods have been developed for incompressible two-phase flows based on the Navier-Stokes–Cahn–Hilliard (NSCH) model in the Eulerian framework, while few investigations have been made in the Lagrangian framework. Smoothed particle hydrodynamics (SPH) is a popular mesh-free Lagrangian method for solving complex fluid flows. In this paper, we present a pioneering study on the energy-stable SPH discretization of the NSCH model for incompressible two-phase flows. We prove that this SPH method inherits mass and momentum conservation and the energy dissipation properties at the fully discrete level. With the projection procedure to decouple the momentum and continuity equations, the numerical scheme meets the divergence-free condition. Some numerical experiments are carried out to show the performance of the proposed energy-stable SPH method for solving the two-phase NSCH model. The inheritance of mass and momentum conservation and the energy dissipation properties are verified numerically.

A unified framework for Navier–Stokes Cahn–Hilliard models with non-matching densities

Over the last decades, many diffuse-interface Navier–Stokes Cahn–Hilliard (NSCH) models with non-matching densities have appeared in the literature. These models claim to describe the same physical phenomena, yet they are distinct from one another. The overarching objective of this work is to bring all of these models together by laying down a unified framework of NSCH models with non-zero mass fluxes. Our development is based on three unifying principles: (1) there is only one system of balance laws based on continuum mixture theory that describes the physical model, (2) there is only one natural energy-dissipation law that leads to quasi-incompressible NSCH models, (3) variations between the models only appear in the constitutive choices. The framework presented in this work now completes the fundamental exploration of alternate non-matching density NSCH models that utilize a single momentum equation for the mixture velocity, but leaves open room for further sophistication in the energy functional and constitutive dependence.

A projection-based, semi-implicit time-stepping approach for the Cahn-Hilliard Navier-Stokes equations on adaptive octree meshes

The Cahn-Hilliard Navier-Stokes (CHNS) system provides a computationally tractable model that can be used to effectively capture interfacial dynamics in two-phase fluid flows. In this work we present a semi-implicit, projection-based finite element framework for solving the CHNS system. We use a projection-based semi-implicit time discretization for the Navier-Stokes equation and a fully-implicit time discretization for the Cahn-Hilliard equation. We use a conforming continuous Galerkin (cG) finite element method in space equipped with a residual-based variational multiscale (RBVMS) formulation. Pressure is decoupled using a projection step, which results in two linear positive semi-definite systems for velocity and pressure, instead of the saddle point system of a pressure-stabilized method. All the linear systems are solved using an efficient and scalable algebraic multigrid (AMG) method. We deploy this approach on a massively parallel numerical implementation using parallel octree-based adaptive meshes. The overall approach allows the use of relatively large time steps with much faster time-to-solve than similar fully-implicit methods. We present comprehensive numerical experiments showing detailed comparisons with results from the literature for canonical cases, including the single bubble rise and Rayleigh-Taylor instability.

A Diffuse-Domain Phase-Field Lattice Boltzmann Method for Two-Phase Flows in Complex Geometries

This study constructs a new diffuse-domain (DD) lattice Boltzmann (LB) method for two-phase flows in complex geometries. We first coupled the DD method with the consistent and conservative phase-field (CCPF) method which can be described by the DD-CC Navier–Stokes–Cahn–Hilliard (NSCH) equations with the consistency of reduction, the consistency of mass and momentum transport, and the consistency of mass conservation. Then we conducted a matched asymptotic analysis and found that the DD-CCNSCH equations would converge to the CCNSCH equations as the thickness of the DD interface approaches to zero. Since the boundary conditions imposed on the complex geometries are incorporated into the DD-CCNSCH equations as the source terms, the system can be implemented more readily in a large and regular domain, and there is no need to treat the complex boundary conditions specially. We further developed a DD-CCPFLB method, and through the Chapman–Enskog analysis, the DD-CCPFLB method can correctly recover the DD-CCNSCH equations for two-phase flows in complex geometries. To quantitatively validate the DD-CCPFLB method, three benchmark problems are considered, and the numerical results are in agreement with those obtained through directly solving the CCNSCH equations combined with the original boundary conditions, the analytical solution, or the reported data. Finally, the present method is used to study the problems of two-phase flows in complex geometries, including the droplet dynamics in the Y-shaped channel and a droplet passing through cylindrical obstacles. The effects of the wettability and viscosity ratio are mainly investigated, and it is found that these two factors have significant impacts on the droplet dynamics.

A fully-coupled framework for solving Cahn-Hilliard Navier-Stokes equations: Second-order, energy-stable numerical methods on adaptive octree based meshes

We present a fully-coupled, implicit-in-time framework for solving a thermodynamically-consistent Cahn-Hilliard Navier-Stokes system that models two-phase flows. In this work, we extend the block iterative method presented in Khanwale et al. [Simulating two-phase flows with thermodynamically consistent energy stable Cahn-Hilliard Navier-Stokes equations on parallel adaptive octree based meshes, J. Comput. Phys. (2020)], to a fully-coupled, provably second-order accurate scheme in time, while maintaining energy-stability. The new method requires fewer matrix assemblies in each Newton iteration resulting in faster solution time. The method is based on a fully-implicit Crank-Nicolson scheme in time and a pressure stabilization for an equal order Galerkin formulation. That is, we use a conforming continuous Galerkin (cG) finite element method in space equipped with a residual-based variational multiscale (RBVMS) procedure to stabilize the pressure. We deploy this approach on a massively parallel numerical implementation using parallel octree-based adaptive meshes. We present comprehensive numerical experiments showing detailed comparisons with results from the literature for canonical cases, including the single bubble rise, Rayleigh-Taylor instability, and lid-driven cavity flow problems. We analyze in detail the scaling of our numerical implementation.

Fully decoupled pressure projection scheme for the numerical solution of diffuse interface model of two-phase flow

We describe a fully decoupled finite difference approach for solving the Cahn–Hilliard Navier–Stokes (CHNS) model of two-phase flow. The governing equations consist of the incompressible Navier–Stokes equations coupled with the nonlinear Cahn–Hilliard equation through surface tension. The coupled system of equations is decoupled with the help of an intermediate velocity-field. The spatial discretization is carried out by finite differences while the explicit Euler’s method is used for the temporal part. The decoupling feature of the method makes the numerical implementation easy. Several two-phase flow problems have been numerically simulated in order to test the method’s efficacy.

Bouncing drop impingement on heated hydrophobic surfaces

We extend a diffuse interface phase-field method for two-phase flow simulations so as to include interfacial heat transfer and the thermal Marangoni effect. The set of governing equations hold non-standard terms, which originally stem from the underlying variational consideration of the total free energy of the two-phase system. It consists of the coupled Cahn-Hilliard Navier-Stokes equations with a temperature-dependent mixing energy term and a temperature transport equation implemented in the OpenFOAM framework. The underlying solver phaseFieldFoam is validated for the test cases of thermocapillary convection in a two-fluid-layer and thermocapillary migration of a drop.In the main part of the paper, hydrodynamics and heat transfer of a droplet impinging on a heated hydrophobic surface with subsequent bouncing are studied in detail. By comprehensive simulations, the effects of impact velocity, droplet diameter (in mm range) and substrate wettability (contact angle) are investigated. The numerical results for spreading ratio, wall contact time and cooling effectiveness are found to compare well with experiments indicating that the developed code is well suited for heat transfer simulation in two-phase flow. For the maximum spreading ratio, a new generalizing correlation is proposed which models the kinetic energy and energy losses at maximum spreading by two coefficients, which are determined from simulation results. A new correlation is also proposed for the contact time, which takes into account surface wettability. For the time evolution of drop mean temperature, a crossover is observed when comparing simulations with constant and temperature-dependent surface tension, indicating that the inclusion of Marangoni effects increases both, the heat transfer between drop and wall during spreading and the heat transfer between drop and surrounding air after rebound.

Numerical analysis of fully discrete energy stable weak Galerkin finite element Scheme for a coupled Cahn-Hilliard-Navier-Stokes phase-field model

The Cahn-Hilliard phase-field model of two-phase incompressible flows, namely the Cahn-Hilliard-Navier-Stokes (CH-NS) problem represents the fundamental building blocks of hydrodynamic phase-field models for multiphase fluid flow dynamics. Since finding solution (numerically and theoretically) of the CH-NS system is non-trivial (owing to the coupling between the Navier-Stokes equation and the Cahn-Hilliard equation), in this paper, we propose and analyze a numerical scheme with the following properties: (1) first-order in time, (2) nonlinear, (3) fully coupled and (4) energy stable; for solving CH-NS system, in the framework of weak Galerkin (WG) method. More precisely, we employ the WG method, which uses discontinuous functions to construct the approximation space, and the first-order backward Euler (implicit) method for space and time discretizations, respectively. We first recall the corresponding variational formulation, and then summarize the main WG method ingredients that are required for our discrete analysis. In particular, in order to define the weak discrete bilinear (and trilinear) form, whose continuous version involves classical differential operators, we propose two well-known alternatives for gradient and divergence operators onto a suitable polynomial subspace. Next, we show that the weak global discrete bilinear form satisfies the hypotheses required by the Lax-Milgram lemma. In this way, we derive the associated a priori error estimates for phase field variable, chemical potential, velocity and further the pressure in L2 norm. Finally, several numerical results confirming the theoretical rates of convergence and illustrating the good performance of the method for simulation influence of surface tension, small density variations, and the driven cavity are presented.

Second-order decoupled energy-stable schemes for Cahn-Hilliard-Navier-Stokes equations

The Cahn-Hilliard-Navier-Stokes (CHNS) equations represent the fundamental building blocks of hydrodynamic phase-field models for multiphase fluid flow dynamics. Due to the coupling between the Navier-Stokes equation and the Cahn-Hilliard equation, the CHNS system is non-trivial to be solved numerically. Traditionally, a numerical extrapolation for the coupling terms is used. However, such brute-force extrapolation usually destroys the intrinsic thermodynamic structures of this CHNS system. This paper proposes a new strategy to reformulate the CHNS system into a constraint gradient flow formulation, where the reversible and irreversible structures are clearly revealed. This guides us to propose operator splitting schemes that have several advantageous properties. First of all, the proposed schemes lead to several decoupled systems in smaller sizes to be solved at each time marching step. This significantly reduces computational costs. Secondly, the proposed schemes still guarantee the thermodynamic laws of the CHNS system at the discrete level. In addition, unlike the recently populated IEQ or SAV approaches using auxiliary variables, our resulting energy laws are formulated in the original variables. This is a significant improvement, as the modified energy laws with auxiliary variables sometimes deviate from the original energy law. Our proposed framework lays a foundation for designing decoupled and energy stable numerical algorithms for hydrodynamic phase-field models. Furthermore, various numerical algorithms can be obtained given different splitting steps, making this framework rather general. The proposed numerical algorithms are implemented. Their second-order temporal and spatial accuracy are verified numerically. Some numerical examples and benchmark problems are calculated to verify the effectiveness of the proposed schemes.

Drop impact onto polarized dielectric surface for controlled coating

Control of surface wettability by means of electrowetting-on-dielectric (EWOD) is among the most effective methods of active enhancement of surface wettability. Here, electrohydrodynamics of drop impact onto a dielectric surface with electrodes embedded in the dielectric (or aligned and attached to it) is experimentally investigated. Drop impact of different liquids (water, n-butanol, and motor oil) onto different substrates (stretched Teflon, parafilm, and polypropylene) is studied. Water drop impact onto stretched Teflon (the only Teflon which revealed significant electrowetting) and un-stretched parafilm surfaces is studied in detail. The results for water drop impact indicate that drop spreading on such non-wettable surfaces can be significantly enhanced by the electric field application. In particular, water drop rebound can be suppressed by the electric force. Furthermore, impact dynamics and spreading of hydrocarbon liquids with electric field are explored. Partial suppression of splash phenomena was also observed with the application of the electric field in addition to enhancement of spreading. In addition, the experimental results for water drops are compared with the Cahn−Hilliard−Navier−Stokes (CHNS) simulations for static contact angles and drop impact dynamics, and the results are in close agreement for water drops. This study demonstrates that electrowetting-on-dielectric holds great promise for coating and spraying technologies.

Cahn-Hilliard Navier-Stokes simulations for marine free-surface flows

The paper is devoted to the simulation of maritime two-phase flows of air and water. Emphasis is put on an extension of the classical Volume-of-Fluid (VoF) method by a diffusive contribution derived from a Cahn-Hilliard (CH) model and its benefits for simulating immiscible, incompressible two-phase flows. Such flows are predominantly simulated with implicit VoF schemes, which mostly employ heuristic downwind-biased approximations for the concentration transport to mimic a sharp interface. This strategy introduces a severe time step restriction and requires pseudo time-stepping of steady flows. Our overall goal is a sound description of the free-surface region that alleviates artificial time-step restrictions, supports an efficient and robust upwind-based approximation framework, and inherently includes surface tension effects when needed. The Cahn-Hilliard Navier-Stokes (CH-NS) system is verified for an analytical Couette-flow example and the bubble formation under the influence of surface tension forces. 2D validation examples are concerned with laminar standing waves reaching from gravity to capillary scale as well as a submerged hydrofoil flow. The final application refers to the 3D flow around an experimentally investigated container vessel at fixed floatation for Re = 1.4 × 107 and Fn = 0.26. Results are compared with data obtained from VoF approaches, supplemented by analytical solutions and measurements. The study indicates the superior efficiency, resharpening capability, and wider predictive realm of the CH-based extension for free-surface flows with a confined spatial range of interface Courant numbers.

A novel Cahn–Hilliard–Navier–Stokes model with a nonstandard variable mobility for two-phase incompressible fluid flow

In this study, we present a novel Cahn–Hilliard–Navier–Stokes (CHNS) system with a nonstandard variable mobility for two-phase incompressible fluid flow. Unlike the classical constant mobility, the developed variable mobility has decreasing values nearby the interface and increasing values away from the interface, which minimizes the dynamics of the Cahn–Hilliard (CH) model nearby the interface. An unconditionally stable convex splitting method is used to solve the CH equation and the projection method is used to solve the NS equation. As benchmark tests, the Rayleigh–Taylor instability, drop deformation, and rising bubble are performed to show the accuracy and practicability of the proposed model. The computational results indicate that the proposed model accurately captures the interfacial position and keeps the interface region from being too much distorted.