Navier Stokes Cahn Hilliard Equations Research Articles

Extension Criterion to the 3D Navier–Stokes–Cahn–Hilliard equations

Extension Criterion to the 3D Navier–Stokes–Cahn–Hilliard equations

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.

Stabilized immersed isogeometric analysis for the Navier–Stokes–Cahn–Hilliard equations, with applications to binary-fluid flow through porous media

Binary-fluid flows can be modeled using the Navier–Stokes–Cahn–Hilliard equations, which represent the boundary between the fluid constituents by a diffuse interface. The diffuse-interface model allows for complex geometries and topological changes of the binary-fluid interface. In this work, we propose an immersed isogeometric analysis framework to solve the Navier–Stokes–Cahn–Hilliard equations on domains with geometrically complex external binary-fluid boundaries. The use of optimal-regularity B-splines results in a computationally efficient higher-order method. The key features of the proposed framework are a generalized Navier-slip boundary condition for the tangential velocity components, Nitsche’s method for the convective impermeability boundary condition, and skeleton- and ghost-penalties to guarantee stability. A binary-fluid Taylor–Couette flow is considered for benchmarking. Porous medium simulations demonstrate the ability of the immersed isogeometric analysis framework to model complex binary-fluid flow phenomena such as break-up and coalescence in complex geometries.

Numerical investigation of the sharp-interface limit of the Navier–Stokes–Cahn–Hilliard equations

In this article, we study the behaviour of the Abels–Garcke–Grün Navier–Stokes–Cahn– Hilliard diffuse-interface model for binary-fluid flows, as the diffuse-interface thickness passes to zero. For the diffuse-interface model to approach a classical sharp-interface model in the limit $\varepsilon \to +0$ , the so-called mobility parameter $m$ in the diffuse-interface model must scale appropriately with the interface-thickness parameter $\varepsilon$ . In the literature various scaling relations in the range $o(1)$ to $O(\varepsilon ^3)$ have been proposed, but the optimal order to pass to the limit has not been explored previously. Our primary objective is to elucidate this optimal order of the $m$ – $\varepsilon$ scaling relation in terms of the rate of convergence of the diffuse-interface solution to the sharp-interface solution. Additionally, we examine how the convergence rate is affected by a sub-optimal parameter scaling. We centre our investigation around the case of an oscillating droplet. To provide reference limit solutions, we derive new analytical expressions for small-amplitude oscillations of a viscous droplet in a viscous ambient fluid in two dimensions. For two distinct modes of oscillation, we probe the sharp-interface limit of the Navier–Stokes–Cahn–Hilliard equations by means of an adaptive finite-element method. The adaptive-refinement procedure enables us to consider diffuse-interface thicknesses that are significantly smaller than other relevant length scales in the droplet-oscillation problem, allowing an exploration of the asymptotic regime.

Numerical solution of coupled Cahn–Hilliard Navier–Stokes equations for two‐phase flows having variable density and viscosity

This work is concerned with the implementation of a decoupled diffuse interface approach for the numerical solution of two‐phase flows with a moving interface. The underlying two‐phase flow model consists of a mass‐preserving Cahn–Hilliard (CH) equation coupled with the incompressible Navier–Stokes equations (NSEs) through surface tension. Due to the higher order nonlinearity and stiff nature of the CH equation, its numerical solution is very challenging. We have used the decoupled pressure projection method for the numerical simulation of the governing equations. The spatial variables are discretized using the finite difference scheme on the staggered grids, while the explicit Euler method is used for time discretization. Four examples include the coalescence of inline rising bubbles, the obliques coalescence of two rising bubbles, and the deformation of single and two different‐sized bubbles in a shear flow field is numerically simulated. It is observed that the scheme is capable of tracking the interface accurately and respecting mass conservation and energy dissipation. Moreover, the scheme is efficient, easily implementable, energy stable, and fully decoupled.

Invariant measure for 2D stochastic Cahn–Hilliard–Navier–Stokes equations

In this paper, we investigate the stochastic Cahn–Hilliard–Navier–Stokes equations in two-dimensional spaces. Applying the Maslowski–Seidler method, we establish the existence of invariant measure in state space [Formula: see text] with the weak topology. We also prove the existence of global pathwise solutions using the stochastic compactness argument.

On the strong solution of 3D non-isothermal Navier–Stokes–Cahn–Hilliard equations

In this paper, we consider the global existence of strong solutions of a thermodynamically consistent diffuse interface model describing two-phase flows of incompressible fluids in a non-isothermal setting. In the diffuse interface model, the evolution of the velocity u is ruled by the Navier–Stokes system, while the order parameter φ representing the difference of the fluid concentration of the two fluids is assumed to satisfy a convective Cahn–Hilliard equation. The effects of the temperature are prescribed by a suitable form of heat equation. By using a refined pure energy method, we prove the existence of the global strong solution by assuming that ‖u0‖H32+‖φ0‖H42+‖θ0‖H32+‖φ02−1‖L22+‖θ0‖L1 is sufficiently small, and higher order derivatives can be arbitrarily large.

A systematic study of a droplet breakup process in decaying homogeneous isotropic turbulence using a mesoscopic simulation approach

ABSTRACT The breakup of a spherical droplet in a decaying homogeneous isotropic turbulence is studied by solving the Cahn–Hilliard–Navier–Stokes equations. This flow provides a great opportunity to study the interactions of turbulent kinetic energy and interfacial free energy and their effects on the breakup dynamics. Three distinct stages of droplet evolution, namely, the deformation stage, the breakup stage, and the restoration stage, are identified and then analysed systematically from several perspectives: a geometric perspective, a dynamic perspective, a global energetic perspective, and a multiscale energy transfer perspective. It is found that the ending time of the breakup stage can be estimated by the Hinze criterion. The kinetic energy of the two-phase flow during the breakup stage is found to have a power-law decay with an exponent , compared to for the single-phase flow, mainly due to the enhanced viscous dissipation generated by the daughter droplets. Energy spectra of the two-phase flow show power-law decay, with a slope between and , at high wave numbers, both in the Fourier spectral space and in the spherical harmonics space.

Elasto-capillary fluid–structure interaction with compound droplets

Computational modeling of elasto-capillarity, i.e., a fluid–structure interaction phenomenon where the solid deformation is driven by capillary forces at fluid–fluid interfaces has recently emerged as an important problem in computational mechanics. However, the high-fidelity simulation of elasto-capillary problems involving three immiscible fluids has remained unexplored. Here, we present a mathematical model and an algorithm to simulate elasto-capillary problems involving compound droplets. Such problems are of pivotal importance in pharmaceutical, biological and food industrial applications. We adopt a phase-field model described by the ternary Navier–Stokes–Cahn–Hilliard equations for the three immiscible fluids and a neo-Hookean model for the solid. We define a fluid–solid surface energy function, which determines the fluid–solid wettability and the tractions transmitted to the solid at the fluid–solid interface. We adopt a boundary-fitted approach for our fluid–structure interaction formulation and Isogeometric Analysis for the spatial discretization. To demonstrate the effectiveness of our computational model and algorithm, we perform elasto-capillary simulations involving various types of compound droplets.

Error analysis of a decoupled, linear and stable finite element method for Cahn–Hilliard–Navier–Stokes equations

In this paper, we carry out the error analysis for a totally decoupled, linear and unconditionally energy stable finite element method to solve the Cahn–Hilliard–Navier–Stokes equations. The fully finite element scheme is based on a stabilization for Cahn–Hilliard equation and projection method for Navier–Stokes equation, as well as the first order Euler method for time discretization. A priori error analysis for phase field, velocity field and pressure variable are derived for the fully discrete scheme.

A pressure-correction and bound-preserving discretization of the phase-field method for variable density two-phase flows

In this paper, we present an efficient numerical algorithm for solving the time-dependent Cahn–Hilliard–Navier–Stokes equations that model the flow of two phases with different densities. The pressure-correction step in the projection method consists of a Poisson problem with a modified right-hand side. Spatial discretization is based on discontinuous Galerkin methods with piecewise linear or piecewise quadratic polynomials. One important contribution of this work is the formulation of flux and slope limiting techniques that successfully eliminate the bulk shift, overshoot and undershoot in the order parameter. The bound-preserving property of the discrete order parameter is proved. Several numerical results demonstrate that the proposed numerical algorithm is effective and robust for modeling two-component immiscible flows in porous structures and digital rocks.

New efficient time-stepping schemes for the Navier–Stokes–Cahn–Hilliard equations

In this paper, we construct and analyze a class of efficient time-stepping schemes for the Navier–Stokes–Cahn–Hilliard equation system stemming from phase field modeling of two-phase incompressible flows. The proposed schemes are based on an auxiliary variable approach for the Cahn–Hilliard equation and a delicate treatment of the terms coupling the Navier–Stokes equation and the Cahn–Hilliard equation. The former has been found to be an efficient tool for numerical integration of the Cahn–Hilliard equation in recent years. A detailed comparison with existing schemes is given, and the advantage of the new schemes are emphasized. In the theoretical aspect we rigorously prove that the designed schemes are unconditionally stable in the sense that some kind of energy remains bounded during the time stepping. In the implementation, we show how the schemes can be reformulated into six linear second-order elliptic equations with constant coefficients. The efficiency of the proposed algorithm is verified through several numerical examples.

The non-Lipschitz stochastic Cahn–Hilliard–Navier–Stokes equations in two space dimensions

The stochastic two-dimensional Cahn–Hilliard–Navier–Stokes equations under non-Lipschitz conditions are considered. This model consists of the Navier–Stokes equations controlling the velocity and the Cahn–Hilliard model controlling the phase parameters. By iterative techniques, a priori estimates and weak convergence method, the existence and uniqueness of an energy weak solution to the equations under non-Lipschitz conditions have been obtained.

Direct numerical simulation of two-phase pipe flow: Influence of the domain length on the flow regime

We investigate the dynamics of a kerosene–water mixture in a vertical pipe flow by solving the Cahn–Hilliard–Navier–Stokes equations. We compute the linear stability of laminar core-annular flow in a vertical pipe and find that it is highly unstable. By performing direct numerical simulations initialized with a slightly perturbed core-annular flow, we show that the system transitions to turbulence and finally relaxes into a turbulent slug flow regime provided that the pipe is sufficiently long. This configuration presents mild turbulence and large scale three-dimensional recirculation patterns. Our work highlights the need for applying nonlinear-dynamics approaches and carefully selecting the domain length to investigate the patterns observed in two-phase pipe flows and demonstrates the capabilities of phase-field methods to reliably simulate flows under realistic experimental conditions.

A Decoupled Energy Stable Adaptive Finite Element Method for Cahn–Hilliard–Navier–Stokes Equations

A Decoupled Energy Stable Adaptive Finite Element Method for Cahn–Hilliard–Navier–Stokes Equations

Advected phase-field method for bounded solution of the Cahn–Hilliard Navier–Stokes equations

Phase-field methods based on the Cahn–Hilliard (CH) equation coupled to the incompressible Navier–Stokes equation are becoming increasingly popular for interface resolving numerical simulations of two-phase flows of immiscible fluids. One major limitation of this approach, however, is that the volume of each phase is not inherently preserved. This is associated with the phase-discriminating order parameter, which in the course of the simulation remains in general not within its initial physical bounds. This shortcoming relates to the fact that the CH equation with standard Ginzburg–Landau chemical potential has no volume-preserving stationary solution for interfaces with uniform (non-zero) curvature. In this paper, a curvature-dependent chemical potential is proposed which allows for bounded stationary solutions of the CH equation for drops/bubbles exhibiting uniform curvature. Numerical solutions of the coupled Cahn–Hilliard Navier–Stokes equations show that the proposed chemical potential significantly improves boundedness and phase volume conservation over the standard one.

Spreading and rebound dynamics of sub-millimetre urea-water-solution droplets impinging on substrates of varying wettability

The interaction of droplets consisting of urea-water solution (UWS) with a wall is of interest for automotive exhaust gas after-treatment of Diesel engines by selective catalytic reduction (SCR). Since the impingement of tiny UWS droplets on the solid substrate is difficult to examine experimentally, little is known about the detailed dynamics of this process. In the present study, the normal impact of single UWS droplets impinging on dry solid substrates of greatly differing wettability is investigated numerically under axisymmetric conditions. Simulations are performed by a diffuse interface phase-field solver developed by the authors where the coupled Cahn–Hilliard Navier–Stokes equations are solved using OpenFOAM. The code is thoroughly validated against a number of experiments from literature considering the rebound of millimetre-sized water droplets from hydrophobic substrates.The numerical simulations on the impact dynamics of UWS droplets cover wide ranges of sub-millimetre droplet sizes and impact velocities that are relevant in technical SCR systems. A strong influence of substrate wettability on droplet dynamics is identified. Reducing wettability from hydrophilic to superhydrophobic conditions reduces spreading and enables drop rebound with reduced drop-surface contact time. The effects of drop diameter, drop impact velocity and equilibrium contact angle on the maximum spreading ratio are quantified, and regime maps on rebound versus non-rebound (deposition) impact outcomes are provided. The results of the present interface-resolving numerical simulations may be useful for development of more advanced drop-wall interaction models as they are required in CFD codes relying on the Euler–Lagrange approach for large-scale computations of UWS sprays.

Stationary Cahn–Hilliard–Navier–Stokes equations for the diffuse interface model of compressible flows

A system of partial differential equations for a diffusion interface model is considered for the stationary motion of two macroscopically immiscible, viscous Newtonian fluids in a three-dimensional bounded domain. The governing equations consist of the stationary Navier–Stokes equations for compressible fluids and a stationary Cahn–Hilliard type equation for the mass concentration difference. Approximate solutions are constructed through a two-level approximation procedure, and the limit of the sequence of approximate solutions is obtained by a weak convergence method. New ideas and estimates are developed to establish the existence of weak solutions with a wide range of adiabatic exponent.

An alternative phase-field interfacial tension force representation for binary fluid systems

The Navier–Stokes/Cahn–Hilliard (NSCH) system of equations has been extensively used for investigating the dynamics of two-phase flows of Newtonian fluids. However, the accurate calculation of interfacial tension via NSCH has been perceptibly doubted, and thus, a successive solution of NSCH equations is rarely not accompanied by mesh adaptation techniques and complex numerical schemes. In this work, it is demonstrated that the cause of such a miscalculation of the interfacial tension is inherent when following the conventional way of coupling the Navier–Stokes with the Cahn–Hilliard equation in their dimensionless form, where the capillary number is defined by assuming that the fluid/fluid interface is flat and at equilibrium. Hence, an alternative NSCH model was developed for the more accurate computation of interfacial tension that does not rely on any such a priori assumptions, and it uses a more abstract coupling by accounting for the distribution of the binary system’s energy on the interfacial region. This model was implemented on two different cases: (i) an investigation of the effects of inertia and capillarity on the deformation of liquid drops in simple shear flow and (ii) a study of an interfacial instability due to viscosity stratification. To solve the set of governing equations, implicit time integration schemes based on finite differences were further developed and implemented. The results regarding the topological evolution of the fluid/fluid interface from both cases were additionally cross-validated with other methods from the literature as well as with the conventional NSCH model. The comparison suggests that our NSCH model indeed remedies the standard NSCH model, without the need of mesh adaptation or any complex numerical scheme, by more accurately computing the interfacial tension for binary systems consisting of incompressible, immiscible, and Newtonian fluids.

Pontryagin maximum principle and second order optimality conditions for optimal control problems governed by 2D nonlocal Cahn–Hilliard–Navier–Stokes equations

Abstract In this paper, we formulate a distributed optimal control problem related to the evolution of two isothermal, incompressible, immiscible fluids in a two-dimensional bounded domain. The distributed optimal control problem is framed as the minimization of a suitable cost functional subject to the controlled nonlocal Cahn–Hilliard–Navier–Stokes equations. We describe the first order necessary conditions of optimality via the Pontryagin minimum principle and prove second order necessary and sufficient conditions of optimality for the problem.