Hilliard Type Equation Research Articles

A convection-diffusion-reaction system with discontinuous flux modelling biofilm growth in slow sand filters

A Slow Sand Filter (SSF) consists of biofilm attached to saturated packed sand with supernatant water on top, through which water flows by gravity. SSFs are used in the production of drinking water to improve water quality and remove pathogens, and, as they use biological filtration via the microbes present in the biofilm, they are simple and cheap to construct. A drawback of SSFs is that the difficult sampling and the complex dynamics associated with biofilm growth of microbes pose an obstacle for analysis and predictions. In order to overcome this issue, a mathematical model consisting of a non-linear convection-diffusion-reaction system with discontinuous flux is developed from first principles as a unified framework for the dynamics in time and depth of the biofilm and microbial community, where a variety of ecological models can be included in the reaction terms, which also contain attachment, detachment and transfer processes. The multi-phase approach considers three main physical volumes: a biofilm matrix, the water volume enclosed by it and the flowing suspension through the SSF surrounding the biofilm. The only input data needed are the inflow concentrations and rate, light irradiation and temperature. The model includes a Cahn–Hilliard-type equation for the biofilm velocity in the supernatant water. A numerical scheme that preserves physical properties of the solution is also presented. This is achieved with a time-splitting scheme using an upwind scheme with an adaptive time-step size that ensures positivity of the solution under a CFL condition and a reasonable assumption on the separate numerical solution of the Cahn–Hilliard-type equation. The latter is solved numerically with a semi-implicit finite-difference scheme using a mixed form of the equation. The flexible model framework with its numerical scheme allows for including a variety of ecological models, making the investigation and development of increasingly complex simulations possible. Simulations with different model parameters are shown to be qualitatively consistent with the literature.

Convergence and stability analysis of energy stable and bound‐preserving numerical schemes for binary fluid‐surfactant phase‐field equations

AbstractIn this article, we develop stable and efficient numerical schemes for a binary fluid‐surfactant phase‐field model which consists of two Cahn–Hilliard type equations with respect to the free energy containing a Ginzburg–Landau double‐well potential, a logarithmic Flory–Huggins potential and a nonlinear coupling entropy. The numerical schemes, which are decoupled and linear, are established by the central difference spatial approximation in combination with the first‐ and second‐order exponential time differencing methods based on the convex splitting of the free energy. For the sake of the linearity of the schemes, the nonlinear terms, especially the logarithmic term, are approximated explicitly, which requires the bound preservation of the numerical solution to make the algorithm robust. We conduct the convergence analysis and prove the bound‐preserving property in details for both first‐ and second‐order schemes, where the high‐order consistency analysis is applied to the first‐order case. In addition, the energy stability is also obtained by the nature of the convex splitting. Numerical experiments are performed to verify the accuracy and stability of the schemes and simulate the dynamics of phase separation and surfactant adsorption.

Martingale solutions to stochastic nonlocal Cahn–Hilliard- Navier–Stokes systems with singular potentials driven by multiplicative noise of jump type

Abstract In the diffuse interface theory, the motion of two incompressible viscous fluids and the evolution of their interface are described by the well-known model H. The model consists of the Navier–Stokes equations, nonlinearly coupled with a convective Cahn–Hilliard type equation. Here we consider a stochastic version of the model driven by a noise of Lévy type, and where the standard Cahn–Hilliard equation is replaced by its nonlocal version with a singular (e.g., logarithmic) potential. The case of smooth potentials with arbitrary polynomial growth has been already analyzed in [G. Deugoué, A. Ndongmo Ngana and T. Tachim Medjo, Martingale solutions to stochastic nonlocal Cahn–Hilliard–Navier–Stokes equations with multiplicative noise of jump type, Phys. D 398 2019, 23–68]. Taking advantage of this previous result, we investigate this more challenging and physically relevant case. Global existence of a martingale solution is proved with no-slip and no-flux boundary conditions in both 2D and 3D bounded domains. In the two-dimensional case, we prove the uniqueness of weak solutions when the viscosity is constant.

On a Cahn–Hilliard system with source term and thermal memory

A nonisothermal phase field system of Cahn–Hilliard type is introduced and analyzed mathematically. The system constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. It couples a Cahn–Hilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the constitutive law of the heat flux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the consequence that the balance law of internal energy becomes a second-order in time equation for the thermal displacement or freezing index, that is, a primitive with respect to time of the temperature. Another particular feature of our system is the presence of the source term in the equation for the order parameter, which entails additional mathematical difficulties because the mass conservation of the order parameter is lost. We provide several mathematical results under general assumptions on the source term and the double-well nonlinearity governing the evolution: existence and continuous dependence results are shown for weak and strong solutions to the corresponding initial-boundary value problem.

Convergence of Solutions to a Convective Cahn–Hilliard-Type Equation of the Sixth Order in Case of Small Deposition Rates

Convergence of Solutions to a Convective Cahn–Hilliard-Type Equation of the Sixth Order in Case of Small Deposition Rates

An efficient linear and unconditionally stable numerical scheme for the phase field sintering model

In this article, the phase field sintering model, which is composed of a Cahn–Hilliard type equation and several Allen–Cahn type equations, has been considered. On the scalar auxiliary variable framework, we propose a theoretically efficient and stable method for solid-state sintering. In order to overcome the nonlinear issues, we define a stabilized scalar auxiliary variable method and reformulate the phase field sintering model. An efficient and accurate numerical scheme is investigated to solve our model. The scheme consist of several decoupled diffusion equations at every time step. Therefore, our scheme is easy to implement. Then we prove the numerical discrete energy is unconditionally stable. Several numerical simulations in two- and three-dimensional spaces are presented to demonstrate the robustness of our method.

Optimal Temperature Distribution for a Nonisothermal Cahn–Hilliard System with Source Term

In this note, we study the optimal control of a nonisothermal phase field system of Cahn–Hilliard type that constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. The system couples a Cahn–Hilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the constitutive law of the heat flux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the consequence that the balance law of internal energy becomes a second-order in time equation for the thermal displacement or freezing index, that is, a primitive with respect to time of the temperature. Another particular feature of our system is the presence of the source term in the equation for the order parameter, which entails additional mathematical difficulties because the mass conservation of the order parameter, typical of the classic Cahn–Hilliard equation, is no longer satisfied. In this paper, we analyze the case that the double-well potential driving the evolution of the phase transition is differentiable, either (in the regular case) on the whole set of reals or (in the singular logarithmic case) on a finite open interval; nondifferentiable cases like the double obstacle potential are excluded from the analysis. We prove the Fréchet differentiability of the control-to-state operator between suitable Banach spaces for both the regular and the logarithmic cases and establish the solvability of the corresponding adjoint systems in order to derive the associated first-order necessary optimality conditions for the optimal control problem. Crucial for the whole analysis to work is the boundedness property stating that the order parameter attains its values in a compact subset of the interior of the effective domain of the nonlinearity. While this property turns out to be generally valid for regular potentials in three dimensions of space, it can be shown for the logarithmic case only in two dimensions.

Cahn–Hilliard–Brinkman model for tumor growth with possibly singular potentials

We analyze a phase field model for tumor growth consisting of a Cahn–Hilliard–Brinkman system, ruling the evolution of the tumor mass, coupled with an advection-reaction-diffusion equation for a chemical species acting as a nutrient. The main novelty of the paper concerns the discussion of the existence of weak solutions to the system covering all the meaningful cases for the nonlinear potentials; in particular, the typical choices given by the regular, the logarithmic, and the double obstacle potentials are admitted in our treatise. Compared to previous results related to similar models, we suggest, instead of the classical no-flux condition, a Dirichlet boundary condition for the chemical potential appearing in the Cahn–Hilliard-type equation. Besides, abstract growth conditions for the source terms that may depend on the solution variables are postulated.

Optimal Control and Parameters Identification for the Cahn–Hilliard Equations Modeling Tumor Growth

This paper is dedicated to the setting and analysis of an optimal control problem for a two-phase system composed of two non-linearly coupled Chan–Hilliard-type equations. The model describes the evolution of a tumor cell fraction and a nutrient-rich extracellular water volume fraction. The main objective of this paper is the identification of the system’s physical parameters, such as the viscosities and the proliferation rate, in addition to the controllability of the system’s unknowns. For this purpose, we introduce an adequate cost function to be optimized by analyzing a linearized system, deriving the adjoint system, and defining the optimality condition. Eventually, we provide a numerical simulation example illustrating the theoretical results. Finally, numerical simulations of a tumor growing in two and three dimensions are carried out in order to illustrate the evolution of such a clinical situation and to possibly suggest different treatment strategies.

The subdivision-based IGA-EIEQ numerical scheme for the binary surfactant Cahn–Hilliard phase-field model on complex curved surfaces

In this paper, we consider numerical approximations of the binary surfactant phase-field model on complex surfaces. Consisting of two nonlinearly coupled Cahn–Hilliard type equations, the system is solved by a fully discrete numerical scheme with the properties of linearity, decoupling, unconditional energy stability, and second-order time accuracy. The IGA approach based on Loop subdivision is used for the spatial discretizations, where the basis functions consist of the quartic box-splines corresponding to the hierarchic subdivided surface control meshes. The time discretization is based on the so-called explicit-IEQ method, which enables one to solve a few decoupled elliptic constant-coefficient equations at each time step. We then provide a detailed proof of unconditional energy stability along with implementation details, and successfully demonstrate the advantages of this hybrid strategy by implementing various numerical experiments on complex surfaces.

A convergent SAV scheme for Cahn–Hilliard equations with dynamic boundary conditions

Abstract The Cahn–Hilliard equation is one of the most common models to describe phase separation processes in mixtures of two materials. For a better description of short-range interactions between the material and the boundary, various dynamic boundary conditions for this equation have been proposed. Recently, a family of models using Cahn–Hilliard-type equations on the boundary of the domain to describe adsorption processes was analysed (cf. Knopf, P., Lam, K. F., Liu, C. & Metzger, S. (2021) Phase-field dynamics with transfer of materials: the Cahn–Hilliard equation with reaction rate dependent dynamic boundary conditions. ESAIM: Math. Model. Numer. Anal., 55, 229–282). This family of models includes the case of instantaneous adsorption processes studied by Goldstein, Miranville and Schimperna (2011, A Cahn–Hilliard model in a domain with non-permeable walls. Phys. D, 240, 754–766) as well as the case of vanishing adsorption rates, which was investigated by Liu and Wu (2019, An energetic variational approach for the Cahn–Hilliard equation with dynamic boundary condition: model derivation and mathematical analysis. Arch. Ration. Mech. Anal., 233, 167–247). In this paper, we are interested in the numerical treatment of these models and propose an unconditionally stable, linear, fully discrete finite element scheme based on the scalar auxiliary variable approach. Furthermore, we establish the convergence of discrete solutions towards suitable weak solutions of the original model. Thereby, when passing to the limit, we are able to remove the auxiliary variables introduced in the discrete setting completely. Finally, we present simulations based on the proposed linear scheme and compare them to results obtained using a stable, nonlinear scheme to underline the practicality of our scheme.

A second-order unconditionally energy stable scheme for phase-field based multimaterial topology optimization

A multimaterial topology optimization problem is solved by introducing a numerical scheme with fast convergence, second-order accuracy and unconditional energy stability. The modeling is based on the energy of multi-phase-field elasticity system including the classical Ginzburg–Landau, the elastic potential and some constraints. The material layout is updated by using the volume constrained gradient flow of the system. In this work, we transform the traditional objective functional of multimaterial topology optimization into the energy functional of multi-phase-field elasticity system, and transform the optimal material layout into the solutions of volume constrained Allen–Cahn type equations. For these Allen–Cahn type equations, we propose the second-order unconditionally energy stable numerical scheme which combines linearly stabilized splitting method and Crank–Nicolson scheme. For the proposed second-order scheme, we give a theoretical proof of unconditional energy stability. Numerical results show that the scheme converges fast compared to traditional Cahn–Hilliard type equations. Some classical benchmarks are performed to verify the feasibility and efficiency of our method.

A fully-decoupled discontinuous Galerkin approximation of the Cahn–Hilliard–Brinkman–Ohta–Kawasaki tumor growth model

In this article, we consider the Cahn–Hilliard–Brinkman–Ohta–Kawasaki tumor growth system, which couples the Brinkman flow equations in the porous medium and the Cahn–Hilliard type equation with the nonlocal Ohta–Kawasaki term. We first construct a fully-decoupled discontinuous Galerkin method based on a decoupled, stabilized energy factorization approach and implicit-explicit Euler method in the time discretization, and strictly prove its unconditional energy stability. The optimal error estimate for the tumor interstitial fluid pressure is further obtained. Numerical results are also carried out to demonstrate the effectiveness of the proposed numerical scheme and verify the theoretical results. Finally, we apply the scheme to simulate the evolution of brain tumors based on patient-specific magnetic resonance imaging, and the obtained computational results show that the proposed numerical model and scheme can provide realistic calculations and predictions, thus providing an in-depth understanding of the mechanism of brain tumor growth.

Linear relaxation schemes for the Allen–Cahn-type and Cahn–Hilliard-type phase field models

This letter introduces novel linear relaxation schemes for solving the phase field models, particularly the Allen–Cahn (AC) type and Cahn–Hilliard (CH) type equations. The proposed schemes differ from existing schemes for the phase field models in the literature. The resulting semi-discrete schemes are linear by discretizing the AC and CH models on staggered time meshes. Only a linear algebra problem needs to be solved at each time marching step after the spatial discretization. Furthermore, our proposed schemes are shown to be unconditionally energy stable, i.e., the numerical solutions respect energy dissipation laws without restriction on the time steps. Several numerical examples are provided to illustrate the power of the proposed linear relaxation schemes for solving phase field models.

Global existence of weak solutions to viscoelastic phase separation part: I. Regular case

We prove the existence of weak solutions to a viscoelastic phase separation problem in two space dimensions. The mathematical model consists of a Cahn–Hilliard-type equation for two-phase flows and the Peterlin–Navier–Stokes equations for viscoelastic fluids. We focus on the case of a polynomial-like potential and suitably bounded coefficient functions. Using the Lagrange–Galerkin finite element method complex behavior of solution for spinodal decomposition including transient polymeric network structures is demonstrated.

A toy model of misfolded protein aggregation and neural damage propagation in neurodegenerative diseases

Neurodegenerative diseases (NDs) result from the transformation and accumulation of misfolded proteins within the nervous system. Several mathematical models have been proposed to investigate the biological processes underlying NDs, focusing on the kinetics of polymerization and fragmentation at the microscale and on the spread of neural damage at a macroscopic level. The aim of this work is to bridge the gap between microscopic and macroscopic approaches proposing a toy partial differential model able to take into account both the short-time dynamics of the misfolded proteins aggregating in plaques and the long-term evolution of tissue damage. Using mixtures theory, we consider the brain as a biphasic material made of misfolded protein aggregates and of healthy tissue. The resulting Cahn–Hilliard type equation for the misfolded proteins contains a growth term depending on the local availability of precursor proteins, that follow a reaction–diffusion equation. The misfolded proteins also possess a chemotactic mass flux driven by gradients of neural damage, that is caused by local accumulation of misfolded protein and that evolves slowly according to an Allen-Cahn equation. The diffuse interface approach is new for NDs and allows both to consider five different time-scales, from phase separation to neural damage propagation, and to reduce the computational costs compared to existing multi-scale models, allowing a time-step adaptivity. We present here numerical simulations in a simple two-dimensional domain, considering both isotropic and anisotropic mobility coefficients of the misfolded protein and the diffusion of the neural damage, finding that the spreading front of the neural damage follows the direction of the largest eigenvalue of the mobility tensor. In both cases, we computed two biomarkers for quantifying the aggregation in plaques and the evolution of neural damage, that are in qualitative agreement with the characteristic Jack curves for many NDs.

Analysis of a tumor model as a multicomponent deformable porous medium

We propose a diffuse interface model to describe a tumor as a multicomponent deformable porous medium. We include mechanical effects in the model by coupling the mass balance equations for the tumor species and the nutrient dynamics to a mechanical equilibrium equation with phase-dependent elasticity coefficients. The resulting PDE system couples two Cahn–Hilliard type equations for the tumor phase and the healthy phase with a PDE linking the evolution of the interstitial fluid to the pressure of the system, a reaction-diffusion type equation for the nutrient proportion, and a quasistatic momentum balance.We prove here that the corresponding initial-boundary value problem has a solution in appropriate function spaces.

Spinodal de-wetting of light liquids on graphene

We demonstrate theoretically the possibility of spinodal de-wetting in heterostructures made of light–atom liquids (hydrogen, helium, and nitrogen) deposited on suspended graphene. Extending our theory of film growth on two-dimensional (2D) materials to include analysis of surface instabilities via the hydrodynamic Cahn–Hilliard-type equation, we characterize in detail the spatial and temporal scales of the resulting spinodal de-wetting patterns. Both linear stability analysis and direct numerical simulations of the surface hydrodynamics show micron-sized (generally material dependent) patterns of ‘dry’ regions. The physical reason for the development of such instabilities on graphene can be traced back to the inherently weak van der Waals interactions between atomically thin materials and atoms in the liquid. Thus 2D materials could represent a new theoretical and technological platform for studies of spinodal de-wetting.

Well-posedness of a hydrodynamic phase-field system for functionalized membrane-fluid interaction

<p style='text-indent:20px;'>We study a hydrodynamic phase-field system modeling the deformation of functionalized membranes in incompressible viscous fluids. The governing PDE system consists of the Navier–Stokes equations coupled with a convective sixth-order Cahn–Hilliard type equation driven by the functionalized Cahn–Hilliard free energy, which describes the phase separation process in mixtures with an amphiphilic structure. In the three dimensional case, we prove existence of global weak solutions provided that the initial total energy is finite. Then we establish uniqueness of weak solutions under suitable regularity assumptions that are only imposed on the velocity field or its gradient. Next, we prove existence and uniqueness of local strong solutions for arbitrary regular initial data and derive some blow-up criteria. Finally, we show the eventual regularity of global weak solutions for large time. The results are obtained in a general setting with variable fluid viscosity and diffusion mobility.</p>

A decoupled, stable, and linear FEM for a phase-field model of variable density two-phase incompressible surface flow

The paper considers a thermodynamically consistent phase-field model of a two-phase flow of incompressible viscous fluids. The model allows for a non-linear dependence of the fluid density on the phase-field order parameter. Driven by applications in biomembrane studies, the model is written for tangential flows of fluids constrained to a surface and consists of (surface) Navier–Stokes–Cahn–Hilliard type equations. We apply an unfitted finite element method to discretize the system and introduce a fully discrete time-stepping scheme with the following properties: (i) the scheme decouples the fluid and phase-field equation solvers at each time step, (ii) the resulting two algebraic systems are linear, and (iii) the numerical solution satisfies the same stability bound as the solution of the original system under some restrictions on the discretization parameters. Numerical examples are provided to demonstrate the stability, accuracy, and overall efficiency of the approach. Our computational study of several two-phase surface flows reveals some interesting dependencies of flow statistics on the geometry.