Diffuse Interface Model Research Articles

On a Cahn–Hilliard–Keller–Segel model with generalized logistic source describing tumor growth

We propose a new type of diffuse interface model describing the evolution of a tumor mass under the effects of a chemical substance (e.g., a nutrient or a drug). The process is described by utilizing the variables φ, an order parameter representing the local proportion of tumor cells, and σ, representing the concentration of the chemical. The order parameter φ is assumed to satisfy a suitable form of the Cahn–Hilliard equation with mass source and logarithmic potential of Flory–Huggins type (or generalizations of it). The chemical concentration σ satisfies a reaction-diffusion equation where the cross-diffusion term has the same expression as in the celebrated Keller–Segel model. In this respect, the model we propose represents a new coupling between the Cahn–Hilliard equation and a subsystem of the Keller–Segel model. We believe that, compared to other models, this choice is more effective in capturing the chemotactic effects that may occur in tumor growth dynamics (chemically induced tumor evolution and consumption of nutrient/drug by tumor cells). Note that, in order to prevent finite time blowup of σ, we assume a chemical source term of logistic type. Our main mathematical result is devoted to proving existence of weak solutions in a rather general setting that covers both the two- and three- dimensional cases. Under more restrictive assumptions on coefficient and data, and in some cases on the spatial dimension, we prove various regularity results. Finally, in a proper class of smooth solutions we show uniqueness and continuous dependence on the initial data in a number of significant cases.

A new method to discretize a model for isothermal flow with a multi-component equation of state

In this paper we will discuss numerical problems in a phase function model with a multi-component equation of state. It is a sub-model of a diffuse interface model, using a phase field equation, that was introduced by Dreyer et al. in 2014. This model was proposed to describe chemically reacting fluid mixtures consisting of N constituents where phase transitions between a liquid and a vapor phase may occur. The phase field indicates the present phase or the transition layer. The discretization of the model with shock capturing methods for the hyperbolic sub-part is a challenge. One difficulty is that the equation of state has a steep gradient in the density. Moreover, numerical viscosity in diffusive interface computations leads to intermediate values that are unphysical. Our sub-model contains these problems in a nutshell. We develop a solution strategy and apply the new method to several test cases.

Mathematical modeling of transport phenomena in compressible multicomponent flows

The present article proposes a diffuse interface model for compressible multicomponent flows with transport phenomena of mass, momentum and energy (i.e., mass diffusion, viscous dissipation and heat conduction). The model is reduced from the seven-equation Baer-Nuziato type model with asymptotic analysis in the limit of instantaneous mechanical relaxations. The main difference between the present model and the Kapila's five-equation model consists in that different time scales for pressure and velocity relaxations are assumed, the former being much smaller than the latter. Thanks to this assumption, the velocity disequilibrium is retained to model the mass diffusion process. Aided by the diffusion laws, the final model still formally consists of five equations. The proposed model satisfy two desirable properties: (1) it respects the laws of thermodynamics, (2) it is free of the spurious oscillation problem in the vicinity of the diffused interface zone. For solution of the model governing equations, we implement the fractional step method to split the model into five physical steps: the hydrodynamic step, the viscous step, the heat transfer step, the heat conduction step and the mass diffusion step. The split governing equations for the hydrodynamic step formally coincide with the Kapila's five-equation model and are solved with the Godunov finite volume method. The mass diffusion, viscous dissipation and heat conduction processes contribute parabolic partial differential equations that are solved with the Chebyshev method of local iterations. Numerical results show that the proposed model maintains pressure, velocity and temperature equilibrium near the diffused interface. Convergence tests demonstrate that the numerical methods achieve second order in space and time. The proposed model and numerical methods are applied to simulate the hydrodynamic instability problems in the inertial confinement fusion. Good agreement with experimental results is observed.

Efficient IMEX and consistently energy-stable methods of diffuse-interface models for incompressible three-component flows

In this study, we consider the numerical approximation of incompressible three-component fluids, in which the fluid interfaces are captured by ternary Cahn–Hilliard equations and the fluid flows are governed by Navier–Stokes equations. This system includes not only nonlinear effects but also coupling among phase-field variables, velocities, and pressure. The ternary Cahn–Hilliard–Navier–Stokes system also satisfies the energy dissipation law, which is a basic physical property. For the appropriate treatment of the nonlinear and coupling terms and preservation of the energy dissipation law in a discrete version, we develop second-order time-accurate, linearly implicit-explicit (IMEX) methods using a variation of the scalar auxiliary variable (SAV) method. To improve the consistency between the original and modified energies, a simple and effective energy relaxation technique is considered. We analytically proved the unique solvability and the relaxed energy dissipation law. The proposed schemes are highly efficient for implementation because only linear elliptic equations need to be solved separately. Extensive computational experiments are performed to validate the accuracy, energy stability, and performance of the proposed method. To facilitate further study, we provide the C codes for the typical numerical simulations at http://github.com/yang521.

Numerical investigation of gas entrapment in metal additive manufacturing using a quasi-sharp-interface particle method

Porosity is a key issue for the parts fabricated by selective laser melting. However, the porosity formation mechanisms have not been fully clarified. In particular, the formation of small pores in fabricated parts from the gas among granules is controversial. The existence of sharp corners among molten granules is a key numerical challenge because the conventional diffusive interface model (DIM) might trigger spurious gas entrapment. To solve the problem, a quasi-sharp-interface model (QSIM) was proposed for the moving particle semi-implicit (MPS) method in this study. A new approach to calculate the interface area based on the local triangularization was proposed to accurately impose the surface-tension force in the QSIM. The bubble-rising simulation verified the capability of the QSIM to simulate topological changes of interfaces. The sharp-corner deformation problem demonstrated that the QSIM could effectively prevent unphysical breakup and the spurious gas entrapment. Thus, the QSIM was indispensable for reliably simulating the gas entrapment from the gaps among granules in SLM. Finally, the QSIM results indicated that the gas among granules was completely discharged when the granules melted. This suggested that the gas among granules did not cause small pores in the fabricated parts. This observation correlated with an experimental composition analysis of gas in pores.

A robust and accurate adaptive approximation method for a diffuse-interface model of binary-fluid flows

We present an adaptive simulation framework for binary-fluid flows, based on the Abels–Garcke–Grün Navier–Stokes–Cahn–Hilliard (AGG NSCH) diffuse-interface model. The adaptive-refinement procedure is guided by a two-level hierarchical a-posteriori error estimate, and it effectively resolves the spatial multiscale behavior of the diffuse-interface model. To improve the robustness of the solution procedure and avoid severe time-step restrictions for small-interface thicknesses, we introduce an ɛ-continuation procedure, in which the diffuse interface thickness (ɛ) are enlarged on coarse meshes, and the mobility is scaled accordingly. To further accelerate the computations and improve robustness, we apply a modified Backward Euler scheme in the initial stages of the adaptive-refinement procedure in each time step, and a Crank–Nicolson scheme in the final stages of the refinement procedure. To enhance the robustness of the nonlinear solution procedure, we introduce a partitioned solution procedure for the linear tangent problems in Newton’s method, based on a decomposition of the NSCH system into its NS and CH subsystems. We conduct a systematic investigation of the conditioning of the monolithic NSCH tangent matrix and of its NS and CH subsystems for a representative 2D model problem. To illustrate the properties of the presented adaptive simulation framework, we present numerical results for a 2D oscillating water droplet suspended in air, and we validate the obtained results versus those of a corresponding sharp-interface model.

A diffuse interface model for low solubility binary flows in porous media

Dissolution-driven convection of low solubility fluids separated by a free interface in a porous medium is analyzed via two-dimensional numerical simulations and scaling considerations. Towards this end, we employ a low solubility Darcy-Cahn-Hilliard model with a Helmholtz free energy. The momentum equations are solved in the stream function-vorticity formulation with a pseudospectral method, so that continuity is automatically satisfied, while the diffusion equation is integrated in time by an explicit Runge-Kutta procedure, combined with high order compact finite differences. We validate the simulation approach by comparing the growth rate of the standard deviation of the surface location with experimental measurements for a CO2-water system in a Hele-Shaw cell. The numerical results for the evolving interfacial shape in density-driven convection flow of low solubility fluids agree significantly better with experimental observations than corresponding fully soluble results. The temporal evolution of the interfacial plumes is accurately captured, and distinct stages with different plume features are identified. The differences in plume features and solute flux of CO2 between low solubility and fully soluble fluids are discussed, and it is found that the low solubility simulation approach offers clear advantages for the investigation of CO2-dynamics in porous rock.

A computational model for interfacial heat and mass transfer in two-phase flows using a phase field method

Two-phase flows involving heat/mass transfer are widespread in industrial and environmental applications such as chemical reactors, bubbly flows, combustion, boiling, carbon sequestration, and ocean-atmosphere exchanges. It is therefore important to accurately predict the rate of heat/mass transfer across capillary interfaces via numerical simulations. Due to the absence of a well-defined interface, modeling interfacial transfer between two phases is particularly challenging for phase field (diffuse interface) models. In the context of second-order conservative phase field models, by assuming a microstructure that is consistent with the interfacial profile, we use perturbation theory and asymptotic analysis to derive interfacial heat/mass exchange terms that are consistent extensions of the underlying phase field equations. The developed two-scalar model is conservative, preserves positivity of total scalar concentration, and correctly predicts the transient and equilibrium solutions in all limits of diffusivity ratio. Additionally, we demonstrate that by assuming thermodynamic equilibrium in the microstructure, a more reduced model in the form of a one-scalar equation is derived. Several canonical and realistic simulations are presented to assess the consistency, accuracy, and convergence of the models. Crucially, while the one-scalar and two-scalar models both perform well when the two phases have comparable diffusivities, the two-scalar model is found to be much more accurate for realistic problems with large diffusivity ratios as it prevents unphysical leakage of heat/mass.

Diffuse-interface lattice Boltzmann modeling of charged particle transport in Poiseuille flow.

In this paper, we developed a coupled diffuse-interface lattice Boltzmann method (DI-LBM) to study the transport of a charged particle in the Poiseuille flow, which is governed by the Navier-Stokes equationsfor fluid field and the Poisson-Boltzmann equationfor electric potential field. We first validated the present DI-LBM through some classical benchmark problems, and then investigated the effect of electric field on the lateral migration of the particle in the Poiseuille flow. The numerical results show that the electric field has a significant influence on the particle migration. When an electric field in the vertical direction is applied to the charged particle initially located above the centerline of the channel, the equilibrium position of the particle would drop suddenly as the electric field is larger than a critical value. This is caused by the wall repulsion due to lubrication, the inertial lift related to shear slip, the lift owing to particle rotation, the lift due to the curvature of the undisturbed velocity profile, and the electric force. On the other hand, when an electric field in the horizontal direction is adopted, the equilibrium position of the particle would move toward the centerline of the channel with the increase of the electric field.

Global existence of weak solutions to viscoelastic phase separation: part II. Degenerate case

The aim of this paper is to prove global in time existence of weak solutions for a viscoelastic phase separation. We consider the case with singular potentials and degenerate mobilities. Our model couples the diffusive interface model with the Peterlin–Navier–Stokes equations for viscoelastic fluids. To obtain the global in time existence of weak solutions we consider appropriate approximations by solutions of the viscoelastic phase separation with a regular potential and build on the corresponding energy and entropy estimates.

High-order discontinuous Galerkin approximation for a three-phase incompressible Navier–Stokes/Cahn–Hilliard model

In this work we introduce the development of a three-phase incompressible Navier–Stokes/Cahn–Hilliard numerical method to simulate three-phase flows, present in many industrial operations. The numerical method is then applied to successfully solve oil transport problems, such as those found in the oil and gas industry. The three-phase model adopted in this work is a Cahn–Hilliard diffuse interface model, which was derived by Boyer and Lapuerta (2006). The Cahn–Hilliard model is coupled to the kinetic-energy stable incompressible Navier–Stokes equations model derived by Manzanero et al. (2020). The spatial discretization uses a high-order discontinuous Galerkin spectral element method which yields highly accurate results in arbitrary geometries. An implicit–explicit (IMEX) method is adopted as temporal scheme for the Cahn–Hilliard equation, while Runge–Kutta 3 (RK3) is used for the Navier–Stokes equations. The developed numerical tool is validated with a manufactured solution test case and used to simulate multiphase flows in pipes, including and a three-phase T-shaped pipe intersection.

Capillary condensation of saturated vapor in a corner formed by two intersecting walls

The dynamics of saturated vapor between two intersecting walls is examined. It is shown that, if the angle ϕ between the walls is sufficiently small, the vapor becomes unstable, and spontaneous condensation occurs in the corner, similar to the so-called capillary condensation of vapor into a porous medium. As a result, an ever-growing liquid meniscus develops near the corner. The diffuse-interface model and the lubrication approximation are used to demonstrate that the meniscus grows if and only if ϕ+2θ<π, where θ is the contact angle corresponding to the fluid/solid combination under consideration. This criterion has a simple physical explanation: if it holds, the meniscus surface is concave—hence, the Kelvin effect causes condensation. Once the thickness of the condensate exceeds by an order of magnitude the characteristic interfacial thickness, the volume of the meniscus starts to grow linearly with time. If the near-vertex region of the corner is smoothed, the instability can be triggered off only by finite-size perturbations, such that it includes enough liquid to cover the smoothed area by a microscopically thin liquid film.

A Diffuse Interface Model for Cell Blebbing Including Membrane-Cortex Coupling with Linker Dynamics

The aim of this paper is to develop suitable models for the phenomenon of cell blebbing, which allow for computational predictions of mechanical effects including the crucial interaction of the cell membrane and the actin cortex. For this sake we resort to a two phase-field model that uses diffuse descriptions of both the membrane and the cortex, which in particular allows for a suitable description of the interaction via linker protein densities. Besides the detailed modelling we discuss some energetic aspects of the models and present a numerical scheme, which allows to carry out several computational studies. In those we demonstrate that several effects found in experiments can be reproduced, in particular bleb formation by cortex rupture, which was not possible by previous models without the linker dynamics.

A phase separation model for binary fluids with hereditary viscosity

We consider a diffuse interface model for an incompressible binary viscoelastic fluid flow. The model consists of the Navier–Stokes–Voigt equations where the instantaneous kinematic viscosity has been replaced by a memory term incorporating hereditary effects coupled with the Cahn–Hilliard equation with Flory–Huggins potential. The resulting system is subject to no‐slip condition for the (volume averaged) fluid velocity and to no‐flux boundary conditions for the order parameter and the chemical potential . We first establish the well‐posedness of the initial and boundary value problem. Then, taking advantage of an Ekman‐type damping, we obtain a dissipative estimate. Thus, we can define a dissipative dynamical system on a suitable phase space. Also, we show that any weak solution is such that regularizes in finite time, and in dimension two, it stays uniformly away from pure phases; that is, the instantaneous strict separation property holds. Finally, we prove the existence of the global attractor, and exploiting the strict separation property, we demonstrate that an exponential attractor exists in dimension two.

T cell therapy against cancer: A predictive diffuse-interface mathematical model informed by pre-clinical studies

T cell therapy has become a new therapeutic opportunity against solid cancers. Predicting T cell behaviour and efficacy would help therapy optimization and clinical implementation. In this work, we model responsiveness of mouse prostate adenocarcinoma to T cell-based therapies. The mathematical model is based on a Cahn-Hilliard diffuse interface description of the tumour, coupled with Keller-Segel type equations describing immune components dynamics. The model is fed by pre-clinical magnetic resonance imaging data describing anatomical features of prostate adenocarcinoma developed in the context of the Transgenic Adenocarcinoma of the Mouse Prostate model. We perform computational simulations based on the finite element method to describe tumor growth dynamics in relation to local T cells concentrations. We report that when we include in the model the possibility to activate tumor-associated vessels and by that increase the number of T cells within the tumor mass, the model predicts higher therapeutic effects (tumor regression) shortly after therapy administration. The simulated results are found in agreement with reported experimental data. Thus, this diffuse-interface mathematical model well predicts T cell behavior in vivo and represents a proof-of-concept for the role such predictive strategies may play in optimization of immunotherapy against cancer.

A phase field model for mass transport with semi-permeable interfaces

In this paper, a thermaldynamical consistent model for mass transfer across permeable moving interfaces is proposed by using the energy variation method. We consider a restricted diffusion problem where the flux across the interface depends on its conductance and the difference of the concentration on each side. The diffusive interface phase-field framework used here has several advantages over the sharp interface method. First of all, explicit tracking of the interface is no longer necessary. Secondly, interfacial conditions can be incorporated with a variable diffusion coefficient. Finally, topological changes of interfaces can be handed easily. A detailed asymptotic analysis confirms the diffusive interface model converges to the existing sharp interface model as the interface thickness goes to zero. An energy stable numerical scheme is developed to solve this highly nonlinear coupled system.Numerical simulations first illustrate the consistency of theoretical results on the sharp interface limit. Then a convergence study and energy decay test are conducted to ensure the efficiency and stability of the numerical scheme. To illustrate the effectiveness of our phase-field approach, several examples are provided, including a study of a two-phase mass transfer problem where droplets with deformable interfaces are suspended in a moving fluid.

Unconditional stability and optimal error estimates of first order semi-implicit stabilized finite element method for two phase magnetohydrodynamic diffuse interface model

This paper is devoted to present the rigorous unconditional stability and optimal error estimates of first order semi-implicit energy stable finite element method developed by Yang et al. (Comput. Methods Appl. Mech. Engrg. 356 (2019) 435–464) for the two phase magnetohydrodynamic(MHD) flows, some numerical results are also provided to show the performances of the considered numerical scheme.

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.

Numerical analysis for a Cahn–Hilliard system modelling tumour growth with chemotaxis and active transport

Abstract A diffuse interface model for tumour growth in the presence of a nutrient consumed by the tumour is considered. The system of equations consists of a Cahn–Hilliard equation with source terms for the tumour cells and a reaction–diffusion equation for the nutrient. We introduce a fully-discrete finite element approximation of the model and prove stability bounds for the discrete scheme. Moreover, we show that discrete solutions exist and depend continuously on the initial and boundary data. We then pass to the limit in the discretization parameters and prove convergence to a global-in-time weak solution to the model. Under additional assumptions, this weak solution is unique. Finally, we present some numerical results including numerical error investigation in one spatial dimension and some long time simulations in two and three spatial dimensions.

A new 3D OpenFoam solver with improved resolution for hyperbolic systems on hybrid unstructured grids

Although the accuracy and the robustness of discretized schemes solving hyperbolic systems have been greatly improved over the last two decades, devising a low-dissipative and non-oscillatory method remains a challenge. Especially, it is algorithmically complicated and computationally expensive to extend and apply high-resolution non-oscillatory schemes to 3D hybrid unstructured grids. Therefore, in this work we develop accurate, practical, and robust 3D hybrid unstructured OpenFoam solvers which can be applied in wide industrial applications. Different from existing 3D high order numerical solvers of which reconstruction schemes are based on polynomials of high degree, the proposed 3D scheme employs a polynomial function and a hyperbolic tangent basis function as two candidates for reconstruction functions. The MUSCL (Monotone Upstream-centered Schemes for Conservation law) scheme with the MLP (Multi-dimensional Limiting Process) slope limiter is adopted as the polynomial function. The multi-dimensional THINC (Tangent of Hyperbola for INterface Capturing) function with quadratic surface representation and Gaussian quadrature, so-called THINC/QQ, is used as the non-polynomial reconstruction candidate. With these reconstruction candidates, a multi-stage boundary variation diminishing (BVD) algorithm which significantly minimizes numerical dissipation is designed on 3D unstructured grids to select the final reconstruction function. With this accurate and robust reconstruction scheme, then we developed new solvers for simulating inviscid high speed compressible single-phase flow and two-phase flow via the implementation of the proposed scheme and algorithm into the open-source computational fluid dynamics (CFD) software, OpenFOAM. The solver with the proposed low-dissipative scheme, named as EulerEquationFoam-M-TQ-BVD, is constructed with the density-based solver for inviscid high speed compressible flow simulations. To simulate compressible multi-material flow with moving material interfaces, a solver named as FiveEquationFoam-M-TQ-BVD is developed based on mechanical-equilibrium diffusive interface model and MieGrüneisen equation of state. The performance of new solvers is evaluated through wide range benchmark tests and are compared with existing high order schemes such as the third-order WENO (Weighted Essentially Non-Oscillatory) scheme. The new solvers produce superior solution across critical points, contact discontinuities and material interfaces. Moreover, in comparison with WENO schemes, the proposed solver enjoys algorithmic simplicity and greater flexibility in 3D hybrid unstructured grids. Thus, this work provides accurate, practical and robust numerical solvers for high speed compressible single- and multi-material flow simulations.