Cross-diffusion Terms Research Articles

A phase-field-based lattice Boltzmann model for multiphase flows involving N immiscible incompressible fluids

In this work, an efficient and accurate lattice Boltzmann (LB) model is developed based on phase-field theory to study multiphase flows involving N (N≥2) immiscible incompressible fluids. In this model, a reduction-consistent physical formulation including a volume-fraction-dependent mobility in the Cahn–Hilliard (C–H) equations is adopted. Usually, the effect of cross-diffusion makes it difficult to solve such equations directly with the classic LB method. To avoid requiring a special treatment on the cross-diffusion terms of the chemical potential gradients, the proposed LB model introduces some non-diagonal collision operators. In addition, the proper auxiliary source terms are constructed to ensure the correct macroscopic equations. Through a direct Taylor expansion, the C–H equations are recovered from the present LB model. Finally, four classical problems including static droplets, the spreading of a liquid lens between two phases, the Kelvin–Helmholtz instability, and the dynamics of droplets in a four-phase system are used to demonstrate the capability of the LB model. The numerical results show that the present model satisfies the reduction-consistent property and produces physically accurate results.

Finite element analysis of a Keller-Segel model with additional cross-diffusion and logistic source. Part II: Time convergence and numerical simulation

In [1], a finite element method for the (Patlak-) Keller-Segel equations, with additional cross-diffusion and logistic source terms in the elliptic equation, has been introduced. Some a priori estimates of the regularized solutions have been derived, independently of the regularization parameters. Moreover, the existence of the finite element solutions has been shown by using the fixed point theorem and some stability bounds on the fully discrete approximations are obtained. The convergence of the approximate solutions in space has been shown. In this article, we have completed what has been achieved in [1], where we have completed the convergence of time, where this article starts from where the study in [1] ended. The current study aims to prove the existence of weak solutions to this model, which includes passing the limit to the remaining regulation parameter and achieving the convergence in time. We firstly show that the solutions, independent of regulation parameter, can be bounded. Then these bounds are used to derive bounds on the time-derivatives which are also independent of regulation parameter. In addition, compactness arguments have been used to discuss the convergence of the finite element the approximate problem. We demonstrated the existence of a weak solution for the model under study. Finally, we use the implicit scheme to conduct simulations in one and two space dimensions.

Mixed convection of nanofluid by two-phase model in an inclined cavity with variable aspect ratio

In this numerical study, the laminar mixed cross-convection of incompressible nanofluid in a two-dimensional inclined cavity with various aspect ratios is investigated. A nonhomogeneous Buongiorno's model is taken into account. Transport mechanisms incorporating the effects of Brownian motion, thermophoresis diffusion, Soret-Dufour effect are analyzed. The energy governing equation includes both regular diffusion and cross-diffusion terms. The numerical analysis is performed by the collocation spectral method (CSM) based on the fast cosine transformation (FCT). The improved projection scheme (IPS) is specially modified with help of FCT to decouple the pressure and the velocities. The combined effects of inclined cavity angle γ, aspect ratio A, thermophoresis parameters Nt, Brownian motion parameter Nb, Dufour diffusicity Df, Soret diffusicity Sr, along with Re, Gr, Pr, Le, Ln and buoyancy ratios Br, Nb, Nr, etc., on the developments of streamlines, isotherms, solutal isoconcentrations and nanoparticle isoconcentrations as well as on the average Nusselt and Sherwood numbers, are analyzed and discussed in detail. The results indicate that the above parameters, especially γ and A, have substantial effects on the flow, heat and mass transfer. The distributions of temperature, solutal concentration and nanoparticle concentration are changed completely. The cross-diffusive effects including Dufour-Soret effect can't be ignored.

A two-relaxation-time lattice Boltzmann study on the Soret and Dufour effects of double-diffusive convection over a rough surface

This study examines the Soret and Dufour effects of double-diffusive convection (DDC) over a rough surface by using a lattice Boltzmann (LB) model with two-relaxation-time (TRT) collision operators, where the cross-diffusion terms are modeled by introducing additional collision operators that can be used to avoid the need for special treatment of the gradient terms. The TRT model is applied to improve the stability of the LB method for flows at high Rayleigh numbers. We investigate how rough surfaces with different distributions and compactness of rough elements affect heat and mass transfer as well as the flow structure of the DDC system with the Soret and Dufour effects. We examined five rough models with different spatial distributions of the triangular rough elements as well as different values of the Rayleigh number (Ra=106 and 109), Prandtl number (Pr=0.7), Lewis number (Le=1 and 2), buoyancy ratio (−2≤Nc≤6.5), Soret factor (0≤STC≤1), Dufour factor (0≤DCT≤1), and the height of the roughness (1/96≤h≤1/2). Specifically, the influence of the buoyancy ratio on the DDC system was investigated, and the relationship of heat and mass transfer with the buoyancy ratio was divided into four stages: conduction-dominated, transition, steady convection-dominated, and oscillatory convection-dominated stages. The rough surfaces reduced heat and mass transfer at low Rayleigh numbers and low height of the roughness, and the third rough model (Model III) yielded the highest reduction. The flow showed periodic behavior after long-term reversals in some cases, which appear to have not been observed in previous research. Moreover, the effects of the Soret and Dufour factors were examined, and it was found that mass and heat transfer could be enhanced in most cases by increasing their values. To enhance heat and mass transfer, the height of the roughness should be increased as well. Model III delivered the best performance, with a scenario completely opposite to that in case of a low roughness height. Finally, approximate values of the critical height of the roughness with different rough models were obtained such that heat and mass transfer were enhanced only when the height of the roughness was greater than a critical value.

Solitary pulses and periodic wave trains in a bistable FitzHugh-Nagumo model with cross diffusion and cross advection.

We describe analytically, and simulate numerically, traveling waves with oscillatory tails in a bistable, piecewise-linear reaction-diffusion-advection system of the FitzHugh-Nagumo type with linear cross-diffusion and cross-advection terms of opposite signs. We explore the dynamics of two wave types, namely, solitary pulses and their infinite sequences, i.e., periodic wave trains. The effects of cross diffusion and cross advection on wave profiles and speed of propagation are analyzed. For pulses, in the speed diagram splitting of a curve into several branches occurs, corresponding to different waves (wave branching). For wave trains, in the dispersion relation diagram there are oscillatory curves and the discontinuous curve of an isola with two branches. The corresponding wave trains have symmetric or asymmetric profiles. Numerical simulations show that for large values of the period there exist two wave trains, which come closer and closer together and are subject to fusion into one when the value of the period is decreasing. Other types of waves are also briefly discussed.

Coexistence states of a Holling type II predator-prey system with self and cross-diffusion terms

<p style='text-indent:20px;'>In this paper, we present results about existence and non-existence of coexistence states for a reaction-diffusion predator-prey model with the two species living in a bounded region with inhospitable boundary and Holling type II functional response. The predator is a specialist and presents self-diffusion and cross-diffusion behavior. We show the existence of coexistence states by combining global bifurcation theory with the method of sub- and supersolutions.</p>

Hopf bifurcations in the full SKT model and where to find them

<p style='text-indent:20px;'>In this paper, we consider the Shigesada–Kawasaki–Teramoto (SKT) model, which presents cross-diffusion terms describing competition pressure effects. Even though the reaction part does not present the activator–inhibitor structure, cross-diffusion can destabilise the homogeneous equilibrium. However, in the full cross-diffusion system and weak competition regime, the cross-diffusion terms have an opposite effect and the bifurcation structure of the system modifies as the interspecific competition pressure increases. The major changes in the bifurcation structure, the type of pitchfork bifurcations on the homogeneous branch, as well as the presence of Hopf bifurcation points are here investigated. Through weakly nonlinear analysis, we can predict the type of pitchfork bifurcation. Increasing the additional cross-diffusion coefficients, the first two pitchfork bifurcation points from super-critical become sub-critical, leading to the appearance of a multi-stability region. The interspecific competition pressure also influences the possible appearance of stable time-period spatial patterns appearing through a Hopf bifurcation point.</p>

Cross-diffusion effects on stationary pattern formation in the FitzHugh-Nagumo model

<p style='text-indent:20px;'>We investigate the formation of stationary patterns in the FitzHugh-Nagumo reaction-diffusion system with linear cross-diffusion terms. We focus our analysis on the effects of cross-diffusion on the Turing mechanism. Linear stability analysis indicates that positive values of the inhibitor cross-diffusion enlarge the region in the parameter space where a Turing instability is excited. A sufficiently large cross-diffusion coefficient of the inhibitor removes the requirement imposed by the classical Turing mechanism that the inhibitor must diffuse faster than the activator. In an extended region of the parameter space a new phenomenon occurs, namely the existence of a double bifurcation threshold of the inhibitor/activator diffusivity ratio for the onset of patterning instabilities: for large values of inhibitor/activator diffusivity ratio, classical Turing patterns emerge where the two species are in-phase, while, for small values of the diffusion ratio, the analysis predicts the formation of out-of-phase spatial structures (named <i>cross-Turing patterns</i>). In addition, for increasingly large values of the inhibitor cross-diffusion, the upper and lower bifurcation thresholds merge, so that the instability develops independently on the value of the diffusion ratio, whose magnitude selects Turing or cross-Turing patterns. Finally, the pattern selection problem is addressed through a weakly nonlinear analysis.</p>

Martingale solutions of stochastic nonlocal cross-diffusion systems

<p style='text-indent:20px;'>We establish the existence of solutions for a class of stochastic reaction-diffusion systems with cross-diffusion terms modeling interspecific competition between two populations. More precisely, we prove the existence of weak martingale solutions employing appropriate Faedo-Galerkin approximations and the stochastic compactness method. The nonnegativity of solutions is proved by a stochastic adaptation of the well-known Stampacchia approach.</p>

On pattern formation in reaction–diffusion systems containing self- and cross-diffusion

In this article we propose a unified framework in order to study reaction–diffusion systems containing self- and cross-diffusion using a free energy approach. This framework naturally leads to the formulation of an energy law, and to a numerical method respecting a discrete version of the latter. It constitutes an alternative method and complements the standard linear stability analysis, as it allows for the numerical study of nonlinear patterns, while monitoring the energy evolution, even in complex geometries. As an application, we propose and study a modified Gray–Scott system augmented with self- and cross-diffusion terms. Numerical simulations unveil original patterns, clearly distinct from those obtained with linear diffusion only.

Finite element approximation of a Keller–Segel model with additional self- and cross-diffusion terms and a logistic source

A fully finite element approximation of the Keller–Segel model, including self- and cross-diffusion terms and a logistic source, has been considered. The existence of a fully finite element approximation of the Keller–Segel model, some stability bounds of the solution and the convergence of the approximated solution have been shown. As a result, the existence of a solution to the nonlinear Keller–Segel model in Rd, d≤3 has been demonstrated. Moreover, an iterative approach to solving the resulted nonlinear discrete system has been introduced. Finally, some numerical results have been shown.

Cross-diffusion waves resulting from multiscale, multi-physics instabilities: theory

Abstract. We propose a multiscale approach for coupling multi-physics processes across the scales. The physics is based on discrete phenomena, triggered by local thermo-hydro-mechano-chemical (THMC) instabilities, that cause cross-diffusion (quasi-soliton) acceleration waves. These waves nucleate when the overall stress field is incompatible with accelerations from local feedbacks of generalized THMC thermodynamic forces that trigger generalized thermodynamic fluxes of another kind. Cross-diffusion terms in the 4×4 THMC diffusion matrix are shown to lead to multiple diffusional P and S wave equations as coupled THMC solutions. Uncertainties in the location of meso-scale material instabilities are captured by a wave-scale correlation of probability amplitudes. Cross-diffusional waves have unusual dispersion patterns and, although they assume a solitary state, do not behave like solitons but show complex interactions when they collide. Their characteristic wavenumber and constant speed define mesoscopic internal material time–space relations entirely defined by the coefficients of the coupled THMC reaction–cross-diffusion equations. A companion paper proposes an application of the theory to earthquakes showing that excitation waves triggered by local reactions can, through an extreme effect of a cross-diffusional wave operator, lead to an energy cascade connecting large and small scales and cause solid-state turbulence.

Finite element analysis of a Keller–Segel model with additional cross-diffusion and logistic source. Part I: Space convergence

We have studied a finite element method for the (Patlak)–Keller–Segel equations in one and two space dimensions with additional cross-diffusion and logistic source terms in the elliptic equation for the chemical signal. Some a priori estimates of the regularized functions have been derived, independently of the regularization parameter, by deriving a well defined entropy inequality of the regularized problem. Moreover, a fixed point theorem has been utilized to prove the existence of the finite element solutions. Some stability bounds on the fully discrete approximations are obtained. The convergence of the approximate solutions in space has been shown.

Nonlinear waves in a quintic FitzHugh-Nagumo model with cross diffusion: Fronts, pulses, and wave trains.

We study a tristable piecewise-linear reaction-diffusion system, which approximates a quintic FitzHugh-Nagumo model, with linear cross-diffusion terms of opposite signs. Basic nonlinear waves with oscillatory tails, namely, fronts, pulses, and wave trains, are described. The analytical construction of these waves is based on the results for the bistable case [Zemskov et al., Phys. Rev. E 77, 036219 (2008) and Phys. Rev. E 95, 012203 (2017) for fronts and for pulses and wave trains, respectively]. In addition, these constructions allow us to describe novel waves that are specific to the tristable system. Most interesting is the pulse solution with a zigzag-shaped profile, the bright-dark pulse, in analogy with optical solitons of similar shapes. Numerical simulations indicate that this wave can be stable in the system with asymmetric thresholds; there are no stable bright-dark pulses when the thresholds are symmetric. In the latter case, the pulse splits up into a tristable front and a bistable one that propagate with different speeds. This phenomenon is related to a specific feature of the wave behavior in the tristable system, the multiwave regime of propagation, i.e., the coexistence of several waves with different profile shapes and propagation speeds at the same values of the model parameters.

A lattice Boltzmann model for the coupled cross-diffusion-fluid system

In this paper, we propose a lattice Boltzmann (LB) model for the generalized coupled cross-diffusion-fluid system. Through the direct Taylor expansion method, the proposed LB model can correctly recover the macroscopic equations. The cross diffusion terms in the coupled system are modeled by introducing additional collision operators, which can be used to avoid special treatments for the gradient terms. In addition, the auxiliary source terms are constructed properly such that the numerical diffusion caused by the convection can be eliminated. We adopt the developed LB model to study two important systems, i.e., the coupled chemotaxis-fluid system and the double-diffusive convection system with Soret and Dufour effects. We first test the present LB model through considering a steady-state case of coupled chemotaxis-fluid system, then we analyze the influences of some physical parameters on the formation of sinking plumes. Finally, the double-diffusive natural convection system with Soret and Dufour effects is also studied, and the numerical results agree well with some previous works.

Modeling the onset of thermosolutal convective instability in a non-Newtonian nanofluid-saturated porous medium layer

The onset of double-diffusive (thermosolutal) convection in horizontal porous layer saturated with an incompressible couple stress nanofluid saturated is studied with thermal conductivity and viscosity dependent on the nanoparticle volume fraction. To represent the momentum equation for porous media, a modified Darcy-Maxwell nanofluid model incorporating the effects of Brownian motion and thermophoresis has been used. The thermal energy equation includes regular diffusion and cross diffusion (Soret thermo-diffusion and Dufour diffuso-thermal) terms. A linear stability analysis depends on the normal mode technique and the onset criterion for stationary and oscillatory convection is derived analytically. The nonlinear theory based on the representation of the Fourier series method is applied to capture the behavior of heat and mass transfer. It is found that the couple stress parameter enhances the stability of the system in both the stationary and oscillatory convection modes. The viscosity ratio and conductivity ratio both enhance heat and mass transfer. Transient Nusselt number is found to be oscillatory when time is small. However, when time becomes very large, all the three transient Nusselt number values approach to their steady state values.

Intricacies of coupled molecular diffusion on double diffusive viscoelastic porous convection

The effect of cross-diffusion on the onset of convective instability in a horizontal porous layer saturated with double diffusive viscoelastic fluid is investigated. A modified Darcy–Oldroyd-B model is used to describe the viscoelastic fluid flow in a porous medium. A linear instability analysis has been performed to obtain the condition for the onset of stationary and oscillatory convection and the numerical results for NaCl–MgCl2 system are presented. Besides, the eigenvalues are computed numerically using a Galerkin method and the results are found to compare well with those obtained analytically. The simultaneous effect of viscoelasticity and the cross diffusion has led to significant changes in the convective pattern. The cross-diffusion terms induce stabilizing or destabilizing influence on the system depending on the magnitude as well as stabilizing/destabilizing impact of one of the stratifying components. The range of strain retardation parameter beyond which oscillatory convection ceases to occur is found to increase with increasing stress relaxation parameter irrespective of cross diffusion effects. The cross over boundary between stationary and oscillatory convection is demarcated by identifying a codimension-two point in the viscoelastic parameters plane. The results of Maxwell fluid are highlighted as a particular case.

Bi-Lipschitz Mané projectors and finite-dimensional reduction for complex Ginzburg–Landau equation

We present a new method of establishing the finite-dimensionality of limit dynamics (in terms of bi-Lipschitz Mané projectors) for semilinear parabolic systems with cross diffusion terms and illustrate it on the model example of three-dimensional complex Ginzburg-Landau equation with periodic boundary conditions. The method combines the so-called spatial-averaging principle invented by Sell and Mallet-Paret with temporal averaging of rapid oscillations which come from cross-diffusion terms.

Existence of weak solutions to a certain homogeneous parabolic Neumann problem involving variable exponents and cross-diffusion

This paper deals with a homogeneous Neumann problem of a nonlinear diffusion system involving variable exponents dependent on spatial and time variables and cross-diffusion terms. We prove the existence of weak solutions using Galerkin’s approximation and we derive suitable energy estimates. To this end, we establish the needed Poincaré type inequality for variable exponents related to the Neumann boundary problem. Furthermore, we show that the investigated problem possesses a unique weak solution and satisfies a stability estimate, provided some additional assumptions are fulfilled. In addition, we show under which conditions the solution is nonnegative.

Oscillatory multipulsons: Dissipative soliton trains in bistable reaction-diffusion systems with cross diffusion of attractive-repulsive type.

One-dimensional localized sequences of bound (coupled) traveling pulses, wave trains with a finite number of pulses, are described in a piecewise-linear reaction-diffusion system of the FitzHugh-Nagumo type with linear cross-diffusion terms of opposite signs. The simplest case of two bound pulses, the paired-pulse waves (pulse pairs), is solved analytically. The solutions contain oscillatory tails in the wave profiles so that the pulse pairs consist of a double-peak core and wavy edges. Several pulse pairs with different profile shapes and propagation speeds can appear for the same parameter values of the model when the cross diffusion is dominant. The more general case of many bound pulses, multipulse waves, is studied numerically. It is shown that, dependent on the values of the cross-diffusion coefficients, the multipulse waves upon collision can pass through one another with unchanged size and shape, exhibiting soliton behavior. Moreover, multipulse collisions with the system boundaries can generate a rich variety of wave transformations: the transition from the multipulse waves to pulse-front waves and further to simple fronts or to annihilation as well the transition to solitary pulses or to multipulse waves with lower numbers of pulses. Analytical and numerical results for the pulse pairs agree well with each other.