Cross-diffusion Terms Research Articles

The onset of double-diffusive convection in a nanofluid saturated porous layer: Cross-diffusion effects

The onset of double-diffusive convection in a horizontal porous layer saturated with a nanofluid with the Soret and Dufour effects is studied using both linear and nonlinear stability analyses in a three-dimensional way. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis, and the modified Darcy model is used for the porous medium that includes the time derivative term to describe the momentum transport. The thermal energy equation includes the diffusion and cross-diffusion terms. The linear theory depends on the normal mode technique, and the nonlinear analysis depends on the minimal representation of double Fourier series. The effects of the Soret and Dufour parameters, solutal Rayleigh number, viscosity ratio, and conductivity ratio on the stationary and oscillatory convections are presented graphically. It is found that for the stationary mode, the Soret and Dufour parameters, viscosity ratio, and thermal conductivity ratio have a stabilizing effect, while the solutal Rayleigh number destabilizes the system. For the oscillatory mode, the Soret and Dufour parameters, viscosity ratio, and Vadász number have a stabilizing effect, while the solutal Rayleigh number and the thermal conductivity ratio destabilize the system. We also study the effects of time on transient Nusselt number, which is found to be oscillatory when the time is small. However, when the time becomes very large, all three transient Nusselt number values approach the steady-state values.

Spatiotemporal Patterns Induced by Cross-Diffusion in a Three-Species Food Chain Model

This paper focuses on a three-species Lotka–Volterra food chain model with cross-diffusion under homogeneous Neumann boundary conditions. The known results indicate that no spatiotemporal patterns happen in the corresponding reaction–diffusion system. When some cross-diffusion terms are introduced in the system, the existence of nonconstant positive steady-states as well as the Hopf bifurcation is studied. Our result shows that cross-diffusion plays a crucial role in the formation of spatiotemporal patterns, that is, it can create not only stationary patterns but also spatially inhomogeneous periodic oscillatory patterns, which is a strong contrast to the case without cross-diffusion.

Solutal natural convection flows in ternary mixtures

It is known that the cross diffusion terms generate four different types of solutions to the one-dimensional unsteady diffusion equations for ternary mixtures. The stability of the fluid column corresponding to these solutions can be classified depending on the sign of the first derivative of density with respect to the direction of the gravity vector (i.e. ∂ρ/∂y) and the sign of (∂2ρ/∂y2)/y, where y=0 is located at the center of the diffusion layer. The type of solution depends on the initial conditions and on the set of diffusion coefficients considered. One type of solution corresponds to a stable fluid column with (∂ρ/∂y<0) and (∂2ρ/∂y2)/y>0. Two types of solutions generate fluid layers with unstable density stratification (∂ρ/∂y>0) and the fourth type shows a fluid layer with (∂2ρ/∂y2)/y<0. We analyzed the unsteady diffusion processes in a ternary mixture under the conditions of experiments carried out to determine the four diffusion coefficients of the ternary system 1,2,3,4-tetrahydronaphthaline-isobutylbenzene-dodecane (THN-IBB-nC12). These measurements; performed in diffusion cells with an initially stable stratification within the cell, with the denser mixture at the bottom and the lighter at the top; are usually based on the validity of the one-dimensional unsteady diffusion mass transfer equations. The linear stability analysis for the onset of convection in the unstable layers with unstable density stratification (∂ρ/∂y>0) indicates that the critical thickness of these layers depends on the Rayleigh numbers, on the diffusion coefficients and on the initial conditions. To illustrate the flow structures that can be generated in these unstable conditions, we performed numerical simulations for selected sets of diffusion coefficients at different Rayleigh numbers. The results of these simulations are in general agreement with the predictions of the linear stability analysis and indicate that, under specific conditions, the convective motions developed in the cell produce significant departures of the concentration distributions from the pure diffusion situation.

Double-diffusive natural convection in an enclosure filled with nanofluid using ISPH method

The double-diffusive natural convection in an enclosure filled with nanofluid is studied using ISPH method. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. In addition the thermal energy equations include regular diffusion and cross-diffusion terms. In ISPH algorithm, a semi implicit velocity correction procedure is utilized and the pressure is implicitly evaluated by solving pressure Poisson equation. The results are presented with flow configurations, isotherms, concentration and nanoparticle volume fraction contours and average Nusselt and Sherwood numbers for different cases. The results from this investigation are well validated and have favorable comparisons with previously published results. It is found that, among all cases, a good natural convection can be obtained by considering the double diffusive case. An increase in Soret number accompanied by a decrease in Dufour number results in an increase in average Nusselt number and a decrease in average Sherwood number.

Super-critical and sub-critical bifurcations in a reaction-diffusion Schnakenberg model with linear cross-diffusion

In this paper the Turing pattern formation mechanism of a two component reaction-diffusion system modeling the Schnakenberg chemical reaction coupled to linear cross-diffusion terms is studied. The linear cross-diffusion terms favors the destabilization of the constant steady state and the mechanism of pattern formation with respect to the standard linear diffusion case, as shown in Madzvamuse et al. (J. Math. Biol. 2014). Since the subcritical Turing bifurcations of reaction-diffusion systems lead to spontaneous onset of robust, finite-amplitude localized patterns, here a detailed investigation of the Turing pattern forming region is performed to show how the diffusion coefficients for both species (the activator and the inhibitor) influence the occurrence of supercritical or subcritical bifurcations. The weakly nonlinear (WNL) multiple scales analysis is employed to derive the equations for the amplitude of the Turing patterns and to distinguish the supercritical and the subcritical pattern region, both in 1D and 2D domains. Numerical simulations are employed to confirm the WNL theoretical predictions through which a classification of the patterns (squares, rhombi, rectangle and hexagons) is obtained. In particular, due to the hysteretic nature of the subcritical bifurcation, we observe the phenomenon of pattern transition from rolls to hexagons, in agreement with the bifurcation diagram.

Analysis of an Incompressible Navier–Stokes–Maxwell–Stefan System

The Maxwell–Stefan equations for the molar fluxes, supplemented by the incompressible Navier–Stokes equations governing the fluid velocity dynamics, are analyzed in bounded domains with no-flux boundary conditions. The system models the dynamics of a multicomponent gaseous mixture under isothermal conditions. The global-in-time existence of bounded weak solutions to the strongly coupled model and their exponential decay to the homogeneous steady state are proved. The mathematical difficulties are due to the singular Maxwell–Stefan diffusion matrix, the cross-diffusion terms, and the different molar masses of the fluid components. The key idea of the proof is the use of a new entropy functional and entropy variables, which allows for a proof of positive lower and upper bounds of the mass densities without the use of a maximum principle.

Buongiorno’s model for double-diffusive mixed convective stagnation-point flow of a nanofluid considering diffusiophoresis effect of binary base fluid

The development of double-diffusive mixed convective boundary layer flow of a nanofluid near stagnation point region over a vertical surface has been investigated. Buongiorno’s model is used to incorporate the effects of Brownian motion and thermophoresis for the nanofluid, when the base fluid of the nanofluid is itself a binary fluid such as salty water. In addition the thermal energy equations include regular diffusion and cross-diffusion terms. Using the local similarity method, it is shown that a set of suitable similarity transformations reduces the non-linear coupled PDEs governing on the problem into a set of non-linear coupled ODEs. The analysis shows that velocity, temperature, solutal concentration and nanoparticle concentration profiles in the respective boundary layers depend on ten dimensionless physical parameters, namely the mixed convection parameter λ, the regular double-diffusive buoyancy parameter Nc, the nanofluid buoyancy ratio Nr, the Brownian motion parameter Nb, the thermophoresis parameter Nt, the modified Dufour parameter Nd, the regular Schmidt number Sc, the nanofluid Schmidt number Scn, Prandtl number Pr, and the Dufour-solutal Lewis number Ld. The results are presented in graphical form illustrating the effects of these parameters on boundary layer behavior. A sensitivity analysis of model is tabulated where the difference in each graph can be quantified in terms of root mean square deviation (RMSD) with respect to that with the default parameters. These results are supplemented with the data for the reduced Nusselt number and the two reduced Sherwood numbers, one for the solute and the other for the nanoparticles. It is found that for assisting flow regime the reduced Nusselt number is a decreasing function of each of Nr,NbandNt, and an increasing function of each of λ,NcandNd. The results demonstrate that the highest heat transfer rate is obtained for the situation that the thermophoresis effects are negligible. Moreover, a reduction in the reduced Sherwood number of nanoparticle is observed with increase in each of NcandNt, while this quantity increases with Nb.

Analytical model for equiaxed globular solidification in multicomponent alloys

We present a solidification model for equiaxed globular microstructure based on an analytical solution of the diffusion fields in the liquid with cross-diffusion terms for multicomponent alloys. It is demonstrated that the solute profiles for any element can be defined after computation of the eigenvalues and the eigenvectors of the liquid diffusion matrix. The interface velocity and the solute profiles are compared and validated with a numerical solution developed with a front tracking approach. A comprehensive application to the Al–7wt.%Si–1wt.%Mg alloy shows the effect of interdiffusion phenomena on the solidification path. This application is developed with an integral approach for the computation of the liquid composition far from the interface. It appears to be the only relevant choice in order to compute the expected driving force. Analytical solutions of the diffusion profile can also be computed for planar and cylindrical geometries. Coupling of this microsegregation model with thermodynamic equilibrium calculations is finally proposed and discussed.

New results for triangular reaction cross diffusion system

We present an approach based on entropy and duality methods for “triangular” reaction cross diffusion systems of two equations, in which cross diffusion terms appear only in one of the equations. Thanks to this approach, we recover and extend many existing results on the classical “triangular” Shigesada–Kawasaki–Teramoto model.

A hyperbolic-equation system approach for magnetized electron fluids in quasi-neutral plasmas

A new approach using a hyperbolic-equation system (HES) is proposed to solve for the electron fluids in quasi-neutral plasmas. The HES approach avoids treatments of cross-diffusion terms which cause numerical instabilities in conventional approaches using an elliptic equation (EE). A test calculation reveals that the HES approach can robustly solve problems of strong magnetic confinement by using an upwind method. The computation time of the HES approach is compared with that of the EE approach in terms of the size of the problem and the strength of magnetic confinement. The results indicate that the HES approach can be used to solve problems in a simple structured mesh without increasing computational time compared to the EE approach and that it features fast convergence in conditions of strong magnetic confinement.

Analysis of a drift–diffusion model with velocity saturation for spin-polarized transport in semiconductors

A system of drift–diffusion equations with electric field under Dirichlet boundary conditions is analyzed. The system of strongly coupled parabolic equations for particle density and spin density vector describes the spin-polarized semi-classical electron transport in ferromagnetic semiconductors. The presence of a nonconstant and nonsmooth magnetization vector, solution of the Landau–Lifshitz equation, causes the diffusion matrix to be dependent on space and time and to have in general poor regularity properties, thus making the analysis challenging. To partially overcome the analytical difficulties the velocity saturation hypothesis is made, which results in a bounded drift velocity. The global-in-time existence and uniqueness of weak solutions is shown by means of a semi-discretization in time, which yields an elliptic semilinear problem, and a quadratic entropy inequality, which allow for the limit of vanishing time step size. The convergence of the weak solutions to the steady state, under some restrictions on the parameters and data, is shown. Finally, the higher regularity of solutions for a smooth magnetization in two space dimensions is shown through a diagonalization argument, which allows to get rid of the cross diffusion terms in the fluid equations, and the iterative application of Gagliardo–Nirenberg inequalities and a generalized version of Aubin lemma.

Stability of a Continuous Reaction-Diffusion Cournot-Kopel Duopoly Game Model

In order to take into account the territory in which the outputs are in the market and the time-depending firms' strategies, the discrete Cournot duopoly game (with adaptive expectations, modeled by Kopel) is generalized through a non autonomous reaction-diffusion binary system of PDEs, with self and cross diffusion terms. Linear and nonlinear asymptotic L 2-stability, via the Liapunov Direct Methot and a nonautonomous energy functional, are investigated.

Effect of double stratification on mixed convection boundary layer flow of a nanofluid past a vertical plate in a porous medium

The effect of thermal and mass stratification on mixed convection boundary layer flow over a vertical flat plate embedded in a porous medium saturated by a nanofluid has been investigated. The vertical plate is maintained at uniform and constant heat, mass and nanoparticle fluxes, and the behavior of the porous medium is described by the Darcy model. The model considered for nanofluids incorporates the effects of Brownian motion and thermophoresis. In addition, the thermal energy equations include regular diffusion and cross-diffusion terms. A suitable coordinate transformation is introduced, and the obtained system of non-similar, coupled and non-linear partial differential equations is solved numerically. The influence of pertinent parameters on the non-dimensional velocity, temperature, concentration and nanoparticle volume fraction are discussed. In addition, the variation of heat, mass and nanoparticle transfer rates at the plate are exhibited graphically for different values of physical parameters.

Non-similar solution for natural convective boundary layer flow of a nanofluid past a vertical plate embedded in a doubly stratified porous medium

This article studies the effects of thermal and mass stratification on natural convection boundary layer flow over a vertical plate embedded in a porous medium saturated by a nanofluid. The plate is maintained at a uniform and constant wall temperature, concentration and nanoparticle volume fraction. The effects of Brownian motion and thermophoresis are incorporated into the model for nanofluids. In addition, the thermal energy equations include regular diffusion and cross-diffusion terms. A suitable coordinate transformation is introduced, and the obtained system of non-similar, coupled and nonlinear partial differential equations are solved by the Keller-box method. The effect of the Brownian motion parameter, thermophoresis parameter, modified Dufour number, Dufour-solutal Lewis number, thermal and mass stratification parameters on the non-dimensional velocity, temperature, concentration, nanoparticle volume fraction, heat and mass transfer rates at the plate are discussed and displayed graphically.

Double‐Diffusive Convective Transport in a Nanofluid‐Saturated Porous Layer with Cross Diffusion and Variation of Viscosity and Conductivity

AbstractThe onset of double‐diffusive nanofluid convection in a fluid‐saturated horizontal porous layer is studied with thermal conductivity and viscosity dependent on the nanoparticle volume fraction. The Darcy model has been used for the porous medium, while the nanofluid incorporates the effects of Brownian motion along with thermophoresis. The nanofluid is assumed to be diluted and this enables the porous medium to be treated as a weakly heterogeneous medium with variation in the vertical direction of conductivity and viscosity. In addition, the thermal energy equation includes regular diffusion and cross diffusion terms. The linear stability analysis is based on the normal mode technique, while for nonlinear analysis, minimal representation of the truncated Fourier series representation involving only two terms has been used. It is found that for the stationary mode the Soret parameter, Dufour parameter, viscosity ratio, and conductivity ratio have a stabilizing effect, while the solutal Rayleigh number destabilizes the system. For the oscillatory mode, the Soret parameter, Dufour parameter, and viscosity ratio have a stabilizing effect while the solutal Rayleigh number and conductivity ratio destabilize the system. For steady finite amplitude motions, the heat and mass transport decreases with an increase in the values of the Dufour parameter and solutal Rayleigh number. The Soret parameter enhances the solute concentration Nusselt number while it retards the thermal Nusselt number and concentration Nusselt number. The viscosity ratio and conductivity ratio enhances the heat and mass transports. We also study the effect of time on transient Nusselt numbers which is found to be oscillatory when time is small. However, when time becomes very large, all three transient Nusselt values approach a steady value. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 43(7): 628–652, 2014; Published online 11 November 2013 in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21102

Mixed convective boundary layer flow over a vertical cylinder embedded in a porous medium saturated with a nanofluid

Purpose – The purpose of this work is to study the mixed convection boundary layer flow past a vertical cylinder in a porous medium saturated with a nanofluid. Numerical results for friction factor, surface heat transfer rate and mass transfer rate have been presented for parametric variations of the buoyancy ratio parameter Nr, Brownian motion parameter Nb, thermophoresis parameter Nt and Lewis number Le. The dependency of the surface heat transfer rate (Nusselt number) and mass transfer rate on these parameters has been discussed. Design/methodology/approach – Solutions of the set of non-similarity equations are obtained by employing the implicit finite difference method together with Keller box elimination method. Findings – It was found that the heat transfer rate decreases and mass transfer rates increase as Lewis number increases. The heat and mass transfer rates increase as the buoyancy ration parameter increases. As the thermophoresis parameter Nt increases, the heat transfer rate decreases where as the mass transfer rate increases. As the Brownian parameter Nb increases, the heat transfer rate decreases. Brownian motion decelerates the flow in the nanofluid boundary layer. Brownian diffusion promotes heat conduction. The heat and mass transfer rates increase as the buoyancy ratio number Nr increases. The Brownian motion and thermophoresis of nanoparticles increases the effective thermal conductivity of the nanofluid. Both Brownian diffusion and thermophoresis give rise to cross diffusion terms that are similar to the familiar Soret and Dufour cross-diffusion terms that arise with a binary fluid. Research limitations/implications – The analysis is valid for steady, mixed convection two-dimensional boundary layer flow in a nanofluid-saturated Darcy porous medium. An extension to non-Darcy porous medium is left as a part of future study. Practical implications – The research is applicable for enhancing heat exchanger effectiveness by employing nanofluids. Originality/value – The study is useful to engineers interested in designing heat exchangers, water and atmospheric pollution.

Natural convection heat and mass transfer from a sphere in non-Newtonian nanofluids

A boundary layer analysis has been presented for the natural convection flow of a non-Newtonian nanofluid past a sphere. Solutions of the set of nonsimilarity equations are obtained by employing the implicit finite difference method together with Keller box elimination method. Numerical results for friction factor, surface heat transfer rate and mass transfer rate have been presented for parametric variations of the material parameters, buoyancy ratio parameter [Formula: see text], Brownian motion parameter NB, thermophoresis parameter NT and Schmidt number Sc. The dependency of the surface heat transfer rate (Nusselt number) and mass transfer rate on these parameters has been discussed. It was found that the heat transfer rate decreases and mass transfer rates increase as Schmidt number increases. The friction factor and heat transfer rates decrease as the cross viscosity parameter [Formula: see text] increases. The heat transfer rates increase and mass transfer rates decrease as the buoyancy ratio parameter N increases. As the thermophoresis parameter NT increases, the heat and mass transfer rates increase. As the Brownian parameter NB increases, the heat and mass transfer rates decrease. Brownian motion decelerates the flow in the nanofluid boundary layer. Brownian diffusion promotes heat conduction. The Brownian motion and thermophoresis of nanoparticles increase the effective thermal conductivity of the nanofluid. Both Brownian diffusion and thermophoresis give rise to cross diffusion terms that are similar to the familiar Soret and Dufour cross diffusion terms that arise with a binary fluid.

Diffusion of two molecular species in a crowded environment: theory and experiments

Diffusion of a two component fluid is studied in the framework of differential equations, but where these equations are systematically derived from a well-defined microscopic model. The model has a finite carrying capacity imposed upon it at the mesoscopic level and this is shown to lead to nonlinear cross diffusion terms that modify the conventional Fickean picture. After reviewing the derivation of the model, the experiments carried out to test the model are described. It is found that it can adequately explain the dynamics of two dense ink drops simultaneously evolving in a container filled with water. The experiment shows that molecular crowding results in the formation of a dynamical barrier that prevents the mixing of the drops. This phenomenon is successfully captured by the model. This suggests that the proposed model can be justifiably viewed as a generalization of standard diffusion to a multispecies setting, where crowding and steric interferences are taken into account.

On the dynamics of a non-linear Duopoly game model

The Cournot duopoly game modeled by Kopel, with adaptive expectations, is generalized by introducing the self-diffusion and cross-diffusion terms. General properties, such as boundedness and uniqueness, are obtained. Non-linear stability results are reached by the analysis of the stability of a ODE system.

A Study of Physical and Numerical Effects of Dissipation on Turbulent Flow Simulations

ABSTRACT: The present work is primarily concerned with studying the effects of artificial dissipation and of certain diffusive terms in the turbulence model formulation on the capability of representing turbulent boundary layer flows. The flows of interest in the present case are assumed to be adequately represented by the compressible Reynolds-averaged Navier-Stokes equations, and the Spalart-Allmaras eddy viscosity model is used for turbulence closure. The equations are discretized in the context of a general purpose, density-based, unstructured grid finite volume method. Spatial discretization is based on the Steger-Warming flux vector splitting scheme and temporal discretization uses a backward Euler point-implicit integration. The work discusses in detail the theoretical and numerical formulations of the selected model. The computational studies consider the turbulent flow over a flat plate at 0.3 freestream Mach number. The paper demonstrates that the excessive artificial dissipation automatically generated by the original spatial discretization scheme can deteriorate boundary layer predictions. Moreover, the results also show that the inclusion of Spalart-Allmaras model cross-diffusion terms is primarily important in the viscous sublayer region of the boundary layer. Finally, the paper also demonstrates how the spatial discretization scheme can be selectively modified to correctly control the artificial dissipation such that the flow simulation tool remains robust for high-speed applications at the same time that it can accurately compute turbulent boundary layers.