Convection Diffusion Reaction Equations Research Articles

Effect of vascular compression and drug injection time on thrombolytic therapy: Numerical simulation and validation

Venous thromboembolism (VTE) has been occurring frequently in human society. There is an urgent need to study the influence of several factors on thrombolytic therapy, such as the effects of vascular pressure levels (VPL) and the drug injection time (DIT). Considering blood as a non-Newtonian fluid, valve as a hyperelastic material, and thrombus as a porous medium, a new numerical simulation model of biofluid mechanics incorporating fluid–solid coupling phenomena and biochemical substance reactions is established based on the N-S equations and the convection–diffusion reaction equations. Then, a unique in vitro experimental platform is established to verify the correctness of the constructed mathematical model. The results showed that vascular compression resulted in significant differences in blood flow status localized within the vessel. Vascular compression causes the blood boosting index to fluctuate and the valve displacement values are 135% and 158% greater than the lower VPL, respectively. At the same time, vascular compression weakened vortex intensity, accelerated material transport and response, and improved the treatment. Compared with low VPL, the therapeutic efficacy increased by 7% and 15%, respectively. In addition, when the dose of the drug is high, different injection times can increase the therapeutic effect to different degrees, with a maximum difference of 12%. Our in vitro experiments are similar to the results obtained by numerical simulation, which can verify the reliability of numerical simulation. The computational model proposed and the experimental platform designed in this study have the potential to assist in clinical medication prediction in different venous thromboembolism patients.

Periodic wave solutions for a generalized reaction–convection–diffusion equation of high-order

In this paper, we investigate the uniqueness of periodic wave solutions for a generalized reaction–convection–diffusion equation with arbitrarily high-order reaction term or convection term. The main technique is to prove the monotonicity of the ratio of two Abelian integrals by a new criterion and Descartes’ rule of signs. A positive answer to a conjecture stated in Wei and Chen (2023) is given and some numeric simulations are carried out to illustrate the obtained theoretical results.

Impact of hypertension and arterial wall expansion on transport properties and atherosclerosis progression

This study explored the impact of hypertension on atheroma plaque formation through a mechanobiological model. The model incorporates blood flow via the Navier–Stokes equation. Plasma flow through the endothelium is determined by Darcy’s law and the Kedem–Katchalsky equations, which consider the three-pore model utilized for substance flow across the endothelium. The behaviour of these substances within the arterial wall is described by convection–diffusion–reaction equations, while the arterial wall itself is modelled as a hyperelastic material using Yeoh’s model. To accurately evaluate hypertension’s influence, adjustments were made to incorporate wall compression-induced wall compaction by radial compression. This compaction impacts three key variables of the transport phenomena: diffusion, porosity, and permeability. Based on the obtained findings, we can conclude that hypertension significantly augments plaque growth, leading to an over 400% increase in plaque thickness. This effect persists regardless of whether wall mechanics are considered. Tortuosity, arterial wall permeability, and porosity have minimal impact on atheroma plaque growth under normal arterial pressure. However, the atheroma plaque growth changes dramatically in hypertensive cases. In such scenarios, the collective influence of all factors—tortuosity, permeability, and porosity—results in nearly a 20% increase in plaque growth. This emphasizes the importance of considering wall compression due to hypertension in patient studies, where elevated blood pressure and high cholesterol levels commonly coexist.

Error Analysis of the Classical Artificial Diffusion Weak Galerkin Finite Element Method for the steady state- convection diffusion-reaction Equation in 2-D

In this study, we modified the error in the weak Galerkin method when solving problems in which diffusion is the dominant convection( h) in two dimensions. This is done by adding the artificial diffusion term (-δΔw, where δ=h-ϵ). The finite element method for discrete functions using a weakly defined gradient operator is presented in this study. The concept of the weak discrete gradient is introduced, which plays an important role when using numerical methods to solve partial differential equations. The goal of this study is to enhance the accuracy and stability of the solutions by studying the ellipticity and stability properties of the method, which works to ensure that the numerical method retains the properties of the original equation while reducing the fluctuations occurring with the weak galerkin finite element method. Specific theories have been used to estimate the error in parameters -norm, and . Practical examples demonstrate how this method improves the handling of partial equations characterized by convection-dominated diffusion, enhancing its potential for advancing numerical simulations in engineering and physics.

Isolated periodic traveling waves of some reaction convection diffusion equations

AbstractIn this paper, we discuss the existence and uniqueness of isolated periodic traveling wave solutions of a family of nonlinear reaction–convection–diffusion equations using the monotonicity of the ratio of Abelian integrals. A numerical study is also presented.

Model-predicted effect of radial flux distribution on oxygen and glucose pericellular concentration in constructs cultured in axisymmetric radial-flow packed-bed bioreactors

Radial flow packed-bed bioreactors (rPBBs) overcome the transport limitations of static and axial-flow perfusion bioreactors and enable development of clinical-scale bioengineered tissues. We developed criteria to design rPBBs with uniform medium radial flux distribution along bioreactor length ensuring uniform construct perfusion. We report a model-based analysis of the effect of non-uniform axial distribution of medium radial flux on pericellular concentration of oxygen and glucose. Albeit pseudo-homogeneous, the model predicts how medium flux, solutes transport and cellular consumption interact and determine the pericellular oxygen and glucose concentrations in the presence of pore transport resistance to design optimal axisymmetric rPBBs and enable control of pericellular environment. Thus, oxygen and glucose supply may match cell requirements as tissue matures. Flow and solute transport in bioreactor empty spaces and construct was described with Navier-Stokes and Darcy-Brinkman equations, and with convection–diffusion and convection–diffusion-reaction equations, respectively. Solute transport in construct accounted for Michaelian cellular consumption and bulk medium-to-cell surface oxygen transport resistance in terms of a transport-equivalent bed of Raschig rings. The effect of relevant dimensionless groups on pericellular and bulk solute concentrations was predicted under typical tissue engineering operation and evaluated against literature data for bone tissue engineering. Axial distribution of medium radial flux influenced the distribution of pericellular solutes concentration, more so at high cell metabolic activity. Increasing medium feed flow rates relieved non-uniform solute concentration distribution and decayed at cell surface for metabolic consumption, also starting from axially non-uniform radial flux distribution. Model predictions were obtained in runtimes compatible with on-line control strategies.

Impact of geometric and hemodynamic changes on a mechanobiological model of atherosclerosis

Background and Objective:In this work, the analysis of the importance of hemodynamic updates on a mechanobiological model of atheroma plaque formation is proposed. Methods:For that, we use an idealized and axisymmetric model of carotid artery. In addition, the behavior of endothelial cells depending on hemodynamical changes is analyzed too. A total of three computational simulations are carried out and their results are compared: an uncoupled model and two models that consider the opposite behavior of endothelial cells caused by hemodynamic changes. The model considers transient blood flow using the Navier–Stokes equation. Plasma flow across the endothelium is determined with Darcy’s law and the Kedem–Katchalsky equations, considering the three-pore model, which is also employed for the flow of substances across the endothelium. The behavior of the considered substances in the arterial wall is modeled with convection–diffusion–reaction equations, and the arterial wall is modeled as a hyperelastic Yeoh’s material. Results:Significant variations are noted in both the morphology and stenosis ratio of the plaques when comparing the uncoupled model to the two models incorporating updates for geometry and hemodynamic stimuli. Besides, the phenomenon of double-stenosis is naturally reproduced in the models that consider both geometric and hemodynamical changes due to plaque growth, whereas it cannot be predicted in the uncoupled model. Conclusions:The findings indicate that integrating the plaque growth model with geometric and hemodynamic settings is essential in determining the ultimate shape and dimensions of the carotid plaque.

A nodally bound-preserving finite element method for reaction–convection–diffusion equations

This paper introduces a novel approach to approximate a broad range of reaction–convection–diffusion equations using conforming finite element methods while providing a discrete solution respecting the physical bounds given by the underlying differential equation. The main result of this work demonstrates that the numerical solution achieves an accuracy of [Formula: see text] in the energy norm, where [Formula: see text] represents the underlying polynomial degree. To validate the approach, a series of numerical experiments had been conducted for various problem instances. Comparisons with the linear continuous interior penalty stabilised method, and the algebraic flux-correction scheme (for the piecewise linear finite element case) have been carried out, where we can observe the favorable performance of the current approach.

Full discretization of the time dependent Navier–Stokes equations with anisotropic slip boundary condition coupled with the convection–diffusion–reaction equation

Full discretization of the time dependent Navier–Stokes equations with anisotropic slip boundary condition coupled with the convection–diffusion–reaction equation

Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid

Abstract This work is devoted to study the convection–reaction–diffusion behavior of contaminant in the recovered fracturing fluid which flows in the wellbore from shale gas reservoir. First, we apply various constitutive laws for generalized non-Newtonian fluids, diffusion principles, and friction relations to formulate the recovered fracturing fluid model. The latter is a partial differential system composed of a nonlinear and nonsmooth stationary incompressible Navier-Stokes equation with a multivalued friction boundary condition, and a nonlinear convection–reaction–diffusion equation with mixed Neumann boundary conditions. Then, we provide the weak formulation of the fluid model which is a hemivariational inequality driven by a nonlinear variational equation. We establish existence of solutions to the recovered fracturing fluid model via a surjectivity theorem for multivalued operators combined with an alternative iterative method and elements of nonsmooth analysis.

Analysis of a two phase flow model of biofilm spread

We consider a quasi-stationary problem describing the status of velocities, pressures and chemicals affecting cell behavior within a biofilm. The model couples stationary transport equations and compressible Stokes systems with convection–reaction–diffusion equations. We establish existence, uniqueness and stability of solutions of the different submodels involved and then obtain well posedness results for the quasi-stationary system. Our analysis relies on the construction of weak solutions for the steady transport equations under sign assumptions and the reformulation of the compressible Stokes problem as an elliptic system with enhanced regularity properties on the pressure. We need to consider velocity fields whose divergence and normal boundary components satisfy sign conditions. Applications include the study of cells, biofilms and tissues, where one phase is a liquid solution, whereas the other one is assorted biomass.

FDM/FEM for nonlinear convection–diffusion–reaction equations with Neumann boundary conditions—Convergence analysis for smooth and nonsmooth solutions

This paper aims to present in a systematic form the stability and convergence analysis of a numerical method defined in nonuniform grids for nonlinear elliptic and parabolic convection–diffusion–reaction equations with Neumann boundary conditions. The method proposed can be seen simultaneously as a finite difference scheme and as a fully discrete piecewise linear finite element method. We establish second convergence order with respect to a discrete H1-norm which shows that the method is simultaneously supraconvergent and superconvergent. Numerical results to illustrate the theoretical results are included.

A numerical method for singularly perturbed convection–diffusion–reaction equations on polygonal meshes

A numerical method for singularly perturbed convection–diffusion–reaction equations on polygonal meshes

Invariant subspace method and exact solutions of the coupled system of time-fractional convection–reaction–diffusion equations

Invariant subspace method and exact solutions of the coupled system of time-fractional convection–reaction–diffusion equations

Modified Characteristic Finite Element Method with Second-Order Spatial Accuracy for Solving Convection-Dominated Problem on Surfaces.

We present a modified characteristic finite element method that exhibits second-order spatial accuracy for solving convection-reaction-diffusion equations on surfaces. The temporal direction adopted the backward-Euler method, while the spatial direction employed the surface finite element method. In contrast to regular domains, it is observed that the point in the characteristic direction traverses the surface only once within a brief time. Thus, good approximation of the solution in the characteristic direction holds significant importance for the numerical scheme. In this regard, Taylor expansion is employed to reconstruct the solution beyond the surface in the characteristic direction. The stability of our scheme is then proved. A comparison is carried out with an existing characteristic finite element method based on face mesh. Numerical examples are provided to validate the effectiveness of our proposed method.

Very high-order finite difference method on arbitrary geometries with Cartesian grids for non-linear convection diffusion reaction equations

An arbitrary order finite difference method for solving non-linear convection Diffusion Reaction equations in curved boundary domains with Cartesian grid is proposed. Ghost points' values are determined with the Reconstruction Off-Site Data based on a polynomial interpolation using the least square method with constraints to enforce the boundary conditions. We propose a second-, fourth-, and sixth-order schemes for linear non-constant coefficients problem in both the conservative and non-conservative scalar equations. Extensions to non-linear scalar problems and systems are then implemented while preserving the optimal orders. Numerical simulations are carried out to provide evidence about the convergence order and the stability of the method.

Identification of discontinuous parameters in contaminant convection–reaction–diffusion model of recovered fracturing fluid

The primary goal of this paper is to identify the discontinuous parameters in a complicated contaminant convection–reaction–diffusion model of recovered fracturing fluid (RFF model, for short), which is modeled by a nonlinear stationary incompressible Navier–Stokes equation with multivalued and nonsmooth friction boundary condition coupled with a nonlinear convection–reaction–diffusion equation involving mixed Neumann boundary conditions. First, we impose the parameter identification problem which is, exactly, a nonlinear and nonsmooth inverse problem. Then, we show the local boundedness and weakly relative compactness of solution map to RFF model with respect to the discontinuous parameters. Moreover, the generalized continuity of solution map of RFF model is proved. Finally, via employing the theory of nonsmooth analysis and optimization, a sufficiency theorem of existence of solutions to the inverse problem under consideration is established.

Stable numerical algorithm with localized radial basis function for solution of fractional convection–diffusion–reaction equation

Designing an efficient high-order numerical discretization is one of the challenges in numerical solution of fractional differential equations. Therefore in this paper to overcome the ill-conditioning that often destroys the convergence rate of global RBF methods, we apply a local meshfree method known as radial basis function-generated finite difference (RBF-FD) method equipped with a greedy algorithm to design stable stencil weights and approximate spatial derivatives for parabolic fractional partial differential equations (FPDEs) of convection–diffusion–reaction type. In RBF-FD method to select stable stencils that make weights with desirable properties for differentiation matrix, we apply a greedy algorithm, then convergence and stability of proposed scheme are investigated theoretically and numerically to show efficiency of developed technique. Finally, numerical results are provided by some FPDEs in regular and irregular shaped domains to illustrate the approximation quality and convergence of our approach in comparison with the results presented in literatures.

An H1-Galerkin Space-Time Mixed Finite Element Method for Semilinear Convection–Diffusion–Reaction Equations

In this paper, the semilinear convection–diffusion–reaction equation is split into a lower-order system by introducing the auxiliary variable q=a(x)ux. An H1-Galerkin space-time mixed finite element method for the lower-order system is then constructed. The proposed method applies the finite element method to discretize the time and space directions simultaneously and does not require checking the Ladyzhenskaya–Babusˇka–Brezzi (LBB) compatibility constraints, which differs from the traditional mixed finite element method. The uniqueness of the approximate solutions u and q are proven. The L2(L2) norm optimal order error estimates of the approximate solution u and q are derived by introducing the space-time projection operator. The numerical experiment is presented to verify the theoretical results. Furthermore, by comparing with the classical H1-Galerkin mixed finite element scheme, the proposed scheme can easily improve computational accuracy and time convergence order by changing the basis function.

Convergence analysis of a class of iterative methods for propagation of reaction fronts in porous media

We present an iterative scheme for the numerical analysis of propagating reaction front problems in porous media satisfying an Arrhenius-type law. The governing equations consist of the Darcy equations for the pressure and flow field coupled to two convection–diffusion–reaction equations for the temperature and depth of conversion. Well-posedness, existence and uniqueness of the weak solution are first studied using a fixed-point approach and then, analysis of the proposed iterative scheme is investigated. Numerical results are also presented in order to validate the theoretical estimates and to illustrate the performance of the proposed scheme. The obtained results are in line with our expectations for a good numerical resolution with high accuracy and stability behaviors.