Convection Diffusion Reaction Research Articles

Invariant subspace method to the initial and boundary value problem of the higher dimensional nonlinear time-fractional PDEs

This paper systematically explains how to apply the invariant subspace method using variable transformation for finding the exact solutions of the (k+1)-dimensional nonlinear time-fractional PDEs in detail. More precisely, we have shown how to transform the given (k+1)-dimensional nonlinear time-fractional PDEs into (1+1)-dimensional nonlinear time-fractional PDEs using the variable transformation procedure. Also, we explain how to derive the exact solutions for the reduced equations using the invariant subspace method. Additionally, in this careful and systematic study, we will investigate how to find the various types of exact solutions of the (3+1)-dimensional nonlinear time-fractional convection–diffusion–reaction equation along with appropriate initial and boundary conditions for the first time. Moreover, the obtained exact solutions of the equation as mentioned above can be written in terms of polynomial, exponential, trigonometric, hyperbolic, and Mittag-Leffler functions. Finally, the discussed method is extended for the (k+1)-dimensional nonlinear time-fractional PDEs with several linear time delays, and the exact solution of the (3+1)-dimensional nonlinear time-fractional delay convection–diffusion–reaction equation is derived.

A Numerical Thermo-Chemo-Flow Analysis of Thermoset Resin Impregnation in LCM Processes

This paper presents a numerical framework for modelling and simulating convection-diffusion-reaction flows in liquid composite moulding (LCM). The model is developed in ANSYS Fluent with customised user-defined-functions (UDFs), user-defined-scalar (UDS), and user-defined memory (UDM) codes to incorporate the cure kinetics and rheological characteristics of thermoset resin impregnation. The simulations were performed adopting volume-of-fluid (VOF)-a multiphase flow solution-based on finite volume method (FVM). The developed numerical approach solves Darcy's law, heat transfer, and chemical reactions in LCM process simultaneously. Thereby, the solution scheme shows its ability to provide information on flow-front, viscosity development, degree of cure, and rate of reaction at once unlike existing literature that commonly focuses on impregnation stage and cure stage in isolation. Furthermore, it allows online monitoring, controlled boundary conditions, and injection techniques (for design of manufacturing) during the mould filling and curing stages. To examine the validity of the model, a comparative analysis was carried out for a simple geometry, in that the numerical results indicate good agreement-3.4% difference in the degree of cure compared with previous research findings.

Supersymmetry and convection–diffusion–reaction equations

In this work, we are concerned with generating solutions of a class of Convection–Diffusion–Reaction (CDR) equation from the solutions of another CDR equation through the Darboux transformations. The method is elucidated by cases with certain types of the reaction coefficients. We have also discussed briefly supersymmetric pairs of CDR equations connected through similarity solutions.

A local projection stabilization for convection-diffusion-reaction equations using biorthogonal systems

We consider a local projection stabilization based on biorthogonal systems for convection–diffusion–reaction differential equations with mixed boundary conditions. The approach based on biorthogonal systems is numerically more efficient than other existing approaches to obtain a uniform approximation for convection dominated problems. We prove optimal a priori error estimates for the proposed numerical technique. Numerical examples are presented to demonstrate the performance of the approach. doi: 10.1017/S1446181123000019

Evaluation of dual-weighted residual and machine learning error estimation for projection-based reduced-order models of steady partial differential equations

Projection-based reduced-order models (pROMs) show great promise as a means to accelerate many-query applications such as forward error propagation, solving inverse problems, and design optimization. In order to deploy pROMs in the context of high-consequence decision making, accurate error estimates are required to determine the region(s) of applicability in the parameter space. The following paper considers the dual-weighted residual (DWR) error estimate for pROMs and compares it to another promising pROM error estimate, machine learned error models (MLEM). In this paper, we show how DWR can be applied to ROMs and then evaluate DWR on two partial differential equations (PDEs): a two-dimensional linear convection–reaction–diffusion equation, and a three-dimensional static hyper-elastic beam. It is shown that DWR is able to estimate errors for pROMs extrapolating outside of their training set while MLEM is best suited for pROMs used to interpolate within the pROM training set.

Bifurcations and Exact Bounded Solutions of Some Traveling Wave Systems Determined by Integrable Nonlinear Oscillators withq-Degree Damping

For a class of nonlinear diffusion–convection–reaction equations, the corresponding traveling wave systems are well-known nonlinear oscillation type of systems. Under some parameter conditions, the first integrals of these nonlinear oscillators can be obtained. In this paper, the bifurcations, exact solutions and dynamical behavior of these nonlinear oscillators are studied by using methods of dynamical systems. Under some parametric conditions, exact explicit parametric representations of the monotonic and nonmonotonic kink and anti-kink wave solutions, as well as limit cycles, are obtained. Most important and interestingly, a new global bifurcation phenomenon of limit bifurcation is found: as a key parameter is varied, so that singular points (except the origin) disappear, a planar dynamical system can create a stable limit cycle.

On the Stability of Solutions to Control Problems for a Nonlinear Reaction–Diffusion–Convection Model

The problems of multiplicative control for the reaction–diffusion–convection model with coefficients that are nonlinearly dependent on the solution and also dependent on spatial variables are studied. In the case of a power-law dependence of the model coefficients on the solution, optimality systems are derived for extremum problems. With their help, estimates of the local stability of solutions of specific control problems with respect to small perturbations of both the cost functionals and one of the given functions of the boundary value problem are obtained.

Control of Chemoconvection in a Rectangular Slot by Changing Its Spatial Orientation

Recently, we found that a two-layer miscible system placed in a vertical slab reactor shows an occurrence of a density shock-wave-like pattern. This wave resembles a turbulent bore separating immobile fluid and an area of intense mixing. It travels away from the convective core of the system and is highly dependent on the intensity of a gravity-dependent chemoconvection in the cocurrent flow. The novelty of this work is that we demonstrate that the change in angle between gravity and wave direction allows controlling the chemoconvection intensity and, consequently, the rate of a spatially-extended reaction. We study both experimentally and numerically the effect of the spatial orientation of a slab reactor to a gravity field on a flow structure induced by a neutralization reaction. In experiments, we use aqueous mixtures of nitric acid and sodium hydroxide. We apply the Fizeau interferometry to visualize the flow and use the PIV method to measure the fluid velocity. The mathematical model includes reaction–diffusion–convection equations that describe 3D flows. We study the flow modifications with a change in the inclination angle from 0 to 90 degrees. At small angles (up to 30), the cocurrent flow becomes spatially heterogeneous, and the fields of salt and acid are separated. If the inclination exceeds 50 degrees, the wavefront is deformed, and the wave breaks up, resulting in a sharp decrease in the reaction rate.

On the Stability of Solutions to Control Problems for a Nonlinear Reaction–Diffusion–Convection Model

On the Stability of Solutions to Control Problems for a Nonlinear Reaction–Diffusion–Convection Model

Nonlinear development of convective patterns driven by a neutralization reaction in immiscible two-layer systems.

This article provides the results of a theoretical and experimental study of buoyancy-driven instabilities triggered by a neutralization reaction in an immiscible two-layer system placed in a vertical Hele-Shaw cell. Flow patterns are predicted by a reaction-induced buoyancy number [Formula: see text], which we define as the ratio of densities of the reaction zone and the lower layer. In experiments, we observed the development of cellular convection ([Formula: see text]), the fingering process with an aligned line of fingertips at a slightly denser reaction zone ([Formula: see text]) and the typical Rayleigh-Taylor convection for [Formula: see text]. A mathematical model includes a set of reaction-diffusion-convection equations written in the Hele-Shaw approximation. The model's novelty is that it accounts for the water produced during the reaction, a commonly neglected effect. The persisting regularity of the fingering during the collapse of the reaction zone is explained by the dynamic release of water, which compensates for the heavy fluid falling and stabilizes the pattern. Finally, we present a stability map on the plane of the initial concentrations of solutions. Good agreement between the experimental data and theoretical results is observed. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

Exponential Tracking and Disturbance Rejection for Nonlinear Reaction Convection Diffusion Equations via Boundary Control

This paper studies the problem of exponential tracking and disturbance rejection for nonlinear reaction convection diffusion equations via Dirichlet or Neumann boundary control. Disturbances are present both in the equation and on the boundary. The tracking problem is simply a nonlinear dynamic regulator problem. If an initial condition is not compatible with a reference, the nonlinear dynamic regulator problem does not have a solution. Thus, we split the tracking problem into two problems to relax the compatibility condition. We choose one problem to be exponentially stable and leave everything else in a dynamic regulator problem. Then a controller is just a solution component of the dynamic regulator problem, while the tracking error is given by the solution of the exponentially stable problem. The controller can be obtained by integrating the dynamic regulator equation and using the reference condition. The dynamic regulator problem with the obtained controller becomes an initial boundary value problem. We then transfer the initial boundary value problem into an equivalent integral equation and prove that the integral equation has a unique global solution if the nonlinearity is globally Lipschitz. In addition, a numerical example is given to illustrate the theoretical results.

Application of Aboodh Homotopy Perturbation Transform Method for Fractional-Order Convection–Reaction–Diffusion Equation within Caputo and Atangana–Baleanu Operators

This article is an analysis of fractional nonlinear convection–reaction–diffusion equations involving the fractional Atangana–Baleanu and Caputo derivatives. An efficient Aboodh homotopy perturbation transform method, which combines the homotopy perturbation method with the Aboodh transformation, is applied to investigate this fractional-order proposed model, analytically. A modified technique known as the Aboodh homotopy perturbation transform method is formulated to approximate these derivatives. The analytical simulation is investigated graphically as well as in tabular form.

Understanding the Parameter Influence on Lesion Growth for a Mechanobiology Model of Atherosclerosis

In this work, we analyse the influence of the parameters of a mathematical model, previously proposed by the authors, for reproducing atheroma plaque in arteries. The model uses Navier–Stokes equations to calculate the blood flow along the lumen in a transient mode. It also uses Darcy’s law, Kedem–Katchalsky equations, and the three-pore model to simulate plasma and substance flows across the endothelium. The behaviours of all substances in the arterial wall are modelled with convection–diffusion–reaction equations, and finally, plaque growth is calculated. We consider a 2D geometry of a carotid artery, but the model can be extrapolated to other geometries or arteries, such as the coronaries or the aorta. A mono-variant sensitivity analysis of the model parameters was performed, with values of ±25% and ±10%, with respect to the values of the previous model. The results were analysed with respect to the volume in the plaque of foam cells (FC), synthetic smooth muscle cells (SSMC), and collagen fibre. It was observed that the volume in the plaque of the different substances (FC, SSMC, and collagen) has a strong influence on the results, so it could be used to analyse the vulnerability of plaque. The stenosis ratio of the plaque was also analysed, showing a strong influence on the results as well. Parameters that influence all the results considered when ranged ±10% are the rate of LDL degradation and the diffusion coefficients of LDL and monocytes in the arterial wall. Furthermore, it was observed that the change in the volume of foam cells in the plaque has a greater influence on the stenosis ratio than the change of synthetic smooth muscle cells or collagen fibre.

Supercloseness analysis of a stabilizer free weak Galerkin finite element method for time dependent convection diffusion reaction equation

This paper considers a stabilizer-free weak Galerkin (SFWG) finite element method for the time-dependent convection diffusion reaction equation. We describe error estimate for both semidiscrete and fully discrete schemes and achieve the supercloseness convergence rate, which is two orders higher than the optimal order associated with SFWG finite element space (Pk(K),Pk+1(∂K),[Pk+1(K)]2). More precisely, we obtain O(hk+2+τ2) in L∞(H1) norm and O(hk+3+τ2) in L∞(L2) norm. Numerous numerical examples are provided to confirm the theoretical findings and efficiency of the proposed method.

A modified-upwind with block-centred finite difference scheme based on the two-grid algorithm for convection-diffusion-reaction equations

ABSTRACT A modified-upwind with block-centred finite difference scheme on the basis of the two-grid algorithm is presented for the convection-diffusion-reaction equations. This scheme can keep second-order accuracy in spatial mesh sizes for both state variables and fluxes in the convection–diffusion–reaction problem. Moreover, the two-grid algorithm is constructed in order to solve semilinear convection-dominated problems efficiently, in which the main idea is to settle an original semilinear equation on the coarse space, and next to settle a linearized equation on the fine space. The error estimate of the method proposed in this paper is given through theoretical analysis. It is indicated that the two-grid algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy . Thus, solving such a large-scale nonlinear problem is as easy as linearized problems. Besides, there are some numerical experiments to corroborate in practice that the algorithm is effective and robust to solve convection–diffusion–reaction problems.

Weak convergence of the finite element method for semilinear parabolic SPDEs driven by additive noise

This paper aims to investigate the finite element weak convergence rate for semilinear parabolic stochastic partial differential equations(SPDEs) driven by additive noise. In contrast to many results in the current scientific literature, we investigate the more general case where the nonlinearity is allowed to be of Nemytskii-type and the linear operator is not necessarily self-adjoint, which is more challenging and models more realistic phenomena such as convection–reaction–diffusion processes. Using Malliavin calculus, Kolmogorov equations and by splitting the linear operator into a self-adjoint and non self-adjoint parts, we prove the convergence of the finite element approximation and obtain a weak convergence rate that is twice the strong convergence rate.

Support vector regression for the temperature-stimulated drug release

This paper presents a new coupled nonlinear convection–diffusion–reaction model for the temperature-stimulated drug release and a numerical algorithm based on a machine learning technique. By incorporating the fractional derivatives into the model, we take advantage of anomalous diffusion in biological tissues. In the proposed technique a least-squares support vector regression is developed for the numerical simulation of the problem. The Bernstein polynomials are used as the kernel function and some training points are considered for the residual system of the coupled problem in an inverse process. Some numerical experiments are carried out to support the method and its spectral convergence.

Investigating and Modelling an Engineered Millifluidic In Vitro Oocyte Maturation System Reproducing the Physiological Ovary Environment in the Sheep Model.

In conventional assisted reproductive technologies (ARTs), oocytes are in vitro cultured in static conditions. Instead, dynamic systems could better mimic the physiological in vivo environment. In this study, a millifluidic in vitro oocyte maturation (mIVM) system, in a transparent bioreactor integrated with 3D printed supports, was investigated and modeled thanks to computational fluid dynamic (CFD) and oxygen convection-reaction-diffusion (CRD) models. Cumulus-oocyte complexes (COCs) from slaughtered lambs were cultured for 24 h under static (controls) or dynamic IVM in absence (native) or presence of 3D-printed devices with different shapes and assembly modes, with/without alginate filling. Nuclear chromatin configuration, mitochondria distribution patterns, and activity of in vitro matured oocytes were assessed. The native dynamic mIVM significantly reduced the maturation rate compared to the static group (p < 0.001) and metaphase II (MII) oocytes showed impaired mitochondria distribution (p < 0.05) and activity (p < 0.001). When COCs were included in a combination of concave+ring support, particularly with alginate filling, oocyte maturation and mitochondria pattern were preserved, and bioenergetic/oxidative status was improved (p < 0.05) compared to controls. Results were supported by computational models demonstrating that, in mIVM in biocompatible inserts, COCs were protected from shear stresses while ensuring physiological oxygen diffusion replicating the one occurring in vivo from capillaries.

A computational method for a two-parameter singularly perturbed elliptic problem with boundary and interior layers

In this article, we investigate a two-dimensional (2-D) singularly perturbed convection–reaction–diffusion elliptic type problem where two parameters ϵ and μ multiply the diffusion and convection terms, respectively. Furthermore, we assume that jump discontinuities exist in the source term along the x- and y-axis. Due to the presence of perturbation parameters, the solutions to such problems show boundary and corner layers. Moreover, the discontinuity in the source term adds the interior layers to the solution whose suitable numerical approach is the important goal of this article. A numerical approach is carried out using an upwind finite-difference technique that includes an appropriate layer-adapted piecewise uniform Shishkin mesh. Some examples are presented which show the good performance of the proposed method and the agreement with the theoretical analysis.

On algebraically stabilized schemes for convection–diffusion–reaction problems

An abstract framework is developed that enables the analysis of algebraically stabilized discretizations in a unified way. This framework is applied to a discretization of this kind for convection–diffusion–reaction equations. The definition of this scheme contains a new limiter that improves a standard one in such a way that local and global discrete maximum principles are satisfied on arbitrary simplicial meshes.