Unsteady Convection-diffusion Equation Research Articles

Transient dispersion of reactive solute transport in electrokinetic microchannel flow

Motivated by emerging applications in bio-microfluidic devices, the present study rigorously examines the generalized Taylor–Gill hydrodynamic dispersion of a point source solute injected into a microchannel, influenced by a constant axial static electric field along the channel and charged surface with different wall potentials. The solute engages in a first-order irreversible chemical reaction at both the microchannel walls. By incorporating different wall potentials and absorptive coefficients at the lower and upper walls, the current transport model for electro-osmotic flows is extended to encompass a wider range of applications. The solute transport phenomenon is intricately modeled using the unsteady convective diffusion equation. Employing Gill's generalized dispersion model, a concentration decomposition technique, up to the third-order accuracy, we meticulously analyze the transport process. Furthermore, a comprehensive comparison between analytical outcomes and numerical simulations using the Brownian Dynamics method is undertaken, enhancing the robustness of the analytical approach. The scattering process is mainly analyzed with the help of exchange, convection, dispersion, and asymmetry coefficients, along with the mean concentration profile. The effect of initial solute release at various vertical locations in the microchannel is shown to exert a considerable impact on all the transport coefficients at initial times.

A Crank-Nicolson WG-FEM for unsteady 2D convection-diffusion equation with nonlinear reaction term on layer adapted mesh

This paper is contributed to explore how a Crank-Nicolson weak Galerkin finite element method (WG-FEM) addresses the singularly perturbed unsteady convection-diffusion equation with a nonlinear reaction term in 2D. The problem and some asymptotic behavior results are given for the exact solution and its derivatives with the parameter ε. These results are essential for proving the uniform convergence of the proposed WG-FEM. A tensor product Shishkin mesh featuring piece-wise discontinuous bilinear polynomials is employed to handle the layers for uniform convergence at different parameter values of ε. An error estimate ‖uN−INu‖WG is presented, where INu denotes the vertex-edge-cell interpolation of the solution u, and ‖⋅‖WG denotes the Weak Galerkin norm. The optimal uniform convergence order is demonstrated for semi-discrete and fully discrete schemes. Various numerical experiments are conducted to validate the optimal order of convergence demonstrated by the proposed method.

Analyzing the effects of suction and injection Reynolds number on the transport process in a hydromagnetic flow through a channel of reactive porous walls

The present paper analyzes the effects of suction and injection Reynolds number on the dispersion process of solute in an incompressible magneto-hydrodynamics flow through a channel. The walls of the channel are permeable and a first order heterogeneous reaction is applied on it. Also, an inclined magnetic field is employed at both walls of the channel. The unsteady convection-diffusion equation is solved semi-analytically to find out the coefficient of dispersion, skewness and kurtosis of the solute distribution for all time period. Using Hermite polynomial representation, the distribution of mean concentration is obtained. The effects of various parameters such as suction Reynolds number (R), injection Reynolds number (R′), boundary absorption parameter (β), inclination angle of the magnetic field (α), Hartmann number (M) and dispersion time (t) on dispersion process of the solute are examined. It is shown that with the increment of suction Reynolds number, the peak of the mean concentration of solute enhances prominently. But, when injection Reynolds number increases, the amplitude of the mean concentration decreases. The current research gives an insight to understand the complex behavior of transport process of the solute in a hydro-magnetic channel flow with reactive porous walls.

Radial Basis Function Based Partition of Unity Method for Two-Dimensional Unsteady Convection Diffusion Equations

Radial Basis Function Based Partition of Unity Method for Two-Dimensional Unsteady Convection Diffusion Equations

Fourth order compact scheme for the Navier–Stokes equations on time deformable domains

In this work, we report the development of a spatially fourth order temporally second order compact scheme for incompressible Navier–Stokes (N–S) equations in time-varying domain. Sen (2013) put forward an implicit compact finite difference scheme for the unsteady convection–diffusion equation. It is now further extended to simulate fluid flow problems on deformable surfaces using curvilinear moving grids. The formulation is conceptualized in conjunction with recent advances in numerical grid deformations techniques such as inverse distance weighting (IDW) interpolation and its hybrid implementation. Adequate emphasis is provided to approximate grid metrics up to the desired level of accuracy and freestream preserving property has been numerically examined. As we discretize the non-conservative form of the N–S equation, the importance of accurate satisfaction of geometric conservation law (GCL) is investigated. To the best of our knowledge, this is the first higher order compact method that can directly tackle the non-conservative form of the N–S equation in single and multi-block time dependent complex regions. Several numerical verification and validation studies are carried out to illustrate the flexibility of the approach to handle high-order approximations on evolving geometries. The method presented here is found to be effective in modeling incompressible flow on deformable domains, including situations of fluid–structure interaction (FSI). Additionally, the necessity of GCL preserving metric computation is eased as a result of the scheme’s adaptation for non-conservative formulation.

The laminar seabed thermal boundary layer forced by propagating and standing free-surface waves

A mathematical model is developed to investigate seabed heat transfer processes under long-crested ocean waves. The unsteady convection–diffusion equation for water temperature includes terms depending on the velocity field in the laminar boundary layer, analogous to mass transfer near the seabed. Here we consider regular progressive waves and standing waves reflected from a vertical structure, which complicate the convective term in the governing equation. Rectangular and Gaussian distributions of seabed temperature and heat flux are considered. Approximate analytical solutions are derived for uniform and trapezoidal currents, and compared against predictions from a numerical solver of the full equations. The effects of heat source profile, location and strength on heat transfer dynamics in the thermal boundary layer are explained, providing insights into seabed temperature forced convection mechanisms enhanced by free-surface waves.

Effect of Catheter and Stenosis on Solute Diffusion in Non-Newtonian Blood Flow through a Catheterized Stenosed Artery

The presence of stenosis at the wall of the artery lead to further cardiovascular diseases such as heart attack, stroke and many more. Treatment of a stenosed artery includes the insertion of a catheter through the artery which affects the blood flow and solute dispersion. This present study focuses on the effect of catheter radius and stenosis height on the blood flow and solute dispersion behavior. The problem is modelled using the Herschel-Bulkley fluid to represent the blood rheology, with catheter and stenosis as the boundary conditions. Analytical solutions in integral form are obtained by solving the momentum equation and Herschel-Bulkley constitutive equation. The integrals are numerically evaluated using the Simpson’s 3/8 rule and Regula-Falsi method to obtain the blood velocity. The obtained velocity is employed into the unsteady convective-diffusion equation and solved using the generalized dispersion model (GDM) to analyse the behaviour of solute diffusion. The influence of catheter radius and stenosis height on the diffusion coefficient and mean concentration of solute are observed. Results show that the diffusion coefficient decreases as the catheter radius and stenosis height increases. A decrease in diffusion coefficient simultaneously increases the solute mean concentration.

Analysis of Unsteady Solute Dispersion in a Blood Flow of Herschel-Bulkley through a Catheterized Stenosed Artery

Blockage of blood flow due to cholesterol deposits at the arterial wall, known as stenosis, can lead to conditions such as heart attack and stroke. Treatment such as balloon angioplasty involves the catheterization of an artery where a stented catheter is inflated at the stenosis site to open the narrowed artery. The catheterization of the stenosed artery affects the surrounding blood flow and dispersion process. The present study analyses the effect of catheter radius and stenosis height on the blood velocity and solute dispersion behavior. Herschel-Bulkley fluid is used to model the problem with stenosis as the boundary condition. The momentum equation and Herschel-Bulkley constitutive equation are solved analytically into integral forms. Simpson’s 3/8 rule and Regula-Falsi method were used to evaluate the integral numerically to obtain the velocity. The velocity was utilized to solve the unsteady convective-diffusion equation using the generalized dispersion model (GDM) to obtain the dispersion function. This present research can potentially help the medical field and industry in determining the suitable catheter radius for patients, calculating drug dosage and improving stent catheter design. Results show that the velocity decreases as the catheter radius and stenosis height increase. A decrease in velocity simultaneously increases the solute dispersion function.

An analytical approximate method for solving unsteady state two-dimensional convection-diffusion equations

In this paper, an analytic approximate method for solving the unsteady two-dimensional convection-diffusion equations is introduced. Also, the convergence of the approximate methods is analyzed. Three test examples are presented, two have exact and one has not exacted solutions. The results obtained show that these methods are powerful mathematical tools for solving linear and nonlinear partial differential equations, moreover, new analytic Taylor method (NATM), reduced differential transform method (RDTM), and homotopy perturbation method (HPM), are more accurate and have less CPU time than the other methods.

Numerical Methods for Unsteady Convection-diffusion Problems Based on Combining Compact Difference Schemes with Runge-Kutta Methods

The convection-diffusion equation is of primary importance in understanding transport phenomena within a physical system. However, the currently available methods for solving unsteady convection-diffusion problems are generally not able to offer excellent accuracy in space and time. The one-dimensional unsteady convection-diffusion equation was solved by combing a compact difference scheme with the Runge-Kutta method. The combined method has fourth-order accuracy in space and time. To check the accuracy of the combined method, numerical experiments were carried out and comparisons were performed with the Crank-Nicolson method. The analysis results indicated that the combined method is numerically stable at low wave numbers and small Courant-Friedrichs-Lewy numbers. The combined method has higher accuracy than the Crank-Nicolson method.

A new implicit finite difference method with a compact correction term for solving unsteady convection diffusion equations

A new implicit finite difference method with a compact correction term is established for solving unsteady convection-diffusion equations. The correction terms by connecting classical and compact finite difference formulas are proposed for improving the accuracies of numerical solutions. This new method with fourth order accuracy can be used for the numerical calculations of convection diffusion equation on both uniform and nonuniform grid systems. It can be extended not only from one-dimensional to multi-dimensional equations, but also from steady to unsteady convection-diffusion equations. Several convection-diffusion problems are stimulated for verifying the flexibility and high accuracy of this method.

A half boundary method for two dimensional unsteady convection–diffusion equations

Convection diffusion equation is widely applied in many fields of science and technology. Many practical engineering problems can be expressed by this equation in unsteady state. However, it is usually difficult and time-consuming to find its solution. In this paper, a boundary type method named half boundary method (HBM) is proposed for two dimensional unsteady convection–diffusion equations. The main idea of HBM is introducing new variables to reduce the order of the equation and build relations of variables between nodes inside the area and nodes at half of the boundaries, namely the unknown variables. Since only variables at half of the boundaries are chosen as unknown variables, using HBM can realize dimensionality reduction and the maximum matrix order is less than that in finite volume method when considering large division number, which makes HBM fast and efficient. After the unknown variables are obtained by the boundary conditions, all variables at nodes can be obtained simultaneously according to the relations mentioned above. Numerical studies are carried out for convection dominated problems, problems with variable or discontinuous coefficient and problems with mixed boundary conditions, which show the validity of HBM for two-dimensional unsteady convection-diffusion problems and high accuracy for convection-domination problems.

Performance of similarity explicit group iteration for solving 2D unsteady convection-diffusion equation

In this paper, a similarity finite difference (SFD) solution is addressed for thetwo-dimensional (2D) parabolic partial differential equation (PDE), specifically on the unsteady convection-diffusion problem. Structuring the similarity transformation using wave variables, we reduce the parabolic PDE into elliptic PDE. The numerical solution of the corresponding similarity equation is obtained using a second-order central SFD discretization schemeto get the second-order SFD approximation equation. We propose a four-point similarity explicit group (4-point SEG) iterative methodasa numericalsolution of the large-scale and sparse linear systems derived from SFD discretization of 2D unsteady convection-diffusion equation (CDE). To showthe 4-point SEG iteration efficiency, two iterative methods, such as Jacobiand Gauss-Seidel (GS) iterations, are also considered. The numerical experiments are carried out using three different problems to illustrate our proposed iterative method's performance. Finally, the numerical results showed that our proposed iterative method is more efficient than the Jacobiand GS iterations in terms of iteration number and execution time.

Multiscale dimension reduction for flow and transport problems in thin domain with reactive boundaries

In this paper, we consider flow and transport problems in thin domains. Modeling problems in thin domains occur in many applications, where thin domains lead to some type of reduced models. A typical example is one dimensional reduced-order model for flows in pipe-like geometries (e.g., blood vessels). In many reduced-order models, the model equations are described apriori by some analytical approaches. In this paper, we propose the use of multiscale methods, which are alternative to reduced-order models and can represent reduced-dimension modeling by using fewer basis functions (e.g., the use of one basis function corresponds to one dimensional approximation).The mathematical model considered in the paper is described by a system of equations for velocity, pressure, and concentration, where the flow is described by the Stokes equations, and the transport is described by an unsteady convection-diffusion equation with non-homogeneous boundary conditions on walls (reactive boundaries). We start with the finite element approximation of the problem on unstructured grids and use it as a reference solution for two and three-dimensional model problems. Fine grid approximation resolves complex geometries on the grid level and leads to a large discrete system of equations that is computationally expensive to solve. To reduce the size of the discrete systems, we develop a multiscale model reduction technique, where we construct local multiscale basis functions to generate a lower-dimensional model on a coarse grid. The proposed multiscale model reduction is based on the Discontinuous Galerkin Generalized Multiscale Finite Element Method (DG-GMsGEM). In DG-GMsFEM for flow problems, we start with constructing the snapshot space for each interface between coarse grid cells to capture possible flows. For the reduction of the snapshot space size, we perform a dimension reduction via a solution of the local spectral problem and use eigenvectors corresponding to the smallest eigenvalues as multiscale basis functions for the approximation on the coarse grid. For the transport problem, we construct multiscale basis functions for each interface between coarse grid cells and present additional basis functions to capture non-homogeneous boundary conditions on walls. Finally, we will present some numerical simulations for three test geometries for two and three-dimensional problems to demonstrate the method's performance.

Transport of reactive species in oscillatory Couette-Poiseuille flows subject to homogeneous and heterogeneous reactions

Abstract The longitudinal dispersion and cross-sectional concentration distribution of chemical species through an annular tube have been studied for oscillatory flows in the presence of heterogeneous reactions between the species and tube wall along with homogeneous reaction in the bulk flow. The species is supposed to undergo kinetic reversible phase exchange and irreversible absorptive reactions at the outer wall material whereas in the bulk of the flow, the species participates in a first-order reaction with the solvent. The velocity distribution has a complex interaction with the reaction parameters and in order to track that, three different kinds of oscillatory flows are considered. For the purpose of estimation of dispersion coefficient, the method of moments Aris (1956)[3] is employed. The unsteady convective-diffusion equation gives rise to integral moment equations and are solved numerically by FDM. The cross-sectional concentration distribution is determined from the relationship between central moments and Hermite polynomials for the unsteady components of the flows. The study reveals the coupled effects of reversible phase exchange, irreversible absorption and bulk flow reaction on the transport of species in a variety of flow situations.

Very high-order method on immersed curved domains for finite difference schemes with regular Cartesian grids

A new very high-order technique for solving conservation laws with curved boundary domains is proposed. A Finite Difference scheme on Cartesian grids is coupled with an original ghost cell method that provide accurate approximations for smooth solutions. The technology is based on a specific least square method with restrictions that enables to handle general Robin conditions. Several examples in two-dimensional geometries are presented for the unsteady Convection–Diffusion equation and the Euler equations. A fifth-order WENO scheme is employed with matching fifth-order reconstruction at the boundaries. Arbitrary high-order reconstruction for smooth flows is achievable independently of the underlying differential equation since the method works as a black-box dedicated to boundary condition treatment.

On transport of reactive solute in a pulsatile Casson fluid flow through an annulus

This paper examines the effect of heterogeneous chemical reaction on the transport of a solute in a Casson fluid flow through an annular pipe under a periodic pressure gradient. Aris–Barton's method of moments is employed to understand the transport process. A finite difference implicit scheme is utilized to solve the momentum equation as well as the integral moment equation arising from the unsteady convective diffusion equation. Using the Hermite polynomial representation, the mean concentration distribution in the axial direction is acquired from the first four central moments. The transport phenomenon is discussed by three transport coefficients, namely exchange, advection, and dispersion coefficients owing to wall reactions, yield stress, radius ratio, etc. The present study can, therefore, be helpful to understand the transport process in blood flow through arteries.

A Compact Higher-Order Scheme for Two-Dimensional Unsteady Convection–Diffusion Equations

In this paper, we study a compact higher-order scheme for the two-dimensional unsteady convection–diffusion problems using the nearly analytic discrete method (NADM), especially, focusing on the convection dominated-diffusion problems. The numerical scheme is constructed and the stability analysis is implemented. We find the order of accuracy of scheme is [Formula: see text], where [Formula: see text] is the size of time steps and [Formula: see text] the size of spacial steps, especially, making clear the relation between [Formula: see text] and [Formula: see text] is according to the different values of diffusion parameter [Formula: see text] through von Neumann stability analysis. The obtained numerical results for several benchmark problems show that our method makes progress in the numerical study of NADM for convection–diffusion equation and is to be effective and helpful particularly in computations for the convection dominated-diffusion equations and, furthermore, valuable in the numerical treatment of many real-world problems such as MHD natural convection flow.

Effect of H2O on char-nitrogen conversion during char-O2/H2O combustion under high-temperature entrained flow conditions

The char-O2/H2O combustion under high-temperature entrained flow conditions was investigated experimentally and numerically to study the effect of H2O on the char-nitrogen (char-N) conversion. A numerical model was established based on the unsteady convection–diffusion equations for mass and heat transfer. Local Random Pore Model (RPM) was adopted to consider the change of specific surface area during char combustion. The Langmuir-Hinshelwood mechanism was used in the rate equations of main heterogeneous reactions to consider the competitions on surface active sites. The numerical results of char conversion and fractional conversion of char-N to NO reproduce the experimental results well. Both of the numerical and experimental results shows that the added H2O promotes the fractional conversion of char-N to NO and the promotion effect becomes weaker with the increase of O2 concentration. Based on numerical results, four crucial factors that affect char-N conversion were decoupled and discussed quantitatively with addition of H2O. The added H2O promotes nitrogen release rate at lower oxygen concentration but inhibits nitrogen release rate at higher oxygen concentration. The added H2O lowers the heterogeneous char-NO reduction rate constant greatly but enlarges the pore surface area slightly, resulting in a lower overall heterogeneous char-NO reduction rate. Char-NO reaction is the main NO reduction pathway, and most of HCN formed in char-N–H2O reaction is more likely to release out of char particle and converts to NO through HCN→NH3→NO pathway in the ambient flow.

Numerical solutions of the coupled unsteady nonlinear convection-diffusion equations based on generalized finite difference method

In this paper, the meshless Generalized Finite Difference Method (GFDM) in conjunction with the second-order explicit Runge-Kutta method (RK2 method) is presented to solve coupled unsteady nonlinear convection-diffusion equations (CDEs). Compared with the conventional Euler method, the RK2 method not only has higher accuracy but also reduces the possibility of numerical oscillation in time discretization, especially for the nonlinear and coupled cases. The generalized finite difference method, which is a localized collocation method, is famous for its simplicity and adaptability in the numerical solution of partial differential equations. Benefiting from Taylor series and moving least squares, its partial derivatives can be formed by a series of surrounding space points. In comparison with traditional finite difference methods, the proposed GFDM is free of mesh and available for irregular discretization nodes. In this study, the stencil selection algorithms are introduced to choose the stencil support of a certain node from the whole discretization nodes. Error analysis and numerical investigations are presented to demonstrate the effectiveness of the proposed GFDM for solving the coupled linear and nonlinear unsteady convection-diffusion equations. Then it is successfully applied to three benchmark examples of the coupled unsteady nonlinear CDEs encountered in the double-diffusive natural convection process, chemotaxis-haptotaxis model of cancer invasion, and thermo-hygro coupling model of concrete.