Steady-state Convection-diffusion Equation Research Articles

Variational principle of the 2-D steady-state convection-diffusion equation with fractal derivatives

The convection-diffusion equation describes a convection and diffusion process, which is the cornerstone of electrochemistry. The process always takes place in a porous medium or on an uneven boundary, and an abnormal diffusion occurs, which will lead to deviations in prediction of the convection-diffusion process. To overcome the problem, a fractal modification is suggested to deal with the ?abnormal? problem, and a 2-D steady-state convection-diffusion equation with fractal derivatives in the fractal space is established. Furthermore, its fractal variational principle is obtained by the semi-inverse method. The fractal variational formula can not only provide the conservation law in the fractal space in the form of energy, but also give the possible solution structure of the equation.

Mathematical Modeling of Cyclic Voltammogram Curves of Copper Deposition Involving Multiple Additives

The manufacturing of interconnects and the packaging of integrated circuits are achieved with electrodeposition of copper or other metals. In order to increase the rate of deposition, especially for the large features in packaging, forced convection is provided with certain agitation mechanisms. Although this reduces deposition time, it leads to non-uniform mass transport within each feature and between different features. Special organic additives are used in the solution during the process in order to tune the nucleation and growth of metal, as well as to modify the deposition rate and improve the uniformity. A mathematical model to describe the behavior of organic additives in conjugation with fluid flow and features of various geometry and dimensions is very much desired to facilitate chemistry and process development. In order to achieve this, the physiochemical kinetics of additive and their influence on the Cu deposition rate need to be described precisely.This presentation focuses on a method to extract the kinetic parameters describing the combined effect of multiple additives during copper deposition using rotating disk electrode (RDE). The one-dimensional steady state convection-diffusion equation for each of the chemical species including copper is solved by a semi-analytical method for a range of potentials. The boundary conditions of these differential equations are coupled on the surface of the RDE through the surface coverage of the absorbed species. The steady state of surface coverage of the species represents a dynamic equilibrium of three key processes i.e., adsorption, desorption, and consumption (incorporation). When equilibrium is achieved, the net rate of adsorption and desorption becomes equal to the rate of consumption. At each value of potential, the surface coverage of the additives is solved. At first, the solution is obtained with only one species known as suppressor and it was found that in a specific range of voltage and kinetic parameter multiple solutions of the surface coverage exist at same applied potential. This mathematically explains the S-shaped negative differential resistance (NDR) feature in experimental Cyclic Voltammogram (CV) curves. Figure 1 shows three such experimental S-shaped curves for different concentration of suppressors. The NDR region obtained in the theoretical CV curve is sensitive to the kinetic parameters of the additives. It is possible to match the theoretical and the experimental CV curves by optimizing the kinetic parameters. Determination of the kinetic parameters by particle swarm optimization using experimental data for multiple additive concentration will be discussed in detail in this talk. Figure 1

Pollution Sources Reconstruction Based on the Topological Derivative Method

The topological derivative method is used to solve a pollution sources reconstruction problem governed by a steady-state convection-diffusion equation. To be more precise, we are dealing with a shape optimization problem which consists of reconstruction of a set of pollution sources in a fluid medium by measuring the concentration of the pollutants within some subregion of the reference domain. The shape functional measuring the misfit between the known data and solution of the state equation is minimized with respect to a set of ball-shaped geometrical subdomains representing the pollution sources. The necessary conditions for optimality are derived with the help of the topological derivative method which consists in expanding the shape functional asymptotically and then truncate it up to the second order term. The resulting expression is trivially minimized with respect to the parameters under consideration which leads to a noniterative second-order reconstruction algorithm. Two different cases are considered. Firstly, when the velocity of the leakages is given and we reconstruct the support of the unknown sources, including their locations and sizes. In the second case, we consider the size of the pollution sources to be known and find out the mean velocity of the leakages and their locations. Numerical examples are presented showing the capability of the proposed algorithm in reconstructing multiple pollution sources in both cases.

Stability estimate for an inverse problem of the convection-diffusion equation

Abstract We consider the inverse boundary value problem for the dynamical steady-state convection-diffusion equation. We prove that the first-order coefficient and the scalar potential are uniquely determined by the Dirichlet-to-Neumann map. More precisely, we show in dimension n ≥ 3 {n\geq 3} a log-type stability estimate for the inverse problem under consideration. The method is based on reducing our problem to an auxiliary inverse problem and the construction of complex geometrical optics solutions of this problem.

Very high-order accurate finite volume scheme on curved boundaries for the two-dimensional steady-state convection–diffusion equation with Dirichlet condition

• Very high-order finite volume scheme for curved geometries. • Reconstruction for Off-site Data to provide the optimal order with curved domain. • No curved element and complex mapping on reference element. • A set numerical experiences with non-trivial geometries to assess the method order. Accuracy may be dramatically reduced when the boundary domain is curved and numerical schemes require a specific treatment of the boundary condition to preserve the optimal order. In the finite volume context, Ollivier-Gooch and Van Altena (2002) has proposed a technique to overcome such limitation and restore the very high-order accuracy which consists in specific restrictions considered in the least-squares minimization associated to the polynomial reconstruction. The method suffers from several drawbacks, particularly, the use of curved elements that requires sophisticated meshing algorithms. We propose a new method where the physical domain and the computational domain are distinct and we introduce the Reconstruction for Off-site Data (ROD) where polynomial reconstructions are carried out on the mesh using data localized outside of the computational domain, namely the Dirichlet condition situated on the physical domain. A series of numerical tests assess the accuracy, convergence rates, robustness, and efficiency of the new method and show that the boundary condition is fully integrated in the scheme with a very high-order accuracy and the optimal convergence rate is achieved.

The Steady-State Convection-Diffusion Equation at High Péclet Numbers for a Cluster of Spheres: An Extension of Levich's Theory

This paper describes an approximate analytical model of competitive effects between members of a dense cluster of absorbing objects, which are modeled as spheres. Neighboring absorbing spheres compete for diffusing species and thereby reduce each other's rate of absorption. Levich's well-known asymptotic (high Peźclet number) theory of convection-diffusion considers only the inner region of the concentration boundary layer; it does not describe the wake zone accurately. An extension of the Levich model is constructed for the wake zone. This is used to model intersphere competitive effects. The model demonstrates that for two neighboring spheres aligned along the flow direction, the absorption of the downstream sphere is substantially reduced vis-aź-vis the upstream sphere. The model is verified by comparison to numerical simulation studies. Both single-sphere simulations (reported in this paper) and multisphere simulations (taken from existing literature) are considered. In the single-sphere case, the discrepancy between the analytical model and the numerical results is maximally 10% at Pe = 10 and much lower at higher Peźclet numbers. An appreciable part of the error stems from the original Levich model itself, rather than from our method of extending the Levich model. In the multisphere case, the difference between the analytical model and the numerical studies is generally less than 30% . At small intersphere separations (say, center-to-center distances $<$ 5 sphere radii), the model tends to overestimate the interference effects. This is related to the fact that flow stagnancy in the space between two closely packed spheres is not taken into account in the model.

A semianalytical method for simulating mass transport at channel electrodes

A method for simulating electrode reactions in channel flow is developed and efficiently implemented in the symbolic algebra program Maple™. The steady-state convective diffusion equation for fully developed 2-D laminar (Poiseuille) flow past one or more electrodes in a channel is considered for a charge-transfer electrode reaction between two soluble species. The case where axial diffusion (along the channel, x direction) is neglected and the diffusivities are equal has an exact solution as an infinite series, in which each term is the product of an exponential in x and a confluent hypergeometric function in y (across the channel). The practical implementation consists of evaluating a finite number of terms and numerically evaluating the two parameters in each term. Sturm–Liouville (eigenfunction) theory is used to reliably find the parameters for arbitrary values of the rate constants. Comparison is made with results from a commercial software package that uses a finite-element method.

A Uniqueness Result for an Inverse Problem of the Steady State Convection-Diffusion Equation

We consider the inverse boundary value problem for the steady state convection-diffusion equation. We prove that a velocity field $V$ is uniquely determined by the Dirichlet-to-Neumann map when $V\in C^{0,\gamma}(\Omega)$, $2/3<\gamma\leq1$, i.e., when $V$ is a Holder continuous vector field with $2/3<\gamma\leq1$.

On the numerical modeling of convection-diffusion problems by finite element multigrid preconditioning methods

During the last decades, multigrid methods have been extensively used in order to solve large scale linear systems derived from the discretization of partial differential equations using the finite difference method. The effectiveness of the multigrid method can be also exploited by using the finite element method. Finite Element Approximate Inverses in conjunction with Richardon's iterative method could be used as smoothers in the multigrid method. Thus, a new class of smoothers based on approximate inverses can be derived. Effectiveness of explicit approximate inverses relies in the fact that they are close approximants to the inverse of the coefficient matrix and are fast to compute in parallel. Furthermore, the proposed class of finite element approximate inverses in conjunction with the explicit preconditioned Richardson method yield improved results against the classic smoothers such as Jacobi method. Moreover, a dynamic relaxation scheme is proposed based on the Dynamic Over/Under Relaxation (DOUR) algorithm. Furthermore, results for multigrid preconditioned Krylov subspace methods, such as GMRES(res), IDR(s) and BiCGSTAB based on approximate inverse smoothing and a dynamic relaxation technique are presented for the steady-state convection-diffusion equation.

Mathematical model of electrodialysis with ion-exchange membranes and inert spacers

A mathematical model describing the two-dimensional concentration field of an electrodialysis device with inert spacers is proposed. The boundary-value problem includes the Navier-Stokes, continuity, and steady-state convective diffusion equations and well-defined conditions and is solved by the control-volume numerical method. Results are expressed in the form of functional relationships of generalized variables. It is shown that when channels of the electrodialysis device are filled with spacers that do not conduct electric current, mass transport increases by several times in comparison to devices with open channels. The possibility is discussed for replacing the inert spacers with ones that conduct ion, not only in the complete demineralization of natural waters, but also in the desalination of brackish ground waters.

Strong and Weak Form of the Governing Differential Equations in Least Squares Finite Element Processes in h,p,k Framework

This paper presents an investigation of performance of the least squares finite element process in hpk mathematical framework utilizing: (i) governing differential equations (GDEs) containing highest order derivatives of the dependent variables (strong form of GDEs); (ii) GDEs containing only first order derivatives of the dependent variables and hence constituting a system of first order equations (weak form of GDEs) derived either directly from conservation laws or derived using auxiliary variables or auxiliary equations. It is shown that while the weak form of the GDEs may appear perfectly legitimate, the auxiliary equations in such forms cause irrecoverable inconsistencies in the resulting computational processes due to the fact that local approximations for the dependent variables and the auxiliary variables always remain inconsistent regardless of the choices p-level and the orders of the approximation spaces. The inconsistency of local approximations in the auxiliary equations may introduce spuriousness in the numerically computed solution that may progressively grow if the theoretical solution gradients are severe and isolated and may even cause failure of the computational process. It is shown that strong forms of the GDEs have no such problems and are highly meritorious over the weak forms in all aspects and hence are always the preferred choice but require the use of higher order local approximation spaces. One dimensional steady state convection diffusion equation (BVP) and one dimensional Riemann shock tube (IVP) are used as model problems to illustrate various concepts and to present numerical studies.

DRBEM for Cauchy convection-diffusion problems with variable coefficients

In this study we present a numerical technique based on the Dual Reciprocity Boundary Element Method (DRBEM) combined with the Tikhonov regularisation method, or with the Truncated Singular Value Decomposition (TSVD) method, in order to solve the Cauchy problem associated with the steady-state convection-diffusion equation with variable coefficients. This is an interesting problem from the practical point of view as it can be used to model certain water pollution problems. The numerical results obtained in the test examples investigated show that the Tikhonov regularisation method is very suitable for the smooth geometries considered, i.e. circular and annular domains, while for the non-smooth rectangular domain the TSVD method is preferable. The choice of the regularisation parameter and of the truncation parameter is based on the L-curve method.

THE k-VERSION OF FINITE ELEMENT METHOD FOR NON-SELF-ADJOINT OPERATORS IN BVP

THE k-VERSION OF FINITE ELEMENT METHOD FOR NON-SELF-ADJOINT OPERATORS IN BVP

Analytical Upstream Collocation Solution of a Forced Steady-State Convection-Diffusion Equation

We give herein formulas for the solution of the Hermite collocation discretization of anonhomogeneoussteady-state convection-diffusion equation in one spatial dimension andwith constant coefficients,defined on a uniform mesh, with Dirichlet boundary conditions.The accuracy ofthe method is enhanced by employing ``upstream weighting'' of the convectiveterm in an optimal way. We discuss also the issue of where to optimally samplethe forcing function. Computational examples illustrate the efficacy of the optimalupstream weighting technique combined with optimal sampling of the forcing function.

A comparison analysis between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of partial differential equations

In this article, we present a thorough numerical comparison between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of boundary value problems for partial differential equations. A series of test examples was solved with these two schemes, different problems with different type of governing equations, and boundary conditions. Particular emphasis was paid to the ability of these schemes to solve the steady-state convection-diffusion equation at high values of the Péclet number. From the examples tested in this work, it was observed that the system of algebraic equations obtained with the symmetric method was in general simpler to solve than the one obtained with the unsymmetric method and that the resulting algorithm performs better. However, the unsymmetric method has the advantage of being simpler to implement. Two main features about the results obtained in this work are worthy of special attention: First, with the symmetric method it was possible to solve convection-diffusion problems at a very high Péclet number without the need of any artificial damping term, and second, with these two approaches, symmetric and unsymmetric, it is possible to impose free boundary conditions for problems in unbounded domains.

Effect of gravity on the deposition of colloidal particles on a planar surface

A steady state convective-diffusion equation is solved using a collocation method to find the concentration profile and flux of adsorbing particles near a particle adsorbed on the plane surface. At small values of the gravity number, NG=πd4Δρ/6kT, the concentration profile and flux vary slowly near the preadsorbed particle, while they are highly nonuniform at large values of NG. The effect of the position of the system boundary on the collocation calculation is discussed and it is shown how the concept of flux balance may be used to improve the accuracy of the results. Finally, we develop two fitting functions at high and low values of NG, respectively.

Spectroelectrochemistry for a coupled chemical reaction in the channel cell

Abstract The in situ spectroelectrochemical steady state convective-diffusion equation of the EC reaction in the flow cell was solved by the implicit difference method. The normalized absorbance of the intermediate R and the final product P were calculated. Several factors affecting the spectroelectrochemical behavior were analyzed, which were represented by the normalized parameter w and normalized rate constant K1 respectively. The relationship between the normalized absorbence of intermediate R and w was discussed and a formula for calculating the rate constant of the following reaction was derived. The suitable range of application of the formula was discussed as well.

COVERAGE AND STRUCTURE OF FILMS OF SPHERICAL PARTICLES DEPOSITED AFTER DIFFUSING IN A GRAVITATIONAL FIELD

We investigate the coverage and structure of a layer of particles deposited on a line after diffusion in a gravitational field. The dynamics of the depositing particles is controlled by the gravity number N G(=πd 4 Δρg/6k B T), where d is the diameter of the particles, Δρ is the density difference between the particles and the solution, g is the acceleration due to gravity, k B is Boltzmann's constant, and T is the temperature. The position-dependent flux of particles in a gap formed by two preadsorbed particles is estimated by superposition of solutions of a steady-state convective diffusion equation for the flux in the presence of a single preadsorbed particle. The saturation coverages are found with a recursion relation and are in good agreement with those obtained from Brownian dynamics simulation. The jamming coverage increases rapidly with increasing particle size, particularly for large values of Δρ. An algorithm is presented to generate adsorbed configurations from which the structure of the deposit is determined.

A computational analysis of FXa generation by TF:FVIIa on the surface of rat vascular smooth muscle cells.

A computational model was developed to investigate the contribution of classical mass transport and flow parameters to factor X (FX) activation by the tissue factor-factor VIIa complex (TF:VIIa) on one wall of a parallel-plate flow chamber. The computational results were compared to previously obtained experimental data for the generation of factor Xa (FXa) by TF:VIIa on the surface of cultured rat vascular smooth muscle cells. In this study, the complete steady-state convection-diffusion equation was solved using the commercial software package, FLUENT (Fluent Inc., Lebanon, New Hampshire). A user-defined subroutine interfaced with FLUENT implemented the surface reaction which was modeled using classical Michaelis-Menten reaction kinetics. The numerical solutions were obtained for 12 cases which used combinations of three wall shear rates and four reaction rates. The numerically obtained fluxes for a given reaction rate displayed a wall shear rate dependence which ranged from classical kinetic reaction control (no dependence) to pure diffusional control (maximum dependence). The experimental data, however, were not represented by numerical data generated using a single reaction rate. The three numerically obtained fluxes which corresponded most closely to the experimental fluxes were determined using three different Vmax values. This finding supports the hypothesis that there may be a direct effect of flow on the TF:VIIa complex or the cell membrane.

Effect of diffusion and convection on the flux of depositing particles near a preadsorbed particle

A steady state convective-diffusion equation is solved using a collocation method to find the concentration profile and flux of adsorbing particles near a particle adsorbed on a line. At small values of the gravity number, NG=πd4 Δρg/6kT, the concentration profile and flux vary slowly near the preadsorbed particle, while they are highly non-uniform at large values of NG. The numerical results are compared with Brownian dynamics simulation for a range of NG values. The effect of the position of the system boundary on the collocation calculation is discussed and it is shown how the concept of flux balance may be used to improve the accuracy of the results. Finally, we develop a truncated power series that accurately fits the numerical data.