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A stable boundary elements method for magnetohydrodynamic channel flows at high Hartmann numbers
Abstract
The article is devoted to extension of boundary element method (BEM) for solving coupled equations in velocity and induced magnetic field for time dependent magnetohydrodynamic (MHD) flows through a rectangular pipe. The BEM is equipped with finite difference approach to solve MHD problem at high Hartmann numbers up to 106. In fact, the finite difference approach is used to approximate partial derivatives of unknown functions at boundary points respect to outward normal vector. It yields a numerical method with no singular boundary integrals. Besides, a new approach is suggested in this article where transforms 2D singular BEM's integrals to 1D nonsingular ones. The new approach reduces computational cost, significantly. Note that the stability of the numerical scheme is proved mathematically when computational domain is discretized uniformly and Hartmann number is 40 times bigger than length of boundary elements. Numerical examples show behavior of velocity and induced magnetic field across the sections.
... Furthermore, several works have been designed recently to vanish these singular domain integrals [7,21,30,31,32]. A very simple and accurate scheme based on second Green's identity is presented in [13,28] which transforms domain integrals to boundary ones. The idea proposed in [28] is used here to calculate the singular BEM's domain integrals. ...
... A very simple and accurate scheme based on second Green's identity is presented in [13,28] which transforms domain integrals to boundary ones. The idea proposed in [28] is used here to calculate the singular BEM's domain integrals. ...
In this paper boundary elements method (BEM) is equipped with finite difference approximation (FDA) to solve two-dimensional Navier-Stokes (N-S) equation. The N-S equation is converted to a system of ordinary differential equations (ODEs) respect to time in streamfunction-vorticity formulation. The constant direct BEM and 9-point stencil FDA are utilized to handle spatial derivatives, and the final system of ODEs is solved via three numerical schemes, Euler's, Runge-Kutta and Newton's methods to find a fast ODE solver. Numerical investigations presented in this article show Newton's method is faster than the others and it is able to solve the N-S equation for Reynolds number up to 40, 000 when grid points are at most 141×141. Thanks to BEM and boundary conditions of lid-driven cavity flow, the final system of ODEs is stable also for higher Reynolds numbers. A new technique is proposed in this article which converts BEM's two-dimensional singular integrals to one-dimensional non-singular ones. The proposed technique reduces computational cost of BEM, significantly, when more accurate results are requested. Numerical experiments show the numerical results are fairly close agreement with that accurate ones available in the literature.
Purpose
This study aims to examine the cilia-driven flow of magnetohydrodynamics (MHD) non-Newtonian fluid through a porous medium. The Jeffrey fluid model is taken into account. The fluid motion in a two-dimensional symmetric channel emphasizes the dominance of viscous properties over inertial properties in the context of long wavelength and low Reynolds number approximations.
Design/methodology/approach
An integrated numerical and analytic results are obtained by hybrid approach. A statistical method analysis of variance along with response surface methodology is used. Sensitivity analysis is used to validate the accuracy of nondimensional numbers.
Findings
The impact of various flow parameters is presented graphically and in numerical tables. It is noted that the velocity slip parameter is the most sensitive flow parameter in velocity and relaxation to retardation time ratio in temperature.
Originality/value
A model on cilia-generated flow of MHD non-Newtonian Jeffrey fluid is proposed.
In recent years, the MHD flow problems has received a significant attention as the behaviour of the flow changes when the Lorentz force (electromagnetic force) acts on the fluid. The aim of this study is to observe numerically the behavior of damped wave-type magnetohydrodynamic (MHD) flow in a sufficiently long rectangular channel having time-varied oblique magnetic field. The mathematical model of the considered problem is coupled convection–diffusion heat-type for the velocity of the fluid and convection–diffusion wave-like for the induced magnetic field. To obtain the numerical simulation, we have used the finite difference method for the discretization of the temporal variable and we also have considered the Galerkin finite element method for the discretization of the spatial variable. In the numerical investigation procedure, the time-varied oblique magnetic field has been considered several functions of the time as polynomial, exponential, and trigonometric e.t.c. We have also considered the several values of the Hartmann number and several directions of the induced magnetic field. In literature, there are many studies about numerical solution of steady and unsteady magnetohydrodynamic flow problems using finite element method, boundary element method and finite difference method e.t.c. For all that, it is the first time in the literature, the damped wave type MHD flow has been considered numerically in this study. The numerical results show Hartmann layer occurs for the high Hartmann numbers in accordance with the nature of the MHD flow. We have obtained numerical results in good agreement with the numerical results available in the literature. The acquired numerical results have been presented by the contour plots.
The magnetohydrodynamics (MHD) equation is one of the practical models that it has different applications in physics and engineering. It describes the magnetic properties and behavior of electrically conducting fluids. This paper investigates the MHD equation numerically. First, the time derivative is approximated by utilizing the Crank–Nicholson scheme. The convergence order and unconditional stability of the proposed time-discrete scheme are analyzed by the energy method. Then, the direct meshless local Petrov–Glerkin (DMLPG) method is applied for the spatial derivative to get the full-discrete scheme. The new numerical procedure is used to simulate this equation in a pipe with regular and irregular cross-sectional area. Numerical results show the efficiency of this method is well.
The flow under an Eyring-Powell description has attracted interest to model different scenarios related with non-Newtonian fluids. The goal of the present study is to provide analysis of solutions to a one-dimensional Eyring-Powell fluid in Magnetohydrodynamics (MHD) with general initial conditions. Firstly, regularity and bounds of solutions are shown as a baseline to support the construction of existence and uniqueness results. The existence analysis is based on the definition of a Hamiltonian that constitutes the underlying theory to obtain stationary profiles of solutions that are validated with a numerical approach. Afterwards, non-stationary profiles of solutions are explored based on an asymptotic approximation to a Hamilton-Jacobi type of equation. To this end, an exponential scaling is considered together with perturbation methods. Finally, a region of validity for such exponential scaling is provided.
In this paper, a high-order accurate numerical method for two-dimensional coupled hyperbolic and parabolic partial differential system arising in unsteady magnetohydrodynamics (MHD) flow of upper-convected Maxwell fluid is presented. We apply Galerkin-Legendre spectral method for discretizing spatial derivatives for MHD-M (MHD flow of Maxwell fluid) equations and then Crank-Nicolson scheme is used for time derivatives. Optimal a priori error bound is derived in the L2 and H1 norm for the semidiscrete formulation. We then proceed to discuss an instability mechanism in polynomial spectral methods and show that filtering suffices to ensure stability. Optimal error bound is derived between the semi-analytical solutions and filtered semi-analytical solutions. The results are illustrated by computational experiments. Furthermore, these results indicate that the proposed method is simple, accurate, efficient, and stable for solving various MHD-M and MHD flow problems or in general coupled hyperbolic-parabolic partial differential equations.
In this study, a fully developed, laminar Magnetohydrodynamic (MHD) flow between concentric cylinders with slipping walls is investigated. The induced magnetic field of the electrically conducting inner cylinder is also studied. The problem is considered on the 2D annular cross-section of the pipe. The inner disk and the fluid are electromagnetically coupled and the outer thin boundary is insulated. Both walls of the annulus are slipping with the same or different slip lengths. The flow is subjected to a horizontally applied uniform magnetic field. The system of governing equations is discretized by the Dual Reciprocity Boundary Element Method (DRBEM). The resultant matrix–vector equations are solved as a whole with coupled induced magnetic field conditions on the disk-fluid boundary. The effects of the slip, the Hartmann number (Ha) and the conductivity ratio are analyzed. Numerical results reveal the continuation of magnetic field through the disk in the no-slip case. An increase in the Ha forces the magnetic field isolines to close themselves in the annulus and decelerates the flow. The wall slip accelerates the flow and diminishes boundary layers near the slipping walls. As the inner wall slip length advances, a uniform flow develops between the cylinders. The slip at the outer wall separates the flow into two vortices and strengthens the influence of the Ha increase. When the fluid conductivity is larger than the disk conductivity, the inner wall starts to behave as if it was insulated and the slip at the same wall enhances this effect. The proposed numerical scheme is capable of capturing the MHD flow characteristics with the slip and the electromagnetically coupled wall conditions with less computational cost due to the boundary only nature of the DRBEM.
The study of the conduction of fluids in magnetic fields has attracted a lot of interest in recent years. In this paper, we use new truly meshless methods for numerical solving of the unsteady 2D coupled magneto-hydrodynamic (MHD) equations with uniform and scattered distributions of nodes on regular and polar domains. These methods are implemented based on a generalization of the moving least squares approximation (GMLS) method and local weak forms of MHD equations. By observing the numerical results obtained from these methods and comparing them with others, one can find the efficiency, accuracy, and speed of these methods.
This paper investigates the flow behavior of a viscous, incompressible and electrically conducting fluid in a long channel subjected to a time-varied oblique magnetic field B0(t)=B0f(t). The time-dependent MHD equations are solved by using the dual reciprocity boundary element method (DRBEM). The transient level velocity and induced magnetic field profiles are presented for moderate Hartmann number values, several direction of applied magnetic field and for several functions f(t) as polynomial, exponential, trigonometric, impulse and step functions. It is observed that, when f(t) is a polynomial or exponential function, the velocity magnitude increases up to a certain time level where the flow shows an elliptical elongation and then starts to decrease for all values of Hartmann numbers. For the trigonometric function f(t), the flow repeats its behavior with a period. An impulse function changes flow behavior with an elongation at the level where the impulse is applied to the magnetic field. Then, it behaves as if a uniform magnetic field is applied. Having a step function f(t), the behavior of the flow shows both impulse and polynomial functions features. This study shows that time-varied magnetic field changes the flow behavior significantly at all the transient levels.
The MHD equation has some applications in physics and engineering. The main aim of the current paper is to propose a new numerical algorithm for solving the MHD equation. At first, the temporal direction has been discretized by the Crank–Nicolson scheme. Also, the unconditional stability and convergence of the time-discrete scheme have been investigated by using the energy method. Then, an improvement of element free Galerkin (EFG) i.e. the interpolating element free Galerkin method has been employed to discrete the spatial direction. Furthermore, an error estimate is presented for the full discrete scheme based on the Crank–Nicolson scheme by using the energy method. We prove that convergence order of the numerical scheme based on the new numerical scheme is O(τ²+δm). In the considered method the appeared integrals are approximated using Gauss Legendre quadrature formula. Numerical examples confirm the efficiency and accuracy of the proposed scheme.
In this article, a decoupled fully discrete scheme for solving 2D magnetohydrodynamics (MHD) equations is proposed. The decoupled scheme is used for time discretization, and the finite element method is used for spatial discretization. Firstly, the almost unconditional stability of this scheme is established. Then optimal and error estimates of numerical solution are provided. Finally, a numerical example is presented to confirm our theoretical results and show the high efficiency of the decoupled scheme.
To validate the diffusion model and the aerosol particle model of the GASFLOW computer code, theoretical solutions of advection diffusion problems are developed by using the Green's function method. The work consists of a theory part and an application part. In the first part, the Green's functions of one-dimensional advection diffusion problems are solved in infinite, semi-infinite and finite domains with the Dirichlet, the Neumann and/or the Robin boundary conditions. Novel and effective image systems especially for the advection diffusion problems are made to find the Green's functions in a semi-infinite domain. Eigenfunction method is utilized to find the Green's functions in a bounded domain. In the case, key steps of a coordinate transform based on a concept of reversed time scale, a Laplace transform and an exponential transform are proposed to solve the Green's functions. Then the product rule of the multi-dimensional Green's functions is discussed in a Cartesian coordinate system. Based on the building blocks of one-dimensional Green's functions, the multi-dimensional Green's function solution can be constructed by applying the product rule. Green's function tables are summarized to facilitate the application of the Green's function. In the second part, the obtained Green's function solutions benchmark a series of validations to the diffusion model of gas species in continuous phase and the diffusion model of discrete aerosol particles in the GASFLOW code. Pe aerosol particles in the GASFLOW code. Perfect agreements are obtained between the GASFLOW simulations and the Green's function solutions in case of the gas diffusion. Very good consistencies are found between the theoretical solutions of the advection diffusion equations and the numerical particle distributions in advective flows, when the drag force between the micron-sized particles and the conveying gas flow meets the Stokes' law about resistance. This situation is corresponding to a very small Reynolds number based on the particle diameter, with a negligible inertia effect of the particles. It is concluded that, both the gas diffusion model and the discrete particle diffusion model of GASFLOW can reproduce numerically the corresponding physics successfully. The Green's function tables containing the building blocks for multi-dimensional problems is hopefully able to facilitate the application of the Green's function method to the future work.
In this paper we propose a development of the finite difference method, called the tailored finite point method, for solving steady magnetohydrodynamic (MHD) duct flow problems with a high Hartmann number. When the Hartmann num-ber is large, the MHD duct flow is convection-dominated and thus its solution may ex-hibit localized phenomena such as the boundary layer. Most conventional numerical methods can not efficiently solve the layer problem because they are lacking in either stability or accuracy. However, the proposed tailored finite point method is capable of resolving high gradients near the layer regions without refining the mesh. Firstly, we devise the tailored finite point method for the scalar inhomogeneous convection-diffusion problem, and then extend it to the MHD duct flow which consists of a cou-pled system of convection-diffusion equations. For each interior grid point of a given rectangular mesh, we construct a finite-point difference operator at that point with some nearby grid points, where the coefficients of the difference operator are tailored to some particular properties of the problem. Numerical examples are provided to show the high performance of the proposed method.
The effects of a constant uniform magnetic field on dendritic solidification were investigated using an enthalpy based numerical model. The interaction between thermoelectric currents on a growing crystal and the magnetic field generates a Lorentz force that creates flow. The need for very high resolution at the liquid-solid boundary where the thermoelectric source originates plus the need to accommodate multiple grains for a realistic simulation, make this a very demanding computational problem. For practical simulations, a quasi 3-dimensional approximation is proposed which nevertheless retains essential elements of transport in the third dimension. A magnetic field normal to the plane of growth leads to general flow circulation around an equiaxed dendrite, with secondary recirculations between the arms. The heat/solute advection by the flow is shown to cause a change in the morphology of the dendrite; secondary growth is promoted preferentially on one side of the dendrite arm and the tip velocity of the primary arm is increased. The degree of approximation introduced is quantified by extending the model into 3-dimensions, where the full Navier-Stokes equation is solved, and compared against the 2-dimensional solution.
In this study, Chebyshev polynomial method is applied to solve magnetohydrodynamic (MHD) flow equations in a rectangular duct in the presence of transverse external oblique magnetic field. In present method, approximate solution is taken as truncated Chebyshev series. The MHD equations are decoupled first and then present method is used to solve for positive and negative Hartmann numbers. Numerical solutions of velocity and induced magnetic field are obtained for steady-state, fully developed, incompressible flow for a conducting fluid inside the duct. The results for velocity and induced magnetic field are visualized in terms of graphics for values of Hartmann numbers M<=1000.
Time dependent MHD or stability problems for the aluminium reduction cells are typically restricted to the mathematical developments without the inclusion of the electrolyte channels. However, according to the well known Moreau-Evans model, the presence of electrolyte channels increases very significantly the stationary interface deformation. In this paper general time dependent theory and numerical modelling of an aluminium cell is extended to the case of variable bottom of aluminium pad and variable thickness of electrolyte to account for the channels. Instructive analysis is presented for multi-physical coupling of the magnetic field, electric current, velocity and wave development by animated examples for the high amperage cells. The results indicate that the 'rotating wave' instability is dominant in simplified cases not accounting for the effect of channels. The effect of channels creates a stabilizing effect, resulting in a 'sloshing', para metrically excited MHD wave development in aluminium reduction cells.
The self sustained waves at the aluminium-electrolyte interface, known as 'MHD noise', are observed in the most of commercial cells under certain conditions. The instructive analysis is presented how a step by step inclusion of different physical coupling factors is affecting the wave development in the electrolysis cells. The early theoretical models for wave development do not account for the current distribution at the cathode, instead assuming a uniform current density J z at the bottom. When the electric current is computed according to the actual electrical circuit, the growth rate is significantly lower, and if a sufficient dissipation is included, does not lead to instability. The inclusion of the horizontal circulation-generated turbulence is essential in order to explain the small amplitude self-sustained oscillations. The horizontal circulation vortices create a pressure gradient contributing to the deformation of the interface. The full time dependent model couples the nonlinear fluid dynamics and the extended electromagnetic field that covers the whole bus bar circuit and the ferromagnetic effects.
The boundary-element method is a powerful numerical technique for solving partial differential equations encountered in applied mathematics, science, and engineering. The strength of the method derives from its ability to solve with notable efficiency problems in domains with complex and possibly evolving geometry where traditional methods can be demanding, cumbersome, or unreliable. This dual-purpose text provides a concise introduction to the theory and implementation of boundary-element methods, while simultaneously offering hands-on experience based on the software library BEMLIB. BEMLIB contains four directories comprising a collection of FORTRAN 77 programs and codes on Green's functions and boundary-element methods for Laplace, Helmholtz, and Stokes flow problems. The software is freely available from the Internet site: http://bemlib.ucsd.edu The first seven chapters of the text discuss the theoretical foundation and practical implementation of the boundary-element method. The material includes both classical topics and recent developments, such as methods for solving inhomogeneous, nonlinear, and time-dependent equations. The last five chapters comprise the BEMLIB user guide, which discusses the mathematical formulation of the problems considered, outlines the numerical methods, and describes the structure of the boundary-element codes. A Practical Guide to Boundary Element Methods with the Software Library BEMLIB is ideal for self-study and as a text for an introductory course on boundary-element methods, computational mechanics, computational science, and numerical differential equations.
MHD equations have many applications in physics and engineering. The model is coupled equations in velocity and magnetic field and has a parameter namely Hartmann. The value of Hartmann number plays an important role in the equations. When this parameter increases, using different meshless methods makes the oscillations in velocity near the boundary layers in the region of the problem. In the present paper a numerical meshless method based on radial basis functions (RBFs) is provided to solve MHD equations. For approximating the spatial variable, a new approach which is introduced by Bozzini et al. (2015) is applied. The method will be used here is based on the interpolation with variably scaled kernels. The methodology of the new technique is defining the scale function on the domain . Then the interpolation problem from the data locations transforms to the new interpolation problem in the data locations (Bozzini et al., 2015). The radial kernels used in the current work are Multiquadrics (MQ), Inverse Quadric (IQ) and Wendland’s function. Of course the latter one is based on compactly supported functions. To discretize the time variable, two techniques are applied. One of them is the Crank–Nicolson scheme and another one is based on MOL. The numerical simulations have been carried out on the square and elliptical ducts and the obtained numerical results show the ability of the new method for solving this problem. Also in appendix, we provide a computational algorithm for implementing the new technique in MATLAB software.
The purpose of this paper is to study the closed-form solutions of peristaltic flow of Jeffery fluid under the simultaneous effects of magnetohydrodynamics (MHD) and partial slip conditions in a rectangular duct. The influence of wave train propagation is also taken into account. The analysis of mathematical model consists of continuity and the momentum equations are carried out under long wavelength and low Reynolds number assumptions. The governing equations are first reduced to the dimensionless system of partial differential equation using the appropriate variables and afterwards exact solutions are obtained by applying the method of separation of variables. The role of pertinent parameters such as Hartmann number , slip parameter , volumetric flow rate , Jeffery parameter and the aspect ratio against the velocity profile, pressure gradient and pressure rise is illustrated graphically. The streamlines have also been presented to discuss the trapping bolus discipline. Comparison with the existing studies is made as a limiting case of the considered problem.at the end.
In this paper, we consider the initial-boundary value problem for the three-dimensional incompressible magnetohydrodynamic equations. We first establish certain regularity results for the solution (u, p, H) under the assumption that u, H is an element of L-4(0, T;H-1 (Omega)(3)). Next, we consider the fully discrete first-order Euler semi-implicit scheme based on the mixed finite element method. Then we obtain the L-2-unconditional convergence of this scheme by using the negative norm technique.
MAGNETOHYDRODYNAMIC AXIAL FLOW IN A TRIANGULAR PIPE UNDER TRANSVERSE MAGNETIC FIELD
In the current paper the boundary elements method (BEM) has been employed to solve linear Helmholtz and semi-linear Poisson׳s equations. In fact a new idea will be presented to solve linear Poisson׳s and Helmholtz equations with variable coefficient by the use of BEM. And after that two iterative schemes based on the fixed point theorem and Newton׳s method have been studied to solve semi-linear Poisson׳s equation via the new idea. The main concerning problem of this paper is omitting singularity from BEM׳s domain integrals. So the improvement will be done by transforming the singularity to boundary integrals which can be calculated easily. The new scheme is implemented and compared with some well known numerical methods on various computational domains for the two-dimensional problems with Dirichlet and mixed boundary conditions. Numerical examples show that the new scheme is able to solve linear Helmholtz and semi-linear Poisson׳s equations, efficiently.
In this research, a variational reformulation for magneto-hydrodynamic (MHD) problem is derived. Existence and uniqueness of the weak solution are discussed. The Galerkin boundary node method is a boundary meshless method which uses MLS basis functions to approximate solution of problems. This paper tries to apply this method for a variational form of the magneto-hydrodynamic (MHD) problem. Numerical experiments reveal that the method is effective and convenient for this problem.
In this paper a meshfree weak-strong (MWS) form method is considered to solve the coupled equations in velocity and magnetic field for the unsteady magnetohydrodynamic flow throFor this modified estimaFor this modified estimaFor this modified estimaugh a pipe of rectangular and circular sections having arbitrary conducting walls. Computations have been performed for various Hartman numbers and wall conductivity at different time levels. The MWS method is based on applying a meshfree collocation method in strong form for interior nodes and nodes on the essential boundaries and a meshless local Petrov–Galerkin method in weak form for nodes on the natural boundary of the domain. In this paper, we employ the moving least square reproducing kernel particle approximation to construct the shape functions. The numerical results for sample problems compare very well with steady state solution and other numerical methods.
In this article the constant and the continuous linear boundary elements methods (BEMs) are given to obtain the numerical solution of the coupled equations in velocity and induced magnetic field for the steady magneto-hydrodynamic (MHD) flow through a pipe of rectangular and circular sections having arbitrary conducting walls. In recent decades, the MHD problem has been solved using some variations of BEM for some special boundary conditions at moderate Hartmann numbers up to 300. In this paper we develop this technique for a general boundary condition (arbitrary wall conductivity) at Hartmann numbers up to 105105 by applying some new ideas. Numerical examples show the behavior of velocity and induced magnetic field across the sections. Results are also compared with the exact values and the results of some other numerical methods.
The aim of the paper is the development of an efficient numerical algorithm for the solution of magnetohydrodynamics (MHD) flow problems with either fully insulating walls or partially insulating and partially conducting walls. Toward this, we first extend the influence domain of the shape function for the element free Galerkin (EFG) method to have arbitrary shape. When the influence factor approaches 1, we find that the EFG shape function almost has the Delta property at the node (i.e. the value of the EFG shape function of the node is nearly equal to 1 at the position of this node) as well as the property of slices in the influence domain of the node (i.e. the EFG shape function in the influence domain of the node is nearly constructed by different functions defined in different slices). Therefore, for MHD flow problems at high Hartmann numbers we follow the idea of the variational multiscale finite element method (VMFEM) to combine the EFG method with the variational multiscale (VM) method, namely the variational multiscale element free Galerkin (VMEFG) method is proposed. Subsequently, in order to validate the proposed method, we compare the obtained approximate solutions with the exact solutions for some problems where such exact solutions are known. Finally, several benchmark problems of MHD flows are simulated and the numerical results indicate that the VMEFG method is stable at moderate and high values of Hartmann number. Another important feature of this method is that the stabilization parameter has appeared naturally via the solution of the fine scale problem. Meanwhile, because this proposed method is a type of meshless method, it can avoid the need for meshing, a very demanding task for complicated geometry problems.
We are concerned with the feedback control of unsteady flow of an electrically conducting fluid, confined to a bounded region of space and driven by a combination of distributed body force and externally generated currents. The flow is governed by the Navier–Stokes equations and Maxwell’s equations, coupled via Ohm’s law and the Lorentz force. The mathematical formulation and numerical resolution of the linear feedback control problem for tracking velocity and magnetic fields of electrically conducting flows in two dimensional domain are presented. Semi-discrete in time and full space–time discrete approximations are also studied. Computational results showing the efficacy of the feedback control are presented and compared with analogous results using optimal control theory.
Preface 1. Introduction 2. Current-sheet formation 3. Magnetic annihilation 4. Steady reconnection: the classical solutions 5. Steady reconnection: new generation of fast regimes 6. Unsteady reconnection: the tearing mode 7. Unsteady reconnection: other approaches 8. Reconnection in three dimensions 9. Laboratory applications 10. Magnetospheric applications 11. Solar applications 12. Astrophysical applications 13. Particle acceleration References Appendices Index.
LAPLACE'S EQUATION IN ONE DIMENSION Green's First and Second Identities and the Reciprocal Relation Green's Functions Boundary-Value Representation Boundary-Value Equation LAPLACE'S EQUATION IN TWO DIMENSIONS Green's First and Second Identities and the Reciprocal Relation Green's Functions Integral Representation Integral Equations Hypersingular Integrals Irrotational Flow Generalized Single- and Double-Layer Representations BOUNDARY-ELEMENT METHODS FOR LAPLACE'S EQUATION IN TWO DIMENSIONS Boundary Element Discretization . Discretization of the Integral Representation The Boundary-Element Collocation Method Isoparametric Cubic-Splines Discretization High-Order Collocation Methods Galerkin and Global Expansion Methods LAPLACE'S EQUATION IN THREE DIMENSIONS Green's First and Second Identities and the Reciprocal Relation Green's Functions Integral Representation Integral Equations Axisymmetric Fields in Axisymmetric Domains BOUNDARY-ELEMENT METHODS FOR LAPLACE'S EQUATION IN THREE DIMENSIONS Discretization Three-Node Flat Triangles Six-Node Curved Triangles High-Order Expansions INHOMOGENEOUS, NONLINEAR, AND TIME-DEPENDENT PROBLEMS Distributed Source and Domain Integrals Particular Solutions and Dual Reciprocity in One Dimension Particular Solutions and Dual Reciprocity in Two and Three Dimensions Convection - Diffusion Equation Time-Dependent Problems VISCOUS FLOW Governing Equations Stokes Flow Boundary Integral Equations in Two Dimensions Boundary-Integral Equations in Three Dimensions Boundary-Element Methods Interfacial Dynamics Unsteady, Navier-Stokes, and Non-Newtonian Flow BEMLIB USER GUIDE General Information Terms and Conditions Directory DIRECTORY: GRIDS grid_2d trgl DIRECTORY: LAPLACE lgf_2d lgf_3d lgf_ax flow_1d flow_1d 1p flow_2d body_2d body_ax tank_2d ldr_ 3d lnm_3d DIRECTORY: HELMHOLTZ flow_1d osc DIRECTORY: STOKES sgf_2d sgf_3d sgf_ax flow_2d prtcl_sw prtcl_2d prtcl_ax prtcl_3d APPENDIX A: MATHEMATICAL SUPPLEMENT APPENDIX B: GAUSS ELIMINATION AND LINEAR SOLVERS APPENDIX C: ELASTOSTATICS REFERENCES INDEX
This article considers numerical implementation of the Crank–Nicolson/Adams–Bashforth scheme for the two-dimensional non-stationary Navier–Stokes equations. A finite element method is applied for the spatial approximation of the velocity and pressure. The time discretization is based on the Crank–Nicolson scheme for the linear term and the explicit Adams–Bashforth scheme for the nonlinear term. Comparison with other methods, through a series of numerical experiments, shows that this method is almost unconditionally stable and convergent, i.e. stable and convergent when the time step is smaller than a given constant. Copyright © 2009 John Wiley & Sons, Ltd.
This volume provides a self-contained and systematic development of an aspect of analysis which deals with the theory of fundamental solutions for differential operators, as well as their applications to boundary value problems of mathematical physics, applied mathematics, and engineering. Features of the text include: extensive applications topics presented in detail, with worked examples; coverage of over 70 different differential operators and derivation of fundamental solutions for them by using Fourier transforms and the theory of distributions; computational components discussed in all relevant topics and applications; an appendix describing numerous algorithms, and programs in C to illustrate solutions methods. All computer-based material is available by FTP from the Birkhauser web site.
In this paper, a fundamental solution for the coupled convection–diffusion type equations is derived. The boundary element method (BEM) application then, is established with this fundamental solution, for solving the coupled equations of steady magnetohydrodynamic (MHD) duct flow in the presence of an external oblique magnetic field. Thus, it is possible to solve MHD duct flow problems with the most general form of wall conductivities and for large values of Hartmann number. The results for velocity and induced magnetic field is visualized in terms of graphics for values of Hartmann number M⩽300.
A survey is given of the problems encountered when investigating a
conducting liquid or gas moving in a msgnetic field. It is considered that
magneto-fluid dynamics include all problems which may arise when a gas or liquid
is interacting with a magnetic field. The various problems are grouped together
according to the magnitude of the mean macroscopic velocity v, defined as the
mean velocity of all particles in a small volume element at a particular time.
The survey includes discussions on: the basic processes of the field; the
situation in which v vanishes and a magnetostatic equilibrium is present;
problems of infinitesimal v, including both waves and instabilities; problems of
finite v for a steady state system and the situation where time derivatives are
not zero; and the most important magnetofluids found in nature and in the
laberatory. (39 references). (B.O.G.)
A boundary element formulation is presented to obtain the solution in terms of velocity and induced magnetic field for the steady MHD (magnetohydrodynamic) flow through a rectangular channel having arbitrarily conducting walls, with an external magnetic field applied transverse to the flow. The fluid is viscous, incompressible, and electrically conducting. The boundary element method (BEM) with constant elements has been used to cast the problem into the form of an integral equation over the boundary and to obatain a system of algebraic equations for the boundary unknown values only. Computations have been carried out for several values of Hartmann number (1 ≤ M ≤ 10) and conductivity parameter (0 ≤ λ ≤ ∞). Selected graphs are given showing the behavior of the velocity field and induced magnetic field, which are of the flattening tendency for an increase in M or decrease in λ, and are in agreement with exact solutions.
The paper presents an analysis of laminar motion of a conducting liquid in a rectangular duct under a uniform transverse magnetic field. The effects of the duct having conducting walls are investigated. Exact solutions are obtained for two cases, (i) perfectly conducting walls perpendicular to the field and thin walls of arbitrary conductivity parallel to the field, and (ii) non-conducting walls parallel to the field and thin walls of arbitrary conductivity perpendicular to the field.
The boundary layers on the walls parallel to the field are studied in case (i) and it is found that at high Hartmann number ( M ), large positive and negative velocities of order MV c are induced, where V c is the velocity of the core. It is suggested that contrary to previous assumptions the magnetic field may in some cases have a destabilizing effect on flow in ducts.
This paper studies the steady motion of an electrically conducting, viscous fluid along channels in the presence of an imposed transverse magnetic field when the walls do not conduct currents. The equations which determine the velocity profile, induced currents and field are derived and solved exactly in the case of a rectangular channel. When the imposed field is sufficiently strong the velocity profile is found to degenerate into a core of uniform flow surrounded by boundary layers on each wall. The layers on the walls parallel to the imposed field are of a novel character. An analogous degenerate solution for channels of any symmetrical shape is developed. The predicted pressure gradients for given volumes of flow at various field strengths are finally compared with experimental results for square and circular pipes.
The solution is obtained to the problem of the steady one-dimensional flow of an incompressible, viscous, electrically conducting fluid through a circular pipe in the presence of an applied (transverse) uniform magnetic field. A no-slip condition on the velocity is assumed at the non-conducting wall. The solution is exact and thus valid for all values of the Hartmann number. Excellent agreement exists between the present theoretical results and the experimental values obtained by Hartmann & Lazarus (1937) in the low to medium Hartmann number range. The high Hartmann number case is treated by Shercliff (1962) in the following paper.
The flow rate of liquid metals is commonly measured by electromagnetic flowmeters. In these the fluid moves through a region of transverse magnetic field, inducing a potential difference between two electrodes on the walls of the pipe. The ratio of signal to flow rate is dependent on the velocity profile, and this is affected by electromagnetic forces.
The paper presents an improved, second approximation for the laminar motion of a conducting liquid at high Hartmann number in non-conducting pipes of arbitrary cross-section under uniform transverse magnetic fields. A satisfactory comparison with the author's previously experimental pressure gradient/flow results is made for the case of a circular pipe.
Finite element methods can be used for determining the flow in a straight channel under a variety of wall conductivity conditions when a uniform magnetic field is imposed perpendicular to the flow direction. At high Hartmann numbers oscillatory solutions are found unless sufficient points are placed within the Hartmann layers. In some one- and two-dimensional problems it appears that it is adequate to place one or two points within the Hartmann layer to remove the oscillations. Central core values can sometimes be predicted with good accuracy even when the Hartmann layers are not resolved adequately. A nine-point Gauss point rule has been used to evaluate the stiffness and other matrices for the eight-node elements. Copyright © 2001 John Wiley & Sons, Ltd.
A finite element method is given to obtain the solution in terms of velocity and induced magnetic field for the steady MHD (magnetohydrodynamic) flow through a rectangular pipe having arbitrarily conducting walls. Linear and then quadratic approximations have been taken for both velocity and magnetic field for comparison and it is found that with the quadratic approximation it is possible to increase the conductivity and Hartmann number M (M ≤ 100). A special solution procedure has been used for the resulting block tridiagonal system of equations. Computations have been carried out for several values of Hartmann number (5 ≤ M ≤ 100) and wall conductivity. It is also found that, if the wall conductivity increases, the flux decreases. The same is the effect of increasing the Hartmann number. Selected graphs are given showing the behaviour of the velocity field and induced magnetic field.
In this study, matrix representation of the Chebyshev collocation method for partial differential equation has been represented and applied to solve magnetohydrodynamic (MHD) flow equations in a rectangular duct in the presence of transverse external oblique magnetic field. Numerical solution of velocity and induced magnetic field is obtained for steady-state, fully developed, incompressible flow for a conducting fluid inside the duct. The Chebyshev collocation method is used with a reasonable number of collocations points, which gives accurate numerical solutions of the MHD flow problem. The results for velocity and induced magnetic field are visualized in terms of graphics for values of Hartmann number H≤1000. Copyright © 2010 John Wiley & Sons, Ltd.
A numerical scheme which is a combination of the dual reciprocity boundary element method (DRBEM) and the differential quadrature method (DQM), is proposed for the solution of unsteady magnetohydrodynamic (MHD) flow problem in a rectangular duct with insulating walls. The coupled MHD equations in velocity and induced magnetic field are transformed first into the decoupled time-dependent convection–diffusion-type equations. These equations are solved by using DRBEM which treats the time and the space derivatives as nonhomogeneity and then by using DQM for the resulting system of initial value problems. The resulting linear system of equations is overdetermined due to the imposition of both boundary and initial conditions. Employing the least square method to this system the solution is obtained directly at any time level without the need of step-by-step computation with respect to time. Computations have been carried out for moderate values of Hartmann number (M⩽50) at transient and the steady-state levels. As M increases boundary layers are formed for both the velocity and the induced magnetic field and the velocity becomes uniform at the centre of the duct. Also, the higher the value of M is the smaller the value of time for reaching steady-state solution. Copyright © 2005 John Wiley & Sons, Ltd.
The finite element method has been applied to the steady-state fully developed magnetohydrodynamic channel flow of a conducting fluid in the presence of transverse magnetic field. Simple elements have been used to obtain the numerical values of velocity and induced magnetic field. To test the efficiency of the method, three different geometries, viz., rectangle, circle and triangle, are taken as the section of the pipe whose walls are non-conducting. Comparison is made with those cases in which exact solutions are available. Apart from giving good results, the FEM makes it possible to solve the problem for a pipe with arbitrary cross-section which was not possible by the other methods.
The magnetohydrodynamic (MHD) flow of an incompressible, viscous, electrically conducting fluid in a rectangular duct with an external magnetic field applied transverse to the flow has been investigated. The walls parallel to the applied magnetic field are conducting while the other two walls which are perpendicular to the field are insulators. The boundary element method (BEM) with constant elements has been used to cast the problem into the form of an integral equation over the boundary and to obtain a system of algebraic equations for the boundary unknown values only. The solution of this integral equation presents no problem as encountered in the solution of the singular integral equations for interior methods. Computations have been carried out for several values of the Hartmann number (1 ⩽ M ⩽ 10). It is found that as M increases, boundary layers are formed close to the insulated boundaries for both the velocity and the induced magnetic field and in the central part their behaviours are uniform. Selected graphs are given showing the behaviours of the velocity and the induced magnetic field.
In this paper, a meshfree point collocation method, with an upwinding scheme, is presented to obtain the numerical solutions of the coupled equations in velocity and magnetic field for the fully developed magnetohydrodynamic (MHD) flow through an insulated straight duct of rectangular section. The moving least-square (MLS) approximation is employed to construct the shape functions in conjunction with the framework of the point collocation method. Computations have been carried out for different applied magnetic field orientations and a wide range of values of Hartmann number from 5 to 106. As the adaptive upwinding local support domain is introduced in the meshless point collocation method, numerical results show that the method may compute MHD problems not only at low and moderate values but also at high values of the Hartmann number with high accuracy and good convergence.
A boundary element solution is implemented for magnetohydrodynamic (MHD) flow problem in ducts with several geometrical cross-section with insulating walls when a uniform magnetic field is imposed perpendicular to the flow direction. The coupled velocity and induced magnetic field equations are first transformed into uncoupled inhomogeneous convection–diffusion type equations. After introducing particular solutions, only the homogeneous equations are solved by using boundary element method (BEM). The fundamental solutions of the uncoupled equations themselves (convection–diffusion type) involve the Hartmann number (M) through exponential and modified Bessel functions. Thus, it is possible to obtain results for large values of M (M≤300) using only the simplest constant boundary elements. It is found that as the Hartmann number increases, boundary layer formation starts near the walls and there is a flattening tendency for both the velocity and the induced magnetic field. Also, velocity becomes uniform at the center of the duct. These are the well-known behaviours of MHD flow. The velocity and the induced magnetic field contours are graphically visualized for several values of M and for different geometries of the duct cross-section.
A Galerkin finite element method and two finite difference techniques of the control volume variety have been used to study magnetohydrodynamic channel flows as a function of the Reynolds number, interaction parameter, electrode length and wall conductivity. The finite element and finite difference formulations use unequally spaced grids to accurately resolve the flow field near the channel wall and electrode edges where steep flow gradients are expected. It is shown that the axial velocity profiles are distorted into M-shapes by the applied electromagnetic field and that the distortion increases as the Reynolds number, interaction parameter and electrode length are increased. It is also shown that the finite element method predicts larger electromagnetic pinch effects at the electrode entrance and exit and larger pressure rises along the electrodes than the primitive-variable and streamfunction–vorticity finite difference formulations. However, the primitive-variable formulation predicts steeper axial velocity gradients at the channel walls and lower axial velocities at the channel centreline than the streamfunction–vorticity finite difference and the finite element methods. The differences between the results of the finite difference and finite element methods are attributed to the different grids used in the calculations and to the methods used to evaluate the pressure field. In particular, the computation of the velocity field from the streamfunction–vorticity formulation introduces computational noise, which is somewhat smoothed out when the pressure field is calculated by integrating the Navier–Stokes equations. It is also shown that the wall electric potential increases as the wall conductivity increases and that, at sufficiently high interaction parameters, recirculation zones may be created at the channel centreline, whereas the flow near the wall may show jet-like characteristics.
In Sezgin1,2 the problems considered are the magnetohydrodynamic (MHD) flows in an electrodynamically conducting infinite channel and in a rectangular duct respectively, in the presence of an applied magnetic field. In the present paper we extend the solution procedure of these papers to two rectangular channels connected by a barrier which is partially conductor and partially insulator. The problem has been reduced to the solution of a pair of dual series equations and then to the solution of a Fredholm's integral equation of the second kind. The infinite series obtained were transformed to finite integrals containing Bessel Junctions of the second kind to avoid the computations of slowly converging infinite series and infinite integrals with oscillating integrands. The results obtained compared well with those of Butsenieks and Shcherbinin3 which were obtained for the perfectly conducting barrier separating the flows.
A finite element method is given to obtain the numerical solution of the coupled equations in velocity and magnetic field for unsteady MHD flow through a pipe having arbitrarily conducting walls. Pipes of rectangular, circular and triangular sections have been taken for illustration. Computations have been carried out for different Hartmann numbers and wall conductivity at various time levels. It is found that if the wall conductivity increases, the flux through a section decreases. The same is the effect of increasing the Hartmann number. It is also observed that the steady state is approached at a faster rate for larger Hartmann numbers or larger wall conductivity. Selected graphs are given showing the behaviour of velocity, induced magnetic field and flux across a section.
The two-dimensional convection–diffusion-type equations are solved by using the boundary element method (BEM) based on the time-dependent fundamental solution. The emphasis is given on the solution of magnetohydrodynamic (MHD) duct flow problems with arbitrary wall conductivity. The boundary and time integrals in the BEM formulation are computed numerically assuming constant variations of the unknowns on both the boundary elements and the time intervals. Then, the solution is advanced to the steady-state iteratively. Thus, it is possible to use quite large time increments and stability problems are not encountered. The time-domain BEM solution procedure is tested on some convection–diffusion problems and the MHD duct flow problem with insulated walls to establish the validity of the approach. The numerical results for these sample problems compare very well to analytical results. Then, the BEM formulation of the MHD duct flow problem with arbitrary wall conductivity is obtained for the first time in such a way that the equations are solved together with the coupled boundary conditions. The use of time-dependent fundamental solution enables us to obtain numerical solutions for this problem for the Hartmann number values up to 300 and for several values of conductivity parameter. Copyright © 2007 John Wiley & Sons, Ltd.
A complete three-dimensional mathematical model has been developed governing the steady, laminar flow of an incompressible fluid subjected to a magnetic field and including internal heating due to the Joule effect, heat transfer due to conduction, and thermally induced buoyancy forces. The thermally induced buoyancy was accounted for via the Boussinesq approximation. The entire system of eight partial differential equations was solved by integrating intermittently a system of five fluid flow equations and a system of three magnetic field equations and transferring the information through source-like terms. An explicit Runge-Kutta time-stepping algorithm and a finite difference scheme with artificial compressibility were used in the general non-orthogonal curvilinear boundary-conforming co-ordinate system. Comparison of computational results and known analytical solutions in two and three dimensions demonstrates high accuracy and smooth monotone convergence of the iterative algorithm. Results of test cases with thermally induced buoyancy demonstrate the stabilizing effect of the magnetic field on the recirculating flows.
This paper presents a convection–diffusion-reaction (CDR) model for solving magnetic induction equations and incompressible Navier–Stokes equations. For purposes of increasing the prediction accuracy, the general solution to the one-dimensional constant-coefficient CDR equation is employed. For purposes of extending this discrete formulation to two-dimensional analysis, the alternating direction implicit solution algorithm is applied. Numerical tests that are amenable to analytic solutions were performed in order to validate the proposed scheme. Results show good agreement with the analytic solutions and high rate of convergence. Like many magnetohydrodynamic studies, the Hartmann–Poiseuille problem is considered as a benchmark test to validate the code. Copyright © 2004 John Wiley & Sons, Ltd.
A new analytic finite element method (AFEM) is proposed for solving the governing equations of steady magnetohydrodynamic (MHD) duct flows. By the AFEM code one is able to calculate the flow field, the induced magnetic field, and the first partial derivatives of these fields. The process of the code generation is rather lengthy and complicated, therefore, to save space, the actual formulation is presented only for rectangular ducts. A distinguished feature of the AFEM code is the resolving capability of the high gradients near the walls without use of local mesh refinement. Results of traditional FEM, AFEM and finite difference method (FDM) are compared with analytic results demonstrating the manifest superiority of the AFEM code. The programs for the AFEM codes are implemented in GAUSS using traditional computer arithmetic and work in the range of low and moderate Hartmann numbersM<1000.