Solution of Magnetohydrodynamic Flow in a Rectangular Duct by Chebyshev Polynomial Method

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 1000 M ≤ .


Introduction
Theoretical study of magnetohydrodynamic flow problems within the ducts are frequently encountered in cooling systems of nuclear reactors, magnetohydrodynamic (MHD) generators, blood flow measurements, pumps and accelerators. Due to coupling of the equations for electrodynamics and fluid mechanics, exact solutions are possible only for some simple situations. By using several numerical techniques such as Finite Difference Method (FDM), Finite Element Method (FEM), Finite Volume Method (FVM) and Boundary Element Method (BEM), approximate solutions for the MHD flow problems can be obtained.
Singh and Lal [1][2][3] have used FDM and FEM to solve MHD flow for small Hartmann numbers M less then 10. Gardner and Gardner [4] used bi-cubic B-spline elements with FEM for Hartmann numbers less than 10. Tezer-Sezgin and Köksal [5] extended these studies to moderate Hartmann numbers up to 100 by using standard FEM with linear and quadratic elements. Further, Demendy and Nagy [6] have used the analytical finite element method to obtain numerical solution in the range of the Hartmann numbers M ≤ 1000.
Barreti [7] obtained FEM solution for the high values of M. However, he indicated a method which is computationally very memory intensive and therefore time consuming. Neslitürk and Tezer-Sezgin [8,9] solved MHD flow equations in rectangular ducts by using stabilized FEM with free bubble functions but their method is also time and memory consuming for large M numbers.
The BEM applications for solving MHD duct flow arise from the difficulties for which the solutions of huge systems are obtained in FEM due to the domain discretization. Singh and Agawal [10], Tezer-Sezgin [11], Liu and Zhu [12], Tezer-Sezgin and Han Aydın [13], Carabineanu at al. [14] and Bozkaya and Tezer-Sezgin's [15] papers are some of the publications on the BEM solutions of MHD duct flow problems for the small and moderate values of Hartmann numbers M 50 ≤ . Bozkaya and Tezer-Sezgin employed the Dual Reciprocity Boundary Element Method (DRBEM) [15,16] for non-conducting walls and also time-domain BEM [17] for arbitrary wall conductivity unsteady MHD flow.
To obtain velocity and induced magnetic field for the stady-state fully developed MHD flow for Hartmann numbers ranging from M=10 to 50, Tezer [18] has used Polynomial based Differential Quadrature (PDQ) and Fourier expansion based Differential Quadrature (FDQ) methods with equal and unequal spaced grid points.
Dehghan and Mirzaei [20] have given the meshless local boundary integral equation (LBIE) method to obtain the numerical solution of the coupled equations in velocity and magnetic field for unsteady magnetohydrodynamic (MHD) flow through a pipe of rectangular and circular sections with non-conducting walls.
Çelik [21] has represented Chebyshev Collocation Method for partial differential equation and applied to solve magnetohydrodynamic (MHD) flow equations in a rectangular duct in the presence of transverse external oblique magnetic field.
Chebyshev series solution has been developed and applied to Linear Integro-Differential Equations by Koroglu [19]. Kesan [22] has been concerned with the Chebyshev polynomial solutions of the second-order linear partial differen-tial equations.
The aim of this paper is to use Chebyshev Polynomial Method for which approximate solution is taken as truncated Chebyshev series towards obtaining velocity and induced magnetic field for the steady-state oblique external magnetic field through a rectangular duct. The MHD equations were decoupled first and then were solved for positive and negative Hartmann numbers since the sign of Hartmann number was the difference in the decoupled equations. Chebyshev Polynomial Method has high potential as an alternative compared to the other solution techniques mentioned above with Hartmann numbers ranging from M=0 to1000

Chebyshev Polynomial Method
Second order partial differential equations with variable coefficients may be expressed in the following form:  can transformed into the basic range 1 , 1 Approximate solution of the above differential equation is expressed as the truncated Chebyshev series: To obtain the solution of Eq. (1) in the form (3), first, Eq. (1) must be reduced to a differential equation whose coefficients are polynomials. For this aim, it is assumed that the functions 1, 2, 3, 4, 5, 6 A A A A A A and G can be expressed in the forms can be express as where , 0,1, 2, ... , The elements of n M are given in [10]. Substituting the expressions (4), (5) and (6) into Eq. (1) and by simplifying the result , the following equation can be obtained.
This equation can be written as  (7) can be transformed into a new matrix representation by using: ( 1) ,

X C S =
Matrix representation of boundary conditions can be obtained by using the following relations: x T I where I identity matrix. Consequently, using algebraic equation system obtained from differential equation and matrix representation of boundary conditions, we can get the main algebraic equation system. This system is solved to obtain the solution of the coefficients of the Chebyshev series.

Application to Magnetohydrodynamic Flow Problem
Basic equations of fluid mechanics and Maxwell equations of electromagnetism are well known as coupled system of equations for velocity and magnetic field. In a rectangular duct   x y in Ω Similar to regulation of matrix equation (7), above boundary conditions can be transformed to the matrix equation 0 WC = . Hence, using algebraic equation system obtained from differential equation and matrix representation of boundary conditions, we have the main algebraic equation system which has 2 N unknowns whose rank is 2 N . From the solution of this system, we can find the coefficients of the Chebyshev series and the solution function as a Chebyshev series.

Numerical Results
The Chebyshev collocation method is applied to solve the equation for

Conclusions
This is the first time that MHD flow equation has been solved in a rectangular duct by using the Chebyshev polynomial method. This method has the capability of producing highly accurate solutions using considerably small numbers for N and it is an efficient method for larges values of Hartmann numbers M ≤ 1000. From the figures, accuracy and efficiency of the method were demonstrated. Most commonly, in the previous studies, because of computational efforts, some difficulties were encountered for higher values of Hartmann numbers. In the Chebyshev polynomial method, considerably small number of N can be used so computational effort is less then others.