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The dual reciprocity boundary element method for magnetohydrodynamic channel flows

Published online by Cambridge University Press:  17 February 2009

Huan-Wen Liu  and
Song-Ping Zhu
Huan-Wen Liu
Affiliation:
Department of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning, Guangxi 530006, PR. China; e-mail: [email protected].
Song-Ping Zhu
Affiliation:
School of Mathematics and Applied Statistics, The University of Wollongong, Wollongong, NSW 2522, Australia; e-mail: [email protected].
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Abstract

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In this paper, we consider the problem of the steady-state fully developed magnetohydrodynamic (MHD) flow of a conducting fluid through a channel with arbitrary wall conductivity in the presence of a transverse external magnetic field with various inclined angles. The coupled governing equations for both axial velocity and induced magnetic field are firstly transformed into decoupled Poisson-type equations with coupled boundary conditions. Then the dual reciprocity boundary element method (DRBEM) [20] is used to solve the Poisson-type equations. As testing examples, flows in channels of three different crosssections, rectangular, circular and triangular, are calculated. It is shown that solutions obtained by the DRBEM with constant elements are accurate for Hartmann number up to 8 and for large conductivity parameters comparing to exact solutions and solutions by the finite element method (FEM).

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 2 , October 2002 , pp. 305 - 322
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Brebbia, C. A., Tells, J. C. F. and Wrobel, L. C. (eds.), Boundary element techniques (Springer, Berlin, 1984).CrossRefGoogle Scholar
[2]Bridges, T. R. and Wrobel, L. C., “A dual reciprocity formulation for elasticity problems with body forces using augmented thin plate splines”, Comm. Numer. Methods Eng. 12 (1996) 209–220.3.0.CO;2-N>CrossRefGoogle Scholar
[3]Brunton, I. L. and Pullan, A. J., “A semi-analytic boundary element method for parabolic problem”, Eng. Anal. Boundary Elements 18 (1996) 253–264.CrossRefGoogle Scholar
[4]Coleman, C. J., Tullock, D. L. and Phan-Thien, N., “An effective boundary element method for inhomogeneous partial differential equations”, TAMP 42 (1991) 730–745.Google Scholar
[5]Duchon, J., “Splines minimizing rotation-invariant semi-norms in Sobolev spaces”, in Constructive theory of functions of several variables (Proc. Conf, Math. Res. Inst., Oberwolfach, 1976), Lecture Notes in Math. 571, (Springer, Berlin, 1977) 85–100.CrossRefGoogle Scholar
[6]Gardner, L. R. T. and Gardner, G. A., “A two dimensional bi-cubic B-spline finite element: used in a study of MHD-duct flow”, Comput. Methods Appl. Mech. Eng. 124 (1995) 365–375.CrossRefGoogle Scholar
[7]Golberg, M. A., “The method of fundamental solutions for Poisson's equation”, Eng. Anal. Boundary Elements 16 (1995) 205–213.CrossRefGoogle Scholar
L[8]Golberg, M. A., Chen, C. S. and Karur, S., “Improved multiquadric approximation for partial differential equations”, Eng. Anal. Boundary Elements 18 (1996) 9–17.CrossRefGoogle Scholar
[9]Gold, R. R., “Magnetohydrodynamic pipe flow, part. I”, J. Fluid Mech. 13 (1962) 505–512.CrossRefGoogle Scholar
[10]Itagaki, M. and Brebbia, C. A., “Generation of higher order fundamental solutions to the twodimensional modified Helmholtz equation”, Eng. Anal. Boundary Elements 11 (1993) 87–90.CrossRefGoogle Scholar
[11]Kerr, A. D., “An extension of the Kantorovich method”, Quart. Appl. Math. 26 (1968) 219.CrossRefGoogle Scholar
[12]Liu, H.-W., “Numerical modeling of the propagation of ocean waves”, Ph. D. Thesis, University of Wollongong, Australia, 2001.Google Scholar
[13]Liu, H.-W., Zhu, S.-P. and Marchant, T. R., “A perturbation DRBEM model for weakly nonlinear wave runups around islands”, accepted in principle.Google Scholar
[14]Madych, W. R. and Nelson, S. A., “Multivariate interpolation and conditionally positive definite functions II”, Math. Comp. 54 (1990) 211–230.CrossRefGoogle Scholar
[15]Nardini, D. and Brebbia, C. A., A new approach to free vibration analysis using boundary elements (Comput. Mech., Southampton, and Springer, Berlin, 1982).CrossRefGoogle Scholar
[16]Neves, A. C. and Brebbia, C. A., “The multiple reciprocity boundary element method in elasticity: A new approach for transforming domain integrals to the boundary”, Int. J. Numer. Methods Eng. 31 (1991) 709–727.CrossRefGoogle Scholar
[17]Nowak, A. J. and Brebbia, C. A., “The multiple-reciprocity method. A new approach for transforming BEM domain integrals to the boundary”, Eng. Anal. Boundary Elements 6 (1989) 164–168.CrossRefGoogle Scholar
[18]Nowak, A. J. and Brebbia, C. A., “Solving Helmholtz equation by multiple reciprocity method”, in Computer and experiments in fluid flow (eds. Carlomagno, G. M. and Brebbia, C. A.), (Comput. Mech., Southampton, 1989) 265–270.Google Scholar
[19]Partridge, P. W. and Brebbia, C. A., “Computer implementation of the BEM dual reciprocity method for the solution of Poisson-type equations”, Software Engrg. Workstations 5 (1989) 199–206.Google Scholar
[20]Partridge, P. W., Brebbia, C. A. and Wrobel, L. C., The dual reciprocity boundary element method (Comput. Mech., Southampton, and Elsevier Appl. Sci., London, 1992).Google Scholar
[21]Partridge, P. W. and Wrobel, L. C., “The dual reciprocity method for spontaneous ignition”, Int. J. Numer. Methods Eng. 30 (1990) 953–963.CrossRefGoogle Scholar
[22]Powell, M. J. D., “The uniform convergence of thin plate splines in two dimensions”, in Univ. of Cambridge Numer. Anal. Report DAMTP 1993/NA 16.Google Scholar
[23]Powell, M. J. D., “The theory of radial basis function approximation in 1990”, in Advances in numerical analysis. Vol. II (Lancaster, 1990), (Oxford Univ. Press, New York, 1992) 105–210.CrossRefGoogle Scholar
[24]Shercliff, J. A., “Steady motion of conducting fluid in pipes under transverse magnetic fields”, Math. Proc. Cambridge Philos. Soc. 49 (1953) 136–144.CrossRefGoogle Scholar
[25]Singh, B. and Lai, J., “Magnetohydrodynamic axial flow in a triangular pipe under transverse magnetic field”, Indian J. Pure Appl. Math. 9 (1978) 101–115.Google Scholar
[26]Singh, B. and Lai, J., “MHD axial flow in a triangular pipe under transverse magnetic field parallel to a side of the triangle”, Indian J. Tech. 17 (1979) 184–189.Google Scholar
[27]Singh, B. and Lai, J., “Finite element method in magnetohydrodynamic channel flow problems”, Int. J. Numer. Methods Eng. 18 (1982) 1104–1111.CrossRefGoogle Scholar
[28]Singh, B. and Lai, J., “Finite element method for MHD channel flow with arbitrary wall conductivity”, J. Math. Phys. Sci. 18 (1984) 501–516.Google Scholar
[29]Singh, B. and Lai, J., “Finite element method for unsteady MHD flow through pipes with arbitrary wall conductivity”, Int. J. Numer. Methods Fluids 4 (1984) 291–302.CrossRefGoogle Scholar
[30]Tezer-Sezgin, M., “BEM solution of MHD flow in a rectangular duct”, Int. J. Numer. Methods Fluids 18 (1994) 937–952.CrossRefGoogle Scholar
[31]Tezer-Sezgin, M. and Dost, S., “Boundary element method for MHD channel flow with arbitrary wall conductivity”, Appl. Math. Modeling 18 (1994) 429–436.CrossRefGoogle Scholar
[32]Tezer-Sezgin, M. and Dost, S., “Boundary element solution of inhomogeneous modified Helmholtz equation”, Scientia Iranica 1 (1994) 157–166.Google Scholar
[33]Tezer-Sezgin, M. and Koksal, S., “Finite element method for solving MHD flow in a rectangular duct”, Int. J. Numer. Methods Eng. 28 (1989) 445–459.CrossRefGoogle Scholar
[34]Wu, Z. and Shaback, R., “Local error estimates for radial basis function interpolation of scattered data”, IMA J. Num. Anal. 13 (1993) 13–27.CrossRefGoogle Scholar
[35]Zhang, Y. L. and Zhu, S.-P., “On the choice of interpolation functions used in the dual-reciprocity boundary-element method”, Eng. Anal. Boundary Elements 13 (1994) 387–396.CrossRefGoogle Scholar
[36]Zheng, R., Coleman, C. J. and Phan-Thien, N., “A boundary element approach for non-homogeneous potential problems”, Comp. Mech. 7 (1991) 279–288.CrossRefGoogle Scholar
[37]Zhu, S.-P. and Liu, H.-W., “On the application of the multiquadric bases in conjunction with the LTDRM method to solve nonlinear diffusion equations”, Appl. Math. Comput. 96 (1998) 161–175.Google Scholar
[38]Zhu, S.-P., Liu, H.-W. and Lu, X.-P., “A combination of the LTDRM and the ATPS in solving linear diffusion problems”, Eng. Anal. Boundary Elements 21 (1998) 285–289.CrossRefGoogle Scholar
[39]Zhu, S.-P., Satravaha, P. and Lu, X.-P., “Solving linear diffusion equations with the dual reciprocity method in Laplace space”, Eng. Anal. Boundary Elements 13 (1994) 1–10.CrossRefGoogle Scholar
[40]Zhu, S.-P. and Zhang, Y. L., “Improvement on dual reciprocity boundary element method for equations with convective terms”, Comm. Numer. Methods Eng. 10 (1994) 361–371.CrossRefGoogle Scholar