Local radial point interpolation method for the fully developed magnetohydrodynamic flow. (English) Zbl 1428.76229
Summary: In this paper, a local radial point interpolation method (LRPIM) is presented to obtain the numerical solutions of the coupled equations in velocity and magnetic field for the fully developed magnetohydrodynamic (MHD) flow through a straight duct of rectangular section with arbitrary wall conductivity and orientation of applied magnetic field. Local weak forms are developed using weighted residual method locally for the governing equations of fully developed MHD flow. The shape functions from LRPIM possess the delta function property. Therefore, essential boundary conditions can be applied as easily as that in the finite-element method. The implementation procedure of LRPIM method is node based, and it doesn’t need any “mesh” or “element”. Computations have been carried out for different Hartmann numbers, wall conductivities and orientations of applied magnetic field.
References:
| [1] | Shercliff, J. A., Steady motion of conducting fluids in pipes under transverse magnetic fields, Proc. Camb. Phil. Soc., 49, 136-144 (1953) · Zbl 0050.19404 |
| [2] | Singh, B.; Lal, J., MHD axial flow in a triangular pipe under transverse magnetic field, Ind. J. Pure Appl. Math., 9, 101-115 (1978) · Zbl 0383.76094 |
| [3] | Singh, B.; Lal, J., MHD axial flow in a triangular pipe under transverse magnetic field parallel to a side of the triangle, Ind. J. Tech., 17, 184-189 (1979) · Zbl 0413.76094 |
| [4] | Singh, B.; Lal, J., Finite element method in MHD channel flow problems, Int. J. Numer. Meth. Eng., 18, 1104-1111 (1982) · Zbl 0489.76119 |
| [5] | Singh, B.; Lal, J., Finite element method for unsteady MHD flow through pipes with arbitrary wall conductivity, Int. J. Numer. Meth. Fluids, 4, 291-302 (1984) · Zbl 0547.76119 |
| [6] | Singh, B.; Lal, J., Finite element method for MHD channel flow with arbitrary wall conductivity, J. Math. Phys. Sci., 18, 501-516 (1984) · Zbl 0574.76117 |
| [7] | Nesliturk, A. I.; Tezer-Sezgin, M., The finite element method for MHD flow at high Hartmann numbers, Comput. Methods Appl. Mech. Eng., 194, 1201-1224 (2005) · Zbl 1091.76036 |
| [8] | Verardi, S. L.L.; Cardoso, J. R.; Motta, C. C., A solution of two-dimensional magnetohydrodynamic flow using the finite element method, IEEE Trans. Magn., 34, 3134-3137 (1998) |
| [9] | Ben Salah, N.; Soulaiman, A., A finite element method for magnetohydrodynamics, Comput. Methods Appl. Mech. Eng., 190, 5867-5892 (2001) · Zbl 1044.76030 |
| [10] | Liu, W. K.; Chen, Y., Generalized multiple scale reproducing kernel particle methods, Comput. Methods Appl. Mech. Eng., 139, 91-157 (1996) · Zbl 0896.76069 |
| [11] | Babuska, I.; Melenk, J. M., The partition of unity method, Int. J. Num. Meth. Eng., 40, 727-758 (1996) · Zbl 0949.65117 |
| [12] | Duarte, C. A.; Oden, J. T., Hp-cloud – a meshless method to solve boundary-value problems, Comput. Meth. Appl. Mech. Eng., 139, 237-262 (1996) · Zbl 0918.73328 |
| [13] | Sukumar, N.; Moran, B.; Belytschko, T., The natural element method in solid mechanics, Int. J. Numer. Meth. Eng., 43, 839-887 (1998) · Zbl 0940.74078 |
| [14] | Dehghan, M.; Shokri, A., A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions, Math. Comput. Simulat., 79, 700-715 (2008) · Zbl 1155.65379 |
| [15] | Dehghan, M.; Shokri, A., Numerical solution of the nonlinear Klein-Gordon equation using radial basis functions, J. Comput. Appl. Math., 230, 400-410 (2009) · Zbl 1168.65398 |
| [16] | Dehghan, M.; Ghesmati, A., Combination of meshless local weak and strong (MLWS) forms to solve the two-dimensional hyperbolic telegraph equation, Eng. Anal. Boundary Elem., 34, 324-336 (2010) · Zbl 1244.65147 |
| [17] | Zhu, T.; Zhang, J.; Atluri, S. N., A meshless local boundary integral equation (LBIE) method for solving nonlinear problems, Comput. Mech., 22, 174-186 (1998) · Zbl 0924.65105 |
| [18] | Dehghan, M.; Mirzaei, D., Meshless local boundary integral equation (LBIE) method for the unsteady magnetohydrodynamic (MHD) flow in rectangular and circular pipes, Comput. Phys. Commun., 180, 1458-1466 (2009) · Zbl 07872387 |
| [19] | Lucy, L. B., Numerical approach to the testing of the fission hypothesis, J. Astron., 82, 1013-1024 (1977) |
| [20] | Gingold, R. A.; Monaghan, J. J., Smoothed particle hydrodynamics: theory and application to non-spherical stars, Mon. Not. R. Astron. Soc., 181, 375-389 (1977) · Zbl 0421.76032 |
| [21] | Jiang, F. M.; Monica, Oliveira S. A.; Sousa, Antonio C. M., SPH simulations for turbulence control of magnetohydrodynamic Poiseuille flow, ASME Conf. Proc., 1, 385-394 (2005) |
| [22] | Børve, S.; Omang, M.; Trulsen, J., Multidimensional MHD shock tests of regularized smoothed particle hydrodynamics, J. Astron., 652, 1306-1317 (2006) · Zbl 1195.76330 |
| [23] | Atlrui, S. N.; Zhu, T., A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput. Mech., 22, 117-127 (1998) · Zbl 0932.76067 |
| [24] | Atluri, S. N.; Shen, S., The Meshless Local Petrov-Galerkin (MLPG) Method (2002), Tech Science Press · Zbl 1012.65116 |
| [25] | Atluri, S. N.; Shen, S., The meshless local Petrov-Galerkin (MLPG) method: a simple and less-costly alternative to the finite element and boundary element methods, Comput. Model. Eng. Sci., 3, 11-51 (2002) · Zbl 0996.65116 |
| [26] | Dehghan, M.; Mirzaei, D., Meshless Local Petrov-Galerkin (MLPG) method for the unsteady magnetohydrodynamic (MHD) flow through pipe with arbitrary wall conductivity, Appl. Numer. Math., 59, 1043-1058 (2009) · Zbl 1159.76034 |
| [27] | Belytschko, T.; Lu, Y. Y.; Gu, L., Element-free Galerkin methods, Int. J. Num. Meth. Engrg., 37, 229-256 (1994) · Zbl 0796.73077 |
| [28] | Verardi, S. L.L.; M Machado, J.; Cardoso, J. R., The element-free Galerkin method applied to the study of fully developed magnetohydrodynamic duct flows, IEEE Trans. Magn., 38, 941-944 (2002) |
| [29] | Liu, G. R., Point interpolation method based on local residual formulation using radial basis functions, Struct. Eng. Mech., 14, 713-732 (2002) |
| [30] | Liu, G. R., A Local Radial point interpolation method (LRPIM) for free vibration analyses of 2D solids, J. Sound Vibr., 246, 29-46 (2001) |
| [31] | Dehghan, M.; Ghesmati, A., Numerical simulation of two-dimensional sine-Gordon solitons via a local weak meshless technique based on the radial point interpolation method (RPIM), Comput. Phys. Commun., 181, 772-786 (2010) · Zbl 1205.65267 |
| [32] | Wang, J. G., A local radial point interpolation method for dissipation process of excess pore water pressure, Int. J. Numer. Methods Heat Fluid Flow, 15, 567-587 (2005) |
| [33] | Tezer-Sezgin, M., Solution of magnetohydrodynamic flow in a rectangular duct by differential quadrature method, Comput. Fluids, 33, 533-547 (2004) · Zbl 1137.76453 |
| [34] | Xiao, J. R.; McCarthy, M. A., A local Heaviside weighted meshless method for two-dimensional solids using radial basis functions, Comput. Mech., 31, 301-315 (2003) · Zbl 1038.74680 |
| [35] | Kansa, E. J., Multiquadrics - A scattered data approximation scheme with application to computational fluid dynamics, Comput. Math. Appl., 19, 147-161 (1990) · Zbl 0850.76048 |
| [36] | Sharan, M.; Kansa, E. J.; Gupta, S., Application of the multiquadric method for numerical solution of elliptic partial differential equations, Appl. Math. Comput., 84, 275-302 (1997) · Zbl 0883.65083 |
| [37] | Frank, C.; Schaback, R., Solving partial differential equations by collocation using radial basis functions, Appl. Math. Comput., 93, 73-82 (1998) · Zbl 0943.65133 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
