The finite element method for MHD flow at high Hartmann numbers. (English) Zbl 1091.76036
Summary: A stabilized finite element method using the residual-free bubble functions (RFB) is proposed for solving the governing equations of steady magnetohydrodynamic duct flow. A distinguished feature of the RFB method is the resolving capability of high gradients near the layer regions without refining mesh. We show that the RFB method is stable by proving that the numerical method is coercive even not only at low values but also at moderate and high values of Hartmann number. Numerical results confirming theoretical findings are presented for several configurations of interest. The approximate solution obtained by the RFB method is also compared with the analytical solution of Shercliff’s problem.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
References:
| [1] | Nesliturk, A.; Harari, I., The nearly-optimal Petrov-Galerkin method for convection-diffusion problems, Comput. Methods Appl. Mech. Engrg., 192, 2501-2519 (2003) · Zbl 1039.76035 |
| [2] | Brezzi, F.; Franca, L. P.; Russo, A., Further considerations on residual-free bubbles for advective-diffusive equations, Comput. Methods Appl. Mech. Engrg., 166, 25-33 (1998) · Zbl 0934.65126 |
| [3] | Brezzi, F.; Hughes, T. J.R.; Marini, D.; Russo, A.; Suli, E., A priori error analysis of a finite element method with residual-free bubbles for advection dominated equations, SIAM J. Numer. Anal., 36, 1933-1948 (1999) · Zbl 0947.65115 |
| [4] | Brezzi, F.; Russo, A., Choosing bubbles for advection-diffusion problems, Math. Models Meth. Appl. Sci., 4, 571-587 (1994) · Zbl 0819.65128 |
| [5] | 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 |
| [6] | 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 |
| [7] | Singh, B.; Lal, J., FEM in MHD channel flow problems, Int. J. Numer. Methods Engrg., 18, 1104-1111 (1982) · Zbl 0489.76119 |
| [8] | Singh, B.; Lal, J., FEM for unsteady MHD flow through pipes with arbitrary wall conductivity, Int. J. Numer. Methods Fluids, 4, 291-302 (1984) · Zbl 0547.76119 |
| [9] | Franca, L. P.; Tobiska, L., Stability of the residual free bubble method for bilinear finite elements on rectangular grids, IMA J. Numer. Anal., 22, 73-87 (2002) · Zbl 1005.65122 |
| [10] | Shercliff, J. A., Steady motion of conducting fluids in pipes under transverse magnetic fields, Proc. Camb. Philos. Soc., 49, 136-144 (1953) · Zbl 0050.19404 |
| [11] | Dragoş, L., Magnetofluid Dynamics (1975), Abacus Press |
| [12] | Pranca, L. P.; Nesliturk, A.; Stynes, M., On the stability of residual-free bubbles for convection-diffusion problems and their approximation by a two-level finite element method, Comput. Methods Appl. Mech. Engrg., 166, 35-49 (1998) · Zbl 0934.65127 |
| [13] | Russo, A., Bubble stabilization of finite element methods for the linearized incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 132, 335-343 (1996) · Zbl 0887.76038 |
| [14] | Singh, B.; Lal, J., FEM for MHD channel flow with arbitrary wall conductivity, J. Math. Phys. Sci., 18, 501-516 (1984) · Zbl 0574.76117 |
| [15] | Tezer, M., Solution of MHD flow in a rectangular duct by differential quadrature method, Comput. Fluids, 33, 533-547 (2004) · Zbl 1137.76453 |
| [16] | Tezer-Sezgin, M., BEM solution of MHD flow in a rectangular duct, Int. J. Numer. Methods Fluids, 18, 937-952 (1994) · Zbl 0814.76063 |
| [17] | Tezer-Sezgin, M.; Koksal, S., FEM for solving MHD flow in a rectangular duct, Int. J. Numer. Methods Engrg., 28, 445-459 (1989) · Zbl 0669.76140 |
| [18] | Demendy, Z.; Nagy, T., A new algorithm for solution of equations of MHD channel flows at moderate Hartmann numbers, Acta Mech., 123, 135-149 (1997) · Zbl 0902.76058 |
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