Magnetohydrodynamic pipe flow in annular-like domains
Keywords:
MHD pipe flow, EDEM, BEM, annular-like domainAbstract
The magnetohydrodynamic (MHD) pipe flow in annular-like domains with electrically conducting walls is investigated using both the extended-domain-eigenfunction method (EDEM) and the boundary element method (BEM). EDEM aims to reformulate the original problem on an extended symmetric domain obtained by transforming the inner boundary to a smaller circle towards the centre of the pipe, so that an eigenfunction solution can be obtained theoretically. By collocating only the inner circular boundary, the solution is transformed back to the original inner wall, which can be regarded as a semi-theoretical solution. On the other hand, BEM is a boundary only nature technique which transforms the differential equation into a boundary integral equation using the fundamental solution of the differential equation. Calculations are carried out for increasing values of Hartmann number (M) in annular-like domains with several shapes of inner wall at various wall conductivities. It is observed that although the results obtained by EDEM and BEM are very compatible for small M, EDEM is computationally less expensive and faster in convergence compared to BEM. However, BEM gives more accurate results than EDEM for large M due to the accumulation of numerical errors close to inner boundary in EDEM.
Downloads
References
Aarão, J., Bradshaw-Hajek, B., Miklavcic, S., & Ward, D. (2010). The extended-domaineigenfunction
method for solving elliptic boundary value problems with annular domains.
Journal of Physics A: Mathematical and Theoretical, 43, 185–202.
Aarão, J., Bradshaw-Hajek, B., Miklavcic, S., &Ward, D. (2011). Numerical implementation
of the EDEM for modified Helmholtz BVPs on annular domains. The Journal of
Computational and Applied Mathematics, 235, 1342–1353.
Barrett, K. E. (2001). Duct flow with a transverse magnetic field at high Hartmann numbers.
The International Journal for Numerical Methods in Engineering, 50, 1893–1906.
Bourantas,G.C., Skouras, E. D.,&Loukopoulos, V. C. (2009). An accurate, stable and efficient
domain-type meshless method for the solution of MHD flow problems. The Journal of
Computational Physics, 228, 8135–8160.
Bozkaya, C., & Tezer-Sezgin, M. (2007). Fundamental solution for coupled
magnetohydrodynamic flow equations. The Journal of Computational and Applied
Mathematics, 203, 125–144.
Carabineanu, A., Dinu, A., & Oprea, I. (1995). The application of the boundary element
method to the magnetohydrodynamic duct flow. ZAMP, 46, 971–981.
Carabineanu, A., & Lungu, E. (2006). Pseudospectral method for MHD pipe flow. The
International Journal for Numerical Methods in Engineering, 68, 173–191.
Cheung, Y. K., Jin, W. G., & Zienkiewicz, O. C. (1991). Solution of Helmholtz equation by
Trefftzmethod. The International Journal for NumericalMethods in Engineering, 32, 63–78.
Drago¸s, L. (1975). Magnetofluid dynamics. Tunbridge Wells, UK: Abacus Press.
Gardner, L. R. T., & Gardner, G. A. (1995). A two-dimensional bi-cubic B-spline finite
element used in a study of MHD duct flow. Computer Methods in Applied Mechanics
and Engineering, 124, 365–375.
Han-Aydın, S., & Tezer-Sezgin, M. (2014). A DRBEM solution for MHD pipe flow in a
conducting medium. The Journal of Computational and Applied Mathematics, 259, 720–
Herrera, I. (2000). Trefftz method: A general theory. Numerical Methods for Partial
Differential Equations, 16, 561–580.
Li, Z. (2008). The Trefftz method for the Helmholtz equation with degeneracy. Applied
Numerical Mathematics, 58, 131–159.
Liu, H. W., & Zhu, S. P. (2002). The dual reciprocity boundary element method for
magnetohydrodynamic channel flows. ANZIAM Journal, 44, 305–322.
Loukopoulos, V. C., Bourantas, G. C., Skouras, E. D., & Nikiforidis, G. C. (2011). Localized
meshless point collocation method for time-dependent magnetohydrodynamic flow
through pipes under a variety of wall conductivity conditions. Computational Mechanics,
, 137–159.
Polyanin, A. (2002). Handbook of linear partial differential equations for engineers and
scientists. Boca Raton, FL: Chapman & Hall/CRC.
Seungsoo, L., & Dulikravich, G. S. (1991). Magnetohydrodynamic steady flow computation
in three dimensions. International Journal for Numerical Methods in Fluids, 13, 917–936.
Shankar, P. (2005). Eigenfunction expansions on arbitrary domains. Proceedings of the Royal
Society of London A: Mathematical, Physical and Engineering Sciences, 461, 2121–2133.
Shankar, P. N. (2006). The embedding method for linear partial differential equations in
unbounded and multiply connected domains. Proceedings of the Indian Academy of Sciences
– Mathematical Sciences, 116, 361–371.
Sheu, T. W. H., & Lin, R. K. (2004). Development of a convection-diffusion-reaction
magnetohydrodynamic solver on nonstaggared grids. International Journal for Numerical
Methods in Fluids, 45, 1209–1233.
Tezer-Sezgin,M.,&Bozkaya, C. (2008).BEMsolution ofMHDflow in a rectangular ductwith
conducting walls parallel to applied magnetic field. ComputationalMechanics, 41, 769–775.
Tezer-Sezgin, M., & Dost, S. (1994). Boundary element method for MHD channel flow with
arbitrary wall conductivity. Applied Mathematical Modelling, 18, 429–436.
Tezer-Sezgin,M., &Han-Aydın, S. (2013). BEM solution ofMHDflow in a pipe coupled with
magnetic induction of exterior region. Computing, 95, 751–770
