Analytical solutions for the unsteady MHD rotating flow over a rotating sphere near the equator

Erik Sweet 1 , Kuppalapalle Vajravelu 1 , and Robert Gorder 1
  • 1 Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

Abstract

In this paper we investigate the three-dimensional magnetohydrodynamic (MHD) rotating flow of a viscous fluid over a rotating sphere near the equator. The Navier-Stokes equations in spherical polar coordinates are reduced to a coupled system of nonlinear partial differential equations. Self-similar solutions are obtained for the steady state system, resulting from a coupled system of nonlinear ordinary differential equations. Analytical solutions are obtained and are used to study the effects of the magnetic field and the suction/injection parameter on the flow characteristics. The analytical solutions agree well with the numerical solutions of Chamkha et al. [31]. Moreover, the obtained analytical solutions for the steady state are used to obtain the unsteady state results. Furthermore, for various values of the temporal variable, we obtain analytical solutions for the flow field and present through figures.

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  • [1] R. Hide, P.H. Roberts, Phys. Chem. Earth 4, 27 (1961) http://dx.doi.org/10.1016/0079-1946(61)90003-9

  • [2] R.H. Dieke, Annu. Rev. Astron. Astr. 88, 297 (1970) http://dx.doi.org/10.1146/annurev.aa.08.090170.001501

  • [3] W.H.H. Banks, Acta Mech. 11, 27 (1971) http://dx.doi.org/10.1007/BF01368116

  • [4] S.N. Singh, J. Appl. Mech. 41, 564 (1974) http://dx.doi.org/10.1115/1.3423349

  • [5] M.H. Rogers, G.N. Lance, J. Fluid Mech. 7, 617 (1960) http://dx.doi.org/10.1017/S0022112060000335

  • [6] D.J. Evans, Q. J. Mech. Appl. Math. 22, 467 (1969) http://dx.doi.org/10.1093/qjmam/22.4.467

  • [7] H. Ockendon, Q. J. Mech. Appl. Math. 25, 291 (1972) http://dx.doi.org/10.1093/qjmam/25.3.291

  • [8] R.J. Badonyi, J. Fluid Mech. 67, 657 (1975) http://dx.doi.org/10.1017/S0022112075000535

  • [9] D. Dijkstra, P.J. Zandbergen, J. Eng. Math. 11, 167 (1977) http://dx.doi.org/10.1007/BF01535696

  • [10] D. Dijkstra, P.J. Zandbergen, Arch. Mech. 30, 411 (1979)

  • [11] S.P. Hastings, Arch. Ration. Mech. An. 38, 308 (1970) http://dx.doi.org/10.1007/BF00281527

  • [12] J.B. McLeod, Q. J. Math. 20, 483 (1969) http://dx.doi.org/10.1093/qmath/20.1.483

  • [13] J.B. McLeod, Mathematika 17, 243 (1970) http://dx.doi.org/10.1112/S0025579300002916

  • [14] J.B. McLeod, P. R. Soc. A 324, 391 (1971) http://dx.doi.org/10.1098/rspa.1971.0146

  • [15] P.J. Bushell, J. London Math. Soc. 4, 701 (1972) http://dx.doi.org/10.1112/jlms/s2-4.4.701

  • [16] P. Hartman, Indiana U. Math. J. 21, 849 (1972) http://dx.doi.org/10.1512/iumj.1972.21.21069

  • [17] K. Stewartson, In: H. Gortler (Ed.), Boundary layer research (Springer, Gottingen Heidelberg, 1957) 59

  • [18] W.H. Banks, Q. J. Mech. Appl. Math. 18, 443 (1965) http://dx.doi.org/10.1093/qjmam/18.4.443

  • [19] W.H. Acta Mech. 24, 273 (1976) http://dx.doi.org/10.1007/BF01190376

  • [20] S.N. Phys. Fluids 13, 2452 (1970) http://dx.doi.org/10.1063/1.1692812

  • [21] S.C.R. Dennis, S.N. Singh, D.B. Ingham, J. Fluid Mech. 101, 257 (1980) http://dx.doi.org/10.1017/S0022112080001656

  • [22] D.B. Ingham, Acta Mech. 42, 111 (1982) http://dx.doi.org/10.1007/BF01176517

  • [23] S.V. Porter, K.R. Rajagopal, Arch. Ration. Mech. An. 36, 305 (1984) http://dx.doi.org/10.1007/BF00280030

  • [24] C.Y. Lai, K.R. Rajagopal, S.Z. Szeri, J. Fluid Mech. 116, 203 (1984) http://dx.doi.org/10.1017/S0022112084001828

  • [25] C.Y. Lai, K.R. Rajagopal, S.Z. Szeri, J. Fluid Mech. 157, 471 (1985) http://dx.doi.org/10.1017/S0022112085002452

  • [26] W.T. Thacker, L.T. Watson, S.K. Kumar, Appl. Math. Model. 14, 527 (1990) http://dx.doi.org/10.1016/0307-904X(90)90185-8

  • [27] A. Slaouti, H.S. Takhar, G. Nath, Acta Mech. 156, 109 (2002) http://dx.doi.org/10.1007/BF01188745

  • [28] A.I. Van de Vooren, E.F.F. Botta, J. Eng. Math. 24, 55 (1990) http://dx.doi.org/10.1007/BF00128846

  • [29] A.I. Van de Vooren, E.F.F. Botta, J. Eng. Math. 24, 261 (1990) http://dx.doi.org/10.1007/BF00058469

  • [30] M.A. Tarek, El Mishkawy, A.A. Hazem, A.M. Adel, Mech. Res. Commun. 25, 271 (1998) http://dx.doi.org/10.1016/S0093-6413(98)00038-X

  • [31] A.J. Chamkha, H.S. Takhar, G. Nath, Acta Mech. 164, 31 (2003) http://dx.doi.org/10.1007/s00707-003-0011-z

  • [32] S.J. Liao, PhD thesis, Shanghai Jiao Tong University, (Shanghai, China, 1992)

  • [33] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, (Chapman & Hall/CRC Press, Boca Raton, 2003) http://dx.doi.org/10.1201/9780203491164

  • [34] S.J. Liao, Int. J. Nonlin. Mech. 34, 759 (1999) http://dx.doi.org/10.1016/S0020-7462(98)00056-0

  • [35] S.J. Liao, Appl. Math. Comput. 147, 499 (2004) http://dx.doi.org/10.1016/S0096-3003(02)00790-7

  • [36] S.J. Liao, Y. Tan, Stud. Appl. Math. 119, 297 (2007) http://dx.doi.org/10.1111/j.1467-9590.2007.00387.x

  • [37] S.J. Liao, Commun. Nonlinear Sci. 14, 983 (2009) http://dx.doi.org/10.1016/j.cnsns.2008.04.013

  • [38] R.A. Van Gorder, K. Vajravelu, Commun. Nonlinear Sci. 14, 4078 (2009) http://dx.doi.org/10.1016/j.cnsns.2009.03.008

  • [39] M. Turkyilmazoglu, Phys. Fluids 21, 106104 (2009) http://dx.doi.org/10.1063/1.3249752

  • [40] S.J. Liao, Stud. Appl. Math. 117, 239 (2006) http://dx.doi.org/10.1111/j.1467-9590.2006.00354.x

  • [41] S.J. Liao, Commun. Nonlinear Sci. 11, 326 (2006) http://dx.doi.org/10.1016/j.cnsns.2004.09.004

  • [42] H. Xu, S.J. Liao, I. Pop, Acta Mech. 184, 87 (2006) http://dx.doi.org/10.1007/s00707-005-0302-7

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