Abstract
We study a coupled nonlinear system of differential equation approximating the rotating MHD flow over a rotating sphere near the equator. In particular, using the Schauder fixed point theorem, we are able to establish existence of solutions. Other results on similar systems show that the question of existence in not obvious and, hence, that the present results are useful. Indeed, the work of McLeod in the 1970s shows some nonexistence results for similar problems. From here, we are also able to discuss some of the features of the obtained solutions. The observed behaviors of the solutions agree well with the numerical simulations present in the literature.
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Hide, R., Roberts, P.H.: The origin of the mean geomagnetic field. In: Physics and Chemistry of the Earth, vol. 4, pp. 27–98. Pergamon Press, New York (1961)
Dieke, R.H.: Internal rotation of the sun. In: Goldberg, L. (ed.) Annual Review of Astronomy and Astrophysics, vol. 88, pp. 297–328, Annual Review (1970)
Banks W.H.H.: A fixed sphere on the axis of an unbounded rotating fluid (R *** 1). Acta Mech. 11, 27–44 (1971)
Singh S.N.: Flow about a stationary sphere in a slowly rotating viscous fluid. J. Appl. Mech. 41, 564–570 (1974)
Rogers M.H., Lance G.N.: The rotationally symmetric flow of a viscous fluid in the presence of an infinite rotating disk. J. Fluid Mech. 7, 617–631 (1960)
Evans D.J.: The rotationally symmetric flow of a viscous fluid in the presence of an infinite rotating disk. Q. J. Mech. Appl. Math. 22, 467–485 (1969)
Ockendon H.: An asymptotic solution for steady flow above an infinite rotating disk with suction. Q. J. Mech. Appl. Math. 25, 291–391 (1972)
Badonyi R.J.: On the rotationally symmetric flow above an infinite disk. J. Fluid Mech. 67, 657–666 (1975)
Dijkstra D., Zandbergen P.J.: Non-unique solutions of the Navier–Stokes equations for the Karman swirling flow. J. Eng. Math. 11, 167–188 (1977)
Dijkstra D., Zandbergen P.J.: Some further investigations of non-unique solutions of the Navier–Stokes equations for the Karman swirling flow. Arch. Mech. 30, 411–419 (1979)
Hastings S.P.: The existence theorem for some problems from boundary layer theory. Arch. Rat. Mech. Anal. 38, 308–316 (1970)
McLeod J.B.: The asymptotic form of solutions of von Karman’s swirling flow problems. Q. J. Math. 20, 483–496 (1969)
McLeod J.B.: A note on rotationally symmetric flow above an infinite rotating disk. Mathematica 17, 243–249 (1970)
McLeod J.B.: The existence of axially symmetric flow above a rotating disk. Proc. Roy. Soc. A 324, 391–414 (1971)
Bushell P.J.: On Von Karman’s equations of swirling flow. J. Lond. Math. Soc. 4, 701–710 (1972)
Hartman P.: On the swirling flow problem. Indiana Univ. Math. J. 21, 849–855 (1972)
Stewartson K.: On rotating laminar boundary layers. In: Boundary layer research, Symposium at Freiburg, pp. 56–71. Springer, Berlin (1958)
Banks W.H.: The boundary layer on a rotating sphere. Q. J. Mech. Appl. Math. 18, 443–454 (1965)
Banks W.H.: The laminar boundary layer on a rotating sphere. Acta Mech. 24, 273–287 (1976)
Singh S.N.: Laminar boundary layer on a rotating sphere. Phys. Fluids 13, 2452–2454 (1970)
Dennis S.C.R., Singh S.N., Ingham D.B.: The steady flow due to a rotating sphere at low and moderate Reynolds number. J. Fluid Mech. 101, 257–279 (1980)
Ingham D.B.: Non-unique solutions of the boundary layer equations for the flow near the equator of a rotating sphere in a rotating fluid. Acta Mech. 42, 111–122 (1982)
Porter S.V., Rajagopal K.R.: Swirling flow between rotating plates. Arch. Rat. Mech. Anal. 36, 305–315 (1984)
Lai C.Y., Rajagopal K.R., Szeri S.Z.: Asymmetric flow between parallel rotating disks. J. Fluid Mech. 116, 203–225 (1984)
Lai C.Y., Rajagopal K.R., Szeri S.Z.: Asymmetric flow above a rotating disk. J. Fluid Mech. 157, 471–492 (1985)
Thacker W.T., Watson L.T., Kumar S.K.: Magnetohydrodynamic free convection from a disk rotating in a vertical plane. Appl. Math. Model. 14, 527–535 (1990)
Slaouti A., Takhar H.S., Nath G.: Spin-up and spin-down of a viscous fluid over a heated disk rotating in a vertical plane in the presence of a magnetic field and a buoyancy force. Acta Mech. 156, 109–129 (2002)
Vande Vooren A.I., Botta E.F.F.: Flow induced by a rotating disk of finite radius. J. Eng. Math. 24, 55–71 (1990)
Vande Vooren A.I., Botta E.F.F.: The torque required for a steady rotation of disk in a quiescent fluid. J. Eng. Math. 24, 261–286 (1990)
Tarek M.A., El Mishkawy, Hazem A.A., Adel A.M.: Asymptotic solution for the flow due to an infinite rotating disk in the case of small magnetic field. Mech. Res. Commun. 25, 271–278 (1998)
Chamkha A.J., Takhar H.S., Nath G.: Unsteady MHD rotating flow over a rotating sphere near the equator. Acta Mech. 164, 31–46 (2003)
Sweet, E., Vajravelu, K., Van Gorder, R.A.: Analytical solutions for the unsteady MHD rotating flow over a rotating sphere near the equator. Cent. Eur. J. Phys. doi:10.2478/s11534-010-0057-1
Evans L.: Partial Differential Equations, Graduate Studies in Mathematics, 2nd edn. American Mathematical Society, Providence (2010)
Necas J.: Les méthodes directes en théorie des équations elliptiques. Masson, New York (1967)
Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, vol. 224. Springer, New York (1983)
Zhang Z.: Self-similar solutions of the magnetohydrodynamic boundary layer system for a non-dilatable fluid. Zeitschrift fur Angewandte Mathematik und Physik (ZAMP) 60, 621–639 (2009)
McLeod B.: Von Karmans swirling flow problem. Arch. Ration. Mech. Anal. 33, 91–102 (1969)
Ascher U.M., Mattheij R.M.M., Russel R.D.: Numerical solution of boundary value problems for ordinary differential equations. SIAM, Philadelphia (1995)
Ascher U., Petzold L.: Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia (1998)
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Bellout, H., Vajravelu, K. & Van Gorder, R.A. Existence results for coupled nonlinear systems approximating the rotating MHD flow over a rotating sphere near the equator. Z. Angew. Math. Phys. 64, 83–100 (2013). https://doi.org/10.1007/s00033-012-0221-0
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DOI: https://doi.org/10.1007/s00033-012-0221-0