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Series solutions of unsteady MHD flows above a rotating disk

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Abstract

The series solutions of unsteady flows of a viscous incompressible electrically conducting fluid caused by an impulsively rotating infinite disk are given by means of an analytic technique, namely the homotopy analysis method. Using a set of new similarity transformations, we transfer the Navier–Stokes equations into a pair of nonlinear partial differential equations. The convergent series solutions are obtained, which are uniformly valid for all dimensionless time 0 ≤  τ <  ∞ in the whole spatial region 0 ≤  η < ∞. To the best of our knowledge, such kind of series solutions have never been reported. The effect of magnetic number on the velocity is investigated.

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References

  1. Von Kármán T (1921) Über läminare und turbulence Reibung. Z Angew Math Mech 1:233–252

    Google Scholar 

  2. Cochran WG (1934) The flow due to a rotating disk. Proc Camb Phil Soc 30:365–375

    Article  MATH  Google Scholar 

  3. Thiriot HK (1940) Über die laminare Anlaufströmung einer Flüssigkeit einem rotierenden Boden bei plötzlicher Änderung des drehungszustandes. Z Angew Math Mech 20:1–13

    MATH  Google Scholar 

  4. Benton ER (1966) On the flow due to a rotating disk. J Fluid Mech 24:781–800

    Article  ADS  MATH  Google Scholar 

  5. Fettis HE (1955) On the integration of a class of differential equations occurring in boundary layer and other hydrodynamic problems. In: Proc 4th Midwestern Conf on Fluid Mech Purdue

  6. Lance GN, Rogers MH (1962) The axial symmetric flow of a viscous fluid between two infinite rotating disks. Proc Roy Soc London Ser A 266:109–121

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Mellor GL, Chapple PJ, Stokes VK (1968) On the flow between a roating and a stationary disk. J Fluid Mech 31:95–112

    Article  ADS  MATH  Google Scholar 

  8. Tam KK (1969) A note on the asymptotic solution of the flow between two oppositely rotating infinite plane disks. SIAM J Appl Math 17:1305–1310

    Article  MATH  Google Scholar 

  9. Schlichting H (1974) Boundary layer theory, 7th edn. McGraw-Hill, New York

    Google Scholar 

  10. Bodonyi RJ (1975) On the rotationlly symmetric flow above an infinite ratating disk. J Fluid Mech 67:657–666

    Article  ADS  MATH  Google Scholar 

  11. Zandbergen PJ, Dijkstra D (1977) Non-unique solutions of the Navier–Stokes equations for the Kármán swirling flow. J Eng Math 11:167–188

    Article  MathSciNet  MATH  Google Scholar 

  12. Dijkstra D (1980) On the relation between adjacent invisid cell type solutions to the rotating-disk equations. J Eng Math 14:133–154

    Article  MathSciNet  MATH  Google Scholar 

  13. Holoniok M, Kubicek M, Hlavacek V (1981) Computation of the flow between two rotating coaxial disks: multiplicity of steady-state solutions. J Fluid Mech 81:227–240

    Article  ADS  Google Scholar 

  14. Dijkstra D, Van Heijst GJF (1983) The flow between two finite rotating disks enclosed by a cylinder. J Fluid Mech 128:123–154

    Article  ADS  MATH  Google Scholar 

  15. Szeri AZ, Schneider SJ, Labbe F, Kaufman HN (1983) Flow between rotating disks, part I: basic flow. J Fluid Mech 134:103–131

    Article  ADS  Google Scholar 

  16. Bodonyi RJ, Ng BS (1984) On the stability of the similarity solutions for swirling flow above an infinite rotating disk. J Fluid Mech 144:311–328

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Yang C, Liao Sj (2006) On the explicit, purely analytic solution of Von Krmn swirling viscous flow. Commun Nonlinear Sci Num Simulat 11:83–93

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Volkan Ersoy H (2003) Unsteady flow due to concentric rotation of eccentric rotating disks. Meccanica 38:325–334

    Article  MATH  Google Scholar 

  19. Zandbergen PJ, Dijkstra D (1987) Kármán swirling flows. Ann Rev Fluid Mech 19:465–491

    Article  ADS  MathSciNet  Google Scholar 

  20. Millspas K, Pohlhausen K (1952) Heat transfer by laminar flow from a rotating disk. J Aeronaut Sci 19:120–136

    MathSciNet  Google Scholar 

  21. Sparrow EM, Cess RD (1962) Magnetohydrodynamic flow and heat transfer about a rotating disk. ASMEJ Appl Mech 29:181–187

    MathSciNet  Google Scholar 

  22. Loffredo MI (1986) Extension of vón Kàrmàn ansatz to magnetohydrodynamics. Meccanica 21:81–86

    Article  ADS  MATH  Google Scholar 

  23. Andersson HI, de Korte E (2002) MHD flow of a power-law fluid over a rotating disk. Eur J Mech B-Fluids 21:317–324

    MATH  Google Scholar 

  24. Takhar HS, Singh AK, Nath G (2002) Unsteady mhd flow and heat transfer on a rotating disk in an ambient fluid. Int J Therm Sci 41:147–155

    Article  Google Scholar 

  25. Hayat T, Hameed MI, Asghar S, Siddiqui AM (2004) Some steady MHD flows of the second order fluid. Meccanica 39:345–355

    Article  MathSciNet  MATH  Google Scholar 

  26. Hilton PJ (1953) An introduction to homotopy theory. Cambridge University Press

  27. Sen S (1983) Topology and geometry for physicists. Academic Press, Florida

    MATH  Google Scholar 

  28. Liao SJ (1992) The proposed homotopy analysis techniques for the solution of nonlinear problems. Ph.D. dissertation (in English), Shanghai Jiao Tong University, Shanghai

    Google Scholar 

  29. Liao SJ (1992) A second-order approximate analytical solution of a simple pendulum by the Process Analysis method. J Appl Mech-Trans ASME 59:970–975

    Article  MATH  Google Scholar 

  30. Liao SJ (1995) A kind of approximate solution technique which does not depend upon small parameters: a special example. Int J Non-Linear Mech 30:371–380

    Article  MATH  Google Scholar 

  31. Liao SJ (1997) A kind of approximate solution technique which does not depend upon small parameters (II): an application in fluid mechanics. Int J Mech 32:815–822

    Article  MATH  Google Scholar 

  32. Liao SJ (1999) An explicit, totally analytic approximation of Blasius viscous flow problems. Int J Non-Linear Mech 34:759–778

    Article  MATH  Google Scholar 

  33. Liao SJ (2004) On the homotopy analysis method for nonlinear problems. Appl Math Comput 147:499–513

    Article  MathSciNet  MATH  Google Scholar 

  34. Lyapunov AM (1892) General problem on stability of motion (English translation). Taylor & Francis, London, 1992

  35. Adomian G (1976) Nonlinear stochastic differential equations. J Math Anal Appl 55:441–452

    Article  MathSciNet  MATH  Google Scholar 

  36. Karmishin AV, Zhukov AT, Kolosov VG (1990) Methods of dynamics calculation and testing for thin-walled structures (in Russian). Mashinostroyenie, Moscow

    Google Scholar 

  37. Liao SJ (2003) Beyond Perturbation: Introduction to Homotopy Analysis Method. Chapman & Hall/ CRC Press, Boca Raton

    Google Scholar 

  38. He JH (1999) Homotopy perturbation technique. Methods Appl Mech Eng 178:257–262

    Article  MATH  Google Scholar 

  39. Liao SJ (2005) Comparison between the homotopy analysis method and homotopy perturbation method. Appl Math Comput 169:1186–1194

    Article  MathSciNet  MATH  Google Scholar 

  40. Liao SJ (1999) A uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat plate. J Fluid Mech 385:101–128

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Liao SJ (2002) An analytic approximation of the drag coefficient for the viscous flow past a sphere. Int J Non-linear Mech 37:1–18

    Article  MATH  Google Scholar 

  42. Liao SJ, Campo A (2002) Analytic solutions of the temperature distribution in Blasius viscous flow problems. J Fluid Mech 453:411–425

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. Li SC, Liao SJ (2005) An analytic approach to solve multiple solutions of a strongly nonlinear problem. Appl Math Comput 169:854–865

    Article  MathSciNet  MATH  Google Scholar 

  44. Xu H (2005) An explicit analytic solution for convective heat transfer in an electrically conducting fluid at a stretching surface with uniform free stream. Int J Eng Sci 43:859–874

    Article  Google Scholar 

  45. Liao SJ (2006) An analytic solution of unsteady boundary-layer flows caused by an impulsively stretching plate. Commun Nonlinear Sci Num Simulat 11:326–339

    Article  ADS  MATH  Google Scholar 

  46. Xu H, Liao SJ (2005) Series solutions of unsteady magnetohydrodynamic flows of non-Newtonian fluids caused by an impulsively stretching plate. J Fluid Mech 159:46–55

    Google Scholar 

  47. Cheng J, Liao SJ, Pop I (2005) Analytic series solution for unsteady mixed convection boundary layer flow near the stagnation point on a vertical surface in a porous medium. Transport Porous Media 61:365–379

    Article  MathSciNet  Google Scholar 

  48. Liao SJ (2005) A new branch of solutions of boundary-layer flows over a permeable stretching plate. Int J Heat Mass Transfer 48:2529–2539

    Article  MATH  Google Scholar 

  49. Liao SJ, Magyari E (in press) Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones. ZAMP

  50. Eringen AC, Maugin GA (1990) Electrodynamics of continua, vol 2. Springer, Berlin

    Google Scholar 

  51. Williams JC, Rhyne TH (1980) Boundary layer development on a wedge impulsively set into motion. SIAM J Appl Math 38:215–224

    Article  ADS  MathSciNet  MATH  Google Scholar 

  52. Cebeci T, Bradshaw P (1988) Physical and computational aspects of convective heat transfer. Springer, New York

    MATH  Google Scholar 

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Authors and Affiliations

  1. School of Naval Architecture, Ocean and Civil Engineering, Shanghai JiaoTong University, Shanghai, 200030, China

    Hang Xu & Shi-Jun Liao

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  1. Hang Xu
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  2. Shi-Jun Liao
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Correspondence to Shi-Jun Liao.

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Xu, H., Liao, SJ. Series solutions of unsteady MHD flows above a rotating disk. Meccanica 41, 599–609 (2006). https://doi.org/10.1007/s11012-006-9006-x

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