Skip to main content
SpringerLink
Log in

Unsteady laminar mixed convection about a spinning sphere with a magnetic field

  • Original
  • Published:
Heat and Mass Transfer Aims and scope Submit manuscript

Abstract

The unsteady laminar boundary-layer flow over an impulsively started translating and spinning isothermal body of revolution in the presence of buoyancy force and magnetic field applied normal to the surface are investigated. Velocity components and temperature are obtained as series of functions in powers of time. Leading and first order functions are obtained analytically and second order functions are determined numerically. The general results are applied to a sphere to investigate the effects of magnetic field and buoyancy force on the velocity and temperature fields and the onset of separation. The magnetic field and buoyancy force are more effective for small rotational speeds and the presence of magnetic field retards the onset of separation. The effect of magnetic field on the temperature field and surface heat flux is weak, indirect and through the velocity field. The magnetic field is observed to initially increase the surface heat flux on the upstream face of the sphere and decrease it on the downstream face.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. Boltze E (1908) Grenzschichten an rotationskörpern in flüssigkeiten mit kleiner reibung. Thesis, Göttingen

  2. Chen TS, Mucoglu A (1977) Analysis of mixed forced and free convection about a sphere. Int J Heat Mass Transfer 20:867–875

    Article  Google Scholar 

  3. Chen TS, Mucoglu A (1978) Mixed convection about a sphere with uniform surface heat flux. ASME J Heat Transfer 100:542–544

    Google Scholar 

  4. Chu ST, Tifford AN (1954) The compressible laminar boundary layer on a rotating body of revolution. JAS 21:345–346

    Google Scholar 

  5. Dennis SCR, Walker JDA (1971) The initial flow past an impulsively started sphere at high Reynolds numbers. J Eng Math 5:263–278

    Article  Google Scholar 

  6. Dennis SCR, Walker JDA (1972) Numerical solutions for time-dependent flow past an impulsively started sphere. Phys Fluids 15:517–525

    Article  Google Scholar 

  7. Ece MC (1992) The initial boundary-layer flow past a translating and spinning rotational symmetric body. J Eng Math 26:415–428

    MathSciNet  Google Scholar 

  8. Ece MC, Öztürk A (1995) Unsteady forced convection heat transfer at the separation point on a spinning sphere In: Proceedings of the ASME 30th national heat transfer conference, vol 8. Oregon, USA pp 53–58

    Google Scholar 

  9. Hatzikonstantinou P (1990) Unsteady mixed convection about a porous rotating sphere. Int J Heat Mass Transfer 33(1):19–27

    Article  CAS  Google Scholar 

  10. Hoskin NE (1955) The laminar boundary layer on a rotating sphere. In: Tollmien W, Görtler H (eds) Fifty Years of boundary layer research. Braunschweig, pp 127–131

  11. Illingworth CR (1953) The laminar boundary layer of a rotating body of revolution. Philos Mag 44:351–389

    Google Scholar 

  12. Kocabiyik S (1996) Boundary layer on a sphere in an oscillating viscous flow. Can Appl Math Q 4:97–107

    Google Scholar 

  13. Le Palec G, Daguenet M (1987) Laminar three-dimensional mixed convection about a rotating sphere in a stream. Int J Heat Mass Transfer 30:1511–1523

    Article  CAS  Google Scholar 

  14. Lien FS, Chen K, Cleaver JW (1986) Mixed and free convection over a rotating sphere with blowing and suction. ASME J Heat Transfer 108:398–404

    Google Scholar 

  15. Luthander S, Rydberg A (1935) Experimentelle untersuchungen über den luftwiderstan bei eine mit der windrichtung parallel achse rotierenden kugel. Phys Z 36:552–558

    Google Scholar 

  16. Öztürk A, Ece MC (1995) Unsteady forced convection heat transfer from a translating and spinning body. J Energ Resour ASME 117:318–323

    Google Scholar 

  17. Öztürk A, Ece MC (1999a) Unsteady laminar mixed convection about a spinning body. J Theo Appl Fluid Mech 2(1–2):25–41

    Google Scholar 

  18. Öztürk A, Ece MC (1999b) Buoyancy effects on unsteady laminar and rotationally symmetrical boundary layer. T Can Soc Mech 23(3–4):397–408

    Google Scholar 

  19. Öztürk A, Ece MC (2002) Heat transfer reduction at the separation point on a spinning sphere in mixed convection. Int J Energ Res 26:291–303

    Article  Google Scholar 

  20. Rajasekaran R, Palekar MG (1985) Mixed convection about a rotating sphere. Int J Heat Mass Transfer 28(5):959–968

    Article  Google Scholar 

  21. Schlichting H (1953) Die laminare strömung um einen axial angeströmten rotierenden drehkörper. Ing Arch 21:227–244

    Article  Google Scholar 

  22. Takhar HS, Nath G (2000) Self similar solution of the unsteady flow in the stagnation point region of a rotating sphere with a magnetic field. Heat Mass Transfer 36:89–96

    Article  Google Scholar 

  23. Takhar HS, Chamkha AJ, Nath G (2001) Unsteady laminar MHD flow and heat transfer in the stagnation region of an impulsively spinning and translating sphere in the presence of buoyancy forces. Heat Mass Transfer 37:397–402

    Article  Google Scholar 

  24. Wang CY (1969) The impulsively starting of a sphere. Q Appl Math 27:273–277

    Google Scholar 

  25. Wang TY, Kleinstreuer C (1989) Mixed convection over rotating bodies with blowing and suction. Int J Heat Mass Transfer 32:1309–1319

    Article  CAS  Google Scholar 

Download references

Acknowledgements

The author acknowledges the valuable suggestions of the reviewers.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Trakya University, 22030, Edirne, Turkey

    Ayşegül Öztürk

Authors
  1. Ayşegül Öztürk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ayşegül Öztürk.

Appendix

Appendix

$$ F^{\prime \prime \prime}_{21} + 2\eta F^{\prime \prime }_{21} - 8F^{\prime }_{21} = 4(2F^{\prime }_0 F^{\prime }_{11} - F^{\prime \prime }_0 F_{11} - F_0 F^{\prime \prime }_{11} ) $$
$$ F^{\prime \prime \prime }_{22} + 2\eta F^{\prime \prime}_{22} - 8F^{\prime }_{22} = 4(F^{\prime }_0 F^{\prime }_{11} - F^{\prime \prime }_0 F_{11} ) $$
$$ F^{\prime \prime \prime }_{23} + 2\eta F^{\prime \prime }_{23} - 8F^{\prime }_{23} = 4(3F^{\prime }_0 F^{\prime }_{12} - F^{\prime \prime }_0 F_{11} - F_0 F^{\prime \prime }_{11} - 2F^{\prime \prime }_0 F_{12} - F_0 F^{\prime \prime }_{12} ) $$
$$ F^{\prime \prime \prime}_{24} + 2\eta F^{\prime \prime}_{24} - 8F^{\prime }_{24} = 4( - F^{\prime }_0 F^{\prime }_{12} - F_0 F^{\prime \prime}_{12} ) $$
$$ F^{\prime \prime \prime }_{25} + 2\eta F^{\prime \prime }_{25} - 8F^{\prime }_{25} = 4(F^{\prime }_0 F^{\prime }_{12} - F^{\prime \prime }_0 F_{12} ) $$
$$ F'''_{26} + 2\eta F''_{26} - 8F'_{26} = 4(F'_0 F'_{13} - 2F''_0 F_{13} - 2G_0 G_{12} - F_0 F''_{13} ) $$
$$F^{\prime \prime \prime }_{27} + 2\eta F^{\prime \prime }_{27} - 8F^{\prime} _{27}=4(F^{\prime }_0 F^{\prime }_{13} - F^{\prime \prime}_0 F_{13})$$
$$F^{\prime \prime \prime }_{28} + 2\eta F^{\prime \prime }_{28} - 8F^{\prime}_{28}=4(F^{\prime }_0 F^{\prime }_{13} - F^{\prime \prime }_{13} F_0 - 2G_0 G_{11})$$
$$F^{\prime \prime \prime }_{29} + 2\eta F^{\prime \prime}_{29} - 8F^{\prime }_{29}=4(F^{\prime }_0 F^{\prime}_{14} - F_0 F^{\prime \prime}_{14} - H_{11})$$
$$F^{\prime \prime \prime }_{210} + 2\eta F^{\prime \prime }_{210} - 8F^{\prime }_{210}=4(F^{\prime }_0 F^{\prime }_{14} - F_{14} F^{\prime \prime }_0)$$
$$F^{\prime \prime \prime }_{211} + 2\eta F^{\prime \prime }_{211} - 8F^{\prime }_{211}=4(- F^{\prime \prime }_0 F_{14} - F_0 F^{\prime \prime }_{14} - H_{11})$$
$$F^{\prime \prime \prime }_{212} + 2\eta F^{\prime \prime }_{212} - 8F^{\prime }_{212}=4(2F^{\prime }_0 F^{\prime }_{15} - F^{\prime \prime }_0 F_{15} - F_0 F^{\prime \prime }_{15} + F^{\prime }_{11})$$
$$F^{\prime \prime \prime }_{213} + 2\eta F^{\prime \prime }_{213} - 8F^{\prime }_{213}=4(- F^{\prime \prime }_0 F_{15} - F_0 F^{\prime \prime }_{15} + F^{\prime }_{12})$$
$$F^{\prime \prime \prime }_{214} + 2\eta F^{\prime \prime }_{214} - 8F^{\prime }_{214}=4(F^{\prime }_{13} - 2G_0 G_{13})$$
$$F^{\prime \prime \prime }_{215} + 2\eta F^{\prime \prime }_{215} - 8F^{\prime }_{215}=4F^{\prime }_{14}$$
$$F^{\prime \prime \prime }_{216} + 2\eta F^{\prime \prime }_{216} - 8F^{\prime }_{216}=4F^{\prime }_{15}$$
$$G^{\prime \prime }_{21} + 2\eta G^{\prime }_{21} - 8G_{21}=4(- G^{\prime }_0 F_{11} - G{\prime }_{11} F_0)$$
$$G^{\prime \prime }_{22} + 2\eta G^{\prime }_{22} - 8G_{22}=4(- G_{11} F^{\prime }_0 - G^{\prime }_0 F_{11})$$
$$G^{\prime \prime }_{23} + 2\eta G^{\prime }_{23} - 8G_{23}=4(2G_{11} F^{\prime }_0 + G_{12} F^{\prime }_0 + 2G)$$
$$G^{\prime \prime }_{24} + 2\eta G^{\prime }_{24} - 8G_{24}=4(G_{12} F^{\prime }_0 + 2G_0 F^{\prime }_{12} - G^{\prime }_{12} F_0)$$
$$G^{\prime \prime }_{25} + 2\eta G^{\prime }_{25} - 8G_{25}=4(G_{12} F^{\prime }_0 - G^{\prime }_0 F_{12})$$
$$G^{\prime \prime }_{26} + 2\eta G^{\prime }_{26} - 8G_{26}=4(2G_0 F^{\prime }_{13} - 2G^{\prime }_0 F_{13})$$
$$G^{\prime \prime }_{27} + 2\eta G^{\prime }_{27} - 8G_{27}=- 4G^{\prime }_0 F_{13}$$
$$G^{\prime \prime }_{28} + 2\eta G^{\prime }_{28} - 8G_{28}=4(2G_0 F^{\prime }_{14} - G^{\prime }_0 F_{14})$$
$$G^{\prime \prime }_{29} + 2\eta G^{\prime }_{29} - 8G_{29}=- 4G^{\prime }_0 F_{14}$$
$$G^{\prime \prime }_{210} + 2\eta G^{\prime }_{210} - 8G_{210}=4(2F^{\prime }_0 G_{13} + 2G_0 F^{\prime }_{15} - G^{\prime }_0 F_{15} - F_0 G^{\prime }_{13} + G_{12})$$
$$G^{\prime \prime }_{211} + 2\eta G^{\prime }_{211} - 8G_{211}=4(- G^{\prime }_0 F_{15} + F_0 G^{\prime }_{13} + G_{11})$$
$$G^{\prime \prime }_{212} + 2\eta G^{\prime \prime }_{212} - 8G_{212}=4G_{13}$$
$$H^{\prime \prime }_{21} + 2\eta H^{\prime }_{21} - 8H_{21}=4(H_{11} F^{\prime }_0 - H^{\prime }_0 F_{12})$$
$$H^{\prime \prime }_{22} + 2\eta H^{\prime }_{22} - 8H_{22}=4(- H_{11} F^{\prime }_0 - H^{\prime }_{11} F_0)$$
$$H^{\prime \prime }_{23} + 2\eta H^{\prime }_{23} - 8H_{23}=4(H_{11} F^{\prime}_0 - H^{\prime }_0 F_{11} - 2H^{\prime }_0 F_{12} - 2H^{\prime }_{11} F_0)$$
$$H^{\prime \prime }_{24} + 2\eta H^{\prime }_{24} - 8H_{24}=4(H_{11} F{\prime }_0 - H^{\prime }_0 F_{11})$$
$$H^{\prime \prime }_{25} + 2\eta H^{\prime }_{25} - 8H_{25}=4(- H^{\prime }_0 F_{11} - H^{\prime }_{11} F_0)$$
$$H^{\prime \prime }_{26} + 2\eta H^{\prime }_{26} - 8H_{26}=- 4H^{\prime }_0 F_{13}$$
$$H^{\prime \prime }_{27} + 2\eta H^{\prime }_{27} - 8H_{27}=- 4H^{\prime }_0 F_{14}$$
$$H^{\prime \prime }_{28} + 2\eta H^{\prime }_{28} - 8H_{28}=- 4H^{\prime }_0 F_{15}$$

Rights and permissions

Reprints and permissions

About this article

Cite this article

Öztürk, A. Unsteady laminar mixed convection about a spinning sphere with a magnetic field. Heat Mass Transfer 41, 864–874 (2005). https://doi.org/10.1007/s00231-004-0584-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00231-004-0584-7

Keywords

Search

Navigation