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Transient natural convection of micropolar fluids in concentric spherical annuli

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Summary

A numerical analysis has been performed to determine the heat transfer of micropolar fluids by natural convection between concentric spheres with isothermal boundary conditions. Calculations were carried out systematically for several radii ratios and a range of Rayleigh numbers broad enough to determine the critical value at which the mode of heat transfer changes from conduction to natural convection. The effects of the micropolar parameter (F) on the flow and temperature fields have been studied graphically. The skin friction stress on the wall has also been studied and discussed. Results indicate that the heat transfer rate of a micropolar fluid is smaller than that of a Newtonian fluid, and the main controlling parameter is the dimensionless vortex viscosity. Expressions were obtained for Nusselt numbers in the transition and convection regions as functions of Rayleigh number and radii ratio. Comparisons between a Newtonian fluid and a micropolar fluid are also made.

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Abbreviations

B :

material parameter,L 2/j

Cj :

skin friction coefficient

F :

material parameter, (κ/μ)

L :

characteristic length,L=r o−ri

\(\bar N\) N :

dimensional and dimensionless angular velocities

Nu:

local Nusselt number,h/Lk

\(\overline {Nu} \) :

average Nusselt number,\(\bar h/Lk\)

Pr:

Prandtl number

Ra:

Rayleigh number, (gβΔTL 3/vα)

Ra cr :

critical Rayleigh number

Ra g :

gap Rayleigh number

Ra gcr :

gap critical Rayleigh number

Rr :

radii ratio,Rr=r o/ri

\(\bar T\),T :

dimensional and dimensionless temperatures,\((\bar T - \bar T_o )/(\bar T_i - \bar T_o )\)

\(\bar V\) :

flow velocity

g :

local gravitational acceleration

h :

heat transfer coefficient

j :

micro-inertia density

\(\bar r\),r :

dimensional and dimensionless coordinates

r i, ro :

inner radii, outer radii

t :

time

α:

thermal diffusivity

β:

thermal expansion coefficient

γ:

spin-gradient viscosity

η:

radial coordinate in transformed plane,(r−r i)/(r o−ri)

\(\bar \theta \) π:

dimensional and dimensionless angular coordinates

κ:

vortex viscosity

λ:

material parameter, γ/jμ

μ:

dynamic viscosity

ν:

kinematic viscosity

ϱ:

fluid density

τ:

dimensionless time

\(\bar \omega \), ω:

dimensional and dimensionless vorticities

\(\bar \psi \), Ψ:

dimensional and dimensionless stream function

i, o :

inner and outer wall

max:

maximum

References

  1. Bishop, E. H., Mack, L. R., Scanlan, J. A.: Heat transfer by natural convection between concentric spheres. Int. J. Heat Mass Transfer9, 649–662 (1966).

    Google Scholar 

  2. Scanlan, J. A., Bishop, E. H., Power, R. H.: Natural convection heat transfer between concentric spheres. Int. J. Heat Mass Transfer13, 1857–1872 (1970).

    Google Scholar 

  3. Mack, L. R., Hardee, H. C.: Natural convection between concentric spheres at low Rayleigh numbers. Int. J. Heat Mass Transfer11, 387–396 (1968).

    Google Scholar 

  4. Astill, K. N., Leong, H., Martorana, R.: A numerical solution for natural convection in concentric spherical annuli. Proceedings of the 19th National Heat Transfer Conference, ASME HTD-Vol. 8 pp. 105–113 (1980).

    Google Scholar 

  5. Garg, V. K.: Natural convection between concentric spheres. Int. J. Heat Mass Transfer35, 1935–1945 (1992).

    Google Scholar 

  6. Chu, H. S., Lee, T. S.: Transient natural convection heat transfer between concentric spheres. Int. J. Heat Mass Transfer36, 3159–3170 (1993).

    Google Scholar 

  7. Chiu, C. P., Chen, W. R.: Transient natural convection heat transfer between concentric and vertically eccentric spheres. Int. J. Heat Mass Transfer39, 1439–1452 (1996).

    Google Scholar 

  8. Chiu, C. P., Chen, W. R.: Transient natural convection between concentric and vertically eccentric spheres with mixed boundary conditions. Heat Mass Transfer31, 137–143 (1996).

    Google Scholar 

  9. Eringen, A. C.: Simple microfluids. Int. J. Eng. Sci.2, 205–217 (1964).

    Google Scholar 

  10. Eringen, A. C.: Theory of micropolar fluids. J. Math. Mech.16, 1–18 (1966).

    Google Scholar 

  11. Eringen, A. C.: Theory of thermomicrofluids. J. Math. Anal. Appl.38, 480–495 (1972).

    Google Scholar 

  12. Ariman, T., Turk, M. A., Sylvester, N. D.: Microcontinuum fluid mechanics — a review. Int. J. Eng. Sci.11, 905–930 (1973).

    Google Scholar 

  13. Jena, S. K., Bhattacharyya, S. P.: The effect of microstructure on the thermal convection in a rectangular box of fluid heated from below. Int. J. Eng. Sci.24, 69–78 (1986).

    Google Scholar 

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Author notes

  1. C. P. Chiu, J. Y. Shich & W. R. Chen

    Present address: Department of Mechanical Engineering, National Cheng Kung University, No. 1, Ta Hsieh Road, Tainan, Taiwan, Republic of China

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    1. C. P. Chiu
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    2. J. Y. Shich
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    3. W. R. Chen
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    Chiu, C.P., Shich, J.Y. & Chen, W.R. Transient natural convection of micropolar fluids in concentric spherical annuli. Acta Mechanica 132, 75–92 (1999). https://doi.org/10.1007/BF01186961

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    • DOI: https://doi.org/10.1007/BF01186961

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