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Two-dimensional bifurcation phenomena in thermal convection in horizontal, concentric annuli containing saturated porous media

Published online by Cambridge University Press:  21 April 2006

K. Himasekhar  and
Haim H. Bau
K. Himasekhar
Affiliation:
Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104-6315, USA
Haim H. Bau
Affiliation:
Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104-6315, USA
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Abstract

A saturated porous medium confined between two horizontal cylinders is considered. As a result of a temperature difference between the cylinders, thermal convection is induced in the medium. The flow structure is investigated in a parameter space (R, Ra) where R is the radii ratio and Ra is the Darcy-Rayleigh number. In particular, the cases of R = 2, 2½, 21/4 and 2½ are considered. The fluid motion is described by the two-dimensional Darcy-Oberbeck-Boussinesq's (DOB) equations, which we solve using regular perturbation expansion. Terms up to O(Ra60) are calculated to obtain a series presentation for the Nusselt number Nu in the form \[ Nu(Ra^2) = \sum_{s=0}^{30} N_sRa^{2s}. \] This series has a limited range of utility due to singularities of the function Nu(Ra). The singularities lie both on and off the real axis in the complex Ra plane. For R = 2, the nearest singularity lies off the real axis, has no physical significance, and unnecessarily limits the range of utility of the aforementioned series. For R = 2½, 2¼ and 21/8, the singularity nearest to the origin is real and indicates that the function Nu(Ra) is no longer unique beyond the singular point.

Depending on the radii ratio, the loss of uniqueness may occur as a result of either (perfect) bifurcations or the appearance of isolated solutions (imperfect bifurcations). The structure of the multiple solutions is investigated by solving the DOB equations numerically. The nonlinear partial differential equations are converted into a truncated set of ordinary differential equations via projection. The steady-state problem is solved using Newton's technique. At each step the determinant of the Jacobian is evaluated. Bifurcation points are identified with singularities of the Jacobian. Linear stability analysis is used to determine the stability of various solution branches. The results we obtained from solving the DOB equations using perturbation expansion are compared with those we obtained from solving the nonlinear partial differential equations numerically and are found to agree well.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 187 , February 1988 , pp. 267 - 300
Copyright
© 1988 Cambridge University Press

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References

Aidun, C. K. & Steen, P. H. 1987 Transition to oscillatory convective heat transfer in a fluid-saturated porous medium. J. Thermophys. Heat Transfer 1, 268–273.Google Scholar
Baker, G. A. 1965 The theory and application of the Padé approximant method. In Advances in Theoretical Physics (ed. K. A. Brueckner), vol. 1, pp. 1–58. Academic.
Baker, G. A., & Gammel, J. L. (eds) 1970 The Padé Approximant in Theoretical Physics. Academic.
Bau, H. H. 1984 Thermal convection in a horizontal, eccentric annulus containing a saturated porous medium-an extended perturbation expansion. Intl J. Heat Mass Transfer 27, 2277–2787.Google Scholar
Bejan, A. 1984 Convection Heat Transfer, p. 355. Wiley.
Bowers, R. G. & Woolf, M. E. 1969 Some critical properties of the Heisenberg model. Phys. Rev. 177, 917–932.Google Scholar
Brailovskaya, V. A., Petrazhitskii, G. B. & Polezhaev, V. I. 1978 Natural convection and heat transfer in porous interlayers between horizontal coaxial cylinders. J. Appl. Mech. Tech. Phys. 19, 781–785.Google Scholar
Burns, P. J. & Tien, C. L. 1979 Natural convection in porous media bounded by concentric spheres and horizontal cylinders. Intl J. Heat Mass Transfer 22, 929–939.Google Scholar
Caltagirone, J. P. 1976 Thermoconvective instabilities in a porous medium bounded by two concentric horizontal cylinders. J. Fluid Mech. 76, 337–362.Google Scholar
Decker, D. W. & Keller, H. B. 1980 Solution branching - a constructive technique. In New Approaches to Nonlinear Problems in Dynamics (ed. P. J. Holmes), pp. 53–69. Philadelphia: SIAM.
Domb, C. & Sykes, M. F. 1957 On the susceptibility of a ferromagnetic above the Curie point. Proc. R. Soc. Lond. A 240, 214–228.Google Scholar
Fulks, W. 1961 Advanced Calculus. Wiley.
Gaunt, D. S. & Guttmann, A. J. 1974 Asymptotic analysis of coefficients. In Phase Transition and Critical Phenomena (ed. C. Domb & M. S. Green), pp. 181–243. Academic.
Henrici, P. 1977 Applied and Computational Complex Analysis. Wiley.
Himasekhar, K. 1987 Thermal convection in a horizontal eccentric annulus filled with a saturated porous medium. Ph.D. dissertation. University of Pennsylvania, Philadelphia.
Hunter, C. & Guerrier, B. 1980 Deducing the properties of singularities of functions from their Taylor series coefficients. SIAM J. Appl. Maths 39, 248–263.Google Scholar
Lang, S. 1985 Complex Analysis. Springer.
Lapwood, E. R. 1948 Convection of a fluid in a porous medium. Proc. Camb. Phil. Soc. 44, 508–521.Google Scholar
Liu, C. Y., Mueller, W. K. & Landis, F. 1961 Natural convection heat transfer in long horizontal cylindrical annuli. International Developments in Heat Transfer, vol. V, pp. 976–984. ASME.
Rao, Y. F., Fukuda, K. & Hasegawa, S. 1986 Steady and transient analyses of natural convection in a horizontal porous annulus with Galerkin method. Proc. AIAA/ASME 4th Thermophysics and Heat Transfer Conference, HTD-Vol. 56 (ed. V. Prasad & N. A. Hussain), pp. 95–104. Boston, MA.
Van Dyke, M. 1974 Analysis and improvement of perturbation series. Q.J. Mech. Appl. Maths 27, 423–450.Google Scholar
Van Dyke, M. 1975 Computer extension of perturbation series in fluid mechanics. SIAM J. Appl. Maths 28, 720–734.Google Scholar