Abstract
Low Prandtl number convection in porous media is relevant to modern applications of transport phenomena in porous media such as the process of solidification of binary alloys. The transition from steady convection to chaos is analysed by using Adomian's decomposition method to obtain an analytical solution in terms of infinite power series. The practical need to evaluate the solution and obtain numerical values from the infinite power series, the consequent series truncation, and the practical procedure to accomplish this task, transform the analytical results into a computational solution evaluated up to a finite accuracy. The solution shows a transition from steady convection to chaos via a Hopf bifurcation producing a 'solitary limit cycle’ which may be associated with an homoclinic explosion. This occurs at a slightly subcritical value of Rayleigh number, the critical value being associated with the loss of linear stability of the steady convection solution. Periodic windows within the broad band of parameter regime where the chaotic solution persists are identified and analysed. It is evident that the further transition from chaos to a high Rayleigh number periodic convection occurs via a period halving sequence of bifurcations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adomian, G.: 1988, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl. 135, 501-544.
Adomian, G.: 1994, Solving Frontier Problems in Physics: The Decomposition Method, Kluwer Academic Publishers, Dordrecht.
Bejan, A.: 1995, Convection Heat Transfer, 2nd edn, Wiley, New York.
Choudhury, R. S.: 1997, Stability conditions for the persistence, disruption and decay of twodimensional dissipative three-mode patterns in moderately extended nonlinear systems and comparisons with simulations, In: L. Debnath and S. R. Choudhury (<nt>eds</nt>), Nonlinear Instability Analysis Advances in Fluid Mechanics, Computational Mechanics Publ., pp. 43-91.
Gheorghita, St. I.: 1966, Mathematical Methods in Underground Hydro-Gaso-Dynamics, Romanian's Academy Ed., Bucharest (in Romanian).
Kimura, S., Schubert, G. and Straus, J. M.: 1986, Route to chaos in porous-medium thermal convection, J. Fluid Mech. 166, 305-324.
Lorenz, E. N.: 1963, Deterministic non-periodic flows, J. Atmos.Sci. 20, 130-141.
Malkus, W. V. R.: 1972, Non-periodic convection at high and low Prandtl number, Mem. Soc. R. Sci. Liege 4(6), 125-128.
Nield, D. A. and Bejan, A.: 1998, Convection in Porous Media, 2nd edn, Springer-Verlag, New York.
Olek, S.: 1994, An accurate solution to the multispecies Lotka-Volterra equations, SIAM Rev. 36, 480-488.
Olek, S.: 1997, Solution to a class of nonlinear evolution equations by Adomian's decomposition method (<nt>manuscript in preparation</nt>).
R´epaci, A.: 1990, Non-linear dynamical systems: on the accuracy of Adomian's decomposition method, Appl. Math. Lett. 3, 35-39.
Sparrow, C.: 1982, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Springer-Verlag, New York.
Vadasz, P. and Olek, S.: 1998a, Transitions and chaos for free convection in a rotating porous layer, Int. J. Heat Mass Transfer 14(11), 1417-1435.
Vadasz, P. and Olek, S.: 1998b, Route to chaos for moderate Prandtl number convection in a porous layer heated from below (<nt>submitted for publication</nt>).
Vadasz, P.: 1998, Local and global transitions to chaos in a porous layer heated from below (submitted for publication).
Schubert, G. and Straus, J. M.: 1982, Transitions in time-dependent thermal convection in fluidsaturated porous media, J. Fluid Mech. 121, 301-313.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Vadasz, P., Olek, S. Weak Turbulence and Chaos for Low Prandtl Number Gravity Driven Convection in Porous Media. Transport in Porous Media 37, 69–91 (1999). https://doi.org/10.1023/A:1006522018375
Issue Date:
DOI: https://doi.org/10.1023/A:1006522018375