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Research Papers: Natural and Mixed Convection

Nonlinear Rayleigh–Bénard Convection With Variable Heat Source

[+] Author and Article Information
P. G. Siddheshwar

Professor
Department of Mathematics,
Bangalore University,
Central College Campus,
Bangalore 560001, India
e-mail: mathdrpgs@gmail.com; pgsiddheshwar@bub.ernet.in

P. Stephen Titus

Associate Professor
Department of Mathematics,
St. Joseph's College,
Lalbagh Road,
Bangalore 560027, India
e-mail: titusteve@gmail.com

Contributed by the Heat Transfer Division of ASME for publication in the Journal of Heat Transfer. Manuscript received January 10, 2013; final manuscript received June 8, 2013; published online October 14, 2013. Assoc. Editor: Zhixiong Guo.

J. Heat Transfer 135(12), 122502 (Oct 14, 2013) (12 pages) Paper No: HT-13-1010; doi: 10.1115/1.4024943 History: Received January 10, 2013; Revised June 08, 2013

Linear and nonlinear Rayleigh–Bénard convections with variable heat source (sink) are studied analytically using the Fourier series. The strength of the heat source is characterized by an internal Rayleigh number, RI, whose effect is to decrease the critical external Rayleigh number. Linear theory involving an autonomous system (linearized Lorenz model) further reveals that the critical point at pre-onset can only be a saddle point. In the postonset nonlinear study, analysis of the generalized Lorenz model leads us to two other critical points that take over from the critical point of the pre-onset regime. Classical analysis of the Lorenz model points to the possibility of chaos. The effect of RI is shown to delay or advance the appearance of chaos depending on whether RI is negative or positive. This aspect is also reflected in its effect on the Nusselt number. The Lyapunov exponents provide useful information on the closing in and opening out of the trajectories of the solution of the Lorenz model in the cases of heat sink and heat source, respectively. The Ginzburg-Landau models for the problem are obtained via the 3-mode and 5-mode Lorenz models of the paper.

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References

Figures

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Fig. 1

(a) Schematic of the physical configuration and (b) temperature profile of quiescent basic state for different values of RI

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Fig. 2

The stream lines of steady convection for RI,= 0, 1, −1 (the corresponding critical value of wave number and Rayleigh number are given in the adjoining table)

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Fig. 3

The stream lines of unsteady convection for RI = 0, 1, −1 and the corresponding wave numbers

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Fig. 4

Hopf bifurcation for different values of RI

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Fig. 5

Projections of phase portraits in the A1B1/AB plane for rE = 28,Pr = 10 and for (a), (b), and (c) 5-mode, (d), (e), and (f) 3-mode

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Fig. 6

Projections of phase portraits in the A1C/AC plane for rE = 28,Pr = 10 and for (a), (b), and (c) 5-mode, (d), (e), and (f) 3-mode

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Fig. 7

Projections of phase portraits in the B1C/BC plane for rE = 28,Pr = 10 and for (a), (b), and (c) 5-mode, (d), (e), and (f) 3-mode

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Fig. 8

Nusselt number variation with τ for rE = 28,Pr = 10 and for (a), (b), and (c) 5-mode, (d), (e), and (f) 3-mode

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