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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

Drazin, P. G., and Reid, D. H., 2004, Hydrodynamic Stability, Cambridge University Press, Cambridge, UK.
Platten, J. K., and Legros, J. C., 1984, Convection in Liquids, Springer, Berlin.
Straughan, B., 2004, The Energy Method, Stability, and Nonlinear Convection, 2nd ed., Springer-Verlag, New York.
Getling, A. V., 2001, Rayleigh-Benard Convection: Structures and Dynamics, World Scientific Press, Singapore.
Roberts, P. H., 1967, “Convection in Horizontal Layers With Internal Heat Generation,” J. Fluid Mech., 30, pp. 33–49. [CrossRef]
Thirlby, R., 1970, “Convection in an Internally Heated Layer,” J. Fluid Mech., 44, pp. 673–693. [CrossRef]
McKenzie, D. P., Roberts, J. M., and Weiss, N. O., 1974, “Convection in the Earth's Mantle: Towards a Numerical Simulation,” J. Fluid Mech., 62, pp. 465–538. [CrossRef]
Tveitereid, M., and Palm, E., 1976, “Convection Due to Internal Heat Sources,” J. Fluid Mech., 76(3), pp. 481–499. [CrossRef]
Clever, R. M., 1977, “Heat Transfer and Stability Properties of Convection Rolls in an Internally Heated Fluid Layer,” Z. Angew Math. Phys., 28, pp. 585–597. [CrossRef]
Riahi, N., 1984, “Nonlinear Convection in a Horizontal Layer With an Internal Heat Source,” J. Phys. Soc. Jpn, 53, pp. 4169–4178. [CrossRef]
Riahi, D. N., and Hsui, A. T., 1986, “Nonlinear Double Diffusive Convecton With Local Heat Source and Solute Sources,” Int. J. Eng. Sci., 24, pp. 529–544. [CrossRef]
Krishnamurti, R., 1997, “Convection Induced by Selective Absorption of Radiation: A Laboratory Model of Conditional Instability,” Dyn. Atmos. Oceans, 27, pp. 367–382. [CrossRef]
Tse, K. L., and Chasnov, J. R., 1998, “A Fourier Hermite Pseudo Spectral Method for Penetrative Convection,” J. Comput. Phys.142, pp. 489–505. [CrossRef]
Chasnov, J. R., and Tse, K. L., 2001, “Turbulent Penetrative Convection With an Internal Heat Source,” Fluid Dyn. Res., 28, pp. 397–421. [CrossRef]
Zhang, K. K., and Schubert, G., 2002, “From Penetrative Convection to Teleconvection,” Astrophys. J., 572, pp. 461–476. [CrossRef]
Straughan, B., 2002, “Sharp Global Nonlinear Stability for Temperature-Dependent Viscosity Convection,” Proc. R. Soc. London A, 458, pp. 1773–1782. [CrossRef]
Hill, A. A., 2004, “Penetrative Convection Induced by the Absorption of Radiation With a Nonlinear Internal Heat Source,” Dyn. Atmos. Oceans, 38, pp. 57–67. [CrossRef]
Siddheshwar, P. G., Sekhar, G. N., and Jayalatha, G., 2010, “Analytical Study of Convection in Jeffreys Liquid With a Heat Source,” Proceedings of the 37th International and 4th National Conference on Fluid Mechanics and Fluid Power, Paper No. FMFP10HT07, 481, pp. 1–10.
Veronis, G., 1966, “Motions at Subcritical Values of the Rayleigh Number in a Rotating Fluid,” J. Fluid Mech., 24, pp. 545–554. [CrossRef]
Siddheshwar, P. G., Sekhar, G. N., and Jayalatha, G., 2010, “Effect of Time-Periodic Vertical Oscillations of the Rayleigh-Bénard System on Nonlinear Convection in Viscoelastic Liquids,” J. Non-Newtonian Fluid Mech., 165, pp. 1412–1418. [CrossRef]
Hill, A. A., and Malashetty, M. S., 1981, “An Operative Method to Obtain Sharp Nonlinear Stability for Systems With Spatially Dependent Coefficients,” Proc. R. Soc. A, 468, pp. 323–336. [CrossRef]
Busse, F. H., 1982, “Thermal Convection in Rotating Systems,” Proceedings of U.S. National Congress of Applied Mechanics, American Society of Mechanical Engineers, pp. 299–305.
Knobloch, E. S., 1998, “Rotating Convection: Recent Developments,” Int. J. Eng. Sci., 36, pp. 1421–1460. [CrossRef]
Krishnamurthi, R., and Howard, L. N., 1981, “Large-Scale Flow Generation in Turbulent Convection,” Proc. Natl. Acad. Sci. U.S.A., 78(4), pp. 1981–1985. [CrossRef] [PubMed]
Rajagopal, K. R., Ruzicka, M., and Srinivasa, A. R., 1996, “On the Oberbeck—Boussinesq Approximation,” Math. Models Meth. Appl. Sci., 06, pp. 1157–1167. [CrossRef]
Chandrasekhar, S., 1961, Hydrodynamic and Hydromagnetic Stability, Oxford University Press, Oxford.
Lorenz, E. N., 1963, “Deterministic Non-Periodic Flow,” J. Atmos. Sci., 20, pp. 130–141. [CrossRef]
Siddheshwar, P. G., and Radhakrishna, D., 2012, “Linear and Nonlinear Electroconvection Under AC Electric Field,” Commun. Nonlinear Sci. Numer. Simul., 17(7), pp. 2883–2895. [CrossRef]
Laroze, D., Siddheshwar, P. G., and Pleiner, H., 2013, “Chaotic Convection in a Ferrofluid,” Commun. Nonlinear Sci. Numer. Simul., 18(9), pp. 2436–2447. [CrossRef]
Simmons, G. F., 1974, Differential Equations With Applications and Historical Notes, McGraw-Hill, Inc., New York.
Chen, Z. M., and Price, W. G., 2006, “On the Relation Between Rayleigh—Bénard Convection and Lorenz System,” Chaos, Solitons Fractals, 28(2), pp. 571–578. [CrossRef]
Cheung, F. B., 1980, “Heat Source-Driven Thermal Convection at Arbitrary Prandtl Number,” J. Fluid Mech., 97(4), pp. 743–768. [CrossRef]
Watson, P. M., 1968, “Classical Cellular Convection With a Spatial Heat Source,” J. Fluid Mech., 32, pp. 399–411. [CrossRef]
Sparrow, C., 1981, The Lorenz Equations: Bifurcations, Chaos and Strange Attractors, Springer, New York.
Siddheshwar, P. G., 2010, “A Series Solution for the Ginzburg-Landau Equation With a Time-Periodic Coefficient,” Appl. Math., 1(6), pp. 542–554. [CrossRef]
Bhadauria, B. S., Siddheshwar, P. G., and Suthar, Om. P., 2012, “Nonlinear Thermal Instability in a Rotating Viscous Fluid Layer Under Temperature/Gravity Modulation,” ASME J. Heat Transfer, 134, p. 102502. [CrossRef]
Bhadauria, B. S., Bhatia, P. K., and Debnath, L., 2009, “Weakly Non-Linear Analysis of Rayleigh-Bénard Convection With Time-Periodic Heating,” Int. J. Non-Linear Mech., 44(1), pp. 58–65. [CrossRef]
Alligood, K. T., Sauer, T. D., and Yorke, J. A., 1997Chaos, Springer, New York.
Kapitaniak, T., 2000, Chaos for Engineers, Springer, New York.

Figures

Grahic Jump Location
Fig. 1

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

Grahic Jump Location
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)

Grahic Jump Location
Fig. 3

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

Grahic Jump Location
Fig. 4

Hopf bifurcation for different values of RI

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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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