In this paper, we investigated the onset of natural convection in a horizontal fluid layer due to nonuniform internal heat generation, which is relevant to a number of geophysical situations. We investigated a number of special cases, which we believe are paradigmatic. Those include linear, quadratic, concentrated, and exponential source strength distributions. Our results show that those situations that lead to a reduction/increase in the size of the region in which the basic temperature gradient is destabilizing lead to an increase/decrease in stability.
Issue Section:
Natural and Mixed Convection
References
1.
Sparrow
, E. M.
, Goldstein
, R. J.
, and Jonsson
, V. K.
, 1964
, “Thermal Instability in a Horizontal Fluid Layer—Effect of Boundary Conditions and Non-Linear Temperature Profile
,” J. Fluid Mech.
, 18
(4
), pp. 513
–528
.2.
Roberts
, P. H.
, 1967
, “Convection in Horizontal Layers With Internal Heat Generation. Theory
,” J. Fluid Mech.
, 30
(01), pp. 33
–49
.3.
Tritton
, D. J.
, and Zarraga
, M. N.
, 1967
, “Convection in Horizontal Layers With Internal Heat Generation. Experiments
,” J. Fluid Mech.
, 30
(01), pp. 21
–31
.4.
Kulacki
, F. A.
, and Goldstein
, R. J.
, 1975
, “Hydrodynamic Instability in Fluid Layers With Uniform Volumetric Energy-Sources
,” Appl. Sci. Res.
, 31
(2
), pp. 81
–109
.5.
Kulacki
, F.
, and Goldstein
, R. J.
, 1972
, “Thermal Convection in a Horizontal Fluid Layer With Uniform Volumetric Energy Sources
,” J. Fluid Mech.
, 55
(02), pp. 271
–287
.6.
Shivakumara
, I. S.
, and Suma
, S. P.
, 2000
, “Effects of Throughflow and Internal Heat Generation on the Onset of Convection in a Fluid Layer
,” Acta Mech.
, 140
(3–4
), pp. 207
–217
.7.
Perekattu
, G. K.
, and Balaji
, C.
, 2009
, “On the Onset of Natural Convection in Differentially Heated Shallow Fluid Layers With Internal Heat Generation
,” Int. J. Heat Mass Transfer
, 52
(19–20
), pp. 4254
–4263
.8.
Yadav
, D.
, Kim
, C. H.
, and Lee
, J.
, 2015
, “Influence of a Magnetic Field on the Onset of Nanofluid Convection Induced by Purely Internal Heating
,” Comput. Fluids
, 121
, pp. 26
–36
.9.
Yücel
, A.
, and Bayazitoğlu
, Y.
, 1979
, “Onset of Convection in Fluid Layers With Nonuniform Volumetric Energy-Sources
,” ASME J. Heat Transfer
, 101
(4
), pp. 666
–671
.10.
Liu
, P.
, 1996
, “Onset of Benard–Marangoni Convection in a Rotating Liquid Layer With Nonuniform Volumetric Energy Sources
,” Int. J. Heat Fluid Flow
, 17
(6
), pp. 579
–586
.11.
Tasaka
, Y.
, and Takeda
, Y.
, 2005
, “Effects of Heat Source Distribution on Natural Convection Induced by Internal Heating
,” Int. J. Heat Mass Transfer
, 48
(6
), pp. 1164
–1174
.12.
Chatterjee
, S.
, Basak
, T.
, and Das
, S. K.
, 2008
, “Onset of Natural Convection in a Rotating Fluid Layer With Non-Uniform Volumetric Heat Sources
,” Int. J. Therm. Sci.
, 47
(6
), pp. 730
–741
.13.
Straughan
, B.
, 1990
, “Continuous Dependence on the Heat-Source and Nonlinear Stability for Convection With Internal Heat-Generation
,” Math. Methods Appl. Sci.
, 13
(5
), pp. 373
–383
.14.
Tritton
, D. J.
, 1975
, “Internally Heated Convection in Atmosphere of Venus and in Laboratory
,” Nature
, 257
(5522
), pp. 110
–111
.15.
Moresi
, L.
, Gurnis
, M.
, and Zhong
, S. J.
, 2000
, “Plate Tectonics and Convection in the Earth's Mantle: Toward a Numerical Simulation
,” Comput. Sci. Eng.
, 2
(3
), pp. 22
–33
.16.
Nield
, D. A.
, and Kuznetsov
, A. V.
, 2015
, “The Onset of Convection in a Horizontal Porous Layer With Spatially Non-Uniform Internal Heating
,” Transp. Porous Media
, 111
(2
), pp. 541
–553
.17.
Chandrasekhar
, S.
, 1961
, Hydrodynamic and Hydromagnetic Stability
, Clarendon Press
, Oxford, UK
.18.
Straughan
, B.
, 2008
, Stability and Wave Motion in Porous Media
, Vol. 165
, Springer, New York, pp. 1
–437
.19.
Rees
, D. A. S.
, and Bassom
, A. P.
, 2000
, “The Onset of Darcy-Benard Convection in an Inclined Layer Heated From Below
,” Acta Mech.
, 144
(1–2
), pp. 103
–118
.20.
Barletta
, A.
, and Storesletten
, L.
, 2011
, “Thermoconvective Instabilities in an Inclined Porous Channel Heated From Below
,” Int. J. Heat Mass Transfer
, 54
(13–14
), pp. 2724
–2733
.21.
Barletta
, A.
, Celli
, M.
, and Kuznetsov
, A. V.
, 2012
, “Heterogeneity and Onset of Instability in Darcy's Flow With a Prescribed Horizontal Temperature Gradient
,” ASME J. Heat Transfer
, 134
(4
), p. 042602
.22.
Driscoll
, T. A.
, Bornemann
, F.
, and Trefethen
, L. N.
, 2008
, “The Chebop System for Automatic Solution of Differential Equations
,” Bit Numer. Math.
, 48
(4
), pp. 701
–723
.23.
Driscoll
, T. A.
, 2010
, “Automatic Spectral Collocation for Integral, Integro-Differential, and Integrally Reformulated Differential Equations
,” J. Comput. Phys.
, 229
(17
), pp. 5980
–5998
.24.
Straughan
, B.
, 2004
, The Energy Method, Stability, and Nonlinear Convection
, 2nd ed., Springer
, New York
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