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.

References

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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