Convection equations
Consider a fluid flow in a layer of uniform depth where the temperature difference, , between the upper and lower bounding planes is kept constant. Such a system has a steady-state solution in which there is no fluid motion and the temperature varies linearly. If this solution is unstable, convection should develop. When all motion is parallel to the
-plane, the governing equations are [a1]:
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where is the height of the layer (in the
-direction),
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stands for the Jacobian determinant, is a stream function for the two-dimensional fluid motion, and
is the deviation of the temperature from the case where no convection occurs. The coefficients
,
,
,
are, respectively, kinematic viscosity, gravity acceleration, thermal expansion, and thermal conductivity. The part of the first equation that does not depend upon
is the third component of the vorticity equation
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where is the velocity vector and
is the vorticity.
See [a4], [a5], and also Curl and Vector product.
By expanding and
in double Fourier series with coefficients depending on
and truncating to three terms, the Lorenz equations result [a2].
References
[a1] | B. Saltzman, "Finite amplitude free convection as an initial value problem. I" J. Atmos. Sci. , 19 (1962) pp. 329–341 |
[a2] | E.N. Lorenz, "Deterministic non-periodic flow" J. Atmos. Sci. , 20 (1963) pp. 130–141 |
[a3] | G.K. Batchelor, "An introduction to fluid dynamics" , Cambridge Univ. Press (1967) |
[a4] | B.K. Shivamoggi, "Theoretical fluid dynamics" , Martinus Nijhoff (1985) pp. 13–14 |
[a5] | "Modern developments in fluid dynamics" S. Goldstein (ed.) , Dover, reprint (1965) pp. 114 |
Convection equations. M. Hazewinkel (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Convection_equations&oldid=12179