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Differential rotation in slowly dissipating systems. (English) Zbl 0105.39101
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[1] The Sun, edited by G. R. Kuiper (University of Chicago Press, Chicago, 1953), p. 20.
[2] This, by the way, furnishes athermodynamicalproof that for a freely rotating body rigid rotation corresponds to thermal equilibrium if differential rotation is associated with a dissipation. For, in this case the internal dissipation will makeEtend to diminish and its entropy increase, untilKis zero; then the body rotates rigidly, the dissipation ceases, and the entropy is a maximum. Hence, as long as there is a dissipation rigid rotation can be reached; moreover, once it has been reached the body will continue in that state, since otherwiseKmust increase again, and this can happen only if heat is transformed back into kinetic energy with a corresponding loss of entropy contrary to the second law.
[3] See, e.g., H. Lamb,Hydrodynamics(Dover Publications, Inc., New York, 1945), p. 617. The derivation of this theorem is given in Sec. V of this paper, since Lamb’s treatment is not sufficiently general for our purposes.
[4] It is likely that for the sun the initial state was an even one. Evisage a thin rotating gaseous layer which is breaking up to form the solar system. If the top and the bottom of the layers are rotating in the same sense (even state) after breaking up, the bodies formed will have their axis of rotation normal to the plane of the original layer. If, however, the top and the bottom of the layer rotate in the opposite direction, the resulting bodies will have their axis of rotation in the plane of the original layer. In the solar system, all planets except for Uranus have their axis of rotation very nearly perpendicular to the plane of their orbit.
[5] How the convexity enters can immediately be seen by the following counterexample. Suppose this proposition is true for a convex body. Construct now a new body by putting two of these together so that the north pole of one should touch the south pole of the other. This will be the new equatorial region of the new compound body which is not convex. Obviously, if originally the equatorial regions rotated faster, for the new body the equatorial region will rotate slower.
[6] If the heat conductivity cannot be neglected we proceed as follows. In Eq. (23) we divide the integrands withT, the absolute temperature, and introduce an additional term \(\kappa\)(T,iT)2d\(\tau\), the dissipation due to heat conduction; \(\kappa\) is the thermal conductivity. Now the variation leads to an additional equation for the determination ofT(x, y, z). If there are also internal heat sources present a further term (qT)d\(\tau\) is needed;qis the rate at which heat is generated in unit volume. Thus the inclusion of heat generation and heat conduction does not invalidate our variational approach; it only makes the solution more difficult.
[7] See R. Courant and D. Hilbert,Methods of Mathematical Physics(Interscience Publishers, Inc., New York, 1953), Vol. I., p. 173. · Zbl 0051.28802
[8] C. Flammer,Spheroidal Wave Functions(Stanford University Press, Stanford, California, 1957), p. 12. · Zbl 0078.05703
[9] A. N. Lowan, P. M. Morse, H. Feshbach, and M. Lax, ”Scattering and radiation from circular cylinders and sphere,” U.S. Navy Dept. (1946), p. 68. · Zbl 0061.30210
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