Analytical Mechanics of Viscous Fluid.

Abstract

As an addendum to the paper “Analytical Mechanics of Viscous Fluid” [Pap. Meteor. Geophys., 42, 51-63 (1991)], the presence of the state variable φ which denotes dissipation in the law of the increase of entropy s=φ is demonstrated in assumed special model motions of viscous fluid where the viscous stress tensor and the gradient of logarithm of fluid temperature are constants. The quantity y which was introduced in the previous paper as the time integral of the entropy flow density divided by entropy density is unsuitable in the formulation because y violates Fourier's law of heat conduction. It is adequate to assign the time integral of specific entropy flow and the strain tensor as dynamical variables on which the quantity φ depends. We must assume the presence of a class of special motions that have a constant value of the viscous stress tensor and a constant value of the gradient of log θ, where θ is the temperature of a fluid particle, because d φ is an exact differential only when this assumption is satisfied. We can apply the analytical mechanics formulated in this paper only to the special motions assumed above in an exact manner. We can not apply the analytical mechanics to general motions of viscous fluid in an exact manner except the special motions. Since we can regard that the constant property of the stress tensor and the gradient of log θ during the relaxation time of molecular collisions is a sufficiently good approximation if the fluid motions are slow enough, we can expect that the analytical mechanics formulated in this paper is a good approximation to the general slow motions of the actual fluid. However, the proof of the assumption mentioned above is an open question left for future study. Since the atmosphere, the ocean and the solid earth are all dissipative systems in the field of geophysics, the extension of analytical mechanics to dissipative systems such as viscous fluid will contribute to the progress of geophysics.