An analysis of a topological criterion for the time irreducibility and stability of hydrodynamic flows



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A topological criterion for analyzing the time irreducibility and the stability of hydrodynamical flows is considered. The criterion is established by considering the propagation, by means of the Lie derivative, of a mass density along a flow vector field. A determinanta1 condition is derived by examining the constraints on the velocity field necessary to leave the mass density invariant. The resulting criterion is $ = 0 ( $ = f*ay/3T - w9f/3T). It is shown that nonzero values of $ may be an indication of diffusive dissipation, and negative values of $ are to be associated with instabilities. The main thrust of this thesis is to supply examples of $ > Oand $ < 0 that give credance to the theory. These tasks are accomplished analytically for the first category, and numerically for the second category.