Matrix formulation of hydrodynamics and extension of ptolemaic flows to three-dimensional motions

A new matrix formulation of Lagrange hydrodynamic equations is proposed. Exact solutions of those equations are obtained in matrix form. It is found that precession of vortex lines around some fixed axis in space is a general property of the flows described by those solutions. Two types of fluid motion are studied. Flows of the first type have straight vortex lines, and their particle trajectories are windings on toroidal surfaces. The other flows have plane particle trajectories, and their vortex lines are arbitrarily shaped plane curves. All these motions are shown to be three-dimensional generalizations of plane Ptolemaic flows 1,2].Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 39, No. 6, pp. 783–796, June, 1996.The authors express their gratitude to the Russian Foundation for Fundamental Research for support of these investigations under Grant No. 96-01-00585 and to INTAS Foundation for support under Grant No. 93-1373.