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Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3 + 1 Spacetime Perspective

1
Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6354, USA
2
Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996-1200, USA
Received: 12 November 2018 / Revised: 17 December 2018 / Accepted: 21 December 2018 / Published: 26 December 2018
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Abstract

A kinetic theory of classical particles serves as a unified basis for developing a geometric 3 + 1 spacetime perspective on fluid dynamics capable of embracing both Minkowski and Galilei/Newton spacetimes. Parallel treatment of these cases on as common a footing as possible reveals that the particle four-momentum is better regarded as comprising momentum and inertia rather than momentum and energy; and, consequently, that the object now known as the stress-energy or energy-momentum tensor is more properly understood as a stress-inertia or inertia-momentum tensor. In dealing with both fiducial and comoving frames as fluid dynamics requires, tensor decompositions in terms of the four-velocities of observers associated with these frames render use of coordinate-free geometric notation not only fully viable, but conceptually simplifying. A particle number four-vector, three-momentum ( 1 , 1 ) tensor, and kinetic energy four-vector characterize a simple fluid and satisfy balance equations involving spacetime divergences on both Minkowski and Galilei/Newton spacetimes. Reduced to a fully 3 + 1 form, these equations yield the familiar conservative formulations of special relativistic and non-relativistic fluid dynamics as partial differential equations in inertial coordinates, and in geometric form will provide a useful conceptual bridge to arbitrary-Lagrange–Euler and general relativistic formulations. View Full-Text
Keywords: Newtonian and non-Newtonian fluids; statistical and kinetic theory of fluids Newtonian and non-Newtonian fluids; statistical and kinetic theory of fluids
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Cardall, C.Y. Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3 + 1 Spacetime Perspective. Fluids 2019, 4, 1.

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