Exterior Calculus: Thermodynamics and Navier-Stokes Dynamics

Abstract:

Mathematical models of dynamics employing exterior calculus are shown to be mathematical representations of the same unifying principle; namely, the description of a dynamic system with a characteristic differential one-form on an odd-dimensional differentiable manifold leads, by analysis with exterior calculus to a set of differential equations and a characteristic tangent vector which define transformations of the system [1-4]. This principle, whose origin is V. I. Arnolds use of exterior calculus to describe Hamiltonian mechanics and geometric optics, is applied to irreversible thermodynamics and Navier-Stokes dynamics. Results include (a) a set of equations for irreversible thermodynamics equivalent to Maxwells equations for reversible thermodynamics, (b) transformation of the incompressible Navier-Stokes equation into a pair of simpler equivalent equations, which are solved and (c) a characteristic tangent vector for each area of dynamics, which indicates the direction of phase flow.

• Periodical: Advanced Materials Research (Volume 747)
• Pages: 761-764
• DOI: https://doi.org/10.4028/www.scientific.net/AMR.747.761
• Online since: August 2013
• Authors: Troy L. Story
• Keywords: Exterior Calculus, Hamiltonian Geometry, Navier-Stokes, Thermodynamic

References

[1] Troy L. Story, RIMS Kokyuroku Bessatsu, B1 (2007), 365- 382.

[2] Troy L. Story, J. Chem. Phys., 88(1988), 1192–1197.

[3] Troy L. Story, J. Math. Chem., 29(2001), 85–96.

[4] V.I. Arnold, Math. Methods of Classical Mechanics (Springer, New York, 2010).