Fluid Gauge Theory

Tsutomu Kambe

doi:10.34257/gjsfravol21is4pg113

Abstract

According to the general gauge principle, Fluid Gauge Theory is presented to cover a wider class of flow fields of a perfect fluid without internal energy dissipation under anisotropic stress field. Thus, the theory of fluid mechanics is extended to cover time dependent rotational flows under anisotropic stress field of a compressible perfect fluid, including turbulent flows. Eulerian fluid mechanics is characterized with isotropic pressure stress fields. The study is motivated from three observations. First one is experimental observations reporting large-scale structures coexisting with turbulent flow fields. This encourages a study of how such structures observed experimentally are possible in turbulent shear flows, Second one is a theoretical and mathematical observation: the ”General solution to Euler’s equation of motion” (found by Kambe in 2013) predicts a new set of four background-fields, existing in the linked 4d-spacetime. Third one is a physical query, ”what symmetry implies the current conservation law ?”. The latter two observations encourage a gauge-theoretic formulation by defining a differential one-form representing the interaction between the fluid-current field jμand a background field aμ.

Highlights

Introduction a) Background of present research Gauge invariance is one of the fundamental symmetries in modern theoretical physics. It took almost a century for transition from the 19th-century recognition of a mathematical invariance existing in classical electromagnetic theory to the 20thcentury recognition of its fundamental physical significance
Real recognition of the gauge symmetry and its physical significance required two new fields developed in the 20th century: the relativity theory for physics of the world structure of linked 4dspacetime and the quantum mechanics for the new dimension of a phase factor in complex representation of wave function
On the basis of this perspective, the present study proposes a set of new fields to be introduced according to the gauge principle, which may be called a fluid gauge theory

Summary

Summary and Discussions

Theory of fluid mechanics is extended to cover time-dependent rotational flows under anisotropic stress field of a compressible perfect fluid, including turbulent flows. The term fkν vν is non-vanishing, and the equations governing the new field aμ should be given a physically reasonable form. This is done by introducing the third action S(F) with the total action given by. The geodesic equation governing our physical system in curved motion is given by the equation (C.23) of Appendix C.2 (c), which reduces to the modified Euler equation (2.11) obtained in Section II c) or (3.16) of Section III c) ii This is the scenario of Gauge Principle. Larger scale corresponds to the wavelength of the resonant acoustic wave, while the smaller one corresponds to an eddy structure generated by the background gauge field

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11. Kambe T 2021a