Abstract
Fluid mechanics is a field theory of Newtonian mechanics of Galilean symmetry, concerned with fluid flows represented by the velocity field such as \({{\varvec{v}}}({{\varvec{x}}}, t)\) in space-time. A fluid is a medium of continuous mass. Its mechanics is formulated by extending discrete system of point masses. Associated with two symmetries (translation and space-rotation), there are two gauge fields: \({{\varvec{E}}}\equiv ({\varvec{v}}\cdot \nabla ){\varvec{v}}\) and \(H\equiv \nabla \times {\varvec{v}}\), which do not exist in the system of discrete masses. One can show that those are analogous to the electric field and magnetic field in the electromagnetism, and fluid Maxwell equations can be formulated for E and H. Sound waves within the fluid is analogous to the electromagnetic waves in the sense that phase speeds of both waves are independent of wave lengths, i.e. non-dispersive.
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Notes
- 1.
From the thermodynamics, \(\mathrm{d}h= (1/\rho ) \mathrm{d}p+T \mathrm{d}s\) where T is the temperature. If \(\mathrm{{d}}s= 0\), we have \(\mathrm{d} h= (1/\rho ) \mathrm{d}p\).
- 2.
- 3.
According to the hydrodynamic theory (e.g. Landau and Lifshitz [6]) when a solid particle moves through the fluid (at rest), the fluid energy induced by the relative particle motion of velocity \({{\varvec{u}}}= (u_i)\) is expressed in the form \(m_{ik}u_{i}u_{k}\)
by using the mass tensor \(m_{ik}\). Additional fluid momentum induced by the particle motion is given by \(m_{ik}u_{k}\).
References
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Kambe T (2010b) A new formulation of equations of compressible fluids by analogy with Maxwell’s equations. Fluid Dyn Res 42:055502
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Kambe T (2010c) Vortex sound with special reference to vortex rings: theory, computer simulations, and experiments. Aeroacoustics 9(1–2):51–89
Landau LD, Lifshitz EM (1987) Fluid mechanics, 2nd edn. Pergamon Press, Oxford
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Kambe, T. (2014). On Fluid Maxwell Equations. In: Sidharth, B., Michelini, M., Santi, L. (eds) Frontiers of Fundamental Physics and Physics Education Research. Springer Proceedings in Physics, vol 145. Springer, Cham. https://doi.org/10.1007/978-3-319-00297-2_29
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DOI: https://doi.org/10.1007/978-3-319-00297-2_29
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