Abstract
In this chapter we derive formally the fundamental equations describing the transport of mass, momentum and energy of a one component, single phase fluid. Here, the balance equations are derived based on both an Eulerian and a Lagrangian approach. In the former, the conservation principles are enforced using a fixed volume element, as opposed to the Lagrangian approach, focused on a material volume element as it moves in time, with the fluid velocity along the fluid path lines. An easier, albeit less rigorous, derivation of the governing equations can be found in Appendix E, where an Eulerian approach is adopted, by performing the balance of mass, momentum and energy.
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Notes
- 1.
It is not a trivial statement: an integral can vanish even if its integrand is not zero. Think, for example, of the integral of x between −1 and +1. When, however, an integral is zero for any choice of the integration interval, then its integrand is identically null.
- 2.
Note that the second law can be applied because, although the material volume translates and deforms continuously as it follows the fluid motion, it corresponds to a well-defined portion of matter, with constant mass.
- 3.
Since momentum is a vector, Eq. (6.2.1) corresponds to 3 equations, with f indicating, v 1, v 2, and v 3.
- 4.
Here and in the following we adopt the Einstein convention, meaning that when an index appears twice in a single term (sometimes called a dummy index) that implies summation over all values of the index. See Appendix F.
- 5.
Named after Tullio Levi-Civita (1873–1941), an Italian mathematician. ε ijk is actually a pseudo-tensor, as it is invariant with respect to a rigid rotation of the Cartesian axes, but it changes sign upon a mirror reflection.
- 6.
In fact, T ij ∇ i v j = T ij S ij + T ij A ij = T ij S ij . In general, the double inner product between a symmetric and an anti-symmetric tensor is identically zero.
- 7.
Named after Claude-Louis Navier (1785–1836), a French engineer and physicist, who was a professor of calculus and mechanics at the École Polytechnique in Paris, and Sir George Gabriel Stokes (1819–1903), an Irish mathematician, physicist, politician and theologian, who for 54 years held the position of Lucasian professor of mathematics at Cambridge.
- 8.
See R. Mauri, Non-Equilibrium Thermodynamics in Multiphase Flows, Springer (2013).
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Mauri, R. (2015). The Governing Equations of a Simple Fluid. In: Transport Phenomena in Multiphase Flows. Fluid Mechanics and Its Applications, vol 112. Springer, Cham. https://doi.org/10.1007/978-3-319-15793-1_6
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DOI: https://doi.org/10.1007/978-3-319-15793-1_6
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