Abstract
The first step in Fluid Mechanics was certainly carried out by Archimedes ! ( − 287, − 212) who was a mathematician and a physicist in Antiquity. He formulated a now well-known theorem which says that a body immersed in a fluid supports an upward push equal to the weight of the displaced fluid.
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- 1.
We shall often use these notations which are very handy. In Chap. 12 “Mathematical complements”, we give a summary of what is needed to go ahead with these notations. Let us recall here that we always use the implicit summation on repeated indices. Thus \(\mathbf{a} \cdot \mathbf{b} =\sum _{ i=1}^{3}a_{i}b_{i}\) is just noted \(a_{i}b_{i}\).
- 2.
One may often find in literature the terminology “specific entropy” which also means entropy per unit mass.
- 3.
This implies in particular that the stress tensor is independent of the surface on which the stress is computed. It is independent of its orientation n and its curvature radii. That would not be the case if the given surface is the seat of surface tension at the interface between a gas and a liquid. Some additional terms must be taken into account (see (1.70)).
- 4.
The existence of internal energy for a fluid element assumes that the fluid is locally at thermodynamic equilibrium. We shall come back on this point thoroughly when we discuss the constitutive relations.
- 5.
The fluids with very low viscosities are extremely interesting experimentally as they allow us to reach very high Reynolds numbers in a small size experiment. This is the reason why many experiments have been realized with helium near its critical point (2.2 bars and 5.2 K). In these conditions indeed, helium reaches its minimum viscosity. It is not a liquid, thus atoms interactions are weak and while still a gas, the velocity of atoms is minimized. The kinematic viscosity obtained in such conditions is ν ≃ 2 10−8 m2/s.
We shall not discuss the case of superfluids which needs to be approached from the side of quantum mechanics and refer the reader to the book of Guyon et al. (2001) for an introduction.
- 6.
Henri Navier (1785–1836) published this equation in 1822 in Mémoire sur les lois du mouvement des fluides, in Mém. de l’Acad. des Sciences.
- 7.
Other processes like gradients of chemical species may also generate a heat flux but these processes usually give a weak effect that will be neglected in this book.
- 8.
This assumption means, among other things, that the solid is impermeable which is not always the case. If the solid is a porous medium, some mass flux may occur through the boundary. Actually, flows through porous media are very much studied because of their numerous applications like oil or gas extraction.
- 9.
A rather complete account of the history of the quest of the correct boundary conditions at a solid wall may be found in Goldstein (1938, 1965). The irony of the story is that scientists are presently looking for materials that let the fluid slipping on the walls. This is especially important when dealing with small pipes in microfluidic (see Tabeling, 2004).
- 10.
In fact such a model rather applies to solids. The rate of strain is then replaced by the strain itself. Kelvin’s solid does not react instantaneously to a stress and reaches its equilibrium after a relaxation time τ r .
- 11.
Note that in the case of a fluid containing solid particles, the relaxation time is the characteristic time needed by a solid particle to reach the local fluid velocity when their initial velocities differ.
- 12.
Named after Eugen C. Bingham (1878–1945) who proposed the first mathematical description of these fluids.
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Rieutord, M. (2015). The Foundations of Fluid Mechanics. In: Fluid Dynamics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-09351-2_1
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