Abstract
At all scales, from galaxies to elementary particles, matter is discrete. Yet, the “continuum” paradigm is still of paramount importance, both in theory and in practice. This chapter briefly elucidates this paradox, with emphasis on fluid mechanics, and presents the basic conceptual tools needed to approach heuristically the physics of fluids. Both the Lagrangian and the Eulerian descriptions of motion are introduced. The link between these two descriptions stems from the request that a property of an Eulerian field, at a given point and at a given time, coincides with the corresponding property of the Lagrangian particle that occupies that point at that time. In the Lagrangian framework, the governing equations of fluid mechanics, the vorticity dynamics and the parameterization of turbulence are discussed. Illustrative applications of these results to simple models (e.g. flow in a channel and 2-D breezes) are also given.
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Notes
- 1.
Recall that the tensor product between any two vectors a and b is the linear operator defined by a ⊗ b : x↦(b ⋅x) a.
- 2.
Equation (1.50) corresponds to Eqs. (1.4.4) and (1.4.5) of Pedlosky (1987).
- 3.
Standard notation is understood.
- 4.
Here, “correlation” means “sign of the product”.
- 5.
Every generalization is a hypothesis; therefore hypothesis has a necessary role, which nobody has ever questioned. But hypothesis must always undergo verification, as soon as possible and as often as possible. (Henri Poincaré, Science et hypothèse)
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Cavallini, F., Crisciani, F. (2013). Basic Continuum Mechanics. In: Quasi-Geostrophic Theory of Oceans and Atmosphere. Atmospheric and Oceanographic Sciences Library, vol 45. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4691-6_1
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DOI: https://doi.org/10.1007/978-94-007-4691-6_1
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