Abstract
Modelling fluid mechanics generally begins with defining a fixed framework with a fixed domain M of the fluid, a space of diffeomorphisms of M, so that the time intervenes as an exterior parameter as a last resort. When the domain of the fluid is time-dependent, we must change the strategy. The time must be taken into account in the framework from the beginning, and we have to use the basic notions of Euler and Lagrange variables. The point of view is in part different from that of Chapter 1, but there will be some repetitions, and the notation does not always agree with the previous notation.
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Notes
- 1.
Or also a pseudogroup; see [Mall, ch. II.2.4]).
- 2.
If the domain M is time-independent, then \(\tilde {M}\) is identified with M × R. Notice that \(\tilde {M}\) may be unknown, giving a free boundary problem. In usual models, the closure of the domain Ms occupied at time s by the fluid is the support of the volume mass ρ(s).
- 3.
With L(M0) trivial, which is identified with the product M0 × Gl(3), the connection form ω0 is the pullback of the Maurer–Cartan form θMC on Gl(3) by the projection p defined by p(x, g) = g ∈ Gl(3), ω0 = p∗ θMC. This implies that the torsion and the curvature are null; see, for instance, [Kob-Nom, ch. II.9, p. 92].
- 4.
We assume that if σ0 is the section of \(L(\tilde {M})\) such that \(\sigma _{0}(x)=(X_{j})=(\frac {\partial }{\partial x^{j}})_{j=1,2,3,4}\), with \(\frac {\partial }{\partial x^{4}}=\frac {\partial }{\partial t}\), then the differential form \(\sigma _{0}^{*}\omega \) on \(\tilde {M}\) is null.
- 5.
It may be viewed as a reduced bundle of \(L(\tilde {M})\) over \(\tilde {M}\); see [Kob-Nom, Vol. 2, ch. VII.8, p. 53].
- 6.
We recall that we have \(h(\tilde {u}\tilde {a})=ua=h(\tilde {u})h_{0}(\tilde {a})\), with h0 the projection from gl(3 + 1, 3) onto gl(3).
- 7.
The space Lx M is identified with the space of linear bijective mappings from R3 onto Tx M (denoted by Isom (R3, Tx M)), by associating to each frame u = (e1, e2, e3) of Tx M the isomorphism \(\lambda =(\lambda _{1},\lambda _{2},\lambda _{3})\in R^{n}\rightarrow u(\lambda )=\sum \lambda _{i}e_{i} \in T_{x}M\).
- 8.
Here the displacement group is R3 × O(3) instead of R3 × SO(3) in Chapter 2.
- 9.
With the notation 𝜖i,j we refer to the components of the tensor \(\epsilon \in T_{2}^{0}(M)\), whereas \(\epsilon _{i}^{j}\) refers to a tensor in \(T_{1}^{1}(M)\), giving, by taking the product with dxi ⊗ dxj and contraction, the tensor 𝜖.
- 10.
This is allowed by the fact that (G Av)a(fY ) = f(G Av)a(Y ) for every function f.
- 11.
See, for instance, [Bour.alg0, ch. AIII]
- 12.
See [Kob-Nom, ch. VI.2, Prop. 2.5].
- 13.
See [Kob-Nom, Vol. 2, ch. VII.3, p. 15].
- 14.
See [Kob-Nom, Vol. 2, ch. VII.3].
- 15.
See [Kob-Nom, Appendix 6].
- 16.
See, for instance, [Bour.var, 9.1.4].
- 17.
- 18.
This is the space of p-integrable functions with their derivatives up to the order 1.
- 19.
Note that we have to consider separately the incompressible fluid case depending on the Navier–Stokes equation, where the trajectories are independent of the pressure, but thermodynamics is not really present in this model.
- 20.
The physical existence of such a field seems natural from a finer modelling, microscopic or (and) random, with a diffusion process, for instance.
- 21.
The internal energy \(\hat {e}\) is identified with (the pullback of) the thermodynamic internal energy e by the section ξF.
- 22.
Which may be a model of a Joule effect induced by an electromagnetic field.
- 23.
In a similar way to the Clausius–Duhem form; see [D-L19, chap. 1]).
- 24.
Beware of the simple notation v2; we recall that this means g(v, v), with g the pullback of the initial metric by the inverse flow.
- 25.
Its intrinsic character is given in (3.69).
- 26.
See Chapter 4 for this notation.
- 27.
With the time-dependent notation on v and ρ.
- 28.
At least for a Newtonian fluid and if the velocity is null on the boundary of M (that is, the Dirichlet condition).
- 29.
Of course, the use of the van der Waals equation may be criticized, but the same result will be obtained with any reasonable state equation.
- 30.
Their frameworks may be specified thanks to the trace theorems.
- 31.
For instance with the state equation P = γρe.
- 32.
In order to simplify; without this hypothesis, we would have vΓ ∈ L2(J, H1∕2( Γ)), and similar results.
- 33.
Especially in stochastic theory (see, for instance, [Par]).
- 34.
We could also consider a domain Ω in Rd, bounded or not.
- 35.
Note that this is possible notably with u ∈ BV (Rd)N the space of bounded variations.
- 36.
Note that this condition implies that the surface Σ is orientable.
- 37.
Note that sometimes we assume that φ(x, t) = 0 is of the form ψ(x) + t = 0, which implies that Σt ∩ Σs is empty if t ≠ s.
- 38.
We use the formula grad P = (grad P) − [P]nδΣ, with grad P taken in the sense of distributions, (grad P) in the classical sense, and n the unit normal to Σ.
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Cessenat, M. (2018). Fluid Mechanics Modelling. In: Mathematical Modelling of Physical Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-94758-7_3
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