Abstract
Inertia is known to play a significant role in the upper part of the respiratory tract. Additionally, an accurate description of the air velocity field in the branches can be useful in many situations, for example if one aims at investigating the deposition process of sprays. On the other hand, a full computation of the velocity field in all the respiratory tract is out of reach, given the geometric complexity of the domain. It makes it necessary to restrict the full resolution of the fluid motion to the very first generations, where inertial effects are significant. We describe in this chapter how the respiratory tract can be decomposed into different zones, and in particular how the upper zone, in which we aim at solving the Navier–Stokes equations, can be connected to the balloon models we introduced in Chapter 2.
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Notes
- 1.
This strong assumption is sometimes ruled out. In some situations it is in particular more relevant to use the so-called Navier conditions, which still preserve the impervious character of the wall, but allow a non zero tangential velocity (see [58] for details).
- 2.
Dowloadable file: bifNS.edp (FreeFem++ software [54]).
- 3.
The value of the Reynolds number, around 1000, is realistic: it is commonly attained during breathing at rest.
- 4.
In the context of lung modeling, this technical problem will actually disappear, as we shall deal with domains with inlet and outlets, so that this indeterminacy shall not be met.
- 5.
It asserts that
$$ ab\le \frac{a^q}{q}+\frac{b^p}{p}\kern0.36em \forall a,b\ge 0,\frac{1}{p}+\frac{1}{q}=1. $$ - 6.
We restrict ourselves to the situation where the pressure is uniform in each open component of
the boundary, but the approach extends to more general situations (see [13]).
- 7.
The term stress is an abuse of language. According to the previous section, we have considered
here a variational formulation based on the non-symmetrized form ∫ ∇ u : ∇ v, which leads to nonphysical
(yet adapted in this context) conditions based on μ ∇ u − p Id, but the same approach can be carried out with the symmetrized form ∫ (∇ u + t ∇ u) : (∇ v + t ∇ v).
- 8.
This may be of particular importance in the case of the respiratory tract. The domain Ω shall be a truncated N-generation dyadic tree, and the outlet Γ out will consists in 2N artficial boundaries, connecting the domain to alveoli through condensated subtrees. As we shall see, even if the global Poiseuille resistance of the tree is small, individual resistances of the subtrees can reach high values as N grows, which will results in a significant increase of the condition number.
- 9.
As a matter of fact, those parameters may vary significantly with the temperature, see Remark 4.4
for more details.
- 10.
Dowloadable file: bifNSballoon.edp (FreeFem++ software [54]).
- 11.
The velocity components and the pressure are taken piecewise affine with respect to the tetrahedral
mesh. This choice does not fulfill the inf-sup condition (see Definition B.10, p. 257). Stability
is recovered by adding an extra term in the variational formulation (see e.g. [146]).
- 12.
Particles present a danger whenever they are inhaled and captured by the respiratory system, i.e.
whenever they deposit on the inner wall of the respiratory tract, or on the alveolar membrane.
- 13.
This approach relies on different scales: the microscopic scale is the size a of the particle, the
macroscopic scale L is the size of the overall fluid domain, which corresponds to significant changes
in the fluid velocity. The mesoscopic scale η is the size of the elementary volume of fluid overwhich
the fluid velocity is fairly uniform (η ≪ L), but which corresponds to an infinite domain from the
particle standpoint (a ≪ η).
- 14.
Dowloadable file: bifPart.edp (FreeFem++ software [54]).
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Maury, B. (2013). Computing velocity fields. In: The Respiratory System in Equations. MS&A — Modeling, Simulation and Applications. Springer, Milano. https://doi.org/10.1007/978-88-470-5214-7_4
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DOI: https://doi.org/10.1007/978-88-470-5214-7_4
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