Abstract
While introducing Fluid Mechanics, we had to introduce also the idea of continuous media, which is the mathematical idealization of real fluids (or solids). In many circumstances, the limits of this approach arose: for instance the rheological laws, which relate strain and stress, are not given by Fluid Mechanics, they need another model. Fluid Mechanics considers these laws as given. In the first chapter we observed that in the limit of small perturbations of the basic thermodynamic equilibrium by the flow, we could derive the functional form of the rheological laws, namely that of Newtonian fluids, but the specificity of the fluid was then condensed in its viscosity or, more generally, in its transport coefficients.
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Notes
- 1.
A taste of this approach may be found in the book of Guyon et al. (2001a).
- 2.
The origin of the \(\sqrt{2}\)-factor may be understood with a simple argument. We first observe that the number of collisions is controlled by the relative velocity of the particles, i.e. N = n π d 2 v rel. But \(\mathbf{v}_{\mathrm{rel}} =\mathbf{ v} -\mathbf{ v}_{0}\) where \(\mathbf{v}_{0}\) is the velocity of the test particle. If we identify v rel with the rms velocity \(\sqrt{\left \langle \mathbf{v}_{\mathrm{rel } }^{2}\right \rangle }\), we get
$$\displaystyle{\left \langle \mathbf{v}_{\mathrm{rel}}^{2}\right \rangle = \left \langle \mathbf{v}^{2}\right \rangle + \left \langle \mathbf{v}_{ 0}^{2}\right \rangle - 2\left \langle \mathbf{v} \cdot \mathbf{ v}_{ 0}\right \rangle }$$The randomness and uncorrelation of velocities imply that \(\left \langle \mathbf{v} \cdot \mathbf{ v}_{0}\right \rangle = 0\). Since the test particle is not different from other particles \(\left \langle \mathbf{v}_{0}^{2}\right \rangle = \left \langle \mathbf{v}^{2}\right \rangle\). Thus \(v_{\mathrm{rel}} = \sqrt{2}v\).
- 3.
Actually, this expansion can be generalized to all the physical quantities that measure the distance to thermodynamic equilibrium like shear, gradient of concentration, current etc. We touch here Onsager’s approach who worked out the general theory of slight deviations from thermodynamic equilibrium. Our purpose here is not that general and we shall consider only situations where fluxes depend only on a single quantity, which is correct in the simple cases that we are considering.
- 4.
Another way of deriving this factor is to count the number of pairs for a set of N particles. There are \(N(N - 1)/2\) pairs. Thus when \(N \gg 1\), there are just N∕2 pairs for each particles. Thus the exclusion volume is that indicated by (11.11).
- 5.
In some textbooks like Vincenti and Kruger (1965), a normalized distribution function is used; it is such that \(\int f(\mathbf{x},t,\mathbf{v})d^{3}\mathbf{v} = 1\) which is more practical in some cases.
- 6.
- 7.
The BGK model was proposed by Bhatnagar et al. (1954) to explore in a simplified way flows where the continuum approximation is no longer relevant, for instance in the case of very diluted gases where the Knudsen number (cf. 1.1) is no longer small compared to unity.
- 8.
A classical example of such flows is the one surrounding a spacecraft entering the atmosphere. At a height of 120 km the mean free path of air molecules is of the order of a metre. The flow around a space shuttle (wing span of 24 m) is no longer computable with solutions of Navier–Stokes equation. Other examples of such flows includes those of epitaxy by molecular jets that is used to make thin metallic layers in micro-electronic components.
- 9.
The reader who prefers a mathematical demonstration of this result may find it in the book of Kennard (1938).
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Rieutord, M. (2015). Beyond Fluid Mechanics: An Introduction to the Statistical Foundations of Gas Dynamics. In: Fluid Dynamics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-09351-2_11
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