Abstract
The study of the stability of flows is one of the cornerstones of Fluid Mechanics: the subject is so large that it would deserve a whole book to be reviewed. Leaving aside such an ambitious goal, we shall concentrate, in this chapter and the following one, on the fundamentals, although, here and there, making some excursions in more specialized topics.
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Notes
- 1.
A solar mass, symbolized by M\(_{\odot }\), is equal to 2 × 1030 kg.
- 2.
Burgers equation is given by (5.85).
- 3.
We may expect that if the diameter of the bottleneck is smaller than 1.7 cm, the equilibrium is stable. However, we should keep in mind that the value was derived for pure water; impurities decrease the surface tension and lead to a smaller value of the critical wavelength.
- 4.
The name of Carlo Marangoni (Pavia 1840–Firenze 1925) is generally associated with this instability as he was the first physicist to describe fluid flows driven by surface tension gradients (with a paper in Annalen der Physik in 1871).
- 5.
Surface tension comes from the binding energy of molecules due to their mutual interactions in a liquid. We may expect that at the critical temperature, which is the temperature where the gas and liquid phases are undistinguishable, the surface tension disappears. This remark lead L. Eötvös (1848–1919) to propose that surface tension varies with temperature like
$$\displaystyle{\gamma = k(T_{c} - T)/V ^{2/3}}$$Here k is a universal constant for the liquids, V is the volume of one mole and T c is the critical temperature. This law, which is known as Eötvös rule, is only approximate, but suggests that γ decreases linearly with temperature, as actually observed experimentally. For instance, the following fit
$$\displaystyle{ \gamma = 7.3\,10^{-2}\left [1 - 0.0023(T - 291)\right ]\;\mathrm{N/m} }$$(6.38)matches rather well the variations of surface tension of water in the range 273–373 K, as illustrated in Fig. 6.4.
- 6.
The Biot number is the ratio of two heat transfer coefficients. The heat transfer coefficient is a flux surface density divided by a temperature; for instance, χ l ∕d is the heat transfer coefficient of the liquid layer, while σ T 3 is that of the vacuum.
- 7.
This means the same boundary conditions on the bottom plate and on the top plate, stress-free and fixed-flux conditions (this is for the case Bi = 0).
- 8.
The equilibrium solutions are also called fixed points in the language of dynamic systems.
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Rieutord, M. (2015). Flows Instabilities. In: Fluid Dynamics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-09351-2_6
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