Abstract
The derivation of the fundamental equations for Newtonian fluids has been provided in many books, e.g. in Chandrasekhar (1961) and many others. Therefore here we only briefly recall the main points of the derivation, with the aim to keep the book self-consistent and set grounds for later chapters.
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Notes
- 1.
The fundamental assumption for Newtonian fluids is that the dissipative part of the stress tensor, say \(\varvec{\tau }_{\mu }\), associated with frictional effects due to the flow, is linearly related to the tensor of deformation rate, that is the flow velocity gradient tensor, since it is the presence of velocity gradients which is necessary and sufficient for frictional forces to appear. Once this assumption is made, the final form of the constitutive relation (1.8) for an isotropic fluid is simply an outcome of symmetry of the stress tensor, which follows from the angular momentum balance (cf. the next Sect. 1.3.1) and very basic properties of tensorial objects known from the group theory. The latter is simply the Curie’s principle 1984 (cf. Chalmers 1970; de Groot and Mazur 1984), namely that in an isotropic system only those tensorial objects, which at rotations of a system of reference transform according to the same irreducible (therefore distinct) representations of the rotation groups can be linearly related. Since the trace and the symmetric traceless part of a tensor transform differently, one arrives at the constitutive relations \(\mathrm {Tr}\varvec{\tau }_{\mu }=-\mu _{b}\mathrm {Tr}\mathbf {G}=-\mu _{b}\nabla \cdot \mathbf {u}\) and \(\varvec{\tau }_{\mu }^{s}-(1/3)(\mathrm {Tr}\varvec{\tau }_{\mu })\mathbf {I}=-2\mu (\mathbf {G}^{s}-(1/3)(\nabla \cdot \mathbf {u})\mathbf {I})\), which are equivalent to (1.8). In the given relations the superfix s denotes a symmetric part of a tensor, \(\mathbf {G}\) is the velocity gradient, as in (1.9), \(\mathbf {I}\) is the unitary matrix and a negative sign was introduced in front of the right hand sides, since the coefficients \(\mu >0\) and \(\mu _{b}>0\) describe frictional, therefore dissipative effects.
- 2.
- 3.
It is well known from the second law of thermodynamics, that the differential of the internal energy for a single component system takes the form
$$ \mathrm {d}\mathcal {E}=T\mathrm {d}S-p\mathrm {d}V, $$where \(\mathcal {E}\), S and V are the “canonical” thermodynamic variables, that is the actual internal energy, the entropy and the volume of the system (as opposed to the mass densities \(\varepsilon \), s and \(\rho \)); division by the total mass \(M=m_{m}N\), where \(m_{m}\) denotes the molecular mass of the fluid particles and N the number of particles, allows to transform the above into the differential for the mass density of the internal energy, which takes the form (1.35).
- 4.
- 5.
In cases, when the hydrostatic temperature gradient at threshold is height-dependent this determination of critical temperature jump across the fluid layer involves taking a vertical average of (1.62).
- 6.
Note, that in the general dissipative case, \(\mu \ne 0\), \(\mu _{b}\ne 0\) and \(k\ne 0\) the sound waves are damped by the all three dissipative processes. In particular in the long-wavelength limit \(\mathcal {K}\ll \rho _{0}C/\mu \) the dispersion relation (1.76) can be solved by means of asymptotic expansions in the wave number and the sound modes are characterized by
$$ \omega =\pm C\mathcal {K}-\mathrm {i}\frac{1}{2\rho _{0}}\left[ \frac{4}{3}\mu +\mu _{b}+k\left( \frac{1}{c_{v}}-\frac{1}{c_{p}}\right) \right] \mathcal {K}^{2}. $$The dissipative damping of sound waves is also often formulated in terms of spatial absorption, that is diminishing of the waves intensity as it travels a certain distance in the fluid; in other words such a formulation involves complex wave vector and real frequency, but the absorption coefficient is essentially the same and involves all three dissipation coefficients (cf. Landau and Lifshitz 1987, Chap. 79 on “Absorption of sound”, Eq. (79.6)).
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Mizerski, K.A. (2021). The Equations of Hydrodynamics. In: Foundations of Convection with Density Stratification. GeoPlanet: Earth and Planetary Sciences. Springer, Cham. https://doi.org/10.1007/978-3-030-63054-6_1
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