Abstract
The Newtonian fluids are special types of linear Newtonian deformable bodies which we shall define in an exact physical sense presently. Practically every Newtonian deformable body may be considered as a fluid with certain properties. Therefore it is better that we continue with the term fluid in a vague manner until we end up with a definition. For the time being we consider a fluid to be an aggregate of a great number of particles (a continuum) which moves under the mutual interaction of its particles and the application of external forces.
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Notes
- 1.
This is not possible for relativistic fluids where universal time does not exist and one has to introduce the concept of synchronization. For this reason we shall consider relativistic fluids separately.
- 2.
We use δt and not dt because this change is measured at the same point P of space. Furthermore we use δ r and not d r because the change in r is defined in terms of the change in t. If it was independent of the t then we should use d r!
- 3.
We enter \(\frac {d}{dt}\) into the integral because the integrating region is independent of time t.
- 4.
A terminology for this volume is material volume.
- 5.
The mass dM remains constant in the volume dV as the fluid moves due to the conservation of mass and the assumption that the volume dV is comoving i.e. at all times contains the same number and type of particles.
- 6.
The velocity of the fluid relative to the sphere is u = ∇ ϕ and in the coordinate system Σ in which the sphere has velocity \(u_{0}\hat {\mathbf {k}}\) the velocity of the fluid equals:
$$\displaystyle \begin{aligned} {\mathbf{u}}_{\Sigma }=\boldsymbol{\nabla }\phi +u_{0}\hat{\mathbf{k}}. \end{aligned}$$ - 7.
Equation \(\nabla \cdot \mathbf {E}= \frac {\rho }{\varepsilon _{0}}\) in general is ∇⋅D = ρ. Also recall that for empty space B = μ 0 H and D = ε 0 E.
- 8.
The current j = ρ v for empty space. For materials the rhs contains more terms.
- 9.
The equation of state is a relation between the isotropic pressure and the energy density. This equation we meet frequently in General Relativity.
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Tsamparlis, M. (2019). Newtonian Fluids. In: Special Relativity. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-27347-7_22
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DOI: https://doi.org/10.1007/978-3-030-27347-7_22
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