§10 Classical equations of hydrodynamics

In order to derive the classicalequations of hydrodynamics, let us first note the difference between the values of the thermodynamic quantities in the relativistic case and those used classically. To begin with, the quantities in the relativistic case are defined with respect to the proper system of reference of the fluid. In classical mechanics these quantities are referred to the laboratory frame. In the relativistic case the thermodynamic quantities, such as the internal energy density $e$, the entropy density $\sigma$ and the enthalpy density $\omega$ are all defined with respect to the proper volume of the fluid. In classical fluid dynamics, these quantities are defined in units of the mass of the fluid element they refer to. For instance, the specific internal energy $\epsilon$, the specific entropy $s$ and the specific enthalpy $w$ are all measured per unit mass in the laboratory frame. When taking the limit in which the speed of light $c$ tends to infinity we must also bear in mind that the internal energy density $e$ includes the rest energy density $n m \ensuremath{\mathsf{c}}^2$, where $m$ is the rest mass of the particular fluid element under consideration. Therefore the following classical limits should be taken in passing from relativistic to classical fluid dynamics:

$$mn \xrightarrow[\ensuremath{\mathsf{c}}\rightarrow \infty]{} \rho \s... ...emath{\mathsf{c}}^2 } \approx \rho \left( 1 - \ensuremath{v}^2 / 2 c^2 \right),$$

$$e \xrightarrow[\ensuremath{\mathsf{c}}\rightarrow \infty]{} nm\ensu... ... \ensuremath{\mathsf{c}}^2 - \frac{1}{2} \rho \ensuremath{v}^2 + \rho \epsilon,$$

$$\frac{ \omega }{ n } \xrightarrow[\ensuremath{\mathsf{c}}\rightarrow \infty]{} m\ensur... ...ac{ p }{ \rho } \right) \approx m \left( \ensuremath{\mathsf{c}}^2 + w \right),$$ (10.1)

where $\rho$ is the mass density of the fluid in the laboratory frame. Using these limiting values, the equation of continuity eq.(9.1), Euler's equation eq.(9.7) and the conservation of entropy eq.(9.4) have their classical analogues respectively:

$$\frac{ \partial \rho }{ \partial t } + \mathrm{div} \left( \rho \boldsymbol{\mathit{v}} \right) = 0,$$ (10.2)

$$\frac{ \partial \boldsymbol{\mathit{v}} }{ \partial t} + \left( \... ...oldsymbol{\mathit{v}} = - \frac{ 1 }{ \rho } \boldsymbol{ \mathrm{grad} } p,$$ (10.3)

$$\frac{ \partial s }{ \partial t } + \left( \ensuremath{v}\cdot \boldsymbol{ \mathrm{grad} } \right) s = 0$$ (10.4)

In the same approximation, eq.(9.8) gives the classical version of Bernoulli's equation, that is,

$$\frac{1}{2} \ensuremath{v}^2 + w = \mathsf{const},$$ (10.5)

for a given streamline. The constant in the right hand side of eq.(10.5) differs from the constant in the right hand side of eq.(9.8) by an unimportant additional term.

When gravitational effects are taken into consideration, Euler's equation, and hence Bernoulli's law, have to be changed appropriately in order to account for the force exerted by the gravity on the flow of a given fluid element:

$$\frac{ \partial \boldsymbol{\mathit{v}} }{ \partial t} + \left( \... ...}{ \rho } \boldsymbol{\mathrm{grad}} p - \boldsymbol{\mathrm{grad}} \phi,$$ (10.6)

$$\frac{1}{2} \ensuremath{v}^2 + w + \phi = const,$$ (10.7)

where the gravitational potential $\phi$ is related to the gravitational acceleration $\mathbf{g}$ by the relation $\mathbf{g} = \! - \boldsymbol{\mathrm{grad}} \phi$.

Footnotes

... classical

Here and in what follows we use the word ``classical'' to mean non-relativistic.