Equations of fluid dynamics 2 General compressible fluid 4 General incompressible fluid

Chapter 3 Newtonian compressible fluid

For a so called newtonian fluid it can be shown, that the stress tensor $\sigma^{\prime}_{ij}$ is of the form

\displaystyle\sigma^{\prime}_{ij}=2\eta S^{*}_{ij}+\zeta\delta_{ij}\frac{% \partial v_{k}}{\partial r_{k}}

(3.1)

with $S^{*}_{ij}$ being the symmetric tracefree part of the tensor $\frac{\partial v_{i}}{\partial x_{j}}$

\displaystyle S^{*}_{ij}={\left[\frac{1}{2}{\left(\frac{\partial}{\partial r_{% j}}v_{i}+\frac{\partial}{\partial r_{i}}v_{j}\right)}-\frac{1}{3}\delta_{ij}% \frac{\partial v_{k}}{\partial r_{k}}\right]}

(3.2)

The parameter $\eta^{\prime}$ is called dynamic viscosity and $\zeta$ is the so called second dynamic viscosity. The second term of equation (3.1) is often considered as small and therefore neglected. This is true in case of a monoatomic gases there it can be shown, that $\zeta=0$ (Landau/Lifschitz 10). In case of a incompressible fluid with constant density the term can also be neglected, because $\frac{\partial v_{k}}{\partial r_{k}}=0$ . Nevertheless for compressible fluids (supersonic regime) $\frac{\partial v_{k}}{\partial r_{k}}$ can be very large (shocks) and the second dynamic viscosity of a n-atomic gas can not be neglected. In this case the second term of the stress tensor additionally contributes to the pressure, which should be considered in the equation of state. This will alter the nature of $p$ as a thermodynamic variable, which should only depend on the local values of $\rho$ and $e$ and not on $\frac{\partial v_{k}}{\partial r_{k}}$ . But since we have a stress tensor we locally do not have a local thermodynamic equilibrium anyway, so one should expect a change in the nature of the thermodynamic variables, which are defined for local thermodynamic equilibrium.

3.1 Balance equations

In the following we will write the stress tensor for a newtonian compressible fluid in the form

\displaystyle\sigma^{\prime}_{ij}=\eta{\left(\frac{\partial}{\partial r_{j}}v_% {i}+\frac{\partial}{\partial r_{i}}v_{j}\right)}+{\left(\zeta-\frac{2}{3}\eta% \right)}\delta_{ij}\frac{\partial v_{k}}{\partial r_{k}}

(3.3)

Inserting this explicitely into the momentum equation for a compressible fluid (2.48) one gets for the term

\displaystyle\begin{split}\displaystyle\frac{\partial}{\partial r_{j}}\sigma^{% \prime}_{ij}&\displaystyle=\frac{\partial}{\partial r_{j}}{\left[\eta{\left(% \frac{\partial v_{i}}{\partial r_{j}}+\frac{\partial v_{j}}{\partial r_{i}}% \right)}\right]}+{\left(\zeta-\frac{2}{3}\eta\right)}\delta_{ij}\frac{\partial% }{\partial r_{j}}{\left(\frac{\partial v_{k}}{\partial r_{k}}\right)}\\ &\displaystyle=\eta{\left[\frac{\partial^{2}}{\partial r_{j}^{2}}v_{i}+\frac{% \partial}{\partial r_{i}}{\left(\frac{\partial v_{j}}{\partial r_{j}}\right)}% \right]}+{\left(\zeta-\frac{2}{3}\eta\right)}\frac{\partial}{\partial r_{i}}{% \left(\frac{\partial v_{k}}{\partial r_{k}}\right)}\\ &\displaystyle=\eta\frac{\partial^{2}}{\partial r_{j}^{2}}v_{i}+{\left(\frac{% \eta}{3}+\zeta\right)}\frac{\partial}{\partial r_{i}}{\left(\frac{\partial v_{% k}}{\partial r_{k}}\right)}.\end{split}

(3.4)

Inserting the stress tensor into the energy equation for a compressible fluid (2.49) one gets for

\displaystyle\begin{split}\displaystyle\frac{\partial}{\partial r_{j}}(v_{i}% \sigma^{\prime}_{ij})&\displaystyle=\frac{\partial}{\partial r_{j}}{\left[\eta v% _{i}{\left(\frac{\partial v_{i}}{\partial r_{j}}+\frac{\partial v_{j}}{% \partial r_{i}}\right)}\right]}+{\left(\zeta-\frac{2}{3}\eta\right)}\frac{% \partial}{\partial r_{j}}{\left(v_{j}\frac{\partial v_{k}}{\partial r_{k}}% \right)}\\ &\displaystyle=\eta\frac{\partial^{2}}{\partial r_{j}^{2}}{\left(\frac{1}{2}v_% {i}^{2}\right)}+\eta\frac{\partial}{\partial r_{j}}{\left(v_{i}\frac{\partial v% _{j}}{\partial r_{i}}\right)}+{\left(\zeta-\frac{2}{3}\eta\right)}\frac{% \partial}{\partial r_{j}}{\left(v_{j}\frac{\partial v_{k}}{\partial r_{k}}% \right)}\end{split}

(3.5)

In the end we get the following balance equations for a compressible, selfgravitating, newtonian fluid

$\displaystyle\frac{\partial}{\partial t}\rho+\frac{\partial}{\partial r_{j}}(v% _{j}\rho)=$	$\displaystyle\ 0$		(3.6)
$\displaystyle\frac{\partial}{\partial t}(\rho v_{i})+\frac{\partial}{\partial r% _{j}}(v_{j}\rho v_{i})=$	$\displaystyle-\frac{\partial}{\partial r_{i}}p++\rho g_{i}+\eta\frac{\partial^% {2}}{\partial r_{j}^{2}}v_{i}+{\left(\frac{\eta}{3}+\zeta\right)}\frac{% \partial}{\partial r_{i}}{\left(\frac{\partial v_{k}}{\partial r_{k}}\right)}$		(3.7)
$\displaystyle\begin{split}\displaystyle\frac{\partial}{\partial t}(\rho e)+% \frac{\partial}{\partial r_{j}}(v_{j}\rho e)=&\displaystyle-\frac{\partial}{% \partial r_{j}}(v_{j}p)+v_{i}\rho g_{i}+\eta\frac{\partial^{2}}{\partial r_{j}% ^{2}}{\left(\frac{1}{2}v_{i}^{2}\right)}+\eta\frac{\partial}{\partial r_{j}}{% \left(v_{i}\frac{\partial v_{j}}{\partial r_{i}}\right)}\\ &\displaystyle+{\left(\zeta-\frac{2}{3}\eta\right)}\frac{\partial}{\partial r_% {j}}{\left(v_{j}\frac{\partial r_{k}}{\partial v_{k}}\right)}\end{split}$		(3.8)

3.2 Global dissipation of kinetic energy $\mathcal{E}$

Using the form (3.3) we get for the term involving the stress tensor in the equation (2.51) for the dissipation $\mathcal{E}$

\displaystyle\begin{split}\displaystyle\sigma^{\prime}_{ij}\frac{\partial v_{i% }}{\partial r_{j}}&\displaystyle=\eta{\left(\frac{\partial}{\partial r_{j}}v_{% i}+\frac{\partial}{\partial r_{i}}v_{j}\right)}\frac{\partial v_{i}}{\partial r% _{j}}+{\left(\zeta-\frac{2}{3}\eta\right)}\delta_{ij}\frac{\partial v_{k}}{% \partial r_{k}}\frac{\partial v_{i}}{\partial r_{j}}\\ &\displaystyle=\frac{\eta}{2}{\left(\frac{\partial}{\partial r_{j}}v_{i}+\frac% {\partial}{\partial r_{i}}v_{j}\right)}^{2}+{\left(\zeta-\frac{2}{3}\eta\right% )}{\left(\frac{\partial v_{k}}{\partial r_{k}}\right)}^{2}\end{split}

(3.9)

where we also made use of the relation (H.4) for the contraction of a symmetric with an unsymmetric tensor. With this result the global dissipation of kinetic energy for a newtonian compressible fluid is

\displaystyle\begin{split}\displaystyle\mathcal{E}=&\displaystyle\frac{1}{V}% \int_{V}p\frac{\partial v_{j}}{\partial r_{j}}dV-\frac{1}{V}\int_{V}\phi\frac{% \partial}{\partial t}\rho dV-\frac{1}{V}\int_{V}\frac{\eta}{2}{\left(\frac{% \partial}{\partial r_{j}}v_{i}+\frac{\partial}{\partial r_{i}}v_{j}\right)}^{2% }dV\\ &\displaystyle-\frac{1}{V}\int_{V}{\left(\zeta-\frac{2}{3}\eta\right)}{\left(% \frac{\partial v_{k}}{\partial r_{k}}\right)}^{2}dV\end{split}

(3.10)

This equation should be compared to equation (79,1) from Landau and Lifschitz (1991) which additionally includes heat conduction. Nevertheless Landau and Lifschitz (1991) seem to forget the term due to the pressure in the equation for the dissipation.

3.3 Divergence equation

Using the equation for the divergence of the stress tensor for a newtonian compressible fluid (3.4) we get for

\displaystyle\begin{split}\displaystyle\frac{\partial}{\partial r_{i}}{\left(% \frac{\partial}{\partial r_{j}}\sigma^{\prime}_{ij}\right)}&\displaystyle=\eta% \frac{\partial}{\partial r_{i}}\frac{\partial^{2}}{\partial r_{j}^{2}}v_{i}+{% \left(\frac{\eta}{3}+\zeta\right)}\frac{\partial^{2}}{\partial r_{i}^{2}}% \theta\\ &\displaystyle=\eta\frac{\partial^{2}}{\partial r_{j}^{2}}\theta+{\left(\frac{% \eta}{3}+\zeta\right)}\frac{\partial^{2}}{\partial r_{i}^{2}}\theta\\ &\displaystyle={\left(\frac{4}{3}\eta+\zeta\right)}\frac{\partial^{2}}{% \partial r_{i}^{2}}\theta\end{split}

(3.11)

By inserting this and equation (3.4) into the equation for the divergence (2.52) we get the divergence equation for a compressible newtonian fluid

\displaystyle\begin{split}\displaystyle\frac{\partial}{\partial t}\theta-% \theta^{2}-v_{i}\frac{\partial}{\partial r_{i}}\theta+\frac{\partial^{2}}{% \partial r_{i}\partial r_{j}}(v_{i}v_{j})=&\displaystyle\frac{1}{\rho^{2}}{% \left(\frac{\partial\rho}{\partial r_{i}}\right)}\cdot{\left[\frac{\partial p}% {\partial r_{i}}-\eta\frac{\partial^{2}}{\partial r_{j}^{2}}v_{i}-{\left(\frac% {\eta}{3}+\zeta\right)}\frac{\partial}{\partial r_{i}}\theta\right]}\\ &\displaystyle-\frac{1}{\rho}{\left[\frac{\partial^{2}}{\partial r_{i}^{2}}p-{% \left(\frac{4}{3}\eta+\zeta\right)}\frac{\partial^{2}}{\partial r_{i}^{2}}% \theta\right]}-4\pi G\rho\end{split}

(3.12)

or in vector notation

\displaystyle\begin{split}\displaystyle\frac{\partial}{\partial t}\theta-% \theta^{2}-(\boldsymbol{v}\cdot\nabla)\theta+\nabla{\left[\nabla\cdot(v_{i}v_{% j})\right]}=&\displaystyle\frac{1}{\rho^{2}}\nabla\rho\cdot{\left[\nabla p-% \eta\Delta v-{\left(\frac{\eta}{3}+\zeta\right)}\nabla\theta\right]}\\ &\displaystyle-\frac{1}{\rho}{\left[\Delta p-{\left(\frac{4}{3}\eta+\zeta% \right)}\Delta\theta\right]}-4\pi G\rho\end{split}

(3.13)

3.4 Vorticity equation

If we plugin the equation for the divergence of the stress tensor for a newtonian compressible fluid (3.4) into the vorticity equation (2.53) we get for a compressible newtonian fluid with $\theta=\frac{\partial v_{k}}{\partial r_{k}}$

\displaystyle\begin{split}\displaystyle\frac{\partial}{\partial t}\omega_{g}-% \epsilon_{ghi}\frac{\partial}{\partial r_{h}}\epsilon_{ijm}v_{j}\omega_{m}=&% \displaystyle-\frac{1}{\rho^{2}}\epsilon_{ghi}\left[{\left(\frac{\partial}{% \partial r_{h}}\rho\right)}{\left(\frac{\partial}{\partial r_{i}}p\right)}-% \eta{\left(\frac{\partial}{\partial r_{h}}\rho\right)}{\left(\frac{\partial^{2% }}{\partial r_{j}^{2}}v_{i}\right)}\right.\\ &\displaystyle\left.-{\left(\frac{\eta}{3}+\zeta\right)}{\left(\frac{\partial}% {\partial r_{h}}\rho\right)}{\left(\frac{\partial}{\partial r_{i}}\theta\right% )}\right]+\frac{\eta}{\rho}\frac{\partial^{2}}{\partial r_{j}^{2}}\omega_{g}\\ &\displaystyle+\frac{1}{\rho}{\left(\frac{\eta}{3}+\zeta\right)}\epsilon_{ghi}% \frac{\partial}{\partial r_{h}}\frac{\partial}{\partial r_{i}}{\left(\frac{% \partial v_{k}}{\partial r_{k}}\right)}\end{split}

(3.14)

The last term on the right hand side vanishes, because it is the curl of a gradient field. Therefore we are left with

\displaystyle\begin{split}\displaystyle\frac{\partial}{\partial t}\omega_{g}-% \epsilon_{ghi}\frac{\partial}{\partial r_{h}}\epsilon_{ijm}v_{j}\omega_{m}=&% \displaystyle-\frac{1}{\rho^{2}}\epsilon_{ghi}\left[{\left(\frac{\partial}{% \partial r_{h}}\rho\right)}{\left(\frac{\partial}{\partial r_{i}}p\right)}-% \eta{\left(\frac{\partial}{\partial r_{h}}\rho\right)}{\left(\frac{\partial^{2% }}{\partial r_{j}^{2}}v_{i}\right)}\right.\\ &\displaystyle\left.-{\left(\frac{\eta}{3}+\zeta\right)}{\left(\frac{\partial}% {\partial r_{h}}\rho\right)}{\left(\frac{\partial}{\partial r_{i}}\theta\right% )}\right]+\frac{\eta}{\rho}\frac{\partial^{2}}{\partial r_{j}^{2}}\omega_{g}% \end{split}

(3.15)

or in vector notation with

\displaystyle\begin{split}\displaystyle\frac{\partial}{\partial t}\boldsymbol{% \omega}-\nabla\times(\boldsymbol{v}\times\boldsymbol{\omega})=&\displaystyle-% \frac{1}{\rho^{2}}\left[(\nabla\rho)\times(\nabla p)-\eta(\nabla\rho)\times(% \Delta\boldsymbol{v})\right.\\ &\displaystyle\left.-{\left(\zeta+\frac{\eta}{3}\right)}(\nabla\rho)\times(% \nabla\theta)\right]+\frac{\eta}{\rho}\Delta\boldsymbol{\omega}\end{split}

(3.16)

3.5 Summary

Balance equations

$\displaystyle\frac{\partial}{\partial t}\rho+\frac{\partial}{\partial r_{j}}(v% _{j}\rho)=$	$\displaystyle\ 0$		(3.17)
$\displaystyle\frac{\partial}{\partial t}(\rho v_{i})+\frac{\partial}{\partial r% _{j}}(v_{j}\rho v_{i})=$	$\displaystyle-\frac{\partial}{\partial r_{i}}p++\rho g_{i}+\eta\frac{\partial^% {2}}{\partial r_{j}^{2}}v_{i}+{\left(\frac{\eta}{3}+\zeta\right)}\frac{% \partial}{\partial r_{i}}{\left(\frac{\partial v_{k}}{\partial r_{k}}\right)}$		(3.18)
$\displaystyle\begin{split}\displaystyle\frac{\partial}{\partial t}(\rho e)+% \frac{\partial}{\partial r_{j}}(v_{j}\rho e)=&\displaystyle-\frac{\partial}{% \partial r_{j}}(v_{j}p)+v_{i}\rho g_{i}+\eta\frac{\partial^{2}}{\partial r_{j}% ^{2}}{\left(\frac{1}{2}v_{i}^{2}\right)}+\eta\frac{\partial}{\partial r_{j}}{% \left(v_{i}\frac{\partial v_{j}}{\partial r_{i}}\right)}\\ &\displaystyle+{\left(\zeta-\frac{2}{3}\eta\right)}\frac{\partial}{\partial r_% {j}}{\left(v_{j}\frac{\partial r_{k}}{\partial v_{k}}\right)}\end{split}$		(3.19)

with Newtonian gravity (Poisson Equation):

\displaystyle\frac{\partial}{\partial r_{j}}g_{j}=4\pi G\rho

(3.20)

and an equation of state dependent on the material of the fluid. ¹²¹²See Appendix A.

Global dissipation of kinetic energy $\mathcal{E}$

\displaystyle\begin{split}\displaystyle\mathcal{E}=&\displaystyle\frac{1}{V}% \int_{V}p\frac{\partial v_{j}}{\partial r_{j}}dV-\frac{1}{V}\int_{V}\phi\frac{% \partial}{\partial t}\rho dV-\frac{1}{V}\int_{V}\frac{\eta}{2}{\left(\frac{% \partial}{\partial r_{j}}v_{i}+\frac{\partial}{\partial r_{i}}v_{j}\right)}^{2% }dV\\ &\displaystyle-\frac{1}{V}\int_{V}{\left(\zeta-\frac{2}{3}\eta\right)}{\left(% \frac{\partial v_{k}}{\partial r_{k}}\right)}^{2}dV\end{split}

(3.21)

Divergence equation

\displaystyle\begin{split}\displaystyle\frac{\partial}{\partial t}\theta-% \theta^{2}-v_{i}\frac{\partial}{\partial r_{i}}\theta+\frac{\partial^{2}}{% \partial r_{i}\partial r_{j}}(v_{i}v_{j})=&\displaystyle\frac{1}{\rho^{2}}{% \left(\frac{\partial\rho}{\partial r_{i}}\right)}\cdot{\left[\frac{\partial p}% {\partial r_{i}}-\eta\frac{\partial^{2}}{\partial r_{j}^{2}}v_{i}-{\left(\frac% {\eta}{3}+\zeta\right)}\frac{\partial}{\partial r_{i}}\theta\right]}\\ &\displaystyle-\frac{1}{\rho}{\left[\frac{\partial^{2}}{\partial r_{i}^{2}}p-{% \left(\frac{4}{3}\eta+\zeta\right)}\frac{\partial^{2}}{\partial r_{i}^{2}}% \theta\right]}-4\pi G\rho\end{split}

(3.22)

Vorticity equation

\displaystyle\begin{split}\displaystyle\frac{\partial}{\partial t}\omega_{g}-% \epsilon_{ghi}\frac{\partial}{\partial r_{h}}\epsilon_{ijm}v_{j}\omega_{m}=&% \displaystyle-\frac{1}{\rho^{2}}\epsilon_{ghi}\left[{\left(\frac{\partial}{% \partial r_{h}}\rho\right)}{\left(\frac{\partial}{\partial r_{i}}p\right)}-% \eta{\left(\frac{\partial}{\partial r_{h}}\rho\right)}{\left(\frac{\partial^{2% }}{\partial r_{j}^{2}}v_{i}\right)}\right.\\ &\displaystyle\left.-{\left(\frac{\eta}{3}+\zeta\right)}{\left(\frac{\partial}% {\partial r_{h}}\rho\right)}{\left(\frac{\partial}{\partial r_{i}}\theta\right% )}\right]+\frac{\eta}{\rho}\frac{\partial^{2}}{\partial r_{j}^{2}}\omega_{g}% \end{split}

(3.23)