Equations of fluid dynamics 5 Newtonian incompressible fluid A Equations of State

Chapter 6 Fluid dynamics in comoving coordinates

6.1 Introduction

On large scales ( $>$ 100Mpc) the distribution of matter in the universe is isotropic (it looks the same in all directions) and homogeneous (it is isotropic at each point). But only the space is assumed to be isotropic and homogenous. The observed expansion of the universe singles out a special direction in time.²⁰²⁰The universe is not a maximally symmetric 4-dimensional manifold, but can be depicted as maximally symmetric 3-dimensional sheets spacelike sheets in 4-dimensional spacetime. The metric on such a manifold is the Robertson-Walker-metric.

The physical distance on large scales²¹²¹This is a very important point. If the space would also expand on small scales we couldn’t measure the expansion, because everything including our distance measurement device would expand. But on small scales the universe is not homogenous. On small scales the metric of the universe is not a Robertson-Walker metric, but more like a Schwarzschild metric, which is isotropic, but not homogenous. between two points in such an expanding universe varies with time like

\displaystyle r_{i}=a(t)x_{i}

(6.1)

The factor $a$ is a dimensionless scale factor greater than zero, which must be the same for each component of the distance vector because of the assumed isotropy. The scale factor can only depend on the time $t$ and not on the position $x_{i}$ because of the assumed homogenity of space.

The change of the distance with time in an expanding universe is then

\displaystyle\dot{r}_{i}=\dot{a}x_{i}+a\dot{x}_{i}

(6.2)

The global velocity of a particle $v_{i}=\dot{r}_{i}$ which does not move relative to the expanding space ( $\dot{x}_{i}=0$ ) is then

\displaystyle\dot{v}_{i}=\dot{a}x_{i}=\frac{\dot{a}}{a}r_{i}=H(t)r_{i}

(6.3)

where $H$ is the so called Hubble parameter. Is a particle moving relative to the expanding space ( $\dot{x}_{i}\neq 0$ ) then we measure the additional local (also called proper) velocity $u_{i}=a\dot{x}_{i}$ of the particle. This local velocity can, according to special relativity, be never greater than the speed of light $c$ . Nevertheless the global velocity (e.g. the measured escape velocities of galaxies at great distances) can be greater than $c$ (Davis and Lineweaver, 2004). Generally the physical velocity of a particle is the the sum of global and local velocity

\displaystyle v_{i}=\dot{a}x_{i}+u_{i}(x_{i},t)

(6.4)

6.1.1 Useful transformations

$\displaystyle r_{i}$	$\displaystyle=a(t)x_{i}$	(6.5)
$\displaystyle v_{i}$	$\displaystyle=a\dot{x}_{i}+\dot{a}x_{i}=u_{i}+\dot{a}x_{i}$	(6.6)
$\displaystyle\frac{\partial}{\partial r_{i}}$	$\displaystyle=\frac{1}{a}\frac{\partial}{\partial x_{i}}$	(6.7)
$\displaystyle\frac{\partial}{\partial r_{i}}v_{i}$	$\displaystyle=\frac{1}{a}\frac{\partial}{\partial x_{i}}u_{i}+3\frac{\dot{a}}{a}$	(6.8)
$\displaystyle\frac{\partial}{\partial r_{i}}v_{j}$	$\displaystyle=\frac{1}{a}\frac{\partial}{\partial x_{i}}u_{j}+\frac{\dot{a}}{a% }\delta_{ij}$	(6.9)
$\displaystyle{\left(\frac{\partial A}{\partial t}\right)}_{r}+v_{j}\frac{% \partial A}{\partial r_{j}}$	$\displaystyle={\left(\frac{\partial A}{\partial t}\right)}_{x}+\frac{1}{a}u_{j% }\frac{\partial A}{\partial x_{j}}$	(6.10)
$\displaystyle A(r_{i},v_{i},t)$	$\displaystyle\neq A(x_{i},u_{i},t)$	(6.11)

The stress tensor for a newtonian fluid in comoving coordinates is²²²²See Appendix L.

\displaystyle\sigma^{\prime}_{ij}=2\eta T^{*}_{ij}+\zeta\delta_{ij}\frac{1}{a}% {\left(\frac{\partial}{\partial x_{k}}u_{k}+n\dot{a}\right)}

(6.12)

with

\displaystyle T^{*}_{ij}=\frac{1}{a}{\left[\frac{1}{2}{\left(\frac{\partial}{% \partial x_{j}}u_{i}+\frac{\partial}{\partial x_{i}}u_{j}\right)}-\frac{1}{n}% \delta_{ij}\frac{\partial}{\partial x_{k}}u_{k}\right]}

(6.13)

6.1.2 Transformed equations

$\displaystyle\frac{\partial}{\partial t}\rho+\frac{1}{a}\frac{\partial}{% \partial x_{j}}(u_{j}\rho)=$	$\displaystyle-3\frac{\dot{a}}{a}\rho$		(6.14)
$\displaystyle\frac{\partial}{\partial t}(\rho u_{i})+\frac{1}{a}\frac{\partial% }{\partial x_{j}}(u_{j}\rho u_{i})=$	$\displaystyle-\frac{1}{a}\frac{\partial}{\partial x_{i}}p+\frac{1}{a}\frac{% \partial}{\partial x_{j}}\sigma^{\prime}_{ij}+\rho g^{*}_{i}-4\frac{\dot{a}}{a% }\rho u_{i}$		(6.15)
$\displaystyle\begin{split}\displaystyle\frac{\partial}{\partial t}(\rho e)+% \frac{1}{a}\frac{\partial}{\partial x_{j}}(u_{j}\rho e)=&\displaystyle-\frac{1% }{a}\frac{\partial}{\partial x_{j}}(u_{j}p)+\frac{1}{a}\frac{\partial}{% \partial x_{j}}(u_{i}\sigma^{\prime}_{ij})+\frac{1}{a}u_{i}\rho g^{*}_{i}\\ &\displaystyle-3\frac{\dot{a}}{a}(\rho e+\frac{1}{3}\rho u^{2}_{i}+p)\end{split}$		(6.16)

with

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Newtonian Gravity in comoving coordinates (Poisson Equation):

$\frac{1}{a}\frac{\partial}{\partial x_{j}}g^{*}_{j}=4\pi G$ with $g_{i}^{*}=\frac{1}{a}\frac{\partial\varphi}{\partial x_{i}}$ and $\varphi=\phi+\frac{1}{2}a\ddot{a}x_{i}^{2}$

The energy equation is the sum of the equation for the kinetic energy and the internal energy

	$\displaystyle\frac{\partial}{\partial t}(\rho e_{k})+\frac{1}{a}\frac{\partial% }{\partial x_{j}}(u_{j}\rho e_{k})$	$\displaystyle=-\frac{1}{a}u_{i}\frac{\partial}{\partial x_{i}}p+\frac{1}{a}u_{% i}\frac{\partial}{\partial x_{j}}\sigma^{\prime}_{ij}-\frac{1}{a}\rho u_{i}g^{% *}_{i}-5\frac{\dot{a}}{a}\rho e_{k}$		(6.17)
	$\displaystyle\frac{\partial}{\partial t}(\rho e_{int})+\frac{1}{a}\frac{% \partial}{\partial x_{j}}(u_{j}\rho e_{int})$	$\displaystyle=-\frac{1}{a}p\frac{\partial}{\partial x_{j}}u_{j}-\frac{1}{a}% \sigma^{\prime}_{ij}\frac{\partial}{\partial x_{j}}u_{i}-3\frac{\dot{a}}{a}(% \rho e_{int}+p)$		(6.18)