2.1.1 Linear advection as special case: density and momentum

In a 1D flow described by Eq. (40) assuming $\bgroup\color{DEFcolor}$ \frac{\partial P}{\partial x} =0$\egroup$ and $\bgroup\color{DEFcolor}$ \frac{\partial v_{}}{\partial x} =0$\egroup$ at $\bgroup\color{DEFcolor}$t=t_0$\egroup$ gives for the mass

$\displaystyle 0$	$\textstyle =$	$\displaystyle \frac{\partial \rho}{\partial t} + \frac{\partial \rho \; v_{}}{\partial x}$	(51)
	$\textstyle =$	$\displaystyle \frac{\partial \rho}{\partial t} + v_{} \frac{\partial \rho}{\partial x} \enspace ,$	(52)

for the momentum

$\displaystyle 0$	$\textstyle =$	$\displaystyle \frac{\partial \rho v_{}}{\partial t} + \frac{\partial \rho v_{} \; v_{}}{\partial x}$	(53)
	$\textstyle =$	$\displaystyle v_{} \left( \frac{\partial \rho}{\partial t} + v_{} \frac{\partial \rho}{\partial x} \right) + \rho \frac{\partial v_{}}{\partial t}$	(54)
	$\textstyle =$	$\displaystyle \rho \frac{\partial v_{}}{\partial t} \enspace .$	(55)