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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)