Euler equations for ideal fluids, approximations

[Niles]

Homework Statement

The Euler equations for ideal compressible flow are given by
[tex]
\partial_t v + (v\cdot \nabla)v = g-\frac{1}{\rho}\nabla p \\
\partial_t \rho + \nabla \cdot(\rho v) = 0
[/tex]
In my book these are written in terms of the small-value expansions [itex]\rho = \rho_0 + \delta \rho[/itex], [itex]p = p_0 + \delta p[/itex] and the equations become
[tex]
\partial_t v = -\frac{1}{\rho_0}\nabla \delta \rho \\
\partial_t (\delta \rho) = -\rho_0 \nabla \cdot v
[/tex]

In the second equation, I don't understand why the RHS becomes [itex]\rho_0 \nabla \cdot v[/itex] instead of [itex](\rho_0+\delta \rho) \nabla \cdot v[/itex]?

Thanks in advance.

 

[bigfooted]

[itex]\delta \rho \nabla \cdot v[/itex] is negligible compared to [itex]\rho_0 \nabla \cdot v[/itex] because [itex]\delta \rho << \rho_0[/itex]