Euler equations for ideal fluids, approximations

[Niles]

Homework Statement

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

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

Thanks in advance.

 

[bigfooted]

\delta \rho \nabla \cdot v is negligible compared to \rho_0 \nabla \cdot v because \delta \rho &lt;&lt; \rho_0