1. The problem statement, all variables and given/known data
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.
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.
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