# Fluid Dynamics, Linearization Question

1. May 9, 2008

### bakra904

1. The problem statement, all variables and given/known data
Consider an ideal fluid large enough to experience its own gravitational attraction. If the fluid is initially at hydrostatic equilibrium with density $$\rho_{0} (r)$$ and pressure $$p_{0}(r)$$ , it can develop small amplitude pressure waves which may be analyzed as follows.

Show that $$\delta p = p - p_{0}$$ and $$\delta\rho = \rho - \rho_{0}$$ obey the linearized equations of motion

$$\frac{\partial\rho}{\partial t} = - \nabla . ( \rho_{0} v)$$

$$\rho_{0} \frac{\partial v}{\partial t} = -\nabla\delta p + \delta\rho g$$

2. Relevant equations

The relevant equations are

$$\rho\frac{\partial v}{\partial t} + \rho (v . \nabla) v = - \nabla p$$ + g
$$\frac{\partial\rho}{\partial t} + \nabla (\rho .v) = 0$$

3. The attempt at a solution

For the solution I began writing
$$p = p_{0} + \delta p$$ (1)
$$\rho = \rho_{0} + \delta\rho$$ (2)

and assumed that $$\delta\rho <<< \rho_{0}$$ (3)
and $$\delta p <<< p_{0}$$ (4)

I then substituted (3) and (4) into (1) and (2) and only kept the lowest order terms (since $$\delta\rho$$ is small compared to $$\rho$$ ).

So then I get $$\rho_{0}\frac{\partial v}{\partial t} = - \nabla\delta p + \delta\rho g$$
and $$\frac{\partial \delta\rho}{\partial t} + \rho_{0}\nabla .v = 0$$

If anyone could help from here that'd be much appreciated!