(adsbygoogle = window.adsbygoogle || []).push({}); 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 [tex]\rho_{0} (r)[/tex] and pressure [tex]p_{0}(r)[/tex] , it can develop small amplitude pressure waves which may be analyzed as follows.

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

[tex]\frac{\partial\rho}{\partial t} = - \nabla . ( \rho_{0} v)[/tex]

[tex]\rho_{0} \frac{\partial v}{\partial t} = -\nabla\delta p + \delta\rho g [/tex]

2. Relevant equations

The relevant equations are

[tex]\rho\frac{\partial v}{\partial t} + \rho (v . \nabla) v = - \nabla p[/tex] + g

[tex]\frac{\partial\rho}{\partial t} + \nabla (\rho .v) = 0 [/tex]

3. The attempt at a solution

For the solution I began writing

[tex] p = p_{0} + \delta p[/tex] (1)

[tex] \rho = \rho_{0} + \delta\rho[/tex] (2)

and assumed that [tex]\delta\rho <<< \rho_{0}[/tex] (3)

and [tex]\delta p <<< p_{0} [/tex] (4)

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

So then I get [tex]\rho_{0}\frac{\partial v}{\partial t} = - \nabla\delta p + \delta\rho g[/tex]

and [tex]\frac{\partial \delta\rho}{\partial t} + \rho_{0}\nabla .v = 0 [/tex]

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

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Fluid Dynamics, Linearization Question

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**