Mass conservation for pulsating sphere

[enc08]

Hi,

I'm trying to understand the mass conservation equation for a pulsating sphere which has thickness dr. Please refer to the attached solution.

\rho = \rho_{0} (ambient density) + \rho' (small deviation)

There are two things I don't follow.

First, is that to obtain the mass, the area of the sphere is multiplied by the speed (and density) at which it is pulsating. Although the units work out, I don't see the physical meaning behind taking the product with speed .

Secondly, on the LHS, the ρ turns into a ρ'. I can't see how this substitution was made.

Thanks for your input!

 

[DEvens]

It's not clear what you mean by "pulsating sphere." What changes and what stays the same? It looks like the density is changing. The drawing makes it look like the sphere is a shell with inside radius r and outside radius r + dr.

Can you make things a bit more clear?

 

[physwizard]

The equation 1) mentioned in your figure doesn't look correct. It may be tentatively correct if the time derivative on the left hand side (L.H.S.) is really a total time derivative and not a partial time derivative(as indicated in the figure) and the deviations of the density are assumed to be sufficiently small.
Also apparently spherical symmetry appears to have been assumed for the density and the velocity.
If you want to figure things out for yourself you can start out with the equation of continuity:
\nabla.(\rho\bar{v}) = -\frac{\partial\rho}{\partial t}