Continuity Equation

Question

I have a bit of a soft question: The continuity equation in fluid dynamics says that $\frac{\partial p}{\partial t} + div(p\vec{v}) = 0$, where p is the fluid density and v is velocity. From elementary physics, we also have the continuity equation for 1-dimensional flow: pvA = p'v'A' where p and v are the density and velocity through area A, and p' and v' are the density and volume through area A'. My question is, how do you derive the second equation from the first one?

Answer 1

My understanding is that the continuity equation tells you what governs each single fluid particle at a specific place $\vec x$ and time $t$. The equation's derivation may be found in most elementary Fluid Mechanics textbook, which should actually use the second equation you mentioned, i.e. $\rho \cdot v \cdot A = \rho' \cdot v' \cdot A'$, assuming it is a 3D (not 1D) steady ($\frac{\partial \rho}{\partial t}=0$) unidirectional flow with constant speed on each cross section (of the pipe, say), then applying the Divergence Theorem.

Check out Wikipedia (http://en.wikipedia.org/wiki/Continuity_equation) and from the derivation you may see the two equations are related the other way round.

Btw, I would suggest using $\rho$ in context of Fluid Dynamics to avoid confusion with pressure $p$.

Hope this helps.