Divergence theorem and continuity equation

Question

A steady $1D$ flow field is flowing within a pipe in the $x$-direction.

Starting with the continuity equation $$\dfrac {d \rho}{dt} + ∇ · (\rho u) = 0$$ where $u = u \underline i + v \underline j + w \underline k$, and using the divergence theorem, show that this leads to the expression $ \rho uA$ = constant where $A$ is the cross-sectional area of the pipe in the $y − z$ plane

I'm struggling with this and some help would be appreciated.

Thanks!

Well, in that case $\rho$ is constant, so $\partial \rho/\partial t = 0$, therefore $\rho {\bf u} $ is also constant — caverac, May 10 '17 at 21:28

score 1 · Accepted Answer · edited May 01 '18 at 18:05

1

hint

steady $\implies \frac {d\rho}{dt}=0$

divergence theorem is

$$\int\int\int \nabla \cdot (\rho \vec {u})=$$ $$\int\int \rho \vec { u}\cdot\vec {n}ds=0 $$

edited May 01 '18 at 18:05

DS08

367

answered May 10 '17 at 20:55

hamam_Abdallah

62,951

Divergence theorem and continuity equation

1 Answers1