Flux — Question 7

Question 7

A box can have outward flow on some faces and zero net flux

A velocity field in a rectangular box $0\le x\le a$, $0\le y\le b$, $0\le z\le c$ is $\mathbf v=(\alpha x,-\alpha y,0)$, with $\alpha>0$ in s$^{-1}$. Use outward normals.

1. Calculate the flux through all six faces and the total.

2. Replace the y component by $-\beta y$. Find the new total flux and the condition for no net volume transport through the box.

Question 7 - Solution

A box can have outward flow on some faces and zero net flux

(a) Method and calculation

$$Q_{x=a}=\alpha abc,\quad Q_{x=0}=0,\quad Q_{y=b}=-\alpha abc,\quad Q_{y=0}=0.$$$$Q_{z=0}=Q_{z=c}=0,\quad Q_{\mathrm{total}}=0.$$

(b) Interpretation and further calculation

$$Q_{\mathrm{total}}=\alpha abc-\beta abc=(\alpha-\beta)V.$$$$Q_{\mathrm{total}}=0\Longleftrightarrow\alpha=\beta.$$Equal expansion and contraction rates cancel in the net outward flux.

Physical check

The inward y flow has a negative dot product with the outward normal on the positive-y face.