Flux — Question 2

Question 2

A parabolic pipe profile makes center speed a poor flow-rate estimate

A circular opening of radius R is normal to the z axis. The axial fluid speed is $v_z(r)=v_c(1-r^2/R^2)$ for $0\le r\le R$, with no reverse flow.

1. Integrate the signed volume flux and obtain the mean speed over the opening.

2. Find the radius of a central subdisk that carries half the total flux. Compare it with the radius that encloses half the area.

Question 2 - Solution

A parabolic pipe profile makes center speed a poor flow-rate estimate

(a) Method and calculation

$$dA=2\pi r\,dr,\quad Q=2\pi v_c\int_0^R\left(1-\frac{r^2}{R^2}\right)r\,dr=\frac{\pi R^2v_c}{2}.$$$$\bar v=Q/(\pi R^2)=v_c/2.$$

(b) Interpretation and further calculation

$$\frac{Q(a)}{Q(R)}=2(a/R)^2-(a/R)^4=\tfrac12.$$$$\frac aR=\sqrt{1-\frac1{\sqrt2}}\approx.541196,\quad\frac{a_{\mathrm{half\ area}}}R=1/\sqrt2\approx.707107.$$The central region carries disproportionately more flow because its speed is larger.

Physical check

The larger algebraic root for $(a/R)^2$ lies outside the opening and must be discarded.