Flux — Question 9

Question 9

Oscillatory flow has zero cycle transport but nonzero throughput

A fixed opening of area A is normal to a spatially uniform flow $v_n(t)=v_0\cos\omega t$, where $v_0,\omega>0$. A complete cycle lasts $T=2\pi/\omega$.

1. Find signed transported volume over one cycle and over the first quarter-cycle.

2. Integrate the absolute crossing rate to obtain the total volume passing either way in one cycle.

Question 9 - Solution

Oscillatory flow has zero cycle transport but nonzero throughput

(a) Method and calculation

$$V_{\mathrm{cycle}}=Av_0\int_0^{2\pi/\omega}\cos\omega t\,dt=0.$$$$V_{\mathrm{quarter}}=Av_0\int_0^{\pi/(2\omega)}\cos\omega t\,dt=\frac{Av_0}{\omega}.$$

(b) Interpretation and further calculation

$$V_{\mathrm{gross}}=\frac{Av_0}{\omega}\int_0^{2\pi}|\cos u|\,du=\frac{4Av_0}{\omega}.$$$$\frac{V_{\mathrm{gross}}}T=\frac{2Av_0}{\pi}.$$Equal positive and negative transport cancels in the signed result, not in throughput.

Physical check

The factor of four follows from four equal quarter-cycle areas under the absolute cosine.