Flux — Question 6

Question 6

Counterflow separates signed flux from total crossing rate

Across a planar opening oriented toward positive z, half the area A has velocity $+3v_0\hat{\mathbf z}$ and the other half has velocity $-v_0\hat{\mathbf z}$. Density is constant, and $v_0>0$.

1. Find signed volume flux and the total volume crossing in either direction per second.

2. If the positive-flow area fraction is f instead of one half, determine f for zero net flux and find the total crossing rate at that f.

Question 6 - Solution

Counterflow separates signed flux from total crossing rate

(a) Method and calculation

$$Q_{\mathrm{net}}=\tfrac A2(3v_0)+\tfrac A2(-v_0)=Av_0.$$$$Q_{\mathrm{gross}}=\tfrac A2|3v_0|+\tfrac A2|-v_0|=2Av_0.$$

(b) Interpretation and further calculation

$$Q_{\mathrm{net}}=Av_0[3f-(1-f)]=Av_0(4f-1)=0\Rightarrow f=\tfrac14.$$$$Q_{\mathrm{gross}}=Av_0[3f+(1-f)]=\tfrac32Av_0.$$Zero net flux permits substantial two-way transport.

Physical check

The integral of an absolute value is generally different from the absolute value of an integral.