Flux — Question 8

Question 8

A moving collection window measures relative rather than laboratory velocity

A planar window of area A has fixed unit normal $\hat{\mathbf n}$ but translates with velocity $\mathbf u$. Fluid has uniform laboratory velocity $\mathbf v$. Ignore changes in window area or orientation.

1. Using the swept volume in time dt, derive the signed crossing rate relative to the window.

2. For $A=.500$ m$^2$, $\mathbf v=4\hat{\mathbf x}$ m/s, $\mathbf u=6\hat{\mathbf x}$ m/s and $\hat{\mathbf n}=\hat{\mathbf x}$, compare relative crossing flux and laboratory flux.

Question 8 - Solution

A moving collection window measures relative rather than laboratory velocity

(a) Method and calculation

$$dV=A[(\mathbf v-\mathbf u)\cdot\hat{\mathbf n}]\,dt.$$$$Q_{\mathrm{relative}}=A(\mathbf v-\mathbf u)\cdot\hat{\mathbf n}.$$

(b) Interpretation and further calculation

$$Q_{\mathrm{relative}}=(.500)(4-6)=-1.00\,\mathrm{m^3/s}.$$$$Q_{\mathrm{lab}}=A\mathbf v\cdot\hat{\mathbf n}=2.00\,\mathrm{m^3/s}.$$The faster window overtakes the fluid, so crossings relative to it occur in the negative-normal direction.

Physical check

If the window moves with the fluid, its relative crossing rate vanishes even when laboratory velocity is nonzero.