Ideal Fluids in Motion — Question 9

Question 9

Transit time uses the whole speed profile

Along the centerline of a steady flow, a marked parcel moves in the positive $x$ direction with speed $v(x)=U(1+x/L)$ for $0\le x\le L$. Take $U=2.0\,\mathrm{m/s}$ and $L=1.0\,\mathrm{m}$. This is a prescribed centerline profile, not a claim of spatially uniform one-dimensional flow.

1. Find the travel time from $x=0$ to $x=L$ and the parcel acceleration at the exit.

2. Compare the exact time with distance divided by the arithmetic mean of endpoint speeds. Why are they unequal?

Question 9 - Solution

Transit time uses the whole speed profile

(a) Derivation and calculation

$$T=\int_0^L\frac{dx}{U(1+x/L)}=\frac LU[\ln(1+x/L)]_0^L.$$$$\boxed{T=(L/U)\ln2=.346574\,\mathrm s}.$$$$a=v\frac{dv}{dx}=\frac{U^2}{L}(1+x/L)\Rightarrow\boxed{a(L)=2U^2/L=8.00\,\mathrm{m/s^2}}.$$

(b) Result and interpretation

$$v(0)=2.0,\quad v(L)=4.0\,\mathrm{m/s},\quad T_{\mathrm{endpoint}}=\frac1{(2+4)/2}=.333333\,\mathrm s.$$$$\bar v_{\mathrm{time}}=L/T=U/\ln2=2.88539\,\mathrm{m/s}<3.0.$$The parcel spends more time per unit distance in the slower region; linear variation with position is not linear variation with time.

Physical check

The exact time exceeds the endpoint-average estimate, consistent with greater residence time in slow flow.