Ideal Fluids in Motion — Question 10

Question 10

A tracer must follow the fluid closely enough to measure it

A tracer particle is used to photograph a nominally horizontal flow over an observation time $T=0.50\,\mathrm{s}$. Relative to the liquid it settles at $w_s=0.0020\,\mathrm{m/s}$. Its velocity response to a sudden change of fluid velocity has a supplied mismatch model $\Delta v(t)=\Delta v(0)e^{-t/\tau}$ with $\tau=0.10\,\mathrm{s}$.

1. Find its settling displacement during the exposure and the fraction of its initial velocity mismatch remaining after $T$.

2. The desired vertical-path accuracy is $0.10\,\mathrm{mm}$. Does this tracer meet it? State what these calculations say about identifying photographed particle tracks with fluid pathlines.

Question 10 - Solution

A tracer must follow the fluid closely enough to measure it

(a) Derivation and calculation

$$\boxed{\Delta y=w_sT=(.0020)(.50)=.00100\,\mathrm m=1.00\,\mathrm{mm}}.$$$$\boxed{\frac{\Delta v(T)}{\Delta v(0)}=e^{-T/\tau}=e^{-5}=.00673795=0.673795\%}.$$

(b) Result and interpretation

$$\frac{\Delta y}{\epsilon_y}=\frac{1.00\,\mathrm{mm}}{.100\,\mathrm{mm}}=\boxed{10}.$$The tracer fails the vertical-path criterion even though its final velocity mismatch is small.$$\text{small response lag}\ \not\Rightarrow\ \text{negligible accumulated slip}.$$A photographed track approximates a fluid pathline only within the permitted buoyancy and inertia errors.

Physical check

At the given settling speed, the maximum exposure for a .100 mm settling error is only .0500 s.