What Is a Fluid? — Question 7

Question 7

What a motion detector can and cannot prove

A top plate is pulled by a constant nonzero tangential force while the bottom plate remains fixed. A detector reports no resolved relative displacement during $120\,\mathrm{s}$ and can resolve a displacement of $0.010\,\mathrm{mm}$ or larger. Assume any fluid-like displacement would be monotonic, with no initial transient, drift or slip.

1. Find the upper bound on the mean relative speed consistent with the report. Can the report prove that the sample is an elastic solid?

2. A candidate fluid would move at $2.0\times10^{-8}\,\mathrm{m/s}$ under this load. Find the minimum duration needed to reach the detector threshold. Explain what a longer null result could and could not establish.

Question 7 - Solution

What a motion detector can and cannot prove

(a) Derivation and calculation

$$|\Delta x|<.010\,\mathrm{mm}=1.0\times10^{-5}\,\mathrm m.$$$$\boxed{|\bar v|<\frac{1.0\times10^{-5}}{120}=8.33\times10^{-8}\,\mathrm{m/s}}.$$This bound permits both zero speed and slower continued deformation; it cannot prove a solid response.

(b) Result and interpretation

For the candidate speed,$$\boxed{t_{\mathrm{threshold}}=\frac{1.0\times10^{-5}}{2.0\times10^{-8}}=500\,\mathrm s}.$$$$x(120)=2.4\times10^{-6}\,\mathrm m=.0024\,\mathrm{mm}<.010\,\mathrm{mm}.$$A null result at 500 s rules out this candidate under the stated assumptions, but not every slower fluid response.

Physical check

The detector resolves displacement at the threshold itself, so the minimum duration is inclusive.