What Is a Fluid? — Question 10

Question 10

A sliding boundary can imitate a flowing sample

Two shear tests give the same top-plate displacement, $x_{\mathrm{top}}=1.0\,\mathrm{mm}$, while the bottom plate is fixed. The gap is $h=10\,\mathrm{mm}$. Tracer displacements inside sample A are $x_A(y)=0.10y$ for $0\le y\le h$, with both quantities measured in millimeters. Tracers inside B all have $x_B(y)=1.0\,\mathrm{mm}$. The displacement is horizontal; $y$ measures height above the bottom plate.

1. Calculate the relative displacement of tracers at $y=2\,\mathrm{mm}$ and $8\,\mathrm{mm}$ for each sample. Which sample has demonstrated internal shear deformation, and where must B have slipped?

2. Does the internal displacement pattern of A at this single instant prove it is a fluid? Specify the added observation needed for a convincing fluid-like response test.

Question 10 - Solution

A sliding boundary can imitate a flowing sample

(a) Derivation and calculation

$$\Delta x_A=x_A(8)-x_A(2)=.10(8-2)=\boxed{.60\,\mathrm{mm}}.$$$$\Delta x_B=1.0-1.0=\boxed0.$$$$\frac{dx_A}{dy}=.10,\qquad\frac{dx_B}{dy}=0.$$A demonstrates internal shear; B translates rigidly.

(b) Result and interpretation

At the lower boundary,$$x_B(0)-x_{\mathrm{bottom}}=1.0-0=\boxed{1.0\,\mathrm{mm}}$$$$\Rightarrow\text{bottom-boundary slip}.$$A single nonzero shear strain does not prove fluid behavior. Maintain a small nonzero load and measure the evolution of internal strain while checking slip:$$\gamma(t)\to\gamma_\infty\ \text{versus continued growth of }\gamma(t).$$

Physical check

An elastic solid can exhibit the same instantaneous sheared shape; the time-dependent response supplies the missing evidence.