What Is a Fluid? — Question 5

Question 5

Motion is not the same thing as shear

Tracer marks inside three samples have the velocity fields below, where $U>0$, $\Omega>0$, and $k>0$ are constants. A is an elastic solid translating rigidly. B is a fluid rotating as a rigid body after it has settled into steady rotation. C is a fluid in planar shear. Use $\mathbf v_A=U\hat{\mathbf i}$, $\mathbf v_B=-\Omega y\hat{\mathbf i}+\Omega x\hat{\mathbf j}$, and $\mathbf v_C=ky\hat{\mathbf i}$.

1. Two marks are initially separated by $(\Delta x,\Delta y)=(a,a)$, with $a>0$. Find the instantaneous rate of change of their squared separation for each field using $d(\Delta\mathbf r\cdot\Delta\mathbf r)/dt=2\Delta\mathbf r\cdot\Delta\mathbf v$.

2. Which motions distort separations? Explain why neither visible motion nor a difference in velocity between two marks is sufficient by itself to identify shear deformation.

Question 5 - Solution

Motion is not the same thing as shear

(a) Derivation and calculation

For $\Delta\mathbf r=(a,a)$,$$\frac{d|\Delta\mathbf r|^2}{dt}=2\Delta\mathbf r\cdot\Delta\mathbf v.$$$$A:\quad\Delta\mathbf v=(0,0)\Rightarrow\boxed0.$$$$B:\quad\Delta\mathbf v=(-\Omega a,\Omega a)\Rightarrow2(-\Omega a^2+\Omega a^2)=\boxed0.$$$$C:\quad\Delta\mathbf v=(ka,0)\Rightarrow\boxed{2ka^2>0}.$$

(b) Result and interpretation

For an arbitrary separation $(u,w)$ under rigid rotation,$$\Delta\mathbf r\cdot\Delta\mathbf v=(u,w)\cdot(-\Omega w,\Omega u)=0.$$Thus A translates and B rotates without changing distances, while C changes shape. Velocity differences may represent rotation rather than shear.

Physical check

The calculated separation rate has units $\mathrm{m^2/s}$; material classification still requires a mechanical response test.