What Is a Fluid? — Question 1

Question 1

A force that never stops deforming the sample

Two samples fill identical gaps between a fixed lower plate and a movable upper plate. There is no slip at either face. Starting from rest, the same small constant horizontal force is applied at $t=0$. After a brief transient ending before $t=10\,\mathrm{s}$, sample A approaches a fixed top-plate displacement of $0.80\,\mathrm{mm}$; sample B continues to move at $0.20\,\mathrm{mm/s}$. Neither sample fractures. Treat these long-time behaviors as exact idealizations.

1. Which sample has solid-like behavior and which has fluid-like behavior? Explain why the size of the displacement is not the deciding criterion.

2. For B, how much additional displacement occurs between $t=10\,\mathrm{s}$ and $t=40\,\mathrm{s}$? A student says B must eventually stop because the forces on its plate balance. Resolve this claim.

Question 1 - Solution

A force that never stops deforming the sample

(a) Derivation and calculation

Define shear strain by $\gamma=x/h$. Under the maintained nonzero force,$$x_A\to.80\,\mathrm{mm},\quad\dot\gamma_A\to0;\qquad\dot x_B=.20\,\mathrm{mm/s},\quad\dot\gamma_B=\frac{.20\,\mathrm{mm/s}}h>0.$$$$\boxed{A:\ \text{solid-like}},\qquad\boxed{B:\ \text{fluid-like}}.$$The distinction is bounded deformation versus continuing deformation, not the size of one displacement.

(b) Result and interpretation

$$\Delta x_B=\int_{10}^{40}\dot x_B\,dt=(.20)(40-10)=\boxed{6.0\,\mathrm{mm}}.$$$$\sum F_x=0\Rightarrow m\ddot x=0\Rightarrow\dot x=\text{constant}.$$The constant can be nonzero; balanced forces do not require the moving plate to stop.

Physical check

A fluid may shear at constant speed while its resistance exactly balances the applied force.