Q. 2.SP.8

A 1-in-wide space between two horizontal plane surfaces is filled with SAE 30 Western lubricating oil at $80^{\circ} \mathrm{F}$ . What force is required to drag a very thin plate of $4-\mathrm{ft}^2$  area through the oil at a velocity of $20 \mathrm{ft} / \mathrm{min}$  if the plate is 0.33  in from one surface?

Verified Solution

Fig. A.1: $\quad \mu=0.0063 \mathrm{lb} \cdot \mathrm{sec} / \mathrm{ft}^2$

Eq. (2.9): $\quad \tau_1=0.0063 \times(20 / 60) /(0.33 / 12)=0.0764 \mathrm{lb} / \mathrm{ft}^2$

Eq. (2.9): $\quad \tau_2=0.0063 \times(20 / 60) /(0.67 / 12)=0.0394 \mathrm{lb} / \mathrm{ft}^2$

From Eq. (2.9): $\quad F_1=\tau_1 A=0.0764 \times 4=0.305 \mathrm{lb}$

From Eq. (2.9): $\quad F_2=\tau_2 A=0.0394 \times 4=0.158 \mathrm{lb}$

Force $=F_1+F_2=0.463 \mathrm{lb} \quad$

$\tau=\frac{F}{A}=\mu \frac{U}{Y}=\mu \frac{d u}{d y}$       (2.9 )