Question

The velocity of water in a river is $18\text{km/hr}$ at the surface. If the river is $5\text{m}$ deep, find the shearing stress between the horizontal layers of water. The viscosity of water is ${10}^{-2}\text{Poise}$.

• A. ${10}^{-3}{\text{N/m}}^2$
• B. $4\times {10}^{-3}{\text{N/m}}^2$
• C. $3\times {10}^{-3}{\text{N/m}}^2$
• D. $2\times {10}^{-3}{\text{N/m}}^2$

Answer 1

The correct option is A ${10}^{-3}{\text{N/m}}^2$

As velocity at the bottom of the river will be zero,
Velocity gradient in vertical direction,
$\frac{dv}{dy}=\frac{18\times {10}^3}{60\times 60\times 5}=1{\text{s}}^{-1}$
Now as the viscous force, $F_v=\eta A\left(\frac{dv}{dy}\right)$
$\therefore$ Shear stress between horizontal layers of water$\tau =\frac{F_v}{A}=\eta \frac{dv}{dy}={10}^{-2}\times 0.1\times 1={10}^{-3}{\text{N/m}}^2$