Question

A circular disc of a diameter ‘d’ is slowly rotated in a liquid of large viscosity '$\eta$' at a small distance ‘h’ from a fixed surface as shown in figure. If an expression for torque ‘T’ necessary to maintain an angular velocity '$\omega$' is $\frac{\pi \eta \omega d^4}{4\lambda h}$ then find $\lambda$.___

Answer 1

For a circular ring of radius r.
Force due to viscosity $=\eta A\frac{dv}{dx}$
$dF=dFr$
$dT=\frac{2\pi \eta }{h}\omega r^{3}dr$
$T=\frac{2\pi \eta }{h}\omega {\int }_0^{\frac{d}{2}}{r}^{3}dr$
$T=\frac{\pi \eta \omega d^4}{32\lambda h}$