Question

the angular velocity of the rotating fluid as a function of radius $r$;

Answer 1

Let us consider an elemental cylinder of radius $r$ and thickness $dr$ then from Newton's formula

$F=2\pi rl\eta r\frac{d\omega }{dr}=2\pi l\eta r^2\frac{d\omega }{dr}$

and moment of this force actin on the element,

$N=2\pi r^{2}l\eta \frac{d\omega }{dr}r=2\pi r^{3}l\eta \frac{d\omega }{dr}$

or, $2\pi l\eta d\omega =N\frac{dr}{r^3}$

As in the previous problem $N$ is constant when conditions are steady

Integrating, $2\pi l\eta {\int }_0^{\omega }d\omega =N{\int }_{R_1}^r\frac{dr}{r^3}$

or, $2\pi l\eta \omega =\frac{N}{2}[\frac{1}{R_1^2}-\frac{1}{r^2}]$

Putting $r=R_2,\omega ={\omega }_2$, we get

$2\pi l\eta {\omega }_2\frac{N}{2}[\frac{1}{R_1^2}-\frac{1}{R_2^2}]$

From $\left(3\right)$ and $\left(4\right)$

$\omega ={\omega }_2\frac{R_1^2{R}_2^2}{R_2^2-R_1^2}[\frac{1}{R_1^2}-\frac{1}{r_2}]$

From Eq (4)

$N_1=\frac{N}{l}=4\pi \eta {\omega }_2\frac{R_1^2{R}_2^2}{R_1^2-R_1^2}$