A long cylinder of radius $R_{1}$ is displaced along its axis with a constant velocity $v_{0}$ inside a stationary co-axial cylinder of radius $R_{2}$ . The space between the cylinders is filled with viscous liquid. The velocity of the liquid as a function of the distance $r$ from the axis of the cylinders is $v = v_{0} \frac{ln \frac{k}{R_{2}}}{ln \frac{R_{1}}{R_{2}}}$ .
What will be $k$ ?

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Solution

Let us consider a coaxial cylinder of radius $r$ and thickness $d r$ , then force of friction or viscous force on this elemental layer,
$F = 2 π r l η \frac{d v}{d r}$ .
This force must be constant from layer to layer so that steady motion may be possible.
or, $\frac{F d r}{r} = 2 π l η d v$ ........................(1)
Integrating,
$F \int_{R_{2}}^{r} \frac{d r}{r} = 2 π l η \int_{0}^{v} d v$
or, $F ln (\frac{r}{R_{2}}) = 2 π l η v$ .......................(2)
Putting $r = R_{1}$ , we get
$F ln \frac{R_{1}}{R_{2}} = 2 π l η v_{0}$ ...................(3)
From (2) / (3) we get,
$v = v_{0} \frac{ln \frac{r}{R_{2}}}{ln \frac{R_{1}}{R_{2}}}$
Therefore, $k = r$

Note: The force $F$ is supplied by the agency which tries to carry the inner cylinder with velocity $v_{0}$ .

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