Class 11 Physics Mechanics Mechanical Properties of Fluids

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}lnR_{2}R_{1} lnR_{2}k $.

What will be $k$?

(a)

r

Correct answer: (a)

Solution: Let us consider a coaxial cylinder of radius $r$ and thickness $dr$, then force of friction or viscous force on this elemental layer,

$F=2πrlηdrdv $.

This force must be constant from layer to layer so that steady motion may be possible.

or, $rFdr =2πlηdv$........................(1)

Integrating,

$F∫_{R_{2}}rdr =2πlη∫_{0}dv$

or, $Fln(R_{2}r )=2πlηv$ .......................(2)

Putting $r=R_{1}$, we get

$FlnR_{2}R_{1} =2πlηv_{0}$ ...................(3)

From (2) / (3) we get,

$v=v_{0}lnR_{2}R_{1} lnR_{2}r $

Therefore, $k=r$

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

$F=2πrlηdrdv $.

This force must be constant from layer to layer so that steady motion may be possible.

or, $rFdr =2πlηdv$........................(1)

Integrating,

$F∫_{R_{2}}rdr =2πlη∫_{0}dv$

or, $Fln(R_{2}r )=2πlηv$ .......................(2)

Putting $r=R_{1}$, we get

$FlnR_{2}R_{1} =2πlηv_{0}$ ...................(3)

From (2) / (3) we get,

$v=v_{0}lnR_{2}R_{1} lnR_{2}r $

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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