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. Find the velocity of the liquid as a function of the distance r from the axis of the cylinders. The flow is laminar.

Explanation

Let us consider a coaxial cylinder of radius r and thickness `dr`, then force of friction or viscous force on this elemental layer, `F=2pi r l eta(dv)/(dr)`.
This force must be constant from layer to layer so that steady motion may be possible.
or, `(Fdr)/(r)=2pi l eta dv` (1)
Integrating,
`Funderset(R_2)overset(r)int(dr)/(r)=2pi l eta underset(0)overset(v)int dr`
or, `F1n((r)/(R_2))=2pi l eta v` (2)
Putting `r=R_1`, we get
`F1n(R_1)/(R_2)=2pi l eta v_0`
From (2) by (3) we get,
`v=v_0(1nr//R_2)/(1nR_1//R_2)`
Note: The force F is supplied by the agency which tries to carry the inner cylinder with velocity `v_0`.