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.

Options

A. `2 v _(0) (ln (r // R _(2)))/( ln (R _(1) //R _(2)))`
B. `(v _(0))/(2) (ln (r // R _(2)))/( (R _(1) // R _(2)))`
C. `v _(0) (ln ( r //R _(2)))/( (R _(1) // R _(2)))`
D. `(2 v _(0))/( 3) (ln (r // R _(2)))/( ln ( R _(1) // R_(2)))`

Answer

C

Explanation

Viscous force `G =- eta (2 pi r l) (dv)/(dr),`
where l = length of the cylinder
`(dr)/(r ) =- ((2pi eta l)/(F))dv`
`int _(R _(1)) ^(R _(2)) (dr)/(r) =- int _(v _(0)) ^(v) ( 2 pi eta l )/(F) dv implies ln ((R _(2))/( R _(1))) = (2 pi eta l )/( F ) v _(0) " "...(i)`
Suppose velocity of liquid at a distacne r is v,
`int _(R _(1)) ^(r) (dr)/(r) =- int _(v 0) ^(v ) (2 pi eta l )/( F) dv implies ln"" (r )/(R _(1)) = (2pi etal )/(F) (v _(0) -v) " "...(ii)`
`ln ((R _(2))/(r )) = (2 pi eta l )/( F )v`
On solving eqs. (i) and (ii),
`v = v _(0) (ln (R _(2) //r))/( ln (R_(2) //R_(1)))implies v = v _(0) (ln (r //R _(2)))/( ln (R _(1)//R _(2)))`