When a viscous liquid flows , adjacent layers oppose their relative motion by applying a viscous force given by
` F = - eta A (dv)/(dz)`
where , ete = coefficient of viscosity ,
A = surface area of adjacent layers in contact ,
`(dv)/(dz)` = velocity gradient
Now , a viscous liquid having coefficient of viscosity `eta ` is flowing through a fixed tube of length l and radius R under a pressure difference P between the two ends of the tube . Now consider a cylindrical vloume of liquid of radius r . Due to steady flow , net force on the liquid in cylindrical volume should be zero .
` - eta 2pirl (dv)/(dr) = Ppir^(2)`
` - int _(v)^(0),dv = P/(2 eta l) int_(tau)^(R) rdr ( :' ` layer in contact with the tube is stationary )
` v = v_(0) (1- (r^(2))/(R^(2)))`, where `v_(0) = (PR^(2))/(4nl)`
` :. " " Q = (piPR^(4))/(8sta l)`
This is called Poisecuille's equation .

The viscous force on the cylindrical volume of the liquid varies as

Options

A. ` F prop r^(2)`
B. `F prop r`
C. `F prop 1//r`
D. `F prop 1//r^(2)`

Answer

A