A cylindrical piston of length ℓ and radius R is moved with constant velocity v in a cylindrical tube filled with oil. If the clearance between the piston and tube is h (

Question

A cylindrical piston of length ℓ and radius R is moved with constant velocity v in a cylindrical tube filled with oil. If the clearance between the piston and tube is h (<< R), assuming η as the coefficient of viscosity of the oil, find the external force required to derive the piston with constant velocity.

TestHub · Accepted Answer

Since, the thickness of the film is small, the velocity profile of the oil can be assumed as linear.
Then,
$\frac{dv}{dy} = \frac{v}{h}$
Substituting $\frac{dv}{dy} = \frac{v}{h}$ in Newton's viscosity formula
$τ = - η \frac{dv}{dy}$ , we have

$τ = \frac{η v}{h}$

Hence, the shearing force on the curved surface of the piston is

$F = τ A = η \frac{vA}{h},$

where

$A = 2 π R ℓ$

This gives

$F = 2 π R η \frac{v ℓ}{h}$

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