A cylindrical-shaped piston of density rhop, diameter D. and length L falls at s

Question

A cylindrical-shaped piston of density rhop, diameter D. and length L falls at speed V inside a cylinder. The inner walls of the cylinder are coated with a thin film of oil of density rhooil. viscosity mu, and thickness h, as sketched. Assuming that the air pressure below the piston is equal to that above the piston, and neglecting any air resistance, generate an expression for terminal speed V as a function of the other variables. Note: Terminal speed is the constant speed at which the piston would fall if the cylinder were tall enough for steady conditions to be achieved.

Explanation / Answer

a) At terminal speed v, accelartion of piston = 0

hence, net force = ma = 0

weight of piston = viscous force

assuming, that the velocity profile in the film is linear

mg = Adv/dy

Vg = (Dh)dv/dy

dv/dy = Vg/[Dh]

v = Vg/D

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