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

Question

A cylindrical-shaped piston of density rho p, 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 rho 0il. 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. Suppose rho p = 7350 kg/m3 (steel). D = 4.20 cm. L = 25.4 cm. h = 0.0150 cm. and the oil is unused engine oil at a temperature of 150 degree C (see the Appendices for the required properties). Calculate the terminal speed V for these conditions, showing all your units and unity conversion ratios.

Explanation / Answer

a) at terminal speed, accelartion = 0

hence, net force = 0

mg = Adv/dy

Vg = (Dh)dv/dy

dv/dy = Vg/[Dh]

v = Vg/D

b)

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