Problem Statement
A piston 796 mm in diameter and 200 mm long works in a cylinder of 800 mm diameter. If the annular space is filled with a lubricating oil of viscosity 5 cp (centipoise), calculate the speed of descent of the piston in a vertical position. The weight of the piston and axial load are 9.81 N.
Given Data
- Piston Diameter, \(D_p = 796 \, \text{mm}\)
- Piston Length, \(L = 200 \, \text{mm}\)
- Cylinder Diameter, \(D_c = 800 \, \text{mm}\)
- Oil Viscosity, \(\mu = 5 \, \text{cp}\)
- Total Downward Force (Weight + Load), \(W = 9.81 \, \text{N}\)
Solution
1. Convert All Units to SI
Diameters and Length to metres:
Viscosity to N·s/m² (1 cp = 0.001 N·s/m²):
2. Calculate Annular Gap and Piston Area
Annular Gap (Oil Film Thickness), \(t\):
Piston Surface Area, \(A\):
3. Apply Force Equilibrium to Find Speed (\(U\))
For the piston to descend at a constant speed, the downward force (\(W\)) must be balanced by the upward viscous drag force (\(F_{drag}\)).
Rearranging to solve for the speed, \(U\):
The speed of descent of the piston is approximately \( U \approx 7.85 \, \text{m/s} \).
Explanation of the Physics
1. Force Balance (Terminal Velocity):
The piston descends at a constant speed, meaning it is not accelerating. According to Newton's First Law, the net force acting on it must be zero. This state of equilibrium is achieved when the downward force (gravity acting on the piston's mass plus any axial load) is perfectly balanced by the upward-acting viscous drag force from the lubricating oil.
2. Viscous Drag in Annular Space:
As the piston moves downward, it shears the oil in the gap between it and the cylinder wall. The oil's internal friction (viscosity) resists this shearing motion. This resistance manifests as a drag force acting on the entire surface area of the piston, opposing its motion.
3. Linear Velocity Profile:
Because the annular gap (\(t\)) is very small compared to the piston diameter, we can simplify the problem by assuming the velocity of the oil changes linearly across the gap—from zero at the stationary cylinder wall to the piston's speed, \(U\), at the piston surface. This allows us to use the simple velocity gradient \(U/t\).
Physical Meaning
The calculated speed of 7.85 m/s is the terminal velocity of the piston under these specific conditions. It is the maximum speed the piston can reach; at this speed, the fluid friction has grown large enough to fully counteract the driving force of the weight.
This principle is the basis for hydraulic dashpots and shock absorbers. By carefully selecting the fluid viscosity and the dimensions of the piston and cylinder, engineers can control the speed at which a component moves, allowing them to dampen vibrations and absorb shocks effectively. If a more viscous oil were used, or if the gap were smaller, the descent speed would be significantly lower.
