A piston having a diameter of 5.48 in. and a length of 9.50...

A piston having a diameter of 5.48 in. and a length of 9.50 in. slides downward with a velocity $V$ through a vertical pipe. The downward motion is resisted by an oil film between the piston and the pipe wall. The film thickness is 0.002 in., and the cylinder weighs 0.5 lb. Estimate $V$ if the oil viscosity is $0.016 \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}^{2}$ Assume the velocity distribution in the gap is linear.

Key Concepts

Viscosity

Viscosity is a measure of a fluid's resistance to deformation and shear. In fluid dynamics, it characterizes how the fluid resists motion due to internal friction. A higher viscosity means more resistance to flow, which directly affects how forces are transmitted through the fluid medium.

Shear Stress

Shear stress in fluid dynamics is the force per unit area acting parallel to a surface. In the context of a thin film between moving surfaces, it is directly proportional to the dynamic viscosity and the velocity gradient. It plays a central role in calculating the resistance offered by the fluid to the motion of a solid body.

Lubrication Theory

Lubrication theory is a simplification used in fluid mechanics for analyzing the flow in thin layers of fluid between surfaces. When the gap is small compared to the other dimensions, the velocity profile can be approximated simply, often as a linear distribution, and the shear effects dominate the behavior of the flow.

Force Equilibrium

Force equilibrium involves balancing all forces acting on a body to determine its motion. In this context, the gravitational force (weight of the piston) is balanced by the viscous drag imposed by the shear stress in the oil film. This balance is essential for calculating the steady descent speed of the piston.

Linear Velocity Profile Assumption

Assuming a linear velocity profile in the thin film simplifies the calculations by allowing the velocity gradient to be taken as a constant across the gap. This approximation is often valid in lubrication flows, where the effects of inertia are negligible compared to viscous forces.

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