How Do You Calculate the Velocity of a Piston in a Viscous Fluid?

[Logan McEntire]

1. piston having a diameter of D = 5.48 in. and a length of L = 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 δ = 0.002 in., and the cylinder weighs W = 0.5 lb. Estimate the velocity V if the oil viscosity is μ = 0.016 lb*s/ft^2. Assume the velocity distribution in the gap is linear.

Homework Equations

I think I need Tau=μ(du/dy) and tau=P/A. I don't know what du/dy is.

The Attempt at a Solution

A=pi*(2.74)^2=23.586 in^2
tau=0.5lb/23.586in^2=0.0212.
I don't know what to substitute for du or dy.

 

[BvU]

Hello Logan,

Without my Bird, Stewart and Lightfoot at hand, I can still guess that dy has something to do with $\delta$: over the oil film there is a velocity difference from 0 to V

 

[Chestermiller]

This is basically shear flow between two "flat parallel plates," where the outside surface of the piston comprises the moving plate and the inside surface of the cylinder comprises the stationary plate. Now, I realize that these are not exactly flat, but, on a local scale on the order of the gap between the cylinders, they are nearly flat. So, if $y=0$ is the inner surface of the cylinder and $y = \delta$ is the outer surface of the piston, the axial velocity profile in the gap between the piston and cylinder is $$v_z=V\frac{y}{\delta}$$and the r-z shear stress is $$\tau_{rz}=\mu\frac{V}{\delta}$$This shear stress is independent of y, and so is the same value at both walls of the gap.