## A Sticky Physics Problem

Posted in Cute Problems with tags , , on May 1, 2014 by telescoper

As I often do when I’m too busy to write anything strenuous I thought I’d post something from my back catalogue of physics problems. I don’t remember where this one comes from but I think you’ll find it interesting…

Oil of viscosity η and density ρ flows downhill in a flat shallow channel of width w which is sloped at an angle θ. The oil is everywhere of the same depth, d, where d<<w. The effect of viscosity on the side walls can be assumed to be negligible.

If x is a coordinate that represents the vertical position within the flow (i.e. x=0 at the bottom and x=d at the top), write down a differential equation for the velocity within the flow  v(x) as a function of x. Use physical arguments to derive appropriate boundary conditions at x=0 and x=d and use these to solve the equation, thereby determining an explicit form for v(x). Hence determine the volume flow rate in terms of η, ρ, θ, d and w as well as the acceleration due to gravity, g.