# Fluid Mechanics (Shear Stress)

1. May 28, 2009

### lfwake2wake

1. Question
A large plate moves with speed (v.o) over a stationary plate on a layer of oil of thickness (d) and viscosity (u). If the velocity profile is that of a parabola, with the oil at the plates having the same velocity as the plates, what is the shear stress on the moving plate from the oil? If a linear profile is assumed, what is the shear stress on the moving plate? (Answers: u*v.o/(2d) and u*v.o/d)

2. Relevant equations
t: shear stress
t=u*(dv)/(dy)
v=md^2+c

3. The attempt at a solution
So I have the answer, but I cannot figure out the solution. If the velocity changes parabolically, then you have the equation v=md^2+c. dv=2md. So when d=0, v=0 and t=u*(2md-0)/(d-0). This is my solution, but obviously is not correct with the above answers. Any help would be greatly appreciated.
Thanks

2. May 28, 2009

### Hao

Parabolically can either mean $$v = a y^2 + b$$ or $$v = a \sqrt{y} + b$$, where a and c are constants to be determined. Think carefully about which is the correct profile.

Furthermore, avoid expressing v immediately as a function where y = d. The velocity profile is $$v(y)$$. The velocity at the moving plate is $$v(d)$$. You will make mistakes with your derivatives otherwise.