1. Navier-Stokes: We All Know What Happens When You Assume
Stephen McMullan
BIEN 301

2. Problem 4.80
Oil of density r and viscosity m, drains steadily down the side of a vertical plate.
After a development region near the top of the plate, the oil film will become independent of z and of constant thickness d.

3. Problem 4.80
Figure 1
Plate
Oil film
Air g d z x

4. Problem
Solve the Navier-Stokes equation for w(x), and sketch its approximate shape.
Suppose that film thickness d and the slope of the velocity profile at the wall are measured with a laser-Doppler anemometer (Chapter 6). Find an expression for oil viscosity m as a function of (r, d, g, [dw/dx]wall).

5. Assumptions Newtonian Viscous Incompressible Liquid Steady
Fully developed
No slip condition at the plate surface
w = w(x)
No shear due to pa

6. Navier-Stokes

7. Navier-Stokes Becomes:
* g is negative because it is pointing in the negative z direction.

8. Navier-Stokes
Equation 4.142
So Equation becomes:

9. Navier-Stokes Remember no slip condition: x = 0 w = 0 Also: x = d
w = wmax
Therefore:

10. Navier-Stokes
Plug C1 back in:
Simplify:
This is the answer!

11. Navier-Stokes
Final Answer:
Or:

12. Navier-Stokes

13. Finding m
At this step only integrate once to isolate [dw/dx]wall

14. Finding m
Rearrange for m
This is the answer!

15. BME Application Design of an artificial vessel Femoral Artery
Gravity
Pumping
Motion
Understand velocity profile to match the natural

16. Questions?