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Navier-Stokes: We All Know What Happens When You Assume

Navier-Stokes: We All Know What Happens When You Assume. Stephen McMullan 1-18-07 BIEN 301. Problem 4.80. Oil of density r and viscosity m , drains steadily down the side of a vertical plate.

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Navier-Stokes: We All Know What Happens When You Assume

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  1. Navier-Stokes: We All Know What Happens When You Assume Stephen McMullan 1-18-07 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 4.142 becomes:

  9. Navier-Stokes Remember no slip condition: x = 0 w = 0 So: 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?

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