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

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Navier-Stokes We All Know What Happens When You
Assume

• Stephen McMullan
• 1-18-07
• BIEN 301

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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.

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

1. Solve the Navier-Stokes equation for w(x), and
   sketch its approximate shape.
2. 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/dxwall).

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Assumptions

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

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Navier-Stokes
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Navier-Stokes

• Becomes

g is negative because it is pointing in the
negative z direction.
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Navier-Stokes

• Equation 4.142

So Equation 4.142 becomes
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Navier-Stokes

• Remember no slip condition
• x 0
• w 0

So
Also x d w wmax
Therefore
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Navier-Stokes

• Plug C1 back in

Simplify
This is the answer!
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Navier-Stokes

• Final Answer

Or
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Navier-Stokes
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Finding m

• At this step only integrate once to isolate
  dw/dxwall

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Finding m

• Rearrange for m

This is the answer!
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BME Application

• Design of an artificial vessel
• Femoral Artery
• Gravity
• Pumping
• Motion
• Understand velocity profile to match the natural

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Questions?