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

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Problem 4.80 Plate Oil film Air g z x Figure 1

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Problem 1. Solve the Navier-Stokes equation for w(x), and sketch its approximate shape. 2. Suppose that film thickness 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 as a function of ( , , g, [ w/ x] wall ).

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

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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 = w = w max Therefore:

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Navier-Stokes Plug C 1 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 At this step only integrate once to isolate [ w/ x] wall At this step only integrate once to isolate [ w/ x] wall

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Finding Rearrange for Rearrange for This is the answer!

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

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Questions?

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