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

Stephen McMullan 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 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).

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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 becomes:

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**Navier-Stokes Remember no slip condition: x = 0 w = 0 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/dx]wall

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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?

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Basic Governing Differential Equations

Basic Governing Differential Equations

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