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

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Presentation transcript:

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. After a development region near the top of the plate, the oil film will become independent of z and of constant thickness d.

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

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).

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

Navier-Stokes

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

Navier-Stokes Equation 4.142 So Equation 4.142 becomes:

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

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

Navier-Stokes Final Answer: Or:

Navier-Stokes

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

Finding m Rearrange for m This is the answer!

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

Questions?