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Vorticity Equation when nu is not a constant

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Old   September 16, 2005, 22:35
Default Vorticity Equation when nu is not a constant
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Sam
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Folks:

I am interested in deriving the Vorticity equation in Cylindrical Coordinates (r,theta,z) with Theta symmetry and u_theta = 0.

The main problem is the viscosity cannot be treated as a constant but varies spatially. For a constant viscosity we get the simple equation,

dw/dt + u.Del(w) = nu Del^2(w)

where u = {u_r,0,u_z} and w = vorticity = {0,w,0}, where w = du_r/dz - du_z/dr.

I am interested in a similar vector equation when nu is not a constant. Could some one please help me or let me know where I can find it. Thanks.

Sam
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Old   September 16, 2005, 23:09
Default Re: Vorticity Equation when nu is not a constant
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Adrin Gharakhani
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You can start with the general Navier-Stokes equations, take the curl, and then do a "simplification" in the cylinderical coordinate system! This is a simple exercise in math.

Note that your question is vague. A variable nu could imply variable density only, variable viscosity only, or variable viscosity and density. Either way, you're going to end up with quite a bit more terms than for the constant nu case

Adrin Gharakhani
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