Deriving navier stokes in polar

AI Thread Summary
The discussion focuses on the derivation of the Navier-Stokes equations in polar coordinates, specifically addressing confusion regarding the theta derivative terms in the equations. Participants highlight that the provided link does not contain a complete derivation and express difficulty in understanding the last terms of the second and third equations. Suggestions include examining the stress tensor equations, which contain similar terms, and reviewing fluid mechanics literature for guidance. The book "Transport Phenomena" by Bird, Stewart, and Lightfoot is recommended for further insights, with a note on their treatment of stress. Overall, the conversation emphasizes the need for clarity in deriving these equations in polar coordinates.
ericm1234
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Can anyone point me to a derivation of the navier stokes equations in polar? I don't see where the single derivative in theta terms are coming from in the first 2 components.
 
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the equations are not derived in the link you gave; my own derivation fails to see where the last terms in those 2nd and 3rd equations in that link are coming from.
 
ericm1234 said:
the equations are not derived in the link you gave; my own derivation fails to see where the last terms in those 2nd and 3rd equations in that link are coming from.

Check out the set of equations for the stress tensor, which have these same terms in them. You can probably figure this out by first writing out your derivation in terms of the components of the stress tensor. You can check your derivation of the differential force balance equations in terms of the stress tensor in cylindrical coordinates in most fluid mechanics books. Check out Transport Phenomena by Bird, Stewart, and Lightfoot (but note that their presentation treats positive stress as compressive, so that their τ's are equal to our -σ's).
 

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