Deriving navier stokes in polar

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Discussion Overview

The discussion revolves around the derivation of the Navier-Stokes equations in polar coordinates, focusing on the specific terms related to the angular derivative in the equations. Participants are seeking clarity on the source of certain terms in the equations and how they relate to the stress tensor in fluid mechanics.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant requests a derivation of the Navier-Stokes equations in polar coordinates, specifically questioning the origin of the single derivative in theta terms in the first two components.
  • Another participant provides a link to a document but is met with criticism that the equations are not derived in that source.
  • One participant expresses frustration with their own derivation, noting a lack of understanding regarding the last terms in the second and third equations from the provided link.
  • A later reply suggests examining the stress tensor equations, indicating that the same terms appear there and recommending a derivation based on the components of the stress tensor in cylindrical coordinates.
  • The same reply references a fluid mechanics textbook, "Transport Phenomena," while cautioning that the treatment of stress in that text may differ from the participants' conventions.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus, as there are multiple competing views regarding the derivation and the interpretation of terms in the equations. The discussion remains unresolved with ongoing questions and critiques of provided resources.

Contextual Notes

Limitations include the unclear derivation steps for the terms in question and potential differences in conventions regarding stress treatment in various references.

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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