Deriving navier stokes in polar

[ericm1234]

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.

 

[Andy Resnick]

http://www.its.caltech.edu/~mefm/me19b/handouts/Equations.pdf

 

[ericm1234]

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.

 

[Chestermiller]

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).

 

[sardar]

The derivation of the Navier-Stokes equations in polar coordinates can be found in many fluid mechanics textbooks, such as "Introduction to Fluid Mechanics" by Robert W. Fox and Alan T. McDonald. In this derivation, the equations are derived from the fundamental equations of fluid mechanics, which include the continuity equation, the momentum equation, and the energy equation.

The single derivative in theta terms arises from the use of the polar coordinate system, where the velocity components are expressed in terms of radial and tangential components. The radial component is a function of the radial coordinate, while the tangential component is a function of both the radial and tangential coordinates. This results in a single derivative in the theta direction.

Additionally, the Navier-Stokes equations are derived from the conservation of momentum, which takes into account the forces acting on a fluid element, including pressure, viscous forces, and body forces. These forces are expressed in terms of the velocity components, resulting in the appearance of the single derivative in theta terms.

Overall, the derivation of the Navier-Stokes equations in polar coordinates is a complex process that takes into account the fundamental equations of fluid mechanics and the use of a specific coordinate system. I suggest referring to a textbook or consulting with a fluid mechanics expert for a more detailed explanation of the derivation.