Derivation of momentum equation for newtonian fluid in cylindrical coordinates

Robin.Kamenicky · December 11, 2017, 10:03

Hi everybody,

Would anyone have some tip for source, where would be complete derivation of momentum equation for newtonian fluid in cylindrical coordinates?

Thank you,
Rob

FMDenaro · December 11, 2017, 12:21

Quote:

Originally Posted by Robin.Kamenicky

Hi everybody,

Would anyone have some tip for source, where would be complete derivation of momentum equation for newtonian fluid in cylindrical coordinates?

Thank you,
Rob

Many basic fluid dynamic textbooks illustrate the NS equation in cylindrical coordinates, do you need only the expression or do you want to understand how the expression is derived from the general vector formulation?

Robin.Kamenicky · December 12, 2017, 09:01

Hi FMDenaro,
First of all, thank you for the reply.

I would like the entire derivation of differential form from scratch.
I would like to see wether the area of the finite element faces are different. I saw different approaches to this. In one source, the faces in r direction were computed r*dangle*dz and (r+dr)*dangle*dz and in another one they assumed to have same areas. Sure, smaller the element, the more similar areas.
Also the derivation of viscous stress components would be interesting to see.

I checked following document where in r direction, the normal stress in the angle direction is added. I do not understand the reason for it.

Overall, I feel I quite understand the derivation but there are some small nuances in which I am not 100% sure. So I would appreciate a source as detailed as possible.

FMDenaro · December 12, 2017, 12:39

You can find some explanations but they are quite equivalent each other.
I generally introduce to my student the general vector notation for the expressing the equation (valid for any type of reference system) and them express the components of the vectors and differential operators in the chosen reference system.
For example:

v= ir *vr+ itheta *vtheta+iz*vz

and the differential operator

nabla = ir *d/dr+ itheta *(1/r)*d/dtheta+iz*d/dz

(see https://en.wikipedia.org/wiki/Del_in...al_coordinates) so that for example the divergence of the velocity is

(ir *d/dr+ itheta *(1/r)*d/dtheta+iz*d/dz).(ir *vr+ itheta *vtheta+iz*vz)

and now you have to extend the expression using the rules for the products and derivatives.
The same procedure is applied for the gradient of both a scalar and vector function.

December 11, 2017, 10:03	Derivation of momentum equation for newtonian fluid in cylindrical coordinates	#1
Robin.Kamenicky Member Robin Kamenicky Join Date: Mar 2016 Posts: 74 Rep Power: 11	Hi everybody, Would anyone have some tip for source, where would be complete derivation of momentum equation for newtonian fluid in cylindrical coordinates? Thank you, Rob

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December 12, 2017, 09:01		#3
Robin.Kamenicky Member Robin Kamenicky Join Date: Mar 2016 Posts: 74 Rep Power: 11	Hi FMDenaro, First of all, thank you for the reply. I would like the entire derivation of differential form from scratch. I would like to see wether the area of the finite element faces are different. I saw different approaches to this. In one source, the faces in r direction were computed rdangledz and (r+dr)dangledz and in another one they assumed to have same areas. Sure, smaller the element, the more similar areas. Also the derivation of viscous stress components would be interesting to see. I checked following document where in r direction, the normal stress in the angle direction is added. I do not understand the reason for it. Overall, I feel I quite understand the derivation but there are some small nuances in which I am not 100% sure. So I would appreciate a source as detailed as possible.

December 12, 2017, 12:39		#4
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,882 Rep Power: 73	You can find some explanations but they are quite equivalent each other. I generally introduce to my student the general vector notation for the expressing the equation (valid for any type of reference system) and them express the components of the vectors and differential operators in the chosen reference system. For example: v= ir vr+ itheta vtheta+izvz and the differential operator nabla* = ir d/dr+ itheta (1/r)d/dtheta+izd/dz (see https://en.wikipedia.org/wiki/Del_in...al_coordinates) so that for example the divergence of the velocity is (ir d/dr+ itheta (1/r)d/dtheta+izd/dz).(ir vr+ itheta vtheta+iz*vz) and now you have to extend the expression using the rules for the products and derivatives. The same procedure is applied for the gradient of both a scalar and vector function.