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Derivation of momentum equation for newtonian fluid in cylindrical coordinates |
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#1 |
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Robin Kamenicky
Join Date: Mar 2016
Posts: 74
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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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#2 |
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Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,830
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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?
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#3 |
Member
Robin Kamenicky
Join Date: Mar 2016
Posts: 74
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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. |
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#4 |
Senior Member
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,830
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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. |
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Tags |
conservation of momentum, cylindrical coordinates, newtonian fluid, theory |
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