# Derivation of momentum equation for newtonian fluid in cylindrical coordinates

 Register Blogs Members List Search Today's Posts Mark Forums Read December 11, 2017, 09:03 Derivation of momentum equation for newtonian fluid in cylindrical coordinates #1 Member   Robin Kamenicky Join Date: Mar 2016 Posts: 74 Rep Power: 10 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   December 11, 2017, 11:21 #2
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Filippo Maria Denaro
Join Date: Jul 2010
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 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?   December 12, 2017, 08:01 #3 Member   Robin Kamenicky Join Date: Mar 2016 Posts: 74 Rep Power: 10 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.   December 12, 2017, 11:39 #4 Senior Member   Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,397 Rep Power: 67   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.  Tags conservation of momentum, cylindrical coordinates, newtonian fluid, theory Thread Tools Search this Thread Show Printable Version Email this Page Search this Thread: Advanced Search Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is OffTrackbacks are Off Pingbacks are On Refbacks are On Forum Rules Similar Threads Thread Thread Starter Forum Replies Last Post xiexing CFX 3 March 29, 2017 10:00 Moinul Haque CFX 4 November 25, 2014 17:30 falopsy Main CFD Forum 44 January 15, 2014 02:17 jbambery Main CFD Forum 1 June 9, 2006 13:27 Abhi Main CFD Forum 12 July 8, 2002 09:11

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