Please Make A Note: 9. Derivation of the Continuity Equation in Spherical Coordinates
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9. Deri vation of t he Conti Conti nuit y Equation Equation in Spherical Spherical Coordinates Coordinates
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We st art by select select ing a sp spherical herical cont rol volume dV. dV. As shown in the f igure below, t his is given by
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where r , θ, and φ st and for t he radius, radius, polar, an and d azimuthal ang angles, les, respectively. respectively. The azimuthal azimuthal angle ang le i s also also referred t o as the zenith or colatit ude ang angle. le.
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We will represen entt the velocity velocity f ield via
In an Eulerian Eulerian ref erence frame mas mass cons conservation ervation is represented represented by accumu accumulat lat ion, net f low, and source terms in a contr contr ol volume volume..
Accumulation The accum accumulati ulati on term is given given by the t ime rat e of change change of mass. We t herefore have have
Please Make A Note: 9. Derivation of the Continuity Equation in Spherical Coordinates
The net f low t hrough the control volume can be divided int o that corr esponding to each directi on.
Radial Flow Start ing with t he radial direction, we have
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The inflow area Ain is a trapezoid whose area is given by
The key term here is the sine term. Not e t hat t he mid segment is t he average of t he bases (parallel sides). Upon expansion of Ain, and in the limit of vanishing d θ, we have substit ution into Ain yields
where high order terms have been dropped. The outflow in the radial direction is
but where and By only keeping the l owest (second & t hird) order t erms in t he result ing expression, we have Note, t hat in t he expression for Aout , we kept both second order and thi rd order t erms. The reason for t his is that t his term wil l be multi plied by "dr" and therefore, t he overall order will be three. In principle, one must carry all t hose terms unt il t he final subst it uti on is made, and only t hen one can compare t erms and keep those wit h t he lowest order. At the outset, t he net f low in t he radial directi on is given by
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The inflow in the polar direction is
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where
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The outf low in the θ direction is
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where Upon expansion, and keeping both second and t hir d order t erms, we get
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Finally, the net f low in the polar direction is
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Azimuthal Flow (φ) The infl ow in the azimut hal directi on is given by
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Please Make A Note: 9. Derivation of the Continuity Equation in Spherical Coordinates
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At the outset, the net fl ow in the polar direction is
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which, upon divi ding by dV and combining ter ms, r educes t o
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which is the cont inuit y equation in spherical coordinates.
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Cite as: Saad, T. "9. Derivation of the Continuit y Equation in Spherical Coordinates". htt p:/ / pleasemakeanote.blogspot. com/2010/ 02/9-derivat ion-of-continuit y-equation-in.html
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3 comments:
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Alex Sep 19, 2010 05:27 PM
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Thanks a lot f or the deri vati on, I was having some issues pict uring t he spherical el ement.
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A bett er way of doing it is by doing the grouping done in t he last line using the product rule when you calculate each diff erence in mass flow r ate. Also if you leave some second order t erms, when you divi de by dV, t hen you can say and as dx_k go to 0, t hose terms go away.
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Anonymous Nov 28, 2010 06:43 AM There is an error i n the last li ne. In polar coordinates, it should be derivati ve of ro*v*sin(t heta) wrt theta instead of derivative of j ust ro*v wrt theta. Reply
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yNot Nov 28, 2010 03:54 PM yes, t hanks for catching that t ypo.
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