Sec. 41.9 



GENERAL LIQUID-FLOW FORMULAS 



21 



I. The first case considered is that of the sphere. 

 The relationships between the stream function, 

 the velocity potential, the radial velocity Ur , 

 and the tangential velocity Ub are, for the 

 axisjonmetric case in spherical coordinates [Milne- 

 Thomson, L. M., TH, 1950, pp. 403-404], 



It is explained at the end of this section, and 

 illustrated in Fig. 41.1, that for this mathematic 

 representation in spherical coordinates the refer- 

 ence for zero stream function for a source or 

 sink is the transverse plane through the origin, 

 nonnal to the axis of the system. Referring to the 

 definition sketch of Fig. 41. H, the addition of the 

 values of and ^ for the sources and the sinks gives 



</> = mir^ ^— ) (41.xxixa) 



MtfjK ilsO' 



\p — m(cos dsK — cos Bso) (41.xxixb) 



where the subscript "SO" applies to the source 

 and "SK" to the sink. For the 3-diml doublet, 

 by the sine rule, 



For the uniform stream of velocity — C/„ 

 again using spherical coordinates, 



Ur = - C/co cos d 



Ue = f/„ sin e 



Rso 



RsK 



2s 



sin dsK sin dso sin {Oso — Osk) 



From 



s 



sin ^{dso - Osk) cos ^{Bso - Osk) 



"V TT "V 



RUe, 

 3-dimi = —U^R cos 6 (41.xxvia) 



Then 



From 

 84' 



d^ 



dR= -^««^'n^ ^' 30 



-r U^R- sin' e = 



UrR'' sin d, 



r> _ r> ^ s(sin dsK - sin dso) 



""° '^"' sin ^{dso - dsK) cos hidso - Bsk) 



^ -2s cos \{eso + Bsk) sin K^ao - Osk) 

 sin |(0so - Bsk) cos ^{6 so - Bsk) 



■t/„|- (41.xxvib) 



RsO RsK — 



-2s cos iJBso + Bsk) 

 COS ^{Bso - Bsk) 



Considering a 3-diml source having a trans- 

 verse plane through its center on which ^ = 0, 

 and following the notation of Sec. 3.9, where 

 m is the strength of a 3-diml source, Q = 4:tR^Ur 

 = iwrn, whereupon m = Q/4x, Ur = m/R", and 

 U, = 0. From Eqs. (41.xxvc) and (41.xxvb), 

 respectively, by integration. 



Let M = 2ms equal the strength of a 3-diml 

 doublet. Then, from the expression for the 

 velocity potential for a source and sink, 



: {Rso Rsk) 



R 



(41.xxviia) 



"Aa-dimi = —m cos 6 (41.xxviib) 



For a 3-dLml sink corresponding to the 3-diml 

 source just mentioned, 



RskRso 



^ -M cos 1(050 + Bsk) 

 RskRso COS 2(^50 ~ Bsk) 



Similarly, for the stream function, 



m(x + s) m(x — s) 



^ = 



Ur - ^2 



+ 



Rsk 

 Rs 



Rso 



R 



^3-diml = -|-Wl COS 



(41.xxviiia) 

 (41.xxviiib) 



Then 



^ = 



■fix COS f 



RskRso COS 



'SO' \RsK RsO' 



IJBso + Bsk) , M / J_ , J_'\ 

 S hi^so - Osk) 2 \RsK Rso) 



