G8 



IIM)R()1)\ \ AMICs I\ Mill' DISICN 



Sic -I J. 12 



the prt-stMit rliiiptor, tlu> Iriict's of the How patterns 

 nroiiiitl tlu-iii, iiiul tlu' local stream veliuitifs ami 

 presjiiiri-s, eaii l)e calcuiatetl by mathematical 

 formulas. This metluxl is often the most con- 

 veiiient anil advantageous, depending upon the 

 eipiipment and facilities available and the use to 

 which the deriveil data are to be put. Specific 

 formulas for accomplishing this, in a few selected 

 cases, are derivtnl in Sees. 41.8 and 41.9 and are 

 (pioted on Figs. 41. E, 41. F, and 41. CI accompany- 

 ing them. These formula.s, and others, are found 

 in many mixlern te.xt and reference books, but 

 more often than not they are without adetiuatc 

 explanation as to the symbols and notation 

 employed. 



Some additional formulas were dcNclopi'd iiy 

 \V. J. M. Rankine [Phil. Trans., Roy Soc, 1871], 

 such as those to establish the loci of points of 

 ma.ximum and minimum velocity around stream 

 forms, but the references which contain them are 

 not readily available and the notation in which 

 they were exprcs.sed threc-{|uartcrs of a century 

 ago re(|uires conversion to nKxlern Mt)tatioii. 



43.12 The Forces Exerted by or on Bodies 

 Around Sources and Sinks in a Stream; Lagally's 

 Theorem. If a i-loscd body is formed by a 

 stream surface around sources and sinks, as for 

 the bodies around the source-sink pairs dcscrilied 

 previously in this chapter, there is a resultant 

 hydrfxiynamic force acting on the stream surface 

 bounding the body. It is explained presently that 

 in a uiiifonn stream the resultant force is zero, 

 conforming to the fl'.Memliert paradox, but in an 

 unsteady or non-uniform flow the force is finite. 

 It can be evaluated very neatly by the use of 

 what is known as the Lagalbj Throrrm. The 

 derivation of Lagally's theorem is beyond the 

 .scope of this book, but at the expense of stretching 

 the mathematical truth temporarily, the physical 

 basLs for it is rather easily explained. 



Consider what happens inside the 2-dLml 

 ov(>id-shape<i boundary which envelops the'2-diml 

 Hource or up.stream singularity in the leading end 

 of .such a brxly, formed aroimd a source and a 

 sink at a gnat dixlancc frinn rnrh nihrr. In diagram 

 I of Fig. 13.1', dei)i(ting the nose of this body, 

 lying in a uniform flow of velocity —l\, all the 

 lifiuid emanating from the after part of the .source 

 flows more or le.ss directly toward the left, away 

 from the oncoming stream. That from the forward 

 part of the source turn.s rather sharply and 

 reverses it.s «lirection, also to flow downstream 

 toward the left. Regardlesw of the direct ion in 



Body M CTo'»cd~Ji Ver^ Loni). ond i4 Synmetixol Unif orm Flow 



~'~ "* — "•~-^~" — 1 About Sink ol - 



me LtU tno- 



I'"iG. i'A.Y' Diagrams Ii.lustratino the Factors 

 InVOI.VKD IN' TUB Lacai.i.v Tiikorkm 



which the li(|uid emanated from the source, it.s 

 ultimate direction is downstream. 



As the "inside" li(|uid moves farther and farther 

 from the source the railial component of its 

 velocity |)rogre.ssivel}' diminishes. At a great 

 distance downstream, on its way to the sink, this 

 li<|uid acfiuires a velocity — U essentially etjual 

 to — f/o. , that of the uniform stream. If a certain 

 volume of li(iuid i.ssues from the source in unit 

 time, .so that Q is the volume or (|uanlit,y rate of 

 this (low, then the mass of li(|uid (lowing o\it in 

 unit time is fA). .Fust before the litpiid i.ssued from 

 the source it had no compniiciit of mulioii p;ir;dlel 



