70 



I1M)R()|1VN \MI(:s IN SIIII' DESIGN 



Stc. n.n 



\\i\w\, with no nttciiipl to iloliiicatc tlic surfare 

 rosultiiip from tlie system of sourcos, sinks, 

 doublets, and vortexes. In his diagrams the 

 result ing forces are shown as emanating directly 

 from the singularities. 



There are indications tliat an important future 

 u.se of the I Jigally Theorem may be in the solution 

 of the problem of determining the forces and the 

 moments exerted on ships when pa.'vsing or meeting 

 each other in confinetl waters. Thi.s will involve a 

 knowletigc of the nature and direction of the 

 forces exerted by the boundaries arounrl nearby 

 .sources and sinks, arranged cither singly or in 

 groups or arrays, and placed in non-uniform and 

 unsteady flow. 



It is interesting in this connection to consider 

 another aspect of the forces involved for one or 

 two simple hypothetical conditions. Assume first 

 that two sources of equal strength, lying opposite 

 each other, are enclosed by two large bo<ly 

 surfaces; also that those portions of the body 

 surfaces adjacent to the sources take the form of 

 a pair of parallel but independent flat surfaces 

 place<l between them, as at 2 in Fig. 43. P. As 

 there is no superposed stream flow in this case, 

 the liquirl flowing from each source toward the 

 adjacent flat portion of body surface Ls deflected 

 anfi pu.shetl backward. For the 2-diml case, the 

 flow from each source is pictured graphicallj' by 

 combining the stream functions of the two .sources, 

 indicated by the light radial lines in the diagram. 

 Because of the momentum imparted to the sui- 

 romiding liquid, in a direction opposite to that 

 in which the other source lies, away from the 

 adjacent flat body surface, a reactive force is 

 created on each body, directed toward the ad- 

 jacent body and the other source. While the 

 analogy' Ls by no means a perfect or a valid one, 

 it can be said that, unlike the similar poles of 

 electromagnets, the two adjacent sources give 

 the appearance of attracting each other. 



If a .source and a sink lie near each other in a 

 unifonn stream who.se direction of motion is 

 parallel to the source-sink axis, as in diagram 3 

 of Fig. '13. P, the action of the "inside" lifpiid is 

 such that equal arul opposite reactive forces are 

 directed away from the sink and the source 

 positions, respectively, and act outward on the 

 upstream and the downstream ends of the IxmIv. 

 .Mthongh the analog,v here i.s perhaps less valid 

 than it wa.s in the ca.se of the adjac<>nt .sources, 

 the source and the sink give the appearance of 

 repelling each other. This is again the opposite 



of the electromagnetic phenomenon, where unlike 

 poles attract each other. 



43.13 Partial Bibliography on Sources and 

 Sinks and Their Application. The literature on 

 the application of the radial-flow characteristics 

 of sources and sinks to problems in hydrodj'namics 

 is extremely extensive. There are listed here a 

 few of the most important references, with which 

 the marine architect may pursue his studies in 

 this fascinating field: 



(1) Rank-ine, W. J. M., "On Piano Wator-lincs in Two 



DimpiisionB," Phil. Trans., Roy. Sor.. 1S03 



(2) Runkino, \V. J. M., "Summary of Proportios of 



Certain Stroam-Linc)," Phil. Mag., Oct ISCt, pp. 

 282-288 



(3) Rankine, W. J. M., "Shipbuilding: Theoretical and 



Practical," 1866, p. 106 



(4) Rankine, W. J. M., "On Stream-Line Surfaces," 



IN.\, 1S70, Vol. XI, pp. 175-181 



(5) Rankine, W. J. M., "On the Mathematical Theory 



of Streamline.'!, Especi.iliy those with Four Foci 

 and Upwards," Phil. Trans., Roy. See, Ix)ndon, 

 1S71, Vol. 161, pp. 267-300; alsoPI. XV 



(6) IJaulc, A., "Note sur les Lignes d'eau Propo.sic8 par 



M. le Profefsseur Rankine (Note on the Waterlines 

 Proposed by Professor Rimkine)," Ann. Soc. Sci., 

 Rrus.'icls, 1885 



(7) Pollard, J., and Dudcbout, .V., "Thfiorie du Naviro 



(Theory of the ShipX" 1S02, Vol. Ill, pp. 401-410, 

 417-418, especially Fig. 135 on p. 408 



(8) Taylor, D. W., "On Ship-Sha[>ed Stream Forms," 



INA, 1894, pp. 38.5-106 



(9) Taylor, D. \V,, "On Solid Stream Forms and the 



Depth of Water Necessary to Avoid Abnormal 

 Resistance of Ships," INA, 1895, pp. 234-247 and 

 l>ls. XV-XVIII 



(10) Fuhrraann, O., "Widerstands- und Druckmessungen 



an Ballonmodellen (Resistance and Pressure 

 Mea.surcmenta on Balloon Models)," Zeit. fiir 

 Flugtcchnik und Motorluft-schiffahrt, 15 Jul 1911, 

 Vol. II, pp. lG.5-106 



(11) A series of papers in French by van Meerten |ATMA, 



1903, pp. 51-60 and PI. I: 1904, pp. 275-293; 1905, 

 pp. 209-289 and Pis. III-VIj. These discuss the 

 application of hydrodynamic studies of the theory 

 of sources and sinks in both two and three dimen- 

 sions. They contain diagrams of a variety of source- 

 sink flows, including several involving line sources 

 perpondlrular to a uniform How. .\ st.'irt is made 

 in the 1905 paper to utilize this method of analysis 

 for predicting the resistance of an actual ship. 



(12) Fdltingor, 11., "Fort.schnitte der Striimungslehro im 



Maschinenbau und Schiffbau (/\(lvance8 in the 

 Theory of Flow in Fngineering and .Shipbuilding)," 

 STfi, 1924, Vol. 25, pp. 295-311; lOnglish version 

 ill TMH Transl. IS, May 1952 



(13) AcnMlynainisc-he Versui-li.Mjuistalt (AV.\), rK'ittingen, 



Report UM 32(M(, dat«-<l 30 Dc- 1911; available in 

 lOnglish a.s TMH Transl. 220, .\pr 1947; sec also 

 Sec. 42.0 



