October i i, 1894J 



NA TURE 



^17> 



captain of the ship at between eleven and twelve feet, and the 

 indicator on the camera showed these wings apparently at full 

 stretch at the instant that I pressed the button. The result is 

 certainly somewhat astonishing, and I shall be glaii to know 

 whether it is worth comment in your paper ; to me it certainly 

 seems to entirely upset the accepted theories as to the flight of 

 this bird. A. KiNGSMILL. 



^tanmore, October 10. 



ON THE DOCTRINE OF DlSCONTrNUITY OF 

 FLUID MOTION, IN CONNECTION WITH 

 ^ THE RESISTANCE AGAINST A SOLID 

 MOVING THROUGH A FLUID} 



III. 



§ II. ' I MIK accompanying diagram (Fig. l) illus- 

 JL trates the application of the doctrine in 

 question, to a disk kept moving through water or air 

 with a constant velocity, V, perpendicular to its own 

 plane. The assumption to which I object as being in- 

 consistent with hydrodynamics, and very far from any 



d 

 d 



A 



approximation to the truth for an inviscid incompressible 

 tluid in any circumstances, and utterly at variance with 

 observation of diNks or blades (as oar blades) caused to 

 move through water ; is, that starting from the edge as 

 represented by the two continuous curves in the diagram, 

 and extending indelinitely rearwards, there is a " surface 

 of discontinuity " on the outside of which the water flows, 

 relatively to the disk, with velocity V, and on the inside 

 there is a rear-less mass of " dead water " - following 

 close after the disk. 



I Continued from p. 540. 



- 'rhis is a tcchnic.-il expression of rr.icticil hydraulics, adopted by the 

 Englisli teachers of the doctrine of finite slip between two parts of a homo- 

 geneous lluid, to designate water at rest relatively to the disk. 



NO. 1302, VOL. 50] 



§ 12. The supposed constancy of the velocity on the 

 outside of the supposed surface of discontinuity entails 

 for the inside a constant pressure, and therefore 

 quiescence relatively to the disk, and rearlessness of the 

 '• dead water. ' How such a state of motion could be 

 produced .' and what it is in respect to rear ? are 

 questions which I may suggest to the teachers of the 

 doctrine, but happily, not going in for an examination 

 in hydrokinetics, I need not try to answer. 



§ 13. But now, supposing the motion of the disk to 

 have been started some finite time, /, ago, and consider- 

 ing the consequent necessity (§ 9) for finiteness of its 

 wake, let ab, M be lines sufficiently far behind the rear, 

 and beyond one side, of the disturbed water, to pass 

 only through water not sensibly disturbed. We thus 

 have a real finite case of motion to deal with, instead 

 of the inexplicably infinite one of §11. Let us 

 try if it is possible that for some finite dis- 

 tance from the edge, and from the disk on each 

 side, the motion could be even approximately, if not 

 rigorously, that described in § 11, and indicated by the 

 diagram. 



§ 14. Let V be the velocity at any point in the axis, 

 A(^, at distance / from the disk, rearwards. Draw eJ 

 perpendicular to the stream lines of the fluid, relatively 

 to the disk supposed at rest. 



The "flow"' 



in the line fd is o ; 

 ,, lii „ V X oW ; 



o ; 





o, by hypothesis. 



in the closed polygon 



Hence for the " circulation " 

 edbake, we have 



V X all - j vify. 



Sitnilarly, for the circulation in the same circuit ^ at a 

 time later by any interval, t, when the line 6a has moved 

 to the position />'a', and ed to e'li', we have 



V X ,/i 



-I XI ay. 



where v' denotes, for the later time, t -\-t, the velocity in 

 Aa, at distance y from A. Hence the circulation in 

 edaAe gains in time t an amount equal to 



-/: 



[v' - t<)dy; 



which is the same as 



J 



(-■' - -Vr, 



This, by the general theorem of " circulation,' ■" must be 

 equal to the gain of ciiculation in time r, of all the vortex- 

 sheet in its growth from the edge according to the state- 

 ment of § II. Hence, with the notation of § 10, 



(2k)' - 2k - - / " (-■ - v),!y. 

 J 



5 If. Remarking now that the fluid has only con- 

 tinuous irrotational motion through a finite space all 

 round each of the lines cd, db, ha, at\ ; and all round \c 

 except the space occupied by the disk and the fluid 

 beyond its front side, we have, for the velocity-potential 

 of this motion, relatively to the disk, 



^> + 'P(-»-, y, =. I) 

 where <l> denotes the velocity-potential of the motion 



1 "Vortex Motion " (Thomson), Traits. R.S.E.,, XS69. 

 = !/•!,{. 



y Kemartc that the circulation in ahb'a' Is zero, and therefore the circula- 

 tion in eiib'a'Af is equnl to that in fiil'afiiC. 

 ■> "Vortex Motion," 1 rans. A' .V./i , 1869. 



