574 



NA TURE 



[OCTOBEK 1 1, 1894 



relative to the infinitely distant fluid all round : and we 

 have 



v = \ + ~<^[o,y,o,0. 

 dy 



With this the equation of § 14 becomes 



(2«)' - Ik =)^(o, o, 0, t +t)| - {<^[fi, o, 0, t)). 



Hence, by taking r infinitely small, 



§ 16. Now in the time from / to / + t, there has been, 

 according to the supposition stated in § 11, a growth of 

 vortex sheet from e, at the rate i\', being the mean 

 between the velocities of the fluid on its two sides,' and 

 the circulation, per unit length, /, of the sheet thus grow- 

 ing is IV. Hence the vortex-circulation of the growing 

 sheet augments, in time r, by AVt X V : and therefore, 

 by §15, 



i <p (o, 0, o,t) = l V=. 



§ 17. Now, if n denotes the pressure of the fluid at 

 great distances, where its velocity, relative to the 

 disk is V, and p the pressure at any point of the rear 

 side of the disk, being the same as the pressure at A, we 

 have, by elementary hydrokinetics, 



/ = n + i V2 - '^ -ib(o, o, o, t) 



because the velocity of the fluid at every point of the rear 

 side of the disk is zero according to the assumption of 

 " dead water." Hence, by § 16, 

 /» = n, 



which, being the same as the pressure on the rear side 

 given by the unmitigated assumption of an endless ever 

 broadening wake of " dead water," proves that our sub- 

 stitution (§ 13) of a finite configuration of motion con- 

 ceivably possible as the consequence of setting the disk 

 in motion at some finite time, /, ago, instead of the 

 inconceivable configuration described in § 11, does not 

 alter the pressure on the rear side of the disk. 



§ 18. Hence were the motion of the fluid for some finite 

 distance from the disk, on both its sides, the same, or 

 very approximately the same, as that described in § 11, 

 the force that must be applied to keep it moving uni- 

 formly would be the same, or very approximately the 

 same, as that calculated by Lord Rayleigh from the 

 motion of the fluid supposed to be wholly as described in 



§ 19. But what reason have we for supposing the 

 velocity of the fluid at the edge, on the front side of the 

 disk, to be exactly or even approximately equal to the 

 undisturbed velocity, V, of the fluid at great distances 

 from the disk.' .None that I can see. It seems to me 

 indeed probable that it is in reality much greater than 

 \', when we consider that, with inviscid incompressible 

 fluid in an unyielding outer boundary, the velocity, in the 

 case considered in § 14, is equal to V at even so far 

 from the edge as '85 of an inch, and increases from V to 

 637 '"■ ^ between that distance from the edge, and the 

 edge with its i 2000 of an inch radius of curvature. 



§ 20. And what of the " dead water "in contact with 

 the whole rear side of the disk which the doctrine of dis- 

 continuity assumes."' Look at the reality and you will 

 see the water in the rear exceedingly lively everywhere 

 except at the very centre of the disk. You will see it 

 eddying round from the edge and returning outwards 

 very close along the rear surface, often I believe with 

 much greater velocity than V, but with no steadiness ; on 

 the contrary, with a turbulent unsteadiness utterly unlike 

 the steady regular motion generally assumed in the 

 doctrine of discontinuity. 



1 Helmholl/ ; WmscnKlnflliche Abhandlungcn ; vol. i., fi>ot of p. 151. 



SO. 1302. VOL. 50] 



§ 21. We may I think safely conclude that on the 

 front side the opposing pressure is less than that calcu- 

 lated by Rayleigh. That this diminution of resistance 

 is p.^rtially compensated or is over-compensated by 

 diminution of pressure on the rear, is more than we are 

 able to say from theory alone, in a problem of motion so 

 complex and so f.ir beyond our powers of calculation : 

 but we are entitled to say so, I believe, by experiment. 

 Rayleigh's investigation of the resistance experienced by 

 an infinitely thin rigid plane blade bounded by two 

 parallel straight edges, when caused to move through an 

 inviscid incompressible fluid, with constant velocity, \', 

 in a direction perpendicular to the edges and inclined at 

 an angle i to the plane, gives a force cutting the plane 

 perpendicularly at a distance from its middle equal to 

 3 cos i 

 4(4 + IT sin i) 



of its breadth, and gives for the amount of this force in 

 gravitation measure, 



4 -I- IT sin « 



where A denotes the area of one side of the blade, and 

 P the weight of a column of the fluid of unit cross- 

 sectional area, and of height equal to the height from 

 which a body must fall to acquire a velocity equal to V. 



§ 22. The assumption (§ ii) on which this investiga- 

 tion is founded admits no velocity of fluid motion 

 relatively to the disk greater anywhere than V. It gives 

 velocity reaching this value only at the edges of the 

 blade ; and at the supposed surface of discontinuity ; and 

 in the fluid at infinite distances all round except in the in- 

 finitely broad wake of " dead water " where the velocity is 

 zero. It makes the pressure equal to IT all through the 

 " dead water," and makes it increase through the moving 

 fluid, from n at an infinite distance and at the " surface of 

 discontinuity," to a maximum value n + P attained at the 

 water-shed line of the disk. If the fluid is air, and if 

 \' be even so great as 120 feet per second (i 10 of the 

 velocity of sound) ' P would be only 7 1000 of IT The 

 corresponding augmentation of density could cause no 

 very serious change of the motion from that assumed : 

 and therefore in Rayleigh's investigation air may be 

 regarded as an incompressible fluid if the velocity of the 

 disk is anything less than 120 feet per second. 



We may therefore test his formula for the resistance, 

 by comparison with results of careful experiments 

 made by Dines- on the resistance of air to disks and 

 blades moved through it at velocities of from 40 to 70 

 statute miles per hour (59 to 103 feet per second). 



§ 23. Dines finds for normal incidence the resistance 

 against a foot-square plate, moving through air at m 

 British statute miles per hour to be equal to -0029 /«' of 

 a pound weight. 



This, if we take the specific gravity of the air as 

 1,800, gives according to our notation of §21, 

 I ii6x PA 



as the resistance to a square plate of area A. At the foot 

 of p. 255 {I'roc. A'.S., June iSyo) Dines says that he finds 

 the resistance to a long narrow blade to be more than 20 

 per cent, greater than to a square plate. For a blade 

 we may there take 



1-34XPA 



as the resistance according to Dines' experiments. This 

 is 1-52 times the resistance calculated from Rayleigh's 

 formula (§ 21 above), which is 



■88 -PA, 

 for normal incidence. 



§ 24. For incidences more and more oblique, the dis- 



'Or ,V X t/rTx^^H, whcrt- H is "ihe heizhl of the homogcntoii- 

 atmospher«'." 

 ^ /'r-vc. /{.S.£., June i£90. 



