Page:EB1911 - Volume 14.djvu/135

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HYDRODYNAMICS]
123
HYDROMECHANICS


With

acceleration dLT/dt, given by

22+b2 dU a2+b2, gg

"f””zF:5far"a1"sM an U4)

liflvid Of density P, this gives rise to a kinetic reaction to if M' denotes the mass of liquid displaced by 'unit length of the; cylinder r=b. In particular, when a =oo, the extra inertia is M'. When the cylinder r=a is moved with velocity U and r=b with velocity U, along Ox, ,

¢= Ugg?<§ ;2+r)cos0-U1£ 37l;(r+'¥)cos0, (15) ~//=-U5;'¥;l;<?~r)sin0»UiI¢ I:p(r-£;)sin0; (II6) A and similarly, with velocity components V and V1 along Oy i A 4>= VE; (lg-l-V) sin0-V|b;€% <1-1-gi) sin 0, (17) 2 2 'l

i//= V5%<% -r) cos0+V1=5, -I-i?(r-§ ;>cos0; (18) and then for the resultant motion

az z a2b2 -l-Vi

iw- (U2 + V2)b'-a”U-I-Vi + b” ~ (12 z l>f 5 a'b'U+V£

(U"+V )b2 - a2U, +v, ¢ ' U9)

The resultant impulse of theiliquid on the cylinder is given by the component, over r=a (§ 36), V A

X =fp¢ C05 was =2p22 U QQ-1§ ¢U, ,§ %); (20) A and over r=b A

x, =fp¢¢0So.z2dv=2, , b2 U Z§ %-u.'§ f-f; C, (21) and the difference X-X1 is the component' momentum of the liquid in the inter space; with similar expressions for Y and Yi. Then, if the outside cylinderiis free to move x, =0, %g% X=»p22Uf§ -Ig. (22)

But if the outside cylinder is moved with' velocity U), and the inside cylinder is solid or filled with liquid of density a, U1 2pb2

X—¥Ua?U1 U =p(b2+a2)+a2(b2 ag 1

U-Ui: (p-'¢r)(b3-az) (2)

U1 P(b'+q')+v(5-¢1”)' 3

and the inside cylinder starts forward or backward with respect to the outside cylinder, according as p> or <¢r. 30. The expression for w in (1) § 29 may be increased by the addition of the term,

im log z=-m0 + im log r, V (1)

representing vortex motion circulating round the annulus 'of li uid.

qConsidered by itself, with the cylinders 'held fixed, the vortex sets up a circumferential velocity m/f on a radius r, so that the angular momentum of a circular fi ament of annular cross section dA is pmdA, and of the whole vortex is pm-r(b'¢-az). Any circular filament can be' started from rest by the application of a circumferential impulse vrpmdr at each end. of a diameter; so that a mechanism attached td the cylinders, which can set up a uniform distributed impulse vrpm across the two parts of a diameter in the liquid, will generate the vortex motion, and react on the cylinder with an impulse couple—pm-ra* and pmfbz, having resultant pm1r(b' »~a'), and this couple is infinite when b=oo, as the angular momentum of the vortex is infinite. Round the cylinder 1 =a held fixed in the U current the liquid streams past with velocity q' =2U sin 0+m/a; (2)

and the loss of head due to this increase of velocity from U to Q' is q”'~ U* =(2U sin 0+m/a.)' - U2 ii

23 23 V y

to

so that cavitation will take place, unless the head at a great distance exceeds this loss.

The resultant hydrostatic thrust across any diametral plane of the cylinder will be modified, but the only term in the loss of head which exerts a -resultant thrust on the whole cylrnclexyis 2mU sin 0/ga, and its thrust is 2-rrpmU absolute units in the direction Cy, to be counteracted by a support at the centre C; the liquid is streaming past r=a with velocity U reversed and the cylinder is surrounded by a vortex. Similarly, the streaming velocity V reversed will give rise to a thrust 21-pmV in the direction xC. Now if the cylinder is released, and the components U and V are reversed so as to become the velocity of the cylinder with respect to space filled with liquid, and at rest at infinity, the cylinder will experience components of force perunit length ' 2 (i.) *2fpmV, , 2-rrpmU, due to the yortex motion: (ii.) -iirpazgéi, -vrpa'%¥, due to the kinetic reaction of the liquid; (iii.) o, —1|-(ir-p)a'g, due to gravity, ' ' V taking Oy vertically upward; and denoting the density of the cylinder by tr; so that the equations of motion are ~ 1r¢ra2%&g = - 1rpl1.%~ 21rpmV, (4) dV dV, ,

1:10225 = -1rpcL' 5—l-21rpmV - 1r(n - p)a-g, (5) or, putting m=a'w, so that the vortex velocity is due to an angular velocity w at a radius a,

(0-I-p)tiU/ritjl-2pwV =O, (6)

(<f+P)¢iV/df'-2pwU +(<1-r>)g =0' (7) Thus with g=o, the cylinder will describe a circle with angular velocity 2P¢»1/(vrl-p), so that the radius is (a+p)v/zpw, if the velocity is v.. With ¢==0, the angular 'velocity of the cylinder is 2¢o; in this way the velocity may be calculated of the propagation-of ripples and waves on the surface of a vertical whirlpool in a sink. Restoring u will make the path of the cylinder a trochoid; and so the swerve can be explained of the ball in tennis, cricket, baseball, or golf.,

Another explanation may be given of the sidelong force, arising from the velocity of liquid past a cylinder, which is encircled by a vortex. Taking two planes x= =|=b, and considering the increase of momentum in the liquid between' them, due to the entry and exit of liquid momentum, the increase across dy in the direction Oy, due to elements at P and P' at opposite ends of the diameter PP', is pdy (U - Uazf” cos 2 0+mr'1 sin 0) (Ua'r“2 sin 2 0-1-mr°1cos0) + pdy (-U-l-Ua2r'“ cos 20-l-m7'x sin 0)(Ua2r°2 sin 20 -mr* cos0) =2pdymUf'1(C0S0 -a'r“2 cos 30), (8) “ and with y=b'ta.n 0, rsb sec 0, this is 2pmUd0(»1 -a'b"'cos 30 cos 0), V (9) and integrating between the limits 6= =1==}1r, the resultant, as before, is 21rpmU.

31. Example 2.-Confocal Elliptic Cylinders.—Employ the elliptic coordinates 11, £, and § '='q-l-Ei, such that z=cch§ ', x=cch1;cosE, y=csh-qsinf; (I) then the curves for which 17 and £ are constant are confocal ellipses and hyperbolas, and

J =V%% = ¢”(Ch'n - C0S'£) C

= %C2(Ch2'l7 -cos2.E) = rlrz = OD2,

if OD is the semi-diameter conjugate to OP, and rl, rg the focal (2)

distances,

V r1, r2=c(ch1;=l=cos Q); ' (3)

rf = x' -l-yi = c” (ch'1; - siniij)

= iv”(Ch 211 -l-COS 2E)- (4)

Consider the streaming motion given by w =mCh(s'-1), via-l-Bi, (5)

¢>=m ch(n-¢)c0S(£-/3), Vlpbtm Sh(n-a)Si11(£-B)- (6) j Then W 30 over the ellipse 'r|=a, and the hyperbola £=H, so that these may be taken as fixed boundaries; and ti/'15 a constant on a C4. Over any ellipse 11, moving with components U and V of velocity, V xl/=1P-I-Uy-Vx'=[msh(11-a)cos;3-l-Ucsh1;]sin§ -[msh(q-a) sinB+Vcchn]cos£; (7)

so that gli' =o, if-U=

-%- cos B, V= -7% sin 5, (8)

having a resultant in the direction PO, where P is the intersection of an ellipse 11 with the hyperbola B; and with this velocity the ellipse 1| can be swimming in the liquid, without distortion for an instant. At infinity

U = -'¥e'°cos B = - 3%c0S B, ~ V

V= -¥e““sin B = »5%~sin B, (9)

a and b denoting the semi-axes of the ellipse a; so that the liq uid is streaming at infinity with velocity Q=m/(a.~l-b) in the direction of the asymptote of the hyperbola 1 1 An ellipse interior to 17=¢ will move in a direction opgosite to the exterior current; and when 71 ==o, U =eo, but V= (m/c) s e. sin 13. Negative values of 1; must be interpreted by a streaming rnotion on a parallel plane at a level slightly different, as on adouble Riemann sheet, the 'stream passing from one sheet to the other across a cut SS' joining the foci S, 5'. A diagram 'has been drawn by Col. R. L.

Hippisley.