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HYDRODYNAMICS]
127
HYBROMECHANICS


rotating about the centre with components of angular velocity E, and then Uthen

U 1¢=-yt-l~2n. v=e-=E+xlT»'" F "“"+9f- (2) (Fi) "la'+¢"5*m“£ “my N ow suppose the liquid to be melted, and additional components of angulanvelocity Sh. Sh. Sh communicated to the ellipsoidal 'case =(-, i f}ni(e+ ~»=>(Qf+n:)-(nf+n, »)=1 the additional velocity communicated to the liquid, will be due to ~ i a i = 6) z,2 2 z J 1 2

a velocity-function I ' s [LM N'+{L -M%-Ng; £4 V bz cz cz azzx*na2 bgx () . I 4 2 62, 2 2 es '“1e'»;?=y= '% '=+.»2 1* 3 " l t t l -'“ 2 l '?i?°';»] =z, (23) as may be verified by considering one term at a time., ,V ' X C 6 C) If u', v', w' denote the components of the velocity of the liquid where Z is a quadratic in ff, so that Q' is an elliptic function of 1, relative to the axes, except when c=a, or 30. u, =u+yR ZQ=aégfp9, y Z§ 4%9a, , (4) V Put sz, =n cos 4», o, = -sz sin ¢~, ~ ( 1-WP -xR=V-2-”' t, , w2'>' at <5> § »~"a1i'=:z'%f>¢fi>»:¢'%'=f1“f—Z2”£Z=<f>1ff+<»»»~>f» (24) " " +202 ' a +2 ' A N+, ,2, , 62{2 g ' A f ~ d . - 2 2 ' "T

w'=w+”Q-yP=2f%='¢”'r5i°3°=“"» i (6) g zi=fi% ~ I (25) Th P=ni+s, Q=n, ~+», R=n, +r. (1) ' 2°“<“"'” us, V, a*-l-52

urf +v/%+w/§ =0' (8) ¢= § 'Q§ ' a2+¢;2 'l' 45; fl .g-df' I ag 2 63 D ' '/Z az c2 (a2+c2)2 '/Z so that a liquid particle remains always on a similar ClllpS0ld. t M+ F (25) The hydrodynamical 'equations with moving axes, taking into ' ' ' ' 'account the mutual gravitation of the liquid, become d d d d

Egg-f-41rpAx+l%-1/R+wQ+u'.i;+vfa%+wf7£=o, . . ., . . ., where A

~ abcdk V

A' B'C'=f> Za'+x. § +>~. ¢=+))P

P°=4(G”+7)(b'+>)(¢'+7)- V V UO); With the values above of u, v, w, u', v', w', the equations become of the form A

7'£+41rnKx+ax+hy+gZ=6, i (H)

f;§§ +4f»By+h»+ay+f2=<>, » ~ (12) lg-g+41rnCZ+gx+fy+12=0» (13)

9

and integrating- " V '

pp" + 3'|'P(Ax2 + By' 'lr Cf) V ° (14)f

+§ (ax2+;Sy*+~/z' +a fya-lfzgzx -I-zhxy) »=»-const., so that the surfaces of equal pressure are similar quadric surfaces, which, symmetry and dynamical considerations show, must be coaxial surfaces; and V f, g, h vanish, as follows also by algebraical reduction; and f

z 2 c2 a2 2, , 1 .

“="%5Jf#l“i 'lame-'I a

4-bw-b=>, ,, <»=-bf = l (15)

—~ - — Sh-r V

(az+b2)2 8 a2+bZ I

with similar equations for B and 7. j lf we can make »,

(NFPA -l-a)x' = (411/JB +l9)b' = (41/'C +1)¢". (16) the surfaces of equal pressure are similar to the external case, which can then be removed ivithout"affecting the motion, provided a, B, ~y onstant. V

regliliiiil is so when the axis of revolution is, a principal axis, say Oz; wh

W en 91:01 Ezov "l=0-If

Q, =o or 03=§ ' in addition, we obtain the solution of Jacobi's ellipsoid of liquid of three unequal axes, rotating bodily about the least axis; and putting.-a=b, Maclaurin's solution is obtained of th t ting spheroid.lriil;

general motion again of the liquid filling a case, when a =b, KZ; may be replaced by zero, and the equations, hydrodynamical and dynamical, reduce to, ' g,

d5 252 @ aa' df; 262 ';,

V -rrrzfgeb at -a'f¢a"1f» m G-"=+¢=@2f 919) “Sl -, ~ 2 I I

%=sm+'§§§ »f, %=-nit-%fr; (19)

af which three integrals are j

2

ew =L»f;%r2, (zo)

Qi +125 =1i+2§ “2, , flip", (21) ols'+mnN é +'f-f;§ r*: ~ ' (22)

which, 'as Z is a quadratic function of ff, are non-elliptic integrals; so also for tl/, where E =w cosilf, 71 = -w sin ip; ' f in a state of steady motion i i » V dl' '$i 94,

I ggfo. Q-7 (27)

¢=v//=nt, suppose, f V (28)

g ld 9i£+S7i'0==9w, A Q(29)

2 2

V ¢§ =§ °%;i '£2 gf! V (30)

V- ed

=-, -i%, § r, (31)

2 2 2

Q an ca 2 ' "(a2 61) (902 cz) QA l5a""z * c2 ""°" 4(a2+c'z')" v (33) and a stateof steady motion is impossible when 3a> c >a. An experiment wasidevised by Lord Kelvin for demonstrating this, in which the difference of steadiness was shown of a copper shell filled with liquid and spun gyroscopic ally, according as the shell was slightly oblate or prolate. According to the theory above the stability is regained when the length is more than three diameters, so that a' modern projectile with"a cavity more than three diameters long, should fly steadily, when filled with water; while the' old-fashioned type, not so elongated, would be highly unsteady; 'and for the same reason the gas bags of a dirigible balloon should be over rather than under three diameters long. 4o. A Liquid I et.4By the use of the complex variable and its conjugate functions, an(attempt can be madeto give a mathematical interpretation of problems such as the efilux of waterin a jet or of smoke from ag chimney, the discharge through a weir, the ilow of water through -the piers of a bridge, or past the side of a ship, the wind blowing on a sail or aeroplane, or against a wall, or impinging jets of gas or water; cases where a surface of discontinuity is observable, moreor less distinct, which separates the running stream from the dead water or air. Uniplanar motion alone is so far amenable' to analysis; the velocity function ¢» and stream function il/ are given as conjugate functions of the coordinates x, y by w=f(2)'§ where 2=x+;vi, w=¢+bi, '(1) and then F » .

-Q-§ =§ ~§ '+f§§ = -uw; i (2)

so that, with u =q cos 0, v=g sin 0, the function dz Q Q Q

§ '= -Q§ v1'=, ,Q, , i=ge(14fl'5'fl f;(cos 9+i Sill 0)- (3) gives g' as a vector representing the reciprocal of the velocity 1 in direction and magnitude, in terms of some standard velocity Q. To determine the motion of a jet which issues from avessel with

plane walls, the vector (must be constructed so as to have a constant