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[HYDRODYNAMICS
HYDROMECHANICS


When the polygon is closed by the walls joining, insteadofreaching back to in nity at'aqx', the liquid motion-inust=be'due'to'a source, and this modification has beenfworked out by B'. 'Hoplcinson in the Proc. Land. Math. Soc., 1898.' ' ' »* ' ' ' ' ff Michell has discussed also the hollow 'vortex stationary inside a polygon'(Phil. Trans., 1890); the solution is givenby > if ch nS)'= sn w, sh n(2=i cn -rv

so that, round' the boundary of the polygon, and on the surface of the vortex ]/ =o, q =Q, andcosn0=sn ¢>, m9=§ 1r-am s]c, (12)

the intrinsic equation of the curve. This is a closed' Sumner line for n = 1, when 'the 'boundary consists of two parallel walls; and n =% gives an Elastica. 44. The Motion of gz Solid through a Liquid:-An. important problem in the motion of a liquid is the determination of the state (11)

Jl¢=K', , sin nH==o;

of velocity set up by the passage- of a solid through it;~and thence j of the pressure and reaction of the liquid ontliesurface of the solid, by which its motion is influenced when it is free. ' ' ' V Beginning with a- single body in liquid extending jto'infinity*, 'an'd denoting by- U, V, W, P, Q, R] the components of linear and angular velocity with respect to axes fixed in the body, the velocityfunctlon O takes the form ' ~ »~

¢=U¢1+V4>2'l'fV4?a'l'PX1+QX2'l"RXa, (X) where the ¢'s and. fs ate functions of,3c, ' y, "z depending on the shape of the body; interpreted dynamically, C-po represents the impulsive pressure required to stop the motion, or C+p4> to start it again from rest.

The terms of ¢ ma be determined one at a time, and this problem is purely kinematicai; thus to determine ¢1, the component U alone is taken to exist, and then l, m, n, denoting the direction cosines of the normal of the surface drawn into the exterior liquid, the function 41, must be determined to satisfy the conditions (i.) V'¢, =o, throughout the liquid; ' ~ .., ~, (ii.) $35 = -l, 'the gradient of ¢ down the normalat the surface of the moving solid; ' ' ' "

(iii.) %' ==o, over a fixed boundary, or at infinity; similarly for np, and os. ' ' ” To determine X, the angular velocity¢P alone is introduced, and the conditions to be wtisfied are, (i.) V'X'=0, throughout the liquid; (ii.) %Zff = mz - ny, at the surface oi-the moving bodyfbut zero: over a fixed surface, and at inlinity; the same for xg and Xa. For a cavity filled with liquid in the-interior of the body, since the liquid' inside moves bodily for a motion of 'translation'only, 4n=-x, d>z='-7. ¢a= '*2?' . (2) but a rotation will stir up the liquidin the cavity, so that the'X's de nd on the shape- of the surface. Tehe ellipsoid was the shape first worked out, by George Green, in his'Research on the Vibration of a Pendulum in a Fluid Me in-rn (1833); the extension to any other surface will form an 'important step in this subject. 1 ~ »

A system of confocal ellipsoids is taken x»+y2+z= I

E2+" x" BM-'> ¢2+x= ' (3)

and a velocity function of the form ' V » ¢ gxlpr

where up is function of Xonlyfiso that rl/ is constant-over an ellipsoid; and we seek to determine the motion set up, and the form of 11/ which will satisfy the equation of continuity; *, Over the ellipsoid, p denoting the length of the perpendicular from the centre on a tangent plane,

= px = v = z

I a'+7' '” " FQFX (5)

2: Ex! Zz?,

<a2+>~>2+<b='+»>=+<c2+»>" (6)

P' = (<1'+7ll'+(5'+7)"if+(C'+))1|'» I (7) = azlz -lwbim' -l-czn' +A,

wa

Thence M d d 1

= x '/

as a;"*+'?ar

»=§ ¢+2ca=+;>§§ z§§ , '“ ~ (cg) I0 that't'he'velocity of the liquid ma be resolved into at component -¢' parallel to Ox, and' -2(a2+7)l1iiZ 7d> alongthe norinalf of lthe ellipsoid; and the liquid flows over an ellipsoid along a line of slope 7 with respect to Ox, treated as the vertical. Along the normal itself ~ ¢3 1

i"-f= ¢+¢<a+n@ 1, no

~ d§ , dk Y

Sothat. overthe surfaconf an ellipsoid where 7 and W are constant. the nozrnal velocxtygs the same as that of the ellipsoid itself, moving as azsolid withwelocxty parallel to Ox ~ V ~, P ' fue-w¢'z<<»”+»>§ . T ml

and sothe boundary condition is satisfied; moreover, any ellipsoidal surface X may be supposed' moving as if rigid with the velocity in (1 I), without disturbing the liquid motion for the moment.

  • The Continuity is secured if the liquid between two ellipsoids 7

and)~, , ~moving with the velocity U and U1 of equation (11), is squeezed* out or sucked in across the plane x =o'at a'rate equal to the integral #ow of the velocity:P across the annular 'area a, -a of the two ell1pqoids xnafl e by x=0; or if, f T tu-¢, U, = iv gat, i (12)

<»=~ <f>“+~f+»>- 1 <1s>

Expressed as a differential relation, with the value off; U from (I 1), gi a-, lf-+2 (a'+A)a% -¢%§ =o, V (14) “ of i T H 4 4 'r

3aE¥+ a(a's{-7)a-X<a-aL£) =o, (15) and ~mte'gratxng .

1 (a'+7$)°”a-§§ =a constant, (16) so that we may put ' ' ' ' O

Md) (175

(13)

y 1 1 'f' ' 'P<a”+»> ' t

P”=4(G'+>)(1>“+>)(¢>'+7),

where M denotes a constant; so that np is an elliptic integral of the second kind.

The quiescent ellipsoidal surface, over which the motion is entirely tangential; is 'the one for which ~ f d ~

2(a'+})a¥ +[/=o, (19)

and this is the infinite boundary ellipsoid if we make the upper limit = oo. V ,

I The velocity of the ellipsoid defined by A =o is then V U = *2G2 *¢q

= M '° Mdk

abc. f, , 'ia2+A)P .

with the notation

Am;A~=f“' abcdh

so that in (4)

M

=f%<1-A.>. ' ' <=<>>

" A (a2-l-ASP

1 =-2abc& ~ AQ, (21)

¢=rb-52xA=iU§ . '§ , i¢|=?§ - %, O (22) in (I) for an ellipsoid. ' »

The impulse required to set-up the motion in liquid of density p is the resultant of an impulsive pressure p¢ over the surface S of the ellipsoid, and is therefore .

ff, ,¢zds =p~/f., f {x1ds T

=p¢0 volume of the ellipsoid) =//uW', (23) where W: denotes the weight of liquid displaced. Denotmg the eliectiveinertia of the liquid parallel to Ox by ¢W', the momentum »

°W'U=¢.»w" s <2 4>

==%3=;;i%; (25),

in this way the air drag was calculated by Green for an ellipsoidal pendulum. .

Similarly, the inertia 'parallel to Oy and Oz is B, , C

BW' -TTEW, 7W — I % W', ~ (26) " ' BA»~C>, '=, ? (27)

and V

A+B+C==abc/=}P, A¢+Bo+Cu=I. (28) For a sphere

0=b=€.Ao"Bo*==Co==i» ¢=~'B=~/=§ » (20)