454 HYDROMECHANICS [HYDRODYNAMICS. On the Motion of a Solid through a Liquid. If we take an origin 0, and axes Ox, Oy, Oz fixed in the body, then, if u, v, w, p, q, r denote the component linear and angular velocities of the body at any instant, the velocity function <f> = u^i l + v^., + tfvJ/3 +p Xl + q x . 2 + rx 3 , where the <J/ s and x s are functions of x, y, z, depending only upon the shape of the body. To determine ^ 1( we may suppose the velocity u only to exist, and thus f/ l must satisfy the conditions (i) V 3 ^ = ; (ii ^b = l the cosine of the angle between the normal to an the surface and the axis of x, at the surface of the moving body ; ( iii} -^ = 0, over a fixed surface. v dn Similarly for >|/ 2 and <f/ 3 . To determine xi> we may suppose the velocity p only to exist, and and then, I, m, n being the direction-cosines of the normal to the surface, xi satisfies the conditions (i) V 2 Xi = <>; (jj) -2ti = ny - mz at the surface of the moving body ; (iii) 1*! = at a fixed surface. dn Similarly for x-z and xa- For a cavity filled with liquid in the interior of a moving body, since the liquid moves as if solid when the moving body has a motion of translation only, therefore ^i = ~> t-2 = y , ^ = z - The only cases practically solved are those where the bounding surfaces are similar or confocal surfaces of the second degree. Ex. 1. Consider the space between the ellipsoid -^+~ + Ar= 1 a* o- c 2 and a similar and similarly situated ellipsoid rigidly connected with 7,2 _ C 2 it ; then obviously xi = . 5 2/s, as for plane motion; and therefore 0" + C 1 <f> = ux + vy + wz +p -j- b* The liquid filling this space will behave therefore like a body of 2 2 2 2 % 2 equal mass, and of principal radii of gyration of the radii of gyration if the liquid were solidified. Ex. 2. Consider the liquid filling the space between the ellipsoids o o > . . . (1) . . . (2), - = 1 c 2 and the ellipsoids being confocal, such that Put A = w where and d C = d (3), so that cr + , b 2 + , c 2 + are the squares of the semi-axis of the confocal ellipsoid passing through xyz. Then and therefore P also, if p be the length of the perpendicular from the centre on the tangent plane to (3), m 2 + (c 2 + A)?i 2 , d _ _d dp dn Suppose the ellipsoid (1) moving with velocity u, and the ellipsoid (2) fixed, then ^/ 1 can be made to satisfv the required con ditions by putting >h Mie+NAar, where M and N are constants. For V if i = 0, and dn dn dn -X -(M + NA-2-^ U Consequently, when A = 0, we must have M-N(B + C ) = 1, and when A = 1 , M-N(B 1 + C 1 ) = 0, where A , B , C are the values of A, B, C, when = 0, aiid A 1} Bj, C : when A = A : . Hence N= -g * M = T^ r, JT FT" >
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and Similarly
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/ i , A P A U
- -0"r "0 ^1 A l
If the inner ellipsoid had been fixed and the outer moved, we should have had Next suppose the outer ellipsoid fixed, and the inner to have the angular velocity^? ; then xi can be made to satisfy the required con ditions by putting where M and N are constants. For then v 2 Xi = 0> and dn = I M + N(B - C) M + N(B-C) f ( -r-z+y~ ) +]S 7 ( -. r 7 ) dn J dn J dn dn ( l n J P~ Y 1 1 b* + ~ ^c- + ) 2 + ; -* - M + N(B-C) j f L- + -J_ which when A =-- must = y- mz = ^-y - -p-z = -^ - -p- jpyz , and when A=Aj must = 0. Therefore M and N must be determined from the equations M + N(B - C ) and ( 1 2 +l 2 )-N(A 1 +B 1 +c 1 /^-i)=o. i c i / u i c / Similarly x 2 and xs an be determined, and also xi> X-2> and xs when the inner ellipsoid is fixed and the outer moved with given angular velocities. When the outer ellipsoid is indefinitely great, then A v B 1( C x are zero, as also is M. Then I 1 (B -< To find the effective inertia of the inner ellipsoid, when the outer ellipsoid is fixed, and first for motion parallel to the axis of x ; when A = A,, r^-0 ; but when A = 0, f-- = L and the i^i for the an dn liquid in the interspace is - ^ -~ , Jp of the ^ for the liquid filling the inner ellipsoid ; and hence, since the kinetic energy = $p/<l>,-dS, it follows that the kinetic energy of the liquid in ^/ dn A -4- B -4- O the interspace is 9 LI L_ of the liquid filling the interior BO + C/o Bj Cj ellipsoid for motion parallel to the axis of x, and therefore the
eil ective inertia parallel to the axis of x is
