HYDRODYNAMICS.] HYDROMECHANICS 453 ..cos 20 x? . / (2+%- ,r0 ^ = |a>?~- r + 2 A ( I 3 cos(2n + l)r , COS 2a n=0 2n + iy a / 2a which satisfies the conditions v 2 if = 0, and fy = ^u>r- when 0= a; in order that i^ = ^w? >2 when r = a, we must have A 2n+1 7T0 cos 2fl cos 2a/ and therefore, by Fourier s theorem, - a-(-r ~ " +1 When all the cylinders present rotate, as if rigidly connected, about the axis of z with angular velocity w at any instant, then ^ = $o>(a: 2 + 7/ 2 ) + constant round the boundary of every cylinder; and if we put % = 4 ~ i w ( il2 + 2/ 2 )> then % is the stream function of the relative motion, relative to the cylinders, and satisfies the con ditions 4 + 4 = - 2w at every point of the liquid, and v = con- dx 2 dy* stant round the boundaries. Since % involves u as a factor, which is a function of t only, it follows that - = constant is the equation of a stream line of the rela te tive motion, and any alteration in o> does not affect the shape of the relative stream lines, the liquid being frictionless, and the motion generated from rest. Ex. 6. Put then ^X + ^X = _ 2a ,. dx- dy 2 and the relative stream lines are similar ellipses. Then <J> = Y + o>(x 2 + v") and therefore for the motion between two similar elliptic cylinders, rotating with angular velocity a. The velocity of any liquid particle is ^- ~ of what it would be if rigidly connected to the cylinders ; hence the effective radius of gyration of the liquid is ^- of the radius of gyration of the homogenous rigid body occupying the space. Ex. 7. Put then V 2 x= -2&>, and x = 0, when , which may therefore be taken as boundaries of the liquid. This problem is due to Mr Ferrers. Again put then V 2 x = - 2o>, and the hyperbolas 2.r(x -y)-a- = 0, 2ij(x + y)-tf = Q may be taken as boundaries, but these hyperbolas are only the previous ones turned through a quarter of a right angle. Ex. 8. When the liquid fills a rectangular cylinder bounded by x ~ , and y= & the conditions and M= . are satisfied by putting dy* dif- ^, = when7/=&, du dy - tan Jv . fc a dy_rfX dx dx- /cn( K . V n where _^L K ~ b (Quarterly Journal of Mathematics, vol. xv.). f= cosh (2i + l)^ cos (2i + l)
- -4|a2(-l) _J?
(2i+l) 3 cosh (2i cosh(2t + l)^ cos 2a
- 2T
cosh then (2) ty = (& 2 - y*) whena;=a; (3) i// = ca(a" - x 1 ) when y = 6 ; and therefore ty satisfies the required conditions, and is therefore the value of -fy required. Ex. 9. Consider liquid filling the interior of a cylinder, whose cross section is an equilateral triangle of altitude k, and let o, , 7 denote the perpendicular distances of a point in the interior from the sides. If we put then ( n< and x is the stream function of the relative motion, supposing the cylinder rotating with angular velocity o>. Therefore the cubic 0/87 = constant is the equation of the path of a liquid particle relative to the cylinder, when it is moved in any manner ; and also for the cylinder bounded by a^y c l and We have supposed the liquid motion to have been generated from rest by the :notion of the moving cylinders, but we might also have supposed the liquid to have been of infinite extent, and streaming past the cylinders as fixed obstacles ; in that case, the stream function of the relative motion x = ^ + ^2/> an( i x sa ti sne s the relations dx* dy* and x = constant, the equation of a stream line, and therefore also of a boundary ; also at infinity dx dy For instance, if in liquid, moving with velocity - V parallel to the axis of x, the fixed circular cylinder r=a be introduced, then re 2 X= - V sin + ~Vy r- sin 6. r J If the elliptic, cylinder r) = a be introduced, then, since =o, -rr -IT i sinh r> - cosh -n . , v=y - c sum a ^ - sin f sinh a - cosh a = Vc sinh 77 sin - Vc sinh a c~ r> ^~ a sin | = Vce a sinh (17 - o) sin | = V(a + &) sinh (77-0) sin . If the axis of z be horizontal, and the liquid supposed of infinite extent, and originally at rest, then a circular cylinder of density IT, projected in any manner perpendicular to its length, will describe a parabola with vertical acceleration - g. If, however, previously to projection, a vortex exist in the liquid, co-axial with the cylinder and of strength m, then any motion of the cylinder will not affect the circulation of the liquid round the cylinder due to the vortex, and inequalities of pressure round the cylinder will arise from the vortex motion. Lord Kayleigh has shown (Messenger of Mathematics, vol. vii.) that, if no forces act, the cylinder will describe a circle in the same direction as the circulation of the vortex in the periodic time <r + P > where the circulation of the vortex is 2irft 2 o>, a being the W p radius of the cylinder. If the axis of the cylinder be horizontal, and the infhunce of the boundaries of the liquid neglected, then the cylinder will describe a trochoid, and for a particular velocity of projection can be made to describe a horizontal straight line (Messenger of Mathematics, vol. ix. p. 113). 1 For the annlopy ho wppn the motion of a liquid in a cylinder and the torsion of nn elastic bar. pointed out by St Vcnnnt, consult Thomson and Tail s Xatural
lotofihu, 704.
