A Treatise on the Motion of Vortex Rings; An Essay to Which the Adams Prize Was Adjudged in 1882, in the University of Cambridge

A Treatise on the Motion of Vortex Rings; An Essay to Which the Adams Prize Was Adjudged in 1882, in the University of Cambridge

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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1883 edition. Excerpt: ...equation 43). And if we pick out the coefficient of cos 2-r arising from the velocity potential O, we shall find that it reduces to ia dh+dvj dh Thus dy. we2. 8a, / d? a d dO, J and this, with the preceding equation connecting-Trand yg, enables us to find a2 and y. We have two exactly analogous equations connecting dfijdt and S2, the only difference being that we substitute-for-jj-, where-g denotes differentiation with respect to an axis passing through the centre and coinciding in direction with the radius of the vortex ring for which ifr = 0. 37. "We can apply these equations to find the motion of a vortex ring which passes by a fixed obstacle. We shall suppose that the distance of the vortex from the obstacle is large compared with the diameter of the vortex, and that the obstacle is a sphere. Let the plane containing the centre of the fixed sphere B, the centre of the vortex A, and a parallel to the direction of motion of the vortex be taken as the plane of xy. Let the axis of x be parallel to the direction of motion of the vortex. Let m be the strength of the vortex, and a its radius. The velocity potential due to the vortex at a point P = im'a'. approximately. Now 4 = 4 + 0. + Q. +....(fig.6), if BP AB, and Qv Qr..are spherical harmonics with AB for axis. At the surface of the sphere the velocity parallel to x., d2 ( 1, 23coss0-l, A = Jma! (j-pj = m a2--h smaller terms, where 0 is the angle AB makes with the axis of x. The velocity parallel to the axis of y., s? / 1., " 3 cos 0 sin 0, = mo d(zpj = ma + smaller terms. Now at the surface of the sphere the velocity must be entirely tangential, hence we must superpose a distribution of velocity, giving a radial velocity over the sphere equal and opposite to the...show more

Product details

  • Paperback | 28 pages
  • 189 x 246 x 2mm | 68g
  • Rarebooksclub.com
  • Miami Fl, United States
  • English
  • Illustrations, black and white
  • 1236583426
  • 9781236583420