HYDEODYXAMICS.J HYDROMECHANICS 451 make the velocity across the boundary zero at every point of the boundary. When the boundaries are nlane surfaces, the images are the opti cal images by reflexion of the original vortex, considered as posi tive or negative, according as formed by an even or odd number of reflexions. Thus the curve described by a vortex inside the angle bounded by the planes = is the Cotes s spiral 2n r cos nd = C, and inside the space bounded by the planes x=0, x = a, y 0, y = b is cot 2 am ( K , k } + cot 2 am ( K - r , k } = constant,
a J b J
, K a where =-. = -r- lv b (Quarterly Journal of Mathematics, vol. xv. ). A single vortex of strength m in a circular cylinder of radius a at a distance c from the centre will move with the velocity due to an image of strength - m at a distance from the centre, and therefore describe a circle of radius c with velocity m a _m c 2ir a" 2it and therefore in the periodic time -IL (a 2 - c 2 ) . m A. single circular vortex in infinite fluid will move with a certain velocity in the direction of its axis ("Vortex Motion," Trans. Ji.S.E., 1869 ; "" Vortex Motion," Helmholtz, Crelle, 1858) ; and, if another I qual circular vortex be projected coaxially after the first, the mo tion of the first must be compounded with that due to the second. Consequently the first vortex will dilate and move slower till the second vortex passes through it, when it will contract and move faster till it passes through the second, and so on. This can be verified experimentally with smoke rings projected from the same circular hole, or with half vortex rings, formed on the surface of water by drawing a semi-circular blade a short distance through the water. The motion of a vortex ring projected perpendicularly against a plane boundary will be determined by compounding it with the motion due to an equal and opposite vortex ring, its optical image in the wall. The vortex ring will therefore spread out and move more slowly in the direction of its axis as it approaches the wall ; at the same time the molecular rotation, being inversely propor tional to the cross section of the vortex, will be seen to increase. Plane Motion of Liquids. When the velocity of the fluid is always parallel to a fixed plane, we take this plane as the plane of xy, and then w = 0, and u and v ure functions of x and y, and the stream lines are plane curves. Considering only the cases where the fluid is incompressible, the equation-of continuity becomes dx dy~ and therefore a function 4, exists, called the stream function, such that dy ^ ~dx and 4- = constant is the equation of a line of flow. The spin at any noint 1 fdn du
- > 2dx dy)
If the motion is irrotational, then f=0, nnd a velocity function rf> exists such that dy dx therefore ^ and </> are conjugate functions of .r and y, and By assigning particular values to this function, Helmholtz and Hirchnoff have discovered the solution of various problems of dis- Rontmuous plane liquid motion, an account of which is triven in Lamb s Motion of Fluids. The kinetic energy of the liquid bounded by two planes perpen dicular to the axis of z at unit distance is T = when ( -$- t c -- are the rates of change of </> and >|/ in the direction of the outward drawn normal to the bounding curve at the element ds. dfy dtj> d<p dL dn ds dn~ ~ds Since therefore T= We can interchange $ and ty, and make <j> the stream function and ty the velocity function ; thus from any given irrotational motion in two dimensions another may be derived by turning the velocity through a right angle without altering its magnitude. For instance, if the axis of z be a line source of delivery m, then, since the flow across any cylinder of radius r is m, the velocity must be ; and therefore where 6 is the angle made by a plane through the axis of z and the point with a fixed plane. If the values of < and i|/ be interchanged, we obtain a vortex round the axis of z, of strength m. Plane Motion in a Liquid due to the Motion of Rigid Cylinders perpendicular to their Axes. Suppose a rigid cylindrical surface moving in the direction of the axis of x with velocity V, and other fixed rigid cylindrical surfaces to be present in the liquid, which is supposed for simplicity to be bounded also by two fixed planes perpendicular to the axis of z at unit distance from each other, the generating lines of the cylin ders being supposed parallel to the axis of z ; then at all points of the boundary of the moving surface ^ = normal velocity of fluid ds = velocity of boundary normal to itself = Y^ ds and therefore 4/= -V?/ + constant; and at all points of the fixed surfaces 1=0, and therefore >!/ = constant, efo We must therefore discover a function i|/ which satisfies the equation d-^ d-^ rfie 3 ^ dtf and is equal to a constant round a fixed boundary, and equal to - VT/ + constant round a moving boundary, moving with velocity V in the direction of the axis of x ; and <p, the conjugate function, can then easily be written down. Ex. 1. The moving cylinder a circular cylinder of radius a. and the fixed cylinder a circular cylinder of radius b, both having the axis of z as axis. Then = - Yffl sin 6 _ _ y and therefore - - V - - + r cos 9. b* a~ " If <f/ denote the velocity function of liquid filling the cylinder r = a, <> = W cos 0, and therefore, when r = a, <t> __ b + a 1 * $ b- - a- In determining the kinetic energy of the liquid intermediate to the cylinders. -~ = when ?==&; and when r = a. -- ; and dr iir dr therefore the kinetic energy of the liquid intermediate to the cylin ders is - of the kinetic energy of the liquid filling the cylinder b 2 - a- r = n. Consequently, if the cylinder r = a be moved, the inertia to
be overcome will be its own inertia, together with the inertia of a
