450 HYDROMECHANICS [HYDRODYNAMICS. On Flow, Circulation, and Vortex Motion. The line integral of the tangential velo ity / ( u-.- + --/- + w or f(udx + vdy + wdz) , from one point to another of a curve, is called thefloiv along the curve from the initial to the final point ; and, if the curve be closed, the line integral round the curve is called the circulation in the curve. If a velocity function <f> exist, then the flow =/<A/> = 0. 2 - <f> 1} where <f> 1 and <p. 2 are the initial and final values of (f> ; and therefore the flow is independent of the curve for all mutually reconcilable curves ; and the circulation in any closed curve, capable of being reduced to a point without leaving space for which <p is single-valued, is zero. If through every point of a small closed curve the vortex lines be drawn, a tube is obtained and the fluid contained is called a vortex filament. By analogy with the spin of a rigid body the component spin of the fluid in any plane at any point is defined as the circulation round any infinitesimal area in the plane enclosing the point divided by twice the area. For in a rigid body, rotating about the axis of x with angular velocity | suppose, the circulation round a curve in the plane of yz is la = o) times twice the area of the curve. Now if, in the fluid at the point xyz, we take the circulation round the elementary area dydz, it is equal to dw dv . , ,- }dz - iv dz dw dv^ Du ~dT Dv_ dt Dw dQ where .then Q=/" + V J P dx Ddy >r +v -dt +w -dT + dt Dv , Dw . Tt dy + ~it dz and therefore, by integration round a closed curve, D /(udx + vdy+wdz)**Q ; dt^/ ,,,,, ii i -1 fdw dv and therefore the component spin in the plane yz is -I O ffii // which we have denoted by |. Similarly the component spins in 1 fdu dw , 1 I dv du ,. the planes of and *y ore-^-g-^ J-, and _(___J-f respectively. Since the circulation round any triangular area is the sum of the circulations round the projections of the area on the coordinate planes, the composition of the component spins f, 77, is accord ing to the vector law. Hence in any infinitesimal part of the fluid the circulation is zero round every small plane curve passing through a certain line, the resultant axis of spin of J, 77, at that point of the fluid. Consequently the circulation round any closed curve drawn on the surface of a vortex filament is zero ; and there fore, if at any two points of a vortex filament we draw the cross sections ABC, A B C , joined by the line AA , then, since the flow in A A in the complete circuit ABCAA B C A A is taken in opposite directions, the resultant flow in AA vanishes, and therefore the cir culations in ABC, A B C , estimated in the same direction, are equal. This is expressed by saying that, at all points of a vortex filament, coa is constant, where o is the sectional area of the filament, and u the spin (Clifford, Kinematic, Book iii.). So far the theorems about vortex motion are kinematical ; but, introducing Euler s equation of motion and therefore the circulation in any circuit composed of the same fluid particles is constant, and, if the motion is differentially irro- tational, is zero round all reconcilable paths. The circulation round any small plane curve passing through the axis of spin at any point being always zero, it follows conversely that a vortex filament is always composed of the same fluid particles ; and, since the circulation round any cross section is constant for different times, it follows from the previous kinematical proposi tions that aco is constant for all the time, and the same at all points of a vortex filament. Professor Clifford (Proc. London Mathematical Society, vol. ix.) has given a simple quaternion proof of the theorem To determine the velocity at any point of a fluid, when the spin is given. If <r denote the velocity and w the spin at any point, then 2o> = Vver; also, if k denote the cubical expansion, k= -Sv<r. Hence the quaternion q or-A + 2w is simply v<r ; consequently the problem to be solved is to determine a from the equation <7 = V(T, q being given. Operating by v, v? = v -V ; therefore cr is the potential of V? ; and therefore where <T O means the value of cr at the point a, di i means an element of volume at the point b, and D a i, the distance between the points a, b. Returning to Euler s equations of motion, du du du da + u -r + v and eliminating Q, D ..du dv ..dw J du dv and, since by the equation of continuity 1 Dp du dv dw p dt dx dy dz
f du Ti dv
_3_ I = _2 ___ i __ I ____ p dx p dx <* dw _*_ - ; p dx therefore and similarly These equations, first given by Professor Stokos for homogeneous liquid, were generalized for any fluid by Professor Nanson, Messenger of Mathematics, 1873. They may also be obtained immediately by the differentiation of Cauchy s integrals (4), (5), and (6), given above. Plane Vortex Motion. When a series of straight vertical vortices (called columnar vor tices by Sir W. Thomson) are present in homogeneous liquid, bounded by two horizontal planes, we can determine the motion of any vortex by supposing it due to the remaining vortices. A single vortex will remain at rest, and cause a velocity at any point perpendicular to the plane through the point and the vortex inversely as the distance from the vortex. If m denote the strength of the vortex, i.e., the circulation in any circuit enclosing the vortex once, then the velocity at a distance r from the vortex will be -^- , and the current function vj/ will be ^- log r, and the velocity function <f> will be - 6, where is the angle between any fixed plane and the plane through the vortex and the point. The surface of equal pressure under gravity will be of the form the axis of the vortex being the axis of z. When there are more than one vortex present, each vortex moving with the velocity due to the other vortices will describe the curve whose equation is 2 - log r = constant, where m is the strength of one of the remaining vortices, _ and r the distance between it and the vortex whose motion is considered ; this equation may also be written IT/-" 1 = constant. AVhen the liquid is bounded by a vertical cylindrical surface, the motion of a vortex may be determined as due to a series of vortices
considered as images of the original vortex, and so arranged as to
