Гидродинамика

ON THE PROBLEM OF CUTTING OF VORTEX FILAMENTS In this article Prof. Joukowsky raises the question of a theo ­ retical investigation of the impossibility of cutting a vortex fila­ ment into parts. This problem, considered in two dimensions, is stated by him as follows: given a straight vortex filament and a wedge immersed in the fluid and having its edge parallel t o the axis of the filament, to investigate the motion of the fila­ ment in the vicinity of the edge. To solve the problemProf. Jou­ kowsky deduces formula (8) giving the velocity of the area of a vortex in a flow defined by a function of a complex variable F{z). This formula enables him to determine the velocity of the vortex near a straight boundary of infinite length in the two following cases, viz, 1) when at infinite distance the fluid is at rest (formula 11, Fig. 1), and 2) when a t infinite distance the fluid is moving along the boundary with a velocity да (formula 16, Fig. 2). When the flow defined by the function F{z) is conformally transformed by formula (17), the velocity of the vortex in the transformed flow will be given by formula (18). This formula is used by Prof. Joukowsky to determine, after transforming the two flows considered above, 1) the velocity of the vortex and the equation of its path (equation 25, Fig. 3) within an obtuse, or a sharp, angle occupied by the fluid, and 2) the velocity and the equation of the path of the vortex (equation 31, Fig. 4) in the presence of the wedge. As it is seen on reference to Fig. 4, the path of the vortex in the second case always deviates from the edge, hence the impossibility of cutting the vortex. Prof. Joukowsky also [deduces the law of motion of the vortex along the path in the two cases of motion considered, i. e. the relation giving the position of the vortex in function of time. This relation is expressed by formulae (26) and (32) refer ring, respectively, to the first fvnd to the second case.