Гидродинамика

А CASE OF MOTION OF A LIQUID LAMINA DUE TO INERTIA In this note Prof. Joukowsky establishes the conditions impo ­ sed on the initial velocities of points of a liquid lamina when every particle moves with constant velocity along a straight line, and the area of each element does not change. Imagine a family of lines {s) whose tangents are directed along t he initial velocity of the liquid, and a family of lines ( / ) constructed so that the initial velocities of all points of every line {s) are parallel. From condition (2), in which x, у are the initial coordinates of a particle, and — its coordinates after a time interval t, in view of the condition of invariability of the element of area, Prof. Joukowsky finds that there are only two possible cases of inertia motion: 1) all particles move in straight parallel lines with arbitrary but constant velocities for each line and 2) lines (/) are straight and the parallel velocities of all points of a line (s') are equal. In the latter case we can arbitra ­ rily assume a family of straight lines (s') and one of the (s) lines, all other (s) lines being determined by the condition that they , must cut the straight lines (s') at the same angles as the repre­ sentative line (s). As regards velocity v, it is determined from equation (4). The following example is considered at the end of the article: all straight lines s' intersect at one point and one of the lines s is an ellipse having this point for its centre.