Circulation and vorticity.

The definition of the circulation round a closed curve in two dimensions (see 4·1) as the integral of the tangential component of the velocity round the circumference of the curve can be extended at once to the more general case of motion in three dimensions by removing the restriction that the curve must lie in a single plane. Also by dividing any surface bounded by this curve into a network by a series of intersecting lines it can be shown that the circulation round the curve is equal to the sum of the circulations round the elementary areas formed by the network.

The vorticity of a fluid element in two-dimensional motion was defined (see 4·3) as twice the angular velocity of the element. This definition is retained in the more general case of three-dimensional motion but the axis of rotation of the fluid element may now point in any direction. By following the direction of the axis of rotation of successive fluid elements it is possible to construct a curved line whose direction coincides at every point of its length with the axis of rotation of the corresponding fluid element. Such a line is called a vortex line.

The vortex lines which pass through the points of the circumference of a small closed curve C will form the surface of a vortex tube, of which the curve C is a cross section.