A Linear Vortex of Finite Length

Examine the linear vortex of finite length AB, shown in Figure 5.24. Let P be an adjacent point located by the angular displacements a and в from A and B respectively. Also, the point P has coordinates r and O with respect to an elemental length Ss of AB. Further, h is the height of the perpendicular from P to AB, and the foot of the perpendicular is at a distance s from Ss.

The velocity induced at P by the element of length Ss, by Equation (5.47), is:

The induced velocity is in the direction normal to the plane ABP, shown in Figure 5.24.[4]

The velocity at P due to the length AB is the sum of induced velocities due to all elements, such as Ss. However, all the variables in Equation (5.48) must be expressed in terms of a single variable before integrating to get the effective velocity. A variable such as ф, shown in Figure 5.24 may be chosen for

this purpose. The limits of integration are:

Фа=-(2-a)toфв=+(2-p), since ф passes through zero while integrating from A to B. Here we have:

sin в = cos ф r2 = h2 sec2 ф

ds = d (h tan ф) = h sec2 фdф.

Thus, we have the induced velocity at P due to vortex AB, by Equation (5.48), as:

This is an important result of vortex dynamics. From this result we obtain the following specific results of velocity in the vicinity of the line vortex.