Forced Vortex Flow

• If there exists a solid body rotation at constant ω (induced by some external mechanism), the flow should be called a forced vortex motion (Fig. 21.3 (b).

Fig 21.3 (b)   Forced Vortex Flow

we can write

and
	(21.10)

Equation (21.10) predicts that

1. The circulation is zero at the origin

2.  It increases with increasing radius.

3.  The variation is parabolic.

It may be mentioned that the free vortex (irrotational) flow at the origin is impossible because of mathematical singularity. However, physically there should exist a rotational (forced vortex) core which is shown by the dotted line ( in Fig. 21.3a ).

 Below are given two statements which are related to Kelvin's circulation theorem (stated in 1869) and Cauchy's theorem on irrotational motion (stated in 1815) respectively

1. The circulation around any closed contour is invariant with time in an inviscid fluid.--- Kelvin's Theorem
2. A body of inviscid fluid in irrotational motion continues to move irrotationally.------------ Cauchy's Theorem