Circulation and Vorticity

(4.76) |

where is an element of , and the integral is taken around the whole curve, is termed the

Let be a surface having the closed curve for a boundary, and let be an element of this surface (see Section A.7) with that direction of the normal which is related to the chosen sense of circulation around by the right-hand circulation rule. (See Section A.8.) According to the curl theorem (see Section A.22),

Thus, we conclude that circulation and vorticity are intimately related to one another. In fact, according to the previous expression, the circulation of the fluid around loop is equal to the net sum of the intensities of the vortex filaments passing through the loop and piercing the surface (with a filament making a positive, or negative, contribution to the sum depending on whether it pierces the surface in the direction determined by the chosen sense of circulation around and the right-hand circulation rule, or in the opposite direction). One important proviso to Equation (4.77) is that the surface must lie entirely within the fluid.