I just need a bit of help with a problem. I'm being asked to evaluate the circulation of a velocity field. I've just finished calculating the curl of the vector field in question and I know from my text that I need $\int_{S}^{ } (\mathbf{curl \ u})\cdot \mathbf{n} \ dA$, but there are no examples in the book of this set up. Can someone help me set this up so I can perform the integration?
The question is below. I can see and visualise the boundaries, I'm just not sure how to go about setting up the integral, I'll be fine doing the actual integration.
Thanks!
Chris
Note that a line integral $\int_C \vec{u} \cdot d\vec{l}$ is the integral of the tangential component of the velocity with respect to arclength along the contour. Traversed anticlockwise we have $$\int_C \vec{u} \cdot d\vec{l} = \int_{C_1} \vec{u} \cdot d\vec{l}+\int_{C_2} \vec{u} \cdot d\vec{l}+ \int_{C_3} \vec{u} \cdot d\vec{l}+\int_{C_4} \vec{u} \cdot d\vec{l}$$
where
$$C_1 = \left\{(r,\theta) : \, r = a , \frac{\pi}6 \leqslant \theta \leqslant \frac{\pi}{2}\right\}, \,\,d \vec{\mathcal{l}} = -a\vec{e}_\theta d\theta\\ C_2 = \left\{(r,\theta) : \, a \leqslant r\leqslant 2a , \,\theta = \frac{\pi}{6}\right\}, \,\,d \vec{\mathcal{l}} = \vec{e}_r dr\\ C_3 = \left\{(r,\theta) : \, r = 2a , \frac{\pi}6 \leqslant \theta \leqslant \frac{\pi}{2}\right\}, \,\,d \vec{\mathcal{l}} = 2a\vec{e}_\theta d\theta\\ C_4 = \left\{(r,\theta) : \, a \leqslant r \leqslant 2a , \theta = \frac{\pi}{2}\right\}, \,\,d \vec{\mathcal{l}} = -\vec{e}_r dr$$
Thus,
$$\int_C \vec{u} \cdot d\vec{l} = -\int_{\pi/6}^{\pi/2} u_\theta(a,\theta) a \, d\theta + \int_{a}^{2a} u_r(r,\pi/6) \, dr + \int_{\pi/6}^{\pi/2} u_\theta(2a,\theta) 2a \, d\theta - \int_{a}^{2a} u_r(r,\pi/2) \, dr$$
Since this is a two-dimensional velocity field, the only non-vanishing component of the curl is $\omega_z$ in the direction normal to the plane. By Stokes' (Green's) theorem, the circulation could also be computed as the double integral over the sector bounded by $C$,
$$\int_{\pi/6}^{\pi/2} \int_{a}^{2a} \omega_z \, r \, dr \, d\theta$$