Past exam of the fluids and waves course of the University of Oxford / 2023 / 2 / c

For $q=W^{\prime }=u-iv$, the Blasius force theorem states that the force per unit length exerted by a steady incompressible flow and irrotational flow on an impermeable body iswhere $C$ runs counterclockwise around the body and excludes external flow singularities. The sign corresponds to the force on the body.

Let $c=a^2/b$ and $A=m{a}^2{e}^{-i\alpha }/b^2$. The complex velocity may be writtenOnly the image point $c$ lies inside the circular contour. The term $A^2/(z-c{)}^4$ has zero residue; $H^2$ is regular there. In the cross term, the linear term of the Taylor expansion of $H$ suppliesThe orientation factors have canceled. The residue theorem now givesSince $b>a$, $X>0$: the obstacle is pulled toward the doublet, independently of $\alpha$. This is a force per unit length perpendicular to the plane of flow.