Past exam of the options II course of the University of Oxford / 2007 / K / K3

Orient the body boundary counterclockwise. The outward normal from the body satisfies $nds=(dy,-dx)$, so pressure givesFor steady irrotational flow, Bernoulli's equation makes $p+\rho |u{|}^2/2$ constant in the connected fluid region. Its constant contribution integrates to zero. Write $q=w^{\prime }=u-iv$. Impermeability says the velocity is tangent to the boundary, so $qdz=\bar{q}d\bar{z}$ there. Consequentlywhich proves Blasius' theorem with this contour convention.

The background stream function is the imaginary part of $w_0(z)=\alpha (z-c{)}^3$. Since its coefficients are real, the circle theorem givesIts derivative isIn $q^2$ the coefficient of $z^{-1}$ comes from the exponent pairs $(2,-3)$ and $(1,-2)$, including both orders. It is $36{\alpha }^{2}c{a}^4+36{\alpha }^2{c}^3{a}^2$. The residue theorem then givesThe vanishing transverse force also agrees with reflection symmetry about the real axis.