Past exam of the fluids and waves course of the University of Oxford / 2015 / 2 / b / Solution

Use the complex velocity $q(z)=w^{\prime }(z)=u-iv$. For this steady, constant-mass density, irrotational flow with no body force, the applicable Bernoulli equation iswith a single constant $P_{\ast }$ throughout the connected regular fluid region. The body's impermeability makes the velocity tangent to $C$. Thus $qdz$ is real there, givingThe complex force from part (a) iswhere the constant-$P_{\ast }$ term vanishes. This proves the Blasius force theorem:For the moment, observe thatTherefore $M=\mathrm{Re}{\oint }_{C}pzd\bar{z}$. The constant pressure contribution has zero real part, because $xdx+ydy=\frac{1}{2}d(x^2+y^2)$. Substituting Bernoulli's principle and the same tangency identity gives the Blasius moment theorem:The real-part operation is essential: $\oint zd\bar{z}$ itself need not vanish, but its real part does.