= Blasius moment theorem
{c}
{title2=$M=-\frac\rho2\operatorname{Re}\oint_Cz(w')^2\,dz$}
For steady <incompressible flow> and <irrotational flow> of constant <mass density>, with no <body force>, this formula gives the <hydrodynamic torque> about the origin per unit span on an impermeable body. Orient its boundary counterclockwise, and let positive <torque> be counterclockwise. The proof uses $M=\operatorname{Re}\oint_Cpz\,d\bar z$, the <Bernoulli equation> $p=P_*-\rho|w'|^2/2$, and the tangency identity $|w'|^2d\bar z=(w')^2dz$. The real part of the constant-pressure term vanishes because $\operatorname{Re}(z\,d\bar z)=d|z|^2/2$. The full complex integral of $z\,d\bar z$ need not vanish.
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