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

Orient $C$ counterclockwise around the body. For an arc element $ds$, the body-outward unit normal vector obeys $\mathbf{n}ds=(dy,-dx)$. The pressure acts normally into the body, so its force per unit length in the suppressed third direction isIntegrating givesTaking positive torque to mean counterclockwise about the origin, the infinitesimal moment is $dM=xd{F}_y-yd{F}_x=p(xdx+ydy)$. HenceAll three expressions are per unit span. They are the normal-pressure contributions assumed by the inviscid model; a viscous traction would require extra terms. A constant pressure contributes zero net force and torque on a closed body, since $\oint dx=\oint dy=\oint d(x^2+y^2)=0$.