Past exam of the mathematics course of the University of Cambridge / 2004 / ib / Paper 4 / 18C / Solution

For steady incompressible flow of constant density without a body force, the Euler equation and mass conservation giveApply the divergence theorem componentwise to a fluid volume. With $\mathbf{n}$ its outward normal, the integral momentum flux balance isIf a solid bounds part of the volume, retain its pressure contribution; that contribution supplies the force on the fluid. This derivation includes pressure as well as transported momentum.

Choose the axis in the incoming jet direction. The printed diagram measures $\alpha$ between that axis and the downstream sheet: it is the cone's half-angle. Along a streamline, the steady Euler equation integrates to the Bernoulli equation $p/\rho +|\mathbf{u}{|}^2/2=\text{constant}$. At the incoming free jet and the outgoing free sheet the pressure is atmospheric. With gravity neglected, the outgoing speed is therefore $u$. This argument uses far free surfaces; it does not assert atmospheric pressure on the wetted sphere.

At radial distance $r$, the sheet's circumference is $2\pi r\sin \alpha$. Its cross-section normal to its velocity is this circumference times its small thickness $d$. Conservation of mass gives $Au=(2\pi r\sin \alpha )du$, henceFor the axial momentum balance, transverse contributions cancel by axial symmetry. The outgoing axial momentum rate is $\rho A{u}^2\cos \alpha$ and the incoming rate is $\rho A{u}^2$. Uniform atmospheric pressure cancels on the closed control surface when expressed as gauge pressure. The sphere's force on the fluid is therefore $\rho A{u}^2(\cos \alpha -1)$ in the axis direction. By action and reaction, the fluid force on the sphere is downstream, with magnitudeThe downstream direction in the original diagram fixes this sign and the factor $1-\cos \alpha$. If one calls the full cone opening its vertex angle, that full opening is $2\alpha$; using $\alpha$ as the full opening would conflict with the printed thickness formula.