Consider a steady inviscid, incompressible fluid of constant density $\rho$ in the absence of external body forces. A cylindrical jet of area $A$ and speed $U$ impinges fully on a stationary sphere of radius $R$ with $A<\pi R^{2}$. The flow is assumed to remain axisymmetric and be deflected into a conical sheet of vertex angle $\alpha>0$.

(i) Show that the speed of the fluid in the conical sheet is constant.

(ii) Use conservation of mass to show that the width $d(r)$ of the fluid sheet at a distance $r \gg R$ from point of impact is given by

$$d(r)=\frac{A}{2 \pi r \sin \alpha}$$

(iii) Use Euler's equation to derive the momentum integral equation

$$\iint_{S}\left(p n_{i}+\rho n_{j} u_{j} u_{i}\right) d S=0$$

for a closed surface $S$ with normal $\mathbf{n}$ where $u_{m}$ is the $m$ th component of the velocity field in cartesian coordinates and $p$ is the pressure.

(iv) Use the result from (iii) to calculate the net force on the sphere.