Fluid Dynamics | Part IB, 2004

Use Euler's equation to derive the momentum integral

S(pni+ρnjujui)dS=0\int_{S}\left(p n_{i}+\rho n_{j} u_{j} u_{i}\right) d S=0

for the steady flow u=(u1,u2,u3)\mathbf{u}=\left(u_{1}, u_{2}, u_{3}\right) and pressure pp of an inviscid,incompressible fluid of density ρ\rho, where SS is a closed surface with normal n\mathbf{n}.

A cylindrical jet of water of area AA and speed uu impinges axisymmetrically on a stationary sphere of radius aa and is deflected into a conical sheet of vertex angle α\alpha as shown. Gravity is being ignored.

Use a suitable form of Bernoulli's equation to determine the speed of the water in the conical sheet, being careful to state how the equation is being applied.

Use conservation of mass to show that the width d(r)d(r) of the sheet far from the point of impact is given by

d=A2πrsinα,d=\frac{A}{2 \pi r \sin \alpha},

where rr is the distance along the sheet measured from the vertex of the cone.

Finally, use the momentum integral to determine the net force on the sphere in terms of ρ,u,A\rho, u, A and α\alpha.

Typos? Please submit corrections to this page on GitHub.