Past exam of the mathematics course of the University of Cambridge / 2007 / ib / Paper 1 / 17D / Solution

For a steady inviscid incompressible fluid of constant density in uniform gravity $\mathbf{g}=-g{\mathbf{e}}_z$, the Euler equation for fluid motion and continuity equation areTaking the scalar product with $\mathbf{u}$ givesThus Bernoulli's quantity is constant along each streamline. For an irrotational connected flow it is constant throughout the flow; constancy across different streamlines is not assumed in general.

Since incompressibility makes $\nabla \cdot (\rho \mathbf{u}\otimes \mathbf{u})=\rho (\mathbf{u}\cdot \nabla )\mathbf{u}$, integrating over a fixed control volume and applying the divergence theorem yields the integral momentum balanceHere $\mathbf{n}$ is outward and the left side is the net outgoing momentum flux. Uniform atmospheric pressure has zero resultant on a closed control surface, so it may be subtracted from all pressures in the jet calculations.