 Solution

For a material volume, conservation of mass and the Reynolds transport theorem give ${\rho }_t+\nabla \cdot (\rho u)=0$. Constant material density, or equivalently incompressibility with the stated constant initial density, reduces this to $\nabla \cdot u=0$.

In an fluid undergoing inviscid flow the stress is the isotropic pressure stress $-pI$. Momentum conservation for a material volume then givesTransport and the divergence theorem localize this toFor a steady flow, $u_t=0$. For an irrotational flow the vorticity is zero, so the supplied identity gives $(u\cdot \nabla )u=\nabla (|u{|}^2/2)$. Uniform downward gravity is $g=-\nabla (gz)$, and constant density permits pulling $1/\rho$ inside the pressure gradient. ThereforeThe Bernoulli invariant is constant throughout each connected fluid region. Without irrotationality the corresponding steady statement generally gives a constant only along each streamline.