## THE BERNOULLI THEOREM

First of all let us examine the consequences of irrotationality of flow on the momentum (Navier-Stokes) equation. Recalling the equivalence

if the velocity field is solenoidal (i.e. divergence free) and irrotational, its Laplacian must vanish

The Navier-Stokes equation then becomes then the so-called Euler equation, in which the viscous term (as in inviscid flow) is absent. If we assume the flow to be steady, we can integrate the Euler equation in space to obtain the Bernoulli equation, valid in the whole flow domain:

Finally, if the plane in which the flow evolves is a horizontal plane (z=cost), we eventually obtain

This nonlinear equation allows for the solution of the pressure field once the velocity field is known.

Note that the hypothesis of irrotational flow does not necessarily involve the concept of inviscid fluid: we can have real fluids (in which viscous stresses are nonzero) in irrotational flow. However, having to satisfy to the same set of equations, this flow will have the same velocity and pressure fields of its inviscid counterpart.