For inviscid fluids, Bernuilli’s theorem states
that an increase in flow velocity is paired
with a decrease in pressure and/or potential energy.
For a qualitative argument, look no further than
one of the Euler equations,
with a material derivative:
Assuming that is constant in ,
it becomes clear that a higher requires a lower .
For an incompressible fluid
with a time-independent velocity field (i.e. steady flow),
Bernoulli’s theorem formally states that the
Bernoulli head is constant along a streamline:
Where is the gravitational potential, such that .
To prove this theorem, we take the material derivative of :
In the first term we insert the Euler equation,
and in the other two we expand the derivatives:
Using the fact that ,
we are left with the following equation:
Assuming that the flow is steady, both derivatives vanish,
leading us to the conclusion that is conserved along the streamline.
In fact, there exists Bernoulli’s stronger theorem,
which states that is constant everywhere in regions with
zero vorticity .
For a proof, see the derivation of ’s equation of motion.
- B. Lautrup,
Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition,