Meditation upon the term
reveals that it is the z component of a vector product between
and the velocity vector:
.
Further meditation reveals that
We see that we may hope to integrate Euler's equation if we require that
.
In words this is expressed as, "The curl of the velocity field is zero." We postpone the very interesting question of the physical meaning of the curl of the velocity field, and in particular how it changes a field to require that its curl be zero.
The argument can be repeated for the y and z components of Euler's equation, and we find that, for zero-curl velocity fields:
With zero curl, we may rewrite Euler's equation:
We cleverly observe that (as long as the +y direction is up) we can rewrite
as
.
Now we make the assumption that density is constant, and collect the entire equation as a gradient:
\[ \vec{\nabla} \left( \frac{1}{2} v^2 + \frac{P}{\rho} + gy \right) = 0 \]
If this equation is to be true in general, it must be true that
is independent of spatial
coordinate. That is, it must be constant so that
and we have Bernoulli's equation once again.
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