Since we are now concentrating on steady flows, . Taking the dot product of with Eq. (2.22), and remembering that , we have
If we now consider displacements along a streamline, since is tangent to the streamline Eq. (2.23) becomes
This is the famous Bernoulli's equation for steady flow in a nonviscous fluid. If, in addition, the fluid is irrotational so that , then we have a stronger form of Bernoulli's equation:
We've derived Bernoulli's equation in a somewhat mathematical fashion, but keep in mind that it is simply a statement of conservation of energy in the fluid. The term is the kinetic energy density of the fluid, and the pressure p can be thought of as a type of potential energy (per unit volume).