Bernoulli's Theorem

Bernoulli's theorem is, at its heart, simply the principle of conservation of energy applied to a moving, ideal fluid. An "ideal" fluid, for this purpose, means one that is incompressible (its density does not change as it flows) and non-viscous (it has no internal friction at all, so no energy is lost to viscous drag as it flows) -- both simplifying assumptions, but ones that are excellent approximations for many real flows of ordinary liquids, provided the flow is also steady (the velocity at any fixed point in the fluid does not change with time) and irrotational (the fluid does not have any overall local spinning motion, i.e. it stays in genuine streamline flow, without turbulence).

Under these assumptions, for a fluid particle moving along a single streamline, Bernoulli's theorem states that the sum of three quantities -- each of them a form of energy per unit volume of the fluid -- stays constant all along that streamline:

where $P$ is the fluid's pressure at the point in question, $\rho$ is its (constant, since the fluid is incompressible) density, $v$ is its flow speed at that point, $g$ is the acceleration due to gravity, and $h$ is the height of that point above some chosen reference level.

Physical meaning of each term. The three terms correspond directly to the three familiar forms of mechanical energy, each expressed per unit volume of fluid rather than per unit mass:

• $P$ is the pressure energy per unit volume -- the work the surrounding fluid does (or has done on it) simply by virtue of the pressure pushing the fluid volume along.
• $\tfrac12\rho v^2$ is the kinetic energy per unit volume of the moving fluid, directly analogous to the familiar $\tfrac12 m v^2$ for a particle of mass $m$, but written per unit volume (using $\rho = m/V$) instead of per particle.
• $\rho g h$ is the gravitational potential energy per unit volume of the fluid at height $h$, directly analogous to $mgh$, again written per unit volume. …