The Bernoulli principle, first formulated by Daniel Bernoulli in 1738, is
one means of expressing Newton’s second law of physics, concerning
conservation of energy. Roughly stated, this principle demonstrates that
the sum of pressure and velocity through or over a device represents is
equal, neglecting the effects of losses due to friction and/or increases by
adding energy with external devices such as pumps. The basic concept of
Bernoulli’s principle can be observed in routine daily activities: A ship’s
sail can push a vessel into the wind; an airplane’s wing produces lift; a
pitcher induces spin on a baseball and generates high- and low-pressure
zones forcing the ball into a curved pattern. Bernoulli’s principle can also
be demonstrated in the flow of fluids through pipe.

Heavier-than-air flight was not achieved until a wing was developed
that engaged the Bernoulli principle. Airplane wings generate lift by
creating high- and low-pressure zones (Figure 1). Ignoring losses for
friction, the total energy at any point along the wing is equal to the sum
of the pressure (P) and the velocity (V). Pressure and velocity are equal
at points A and C, that is, P1+V1=P2+V2. Because aircraft wings are
curved on top, air travels farther and thus moves faster above the wing
than underneath it. Therefore, velocity at point B is greater above
the wing than below it. The law of conservation of energy indicates
that pressure is affected inversely: If V increases, then P decreases. This
creates a differential pressure, or ΔP: higher pressure beneath the wing
adds lift. As speed increases, so does lift.

Airplane wing cross section.
figure 1. Airplane wing cross section.

Relationship of Pressure, Velocity, and Head

Likewise, in piping systems, velocity and pressure are measured as fluid
flows internally in the pipe, rather than externally, as over a wing. Again
assuming that there is no friction loss and no energy added to the system
(e.g., pumps), the sum of pressure h, or head of fluid, and velocity will be
a constant at any point in the fluid. Consider an example with an ideal
fluid and frictionless pipe. Figure 2. shows the relationship between
pressure and velocity under steady flow conditions. Remember, ignore
losses caused by friction. Attaching manometers to the pipe will indicate
h, or head levels, at three points in the pipe. Pressure gauges will also
indicate the level of head. Notice that at points A and C, the levels of h
are equal, while at point B it is lower. This is because pipe size is reduced;
therefore, velocity (v) is higher. Once the pipe expands again, v will
decrease, and h will again increase. In actual practice, losses or energy
increases or decreases are encountered and must be included in the
Bernoulli equation.

Relationship of pressure to velocity
figure 2. Relationship of pressure to velocity

This quantity is constant for all points within the pipe, and this is
Bernoulli’s theorem. Although it is not a new principle, it is an expression
of the law of conservation of mechanical energy in a convenient
form for fluid mechanics.