Bernoulli's principle is a key concept in fluid dynamics that relates pressure, speed and height.
Bernoulli's
principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static
pressure or the fluid's potential energy.[1]: Ch.3 [2]: 156–164, § 3.5 The principle is named after the Swiss
mathematician and physicist Daniel Bernoulli, who published it in his book Hydrodynamica in 1738.[3]
Although Bernoulli deduced that pressure decreases when the flow speed increases, it was Leonhard
Euler in 1752 who derived Bernoulli's equation in its usual form.[4][5]
Bernoulli's principle can be derived from the principle of conservation of energy. This states that, in a
steady flow, the sum of all forms of energy in a fluid is the same at all points that are free of viscous
forces. This requires that the sum of kinetic energy, potential energy and internal energy remains
constant.[2]: § 3.5 Thus an increase in the speed of the fluid—implying an increase in its kinetic energy—
occurs with a simultaneous decrease in (the sum of) its potential energy (including the static pressure)
and internal energy. If the fluid is flowing out of a reservoir, the sum of all forms of energy is the same
because in a reservoir the energy per unit volume (the sum of pressure and gravitational potential ρ g h)
is the same everywhere.[6]: Example 3.5 and p.116
Bernoulli's principle can also be derived directly from Isaac Newton's second Law of Motion. If a small
volume of fluid is flowing horizontally from a region of high pressure to a region of low pressure, then
there is more pressure behind than in front. This gives a net force on the volume, accelerating it along
the streamline.[a][b][c]
Fluid particles are subject only to pressure and their own weight. If a fluid is flowing horizontally and
along a section of a streamline, where the speed increases it can only be because the fluid on that
section has moved from a region of higher pressure to a region of lower pressure; and if its speed
decreases, it can only be because it has moved from a region of lower pressure to a region of higher
pressure. Consequently, within a fluid flowing horizontally, the highest speed occurs where the pressure
is lowest, and the lowest speed occurs where the pressure is highest.[10]
Bernoulli's principle is only applicable for isentropic flows: when the effects of irreversible processes (like
turbulence) and non-adiabatic processes (e.g. thermal radiation) are small and can be neglected.
However, the principle can be applied to various types of flow within these bounds, resulting in various
forms of Bernoulli's equation. The simple form of Bernoulli's equation is valid for incompressible flows
(e.g. most liquid flows and gases moving at low Mach number). More advanced forms may be applied to
compressible flows at higher Mach numbers.
