The relationship between pressure and velocity in fluids is described quantitatively by Bernoulli’s equation, named after its discoverer, the Swiss scientist Daniel Bernoulli (1700–1782). Bernoulli’s equation states that for an incompressible, frictionless fluid, the following sum is constant:

P+12ρv2+ρgh=constant,P+12ρv2+ρgh=constant, size 12{P+ { {1} over {2} } ρv rSup { size 8{2} } +ρ ital "gh"="constant,"} {}

(2)where PP size 12{P} {} is the absolute pressure, ρρ size 12{ρ} {} is the fluid density, vv size 12{v} {} is the velocity of the fluid, hh size 12{h} {} is the height above some reference point, and gg size 12{g} {} is the acceleration due to gravity. If we follow a small volume of fluid along its path, various quantities in the sum may change, but the total remains constant. Let the subscripts 1 and 2 refer to any two points along the path that the bit of fluid follows; Bernoulli’s equation becomes

P
1
+
1
2
ρv
1
2
+
ρ
gh
1
=
P
2
+
1
2
ρv
2
2
+
ρ
gh
2
.
P
1
+
1
2
ρv
1
2
+
ρ
gh
1
=
P
2
+
1
2
ρv
2
2
+
ρ
gh
2
.
size 12{P rSub { size 8{1} } + { {1} over {2} } ρv rSub { size 8{1} } "" lSup { size 8{2} } +ρ ital "gh" rSub { size 8{1} } =P rSub { size 8{2} } + { {1} over {2} } ρv rSub { size 8{2} } "" lSup { size 8{2} } +ρ ital "gh" rSub { size 8{2} } "." } {}

(3)Bernoulli’s equation is a form of the conservation of energy principle. Note that the second and third terms are the kinetic and potential energy with mm size 12{m} {} replaced by ρρ size 12{ρ} {}. In fact, each term in the equation has units of energy per unit volume. We can prove this for the second term by substituting ρ=m/Vρ=m/V size 12{ρ=m/V} {} into it and gathering terms:

12ρv2=12mv2V=KEV.12ρv2=12mv2V=KEV. size 12{ { {1} over {2} } ρv rSup { size 8{2} } = { { { {1} over {2} } ital "mv" rSup { size 8{2} } } over {V} } = { {"KE"} over {V} } "."} {}

(4)So 12ρv212ρv2 size 12{ { { size 8{1} } over { size 8{2} } } ρv rSup { size 8{2} } } {} is the kinetic energy per unit volume. Making the same substitution into the third term in the equation, we find

ρgh=mghV=PEgV,ρgh=mghV=PEgV, size 12{ρ ital "gh"= { { ital "mgh"} over {V} } = { {"PE" rSub { size 8{"g"} } } over {V} } "."} {}

(5)so ρ ghρ gh size 12{ρ ital "gh"} {} is the gravitational potential energy per unit volume. Note that pressure PP size 12{P} {} has units of energy per unit volume, too. Since *P=F/AP=F/A size 12{P=F/A} {}*, its units are N/m2N/m2 size 12{"N/m" rSup { size 8{2} } } {}. If we multiply these by m/m, we obtain N⋅m/m3=J/m3N⋅m/m3=J/m3 size 12{N cdot "m/m" rSup { size 8{3} } ="J/m" rSup { size 8{3} } } {}, or energy per unit volume. Bernoulli’s equation is, in fact, just a convenient statement of conservation of energy for an incompressible fluid in the absence of friction.

Conservation of energy applied to fluid flow produces Bernoulli’s equation. The net work done by the fluid’s pressure results in changes in the fluid’s KEKE size 12{"KE"} {} and PEgPEg size 12{"PE" rSub { size 8{g} } } {} per unit volume. If other forms of energy are involved in fluid flow, Bernoulli’s equation can be modified to take these forms into account. Such forms of energy include thermal energy dissipated because of fluid viscosity.

The general form of Bernoulli’s equation has three terms in it, and it is broadly applicable. To understand it better, we will look at a number of specific situations that simplify and illustrate its use and meaning.