Bernoulli's Equation

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Also called the Bernoulli equation, or the Bernoulli relation.


Equation
p0 = p + (1/2)ρv2 + ρgZ Bernoulli equation for:
• steady flow
• nearly inviscid fluid
• nearly incompressible fluid
• negligible heat transfer and shaft work
(Eq?)    
p1
ρ
 +
1
2
v12 + gz1 =
p2
ρ
 +
1
2
v22 + gz2 = const
 Bernoulli equation for:
• steady flow
• incompressible flow
• frictionless flow
• flow along a single streamline
• no shaft work between states 1 and 2
• no heat transfer between states 1 and 2


Nomenclature
p0stagnation pressure
ppressure in moving fluid
vvelocity
ρdensity
Zaltitude
ggravitational acceleration


Details

The following assumptions must be made about a given fluid in order for the Bernoulli equation to be valid:
• steady flow
• incompressible flow
• irrotational flow
• negligible frictional effects

The Bernoulli equation is an approximate relation between pressure, velocity, and elevation, and is valid in regions of steady, incompressible flow where net frictional forces are negligible.
The Bernoulli equation is a useful theoretical equation for computing the relation between pressure, velocity, and altitude in a steady flow of a nearly inviscid, nearly incompressible fluid with negligible heat transfer and shaft work and is the following:

p0 = p + (1/2)ρv2 + ρgZ

The Bernoulli equation is a relation between pressure, velocity, and elevation in a frictionless flow, and is closely related to the steady flow energy equation.

The Bernoulli equation has restrictions — all fluids are viscous and thus all flows have friction to some extent. To use the Bernoulli equation correctly, it must be confined to regions of the flow that are nearly frictionless.


Examples

So what kinds of things can the Bernoulli equation be used for?

The following figures show some examples of that in which the Bernoulli equation can be used:

burst water pipe
see the following problem		water from a water tank filling a cup

As long as the following assumptions are satisfied for any example in mind, the Bernoulli equation can be used:
• steady flow
• incompressible flow
• irrotational flow
• negligible frictional effects