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Bernoulli Equation
The procedure given in the example for Euler’s equation is simple but lengthy. It will be
nice if we can develop a formula, which will yield the pressure field but will not have to
involve the usual integration process. For this, we will integrate the Euler’s equation by
making some simplifications. The resulting equation is called the inviscid Bernoulli
equation (or simply the Bernoulli equation). It is very popular because of its simplicity
and ease of use. However, there are restrictions to use this equation as will be apparent
during the derivation.
Given below is the inviscid Bernoulli equation. As we have realized during the
derivation, it is applicable for incompressible, steady, frictionless flows along a
streamline.
2
p Vs

 gz  const
 2
The inviscid Bernoulli equation has many uses. Construction of Pitot tubes, or pitotstatic probes utilize this equation. The concept of stagnation pressure can be defined
right from this equation.
Imagine a flow, which fits the requirements of the inviscid Bernoulli equation. Let us
choose two points on the same streamline, (1) and (2).
z
(1)
(2)
Streamline
The points are close by such that z2  z1  0.
2
Thus,
p1 V1
p
V

 2 2

2

2
2
Now, if we make the point (1) general point by dropping the subscript. On the other hand,
we make the point (2) a special point where the fluid velocity is zero, (i.e., V2 = 0). Then
we call the second point a stagnation point and define the pressure then by a special
symbol, po.
p V 2 po

Then, 

2

or,
1
p 0  p  V 2
2
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The boxed expression gives us the measure of stagnation pressure. It is the sum total of
1
the static pressure, p, and another term, V 2 , which we call the dynamic pressure.
2
We know that static pressure, p, is the pressure measured in a static fluid. For a flowing
fluid, this pressure may be measured in the direction perpendicular to the flow direction.
On the other hand, both dynamic and stagnation pressures must be measured along the
flow direction. Dynamic pressure is obtained by moving with the fluid velocity, whereas
the stagnation pressure is obtained if you decelerate the fluid particles to a zero
velocity isentropically. The construction of a pitot tube illustrates the use of stagnation
pressure. By bending the tube 90 at the neck, fluid particles are brought to a halt in the
flow direction.
Static Holes
Neck
Flow
Direction
Static Channel
Pitot Channel
Pitot-Static Tube
On the other hand, a Pitot-static tube uses two channels, one facing the flow and the other
perpendicular to the flow.
An interesting theorem by Crocco gives:
 
1
p o  V  

This theorem provides a measure of the changes in stagnation pressure in an
incompressible flow. It can be related to the vector (cross) product of velocity and

vorticity vectors. In an irrotational flow,   0 . Thus, p o  0 in an irrotational flow.
Since the gradient operator involves spatial derivatives, we may write:
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p
p
p
 0,  0,  0
x
y
z
p
p
p
 dp  dx  dy  dz  0 Or, p o  const
x
y
z
Therefore for irrotational flows, the stagnation pressure, po, holds constant not only
between two points on a streamline but between any two points in the fluid flow.
Thus, the use of the inviscid Bernoulli equation becomes less restrictive. For rotational
flows, however, Bernoulli equation can only be applied between any two points on a
streamline.
Examples
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