Bernoulli's Equation In Irrotational Flow
In the previous
lecture (lecture 13) we have obtained Bernoulli’s equation

This equation was obtained by integrating the
Euler’s equation (the equation of motion) with respect
to a displacement 'ds' along a streamline. Thus,
the value of C in the above equation is constant only along a streamline
and should essentially vary from streamline to streamline.

The equation can be used to define
relation between flow variables at point B on the streamline
and at point A, along the same streamline. So, in order to
apply this equation, one should have knowledge of velocity
field beforehand. This is one of the limitations of
application of Bernoulli's equation.
Irrotationality of flow field
Under some special condition, the constant
C becomes invariant from streamline to streamline and the
Bernoulli’s equation is applicable with same value of
C to the entire flow field.
The typical
condition is the irrotationality of flow field.
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Proof:
Let us consider a steady two dimensional
flow of an ideal fluid in a rectangular Cartesian coordinate
system. The velocity field is given by
hence the condition of irrotationality is



(14.1) 
The steady state Euler's equation can be written as

(14.2a) 

(14.2b) 
We consider the yaxis
to be vertical and directed positive upward. From the condition
of irrotationality given by the Eq. (14.1), we substitute
in place of
in the Eq. 14.2a and
in
place of
in the Eq. 14.2b. This results in

(14.3a) 

(14.3b) 
Now multiplying Eq.(14.3a)
by 'dx' and Eq.(14.3b) by 'dy' and then adding these two equations
we have

(14.4) 
The Eq. (14.4) can be physically
interpreted as the equation of conservation of energy for
an arbitrary displacement
.
Since, u, v and p are functions of x and y, we can write
With the help of Eqs (14.5a),
(14.5b), and (14.5c), the Eq. (14.4) can be written as
The integration of Eq. 14.6
results in

(14.7a) 
For an incompressible flow,

(14.7b) 
The constant C in Eqs
(14.7a) and (14.7b) has the same value in the entire flow
field, since no restriction was made in the choice of dr which
was considered as an arbitrary displacement in evaluating
the work.
Note:
In deriving Eq. (13.8) the displacement ds was considered
along a streamline. Therefore, the total mechanical energy
remains constant everywhere in an inviscid and irrotational
flow, while it is constant only along a streamline for an
inviscid but rotational flow.
The equation of motion for
the ﬂow of an inviscid fluid can be written in a vector
form as
where is the body force vector per unit mass
