Bernoulli's Equation in Irrotational Flow
By Bernoulli’s equation we have,
• 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.
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
The steady state Euler's equation can be written as
We consider the y-axis to be vertical and directed positive upward. From the condition of irrotationality given by the Eq. (1), we substitute 
in place of 
in the Eq. 2a and 
in place of 
in the Eq. 2b. This results in
Now multiplying Eq.(3a) by 'dx' and Eq.(3b) by 'dy' and then adding these two equations we have
The Eq. (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 (5a), (5b), and (5c), the Eq. (4) can be written as
The integration of Eq. 6 results in
For an incompressible flow,
The constant C in Eqs (7a) and (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. (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
Plane Circular Vortex Flows
• Plane circular vortex flows are defined as flows where streamlines are concentric circles. Therefore, with respect to a polar coordinate system with the centre of the circles as the origin or pole, the velocity field can be described as
Vθ ≠ 0 Vr = 0
where Vθ and Vr are the tangential and radial component of velocity respectively.
• The equation of continuity for a two dimensional incompressible flow in a polar coordinate system is
which for a plane circular vortex flow gives 
i.e. Vθ is not a function of θ. Hence, Vθ is a function of r only.
• We can write for the variation of total mechanical energy with radius as
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