Bernoulli’s equation cannot be applied between

The correct option is: Any two arbitrary point in the flow field for steady, incompressible and rotational flow

To understand why Bernoulli's equation cannot be applied between any two arbitrary points in a rotational flow field, let us break down the assumptions and derivation of Bernoulli's equation step-by-step.

1. Understanding Bernoulli's Equation:
Bernoulli's equation is a statement of the conservation of energy for a flowing fluid. Along a streamline, it is expressed as:
$\frac{p}{\rho }+\frac{v^2}{2}+gz=\text{Constant}$

2. Application Along a Streamline:
Euler's equation of motion along a streamline (s-direction) is:
$\frac{\partial p}{\partial s}+\rho v\frac{\partial v}{\partial s}+\rho g\frac{\partial z}{\partial s}=0$
Integrating this along a single streamline for steady and incompressible flow yields the Bernoulli constant. Therefore, Bernoulli's equation is always applicable between any two points along the same streamline, regardless of whether the overall flow is rotational or irrotational (since fluid particles on that streamline follow a continuous energy path).

3. Application Between Arbitrary Points (Across Streamlines):
For Bernoulli's equation to have the exact same constant value across different streamlines (allowing us to apply it between any two arbitrary points in the entire flow field), the flow must be irrotational ($\nabla \times \overrightarrow{V}=0$). In an irrotational flow, the velocity potential exists, and the constant of integration is identical throughout the entire flow field.

4. Why it fails for Rotational Flow:
If the flow is rotational, different streamlines will have different Bernoulli constants due to shear, vorticity, or varying energy levels between distinct fluid layers. Therefore, while we can still apply Bernoulli's equation between two points on the same streamline, we cannot apply it between any two arbitrary points belonging to different streamlines in a rotational flow field.