Velocity potential and Stream function are two important concepts in fluid mechanics. They are mathematical tools that help describe and analyze fluid flow. In this question, we are given the stream function (x^2 - y^2) and we need to find the corresponding velocity potential function.

1. Understanding the Stream Function:
- The stream function is a scalar function that gives the value of the streamlines at any point in the flow field.
- It is defined as the streamlines' equation, where the streamlines are the curves that are tangent to the velocity vector at every point.
- In two-dimensional flow, the stream function is given by the relation:
ψ = ∂ψ/∂y = -∂ψ/∂x

2. Finding the Velocity Potential Function:
- The velocity potential is another scalar function that gives the value of the velocity vector at any point in the flow field.
- It is defined as the negative of the potential function, i.e.,
φ = -ψ

3. Applying the given Stream Function:
- Given the stream function ψ = x^2 - y^2, we need to find the corresponding velocity potential function φ.
- To find φ, we take the negative of the stream function:
φ = -(x^2 - y^2)
= -x^2 + y^2

4. Comparing the Result:
- Comparing the obtained velocity potential function φ = -x^2 + y^2 with the given options, we find that it matches with option B, which is 2xy constant.

5. Explanation of Option B:
- The velocity potential function φ = -x^2 + y^2 represents a flow field where the velocity vectors are perpendicular to the streamlines.
- To find the velocity components, we take the partial derivatives of the velocity potential function with respect to x and y:
Vx = -∂φ/∂x = 2x
Vy = -∂φ/∂y = -2y

- These velocity components represent a flow field where the x-component of velocity is proportional to x and the y-component is proportional to -y.
- The constant factor of proportionality is 2, which matches with option B, 2xy constant.

Hence, the correct answer is option B) 2xy constant, as it matches with the derived velocity potential function -x^2 + y^2.