Velocity Potential Function

The velocity potential function is a scalar field that describes the velocity of a fluid flow. It is defined as the scalar function whose partial derivatives with respect to the Cartesian coordinates give the components of the velocity vector.

The velocity potential function, denoted by Φ, can be used to determine the velocity vector field of a fluid flow using the following relationship:

V = ∇Φ

where V is the velocity vector, and ∇ is the del operator.

Understanding the Options

Let's analyze each of the given options and determine if they are valid velocity potential functions:

a) Φ = Axy
- This function is a valid velocity potential function because it is a scalar function of the Cartesian coordinates x and y. It can be differentiated with respect to x and y to obtain the x and y components of the velocity vector.

b) Φ = A(x^2 - y^2)
- This function is not a valid velocity potential function because it depends on both x and y, but the velocity potential function should only depend on the Cartesian coordinates.

c) Φ = mln(x)
- This function is not a valid velocity potential function because it depends on only one Cartesian coordinate, x. The velocity potential function should depend on both x and y.

d) Φ = cos(x)
- This function is not a valid velocity potential function because it is a trigonometric function of x. The velocity potential function should be a scalar function of the Cartesian coordinates.

Conclusion

Among the given options, only option a) Φ = Axy is a valid velocity potential function. It satisfies the requirement of being a scalar function of the Cartesian coordinates x and y, which can be differentiated to obtain the velocity vector components. Options b), c), and d) do not satisfy these requirements and are therefore not valid velocity potential functions.