Verify whether the following functions are valid potential functions.

(i) \(ϕ=A(X^2-Y^2)\)

(ii) ϕ = Acos x

Concept:

Properties of Potential functions:

1. If velocity potential (ϕ) exists, the flow should be irrotational.

2. If velocity potential (ϕ) satisfies the Laplace equation, it represents the possible steady incompressible irrotational flow.

Calculation:

Given:

(i) \(ϕ=A(X^2-Y^2)\), (ii) ϕ = Acos x.

(I) \(ϕ=A(X^2-Y^2)\)

\(\frac{\partial^2 \phi}{\partial x^2}=\frac{\partial^2 }{\partial x^2}[A(X^2-Y^2)]\)

\(\frac{\partial^2 \phi}{\partial x^2}=2A\)

\(\frac{\partial^2 \phi}{\partial y^2}=\frac{\partial^2 }{\partial y^2}[A(X^2-Y^2)]\)

\(\frac{\partial^2 \phi}{\partial y^2}=-2A\)

\(\frac{\partial^2 \phi}{\partial x^2}+\frac{\partial^2 \phi}{\partial y^2}=2A-2A=0\)

(ii) ϕ = Acos x

\(\frac{\partial^2 \phi}{\partial x^2}=\frac{\partial^2 }{\partial x^2}(A\cos x) \)

\(\frac{\partial^2 \phi}{\partial x^2}=-A \cos x\)

\(\frac{\partial^2 \phi}{\partial y^2}=\frac{\partial^2 }{\partial y^2}(A\cos x) \)

\(\frac{\partial^2 \phi}{\partial y^2}=0\)

\(\frac{\partial^2 \phi}{\partial x^2}+\frac{\partial^2 \phi}{\partial y^2}=-A\cos x-0=-A\cos x\)

Thus (i) is a valid potential function as it satisfies the Laplace equation, whereas (ii) is not a valid potential function as it does not satisfy Laplace equation.