For a two-dimensional irrotational flow, the velocity potential

For a two-dimensional irrotational flow, the velocity potential is defined as f = log_e (x² + y²). Which of the following is a possible stream function, Ψ , for this flow?

1. 1/ 2 tan^–1 (y – x)
2. tan^–1 (y / x)
3. 2 tan^–1 (y / x)
4. 2 tan^–1 (x / y)

We know,

∂Ψ	=	∂φ
∂y	=	∂x

=	∂	[ log_e (x² + y²) ]
	∂x

1		d	(x² + y²)
x² + y²		dx

=	1	× 2x
	x² + y²

Integrating both sides,

∫∂Ψ = ∫	2x	∂y
	x² + y²

⇒Ψ = 2x.	1	tan^-1		y		+ c
	x			x

⇒Ψ = 2tan^-1		y		+ c
		x

We know,

∂Ψ	=	∂φ
∂y	=	∂x

=	∂	[ log_e (x² + y²) ]
	∂x

1		d	(x² + y²)
x² + y²		dx

=	1	× 2x
	x² + y²

Integrating both sides,

∫∂Ψ = ∫	2x	∂y
	x² + y²

⇒Ψ = 2x.	1	tan^-1		y		+ c
	x			x

⇒Ψ = 2tan^-1		y		+ c
		x