Following is the velocity potential function for the two dimensional irrotational flow in the cylinder coordinates: - $ϕ=\dfrac{m \cosθ}{r}$ - Determine the conjugate function (Stream function) -

Consider the velocity potential function for the two dimensional irrotational flow,

$$ϕ=\dfrac{m \cosθ}{r}$$

We know that by the definition of velocity potential function in cylindrical coordinates,

$$\dfrac{∂ϕ}{∂r}=u_r$$

$∴u_r=\dfrac{∂ϕ}{∂r} \dfrac{m \cosθ}{r} \\ ∴u_r=-\dfrac{m \cosθ}{r^2}$

And

$\dfrac 1r \dfrac{∂ϕ}{∂θ}=u_θ \\ ∴u_θ=\dfrac 1r \dfrac{∂}{∂θ}\bigg(\dfrac{m \cosθ}{r}\bigg) \\ ∴u_θ=-\dfrac{m \cosθ}{r^2}$

We also know that by …