Question

For a two-dimensional irrotational flow, the velocity potential is defined as $\varphi =lo{g}_e(x^2+y^2)$. Which of the following is a possible stream function, $\psi$, for this flow?

• A. $\frac{1}{2}ta{n}^{-1}\left(y/x\right)$
• B. $ta{n}^{-1}\left(y/x\right)$
• C. $2ta{n}^{-1}\left(y/x\right)$
• D. $2ta{n}^{-1}\left(x/y\right)$

Answer 1

The correct option is C $2ta{n}^{-1}\left(y/x\right)$

We know that

$\frac{\partial \psi }{\partial y}=\frac{\partial \varphi }{\partial x}$

$\Rightarrow \frac{\partial \psi }{\partial y}=\frac{\partial }{\partial x}lo{g}_e(x^2+y^2)$

$\Rightarrow \frac{\partial \psi }{\partial y}=\frac{2x}{x^2+y^2}$

Integrating both sides, we get

$\psi =\int \frac{2x}{x^2+y^2}dy+C$

$\Rightarrow \psi =2x\times \frac{1}{x}ta{n}^{-1}\left(\frac{y}{x}\right)+C$

$\Rightarrow \psi =2ta{n}^{-1}\left(\frac{y}{x}\right)+C$

However, it may be noted that 'C' is a function of x.