Question

If the fluid velocity for a potential flow is given by V(x,y) = u(x,y) i + v(x,y)j with usual notations, then the slope of the potential line at (x,y) is

• A. $\frac{v}{u}$
• B. $-\frac{u}{v}$
• C. $-\frac{v^2}{u_2}$
• D. $\frac{u}{v}$

Answer 1

The correct option is B $-\frac{u}{v}$

$d\varphi =\frac{\partial \varphi }{\partial x}dx+\frac{\partial \varphi }{\partial y}dy$

$\therefore \frac{\partial \varphi }{\partial x}=-u$

$\frac{\partial \varphi }{\partial y}=-v$

$\therefore d\varphi =-udx-vdy$
$If\varphi =constant\therefore d\varphi =0$

$\therefore {\left(\frac{dy}{dx}\right)}_{\varphi }=-\frac{u}{v}$