Question

For a potential flow, the fluid velocity is given by $\vec V(x,y)=u\,\hat{\imath}+v\,\hat{\jmath}$. The slope of the {potential line} at $(x,y)$ is

• A. $\dfrac{u}{v}$
• B. $\dfrac{v}{u}$
• C. $\dfrac{u}{-v}$
• D. $-\dfrac{v}{u}$

Hint:

Streamlines satisfy $dy/dx=v/u$; equipotentials satisfy $dy/dx=-u/v$. In potential flow, streamlines and equipotentials are orthogonal since $(v/u) . (-u/v)=-1$.

Answer 1

The Correct Option is C

Step 1: Definition of potential line.
In potential flow, $u=\dfrac{\partial \phi}{\partial x}$, $v=\dfrac{\partial \phi}{\partial y}$. An {equipotential} (potential line) satisfies $d\phi=0$: \[ d\phi=\frac{\partial \phi}{\partial x}dx+\frac{\partial \phi}{\partial y}dy =u\,dx+v\,dy=0. \]
Step 2: Slope of the potential line.
\[ \frac{dy}{dx}=-\frac{u}{v}=\frac{u}{-v}. \] Thus the slope equals $\boxed{\dfrac{u}{-v}}$. Final Answer:\fbox{(C) $\dfrac{u}{-v}$}