Question

In a flow field the stream lines and equipotential lines

• A. Are parallel
• B. Cut at any angle
• C. Are orthogonal every where in the field
• D. Cut orthogonal expect at the stagnation points

Answer 1

The correct option is D Cut orthogonal expect at the stagnation points

For equi-potential line:
$d\varphi =0$
$-udx-vdy=0$
$\frac{dy}{dx}=-\frac{u}{v}$
Hence, slope of equipotential line,
${\left(\frac{dy}{dx}\right)}_{\varphi =constant}=\frac{-u}{v}.....\left(i\right)$
For equi-stream function line:
$d\Psi =0$
vdx-udy=0 [stream line]
$\frac{dx}{u}=\frac{dy}{v}$
$and\frac{dy}{dx}=\frac{v}{u}$
Hence slope of equi-stream line,
${\left(\frac{dy}{dx}\right)}_{\Psi =constant}=\frac{v}{u}....\left(ii\right)$
From equation (i)and (ii)
${\left(\frac{dy}{dx}\right)}_{\varphi =constant}\times {\left(\frac{dy}{dx}\right)}_{\Psi =constant}$
$=\frac{-u}{v}\times \frac{v}{u}=-1$
Hence, stream lines and equipotential lines are orthogonal to each other expect at the stagnation points.