Question

If the fluid velocity for a potential flow is given by V(x,y) = u(x,y)i + v(x,y)ĵ  with usual notations, then the slope of the potential line at (x,y) is
1. \(\frac{v}{u}\)
2. \(- \frac{u}{v}\)
3. \(\frac{{{v^2}}}{{{u^2}}}\)
4. \(\frac{u}{v}\)

Answer 1

Correct Answer - Option 2 : \(- \frac{u}{v}\)

Explanation:

From potential function definition,

\(u = - \frac{{\partial ϕ }}{{\partial x}}\;and\;v = - \frac{{\partial ϕ }}{{\partial y}}\)

when, ϕ = ϕ(x,y) 

\(dϕ = \frac{{\partial ϕ }}{{\partial x}}dx + \frac{{\partial ϕ }}{{\partial y}}dy\)

dϕ = - u dx - v dy

Since along potential line, ϕ is constant, so dϕ = 0

0 = - udx - vdy

slope = \(\frac{{dy}}{{dx}} = - \frac{u}{v}\)