Question

Which of these conditions is not met at a point for irrotational flow?

(a) \(\frac {∂w}{∂y} = \frac {∂v}{∂z}\)

(b) \(\frac {∂w}{∂x} = \frac {∂u}{∂z}\)

(c) \(\frac {∂v}{∂x} = \frac {∂u}{∂z}\)

(d) \(\frac {∂v}{∂x} = \frac {∂u}{∂y}\)

This question was posed to me in an internship interview.

My question is taken from Irrotational Flow in division Velocity Potential Equation of Aerodynamics

Answer 1

Correct choice is (c) \(\frac {∂v}{∂x} = \frac {∂u}{∂z}\)

For explanation: The Cartesian form of irrotational flow is given by:

∇ × V = \(\begin{vmatrix}

   i & j & k  \\

   \frac {\partial }{\partial x} & \frac {\partial }{\partial y} & \frac {\partial }{\partial z}  \\

   u & v & w  \\

   \end{vmatrix} \)

On expanding this we get,

i(\(\frac {∂w}{∂y} – \frac {∂v}{∂z}\)) – j(\(\frac {∂w}{∂x} – \frac {∂u}{∂z}\)) + k(\(\frac {∂v}{∂x} – \frac {∂u}{∂y}\)) = 0

For irrotational flow since vorticity = 0, ∇ × V = 0

\(\frac {∂w}{∂y} = \frac {∂v}{∂z}\) and \(\frac {∂w}{∂x} = \frac {∂u}{∂z}\) and \(\frac {∂v}{∂x} = \frac {∂u}{∂y}\)