1.

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»

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 \(\frac {∂v}{∂x} = \frac {∂u}{∂y}\)


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