Question

For an irrotational flow, the equation \(\frac{{{\partial ^2}\phi }}{{\partial {x^2}}} + \frac{{{\partial ^2}\phi }}{{\partial {y^2}}}\) is known as

1. Bernoulli’s equation
2. Cauchy Riemann’s equation
3. Eular’s equation
4. Laplace equation

Answer 1

Correct Answer - Option 4 : Laplace equation

Explanation:

Velocity Potential Function:

It is defined as the scalar function of space and time, such that its negative derivative with respect to any direction gives the velocity in that direction. If velocity potential exists, there will be a flow.

It is denoted by ϕ and defined for two-dimensional as well as three-dimensional flow

\(u = - \frac{{\partial \phi }}{{\partial x}};v = - \frac{{\partial \phi }}{{\partial y}};w = - \frac{{\partial \phi }}{{\partial z}}\)

Laplace equation : \(\triangledown^2\phi = 0\)

• \(\triangledown^2\phi = \frac{d^2\phi}{dx^2} + \frac{d^2\phi}{dy^2}\) = 0
• If the velocity potential function satisfies the Laplace equation it is a case of steady incompressible irrotational flow

 Irrotational flow:

• If velocity potential function exists, the flow should be irrotational whether compressible or incompressible
• \([\triangledown \times \vec{V}] = 0\)

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Stream function: It is the scalar function of space and time. The partial derivative of stream function with respect to any direction gives the velocity component perpendicular to that direction. Hence it remains constant for a streamline

\(u = - \frac{{\partial \psi }}{{\partial y}}\) ,  \(v = \frac{{\partial \psi }}{{\partial x}}\)

Cauchy Riemann’s: it is the relation between Stream and potential function.

• \(\frac{{\partial \emptyset }}{{\partial {\rm{x}}}} = {\rm{\;}}\frac{{\partial {\rm{\psi }}}}{{\partial {\rm{y}}}}\)
• \(\frac{{\partial \emptyset }}{{\partial {\rm{y}}}} = - \frac{{\partial {\rm{\psi }}}}{{\partial {\rm{x}}}}\)