Question

What will be the stream function of flow of an incompressible fluid is defined by u = 2, v = 8x.
1. Ψ = 2x² + 2xy + c.
2. Ψ = 2x + 8y² + c.
3. Ψ = 2y + 8x + c
4. Ψ = - 4x² + 2y + c.

Answer 1

Correct Answer - Option 4 : Ψ = - 4x² + 2y + c.

Concept:

The stream function in a two-dimensional flow automatically satisfies the continuity equation.

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

\(u = \frac{{∂ ψ }}{{∂ y}}\) & \(v = - \frac{{∂ ψ }}{{∂ x}}\)

Stream function defines only for the two-dimensional flow which is steady and incompressible..

Properties of stream function:

1. If ψ exists, it follows continuity equation and the flow may be rotational or irrotational.
2. If ψ satisfies the Laplace equation, then the flow is irrotational.

Calculation:

Given u = 2, v = 8x;

From stream function equations,

∂ψ/∂y = 2 ⇒ ψ = 2y + f(x);

partially differentiating ψ with respect t0 x,

∂ψ/∂x = f'(x);

From the given equations,

∂ψ/∂x = - 8x ⇒ f'(x) = - 8x

⇒ f(x) = - 4x² + c;

Now the complete stream function will be 

ψ = 2y - 4x² + c