Question

What will be the circulation around rectangle defined by x = 0, y = 0, x = 1, y = 1 for a velocity field u = x and v = x + y ?
1. Infinity
2. 0 
3. 1 
4. 4

Answer 1

Correct Answer - Option 3 : 1 

Concept:

Circulation = Vorticity × Area

where, 

Vorticity is defined as the value twice of the rotation.

ζ = 2ω

Rotation (ω) =  \({\omega _z} = \frac{1}{2}\left( {\frac{{∂ v}}{{∂ x}} - \frac{{∂ u}}{{∂ y}}} \right)\)

Now,

∴ Vorticity = \(\zeta = \frac{{∂ v}}{{∂ x}} - \frac{{∂ u}}{{∂ y}} \)

Calculation:

Given,

Velocity field, u = x, v = x + y

∴  ∂u/∂y = 0, ∂v/∂x = 1

Rectangular Field x = 0, y = 0, x = 1, y = 1 ( i.e Dimension of rectangular are 1 × 1 )

 Area of Recatangle = 1 × 1 = 1

Circulation = Vorticity × Area

\(Circulation= \left( {\frac{{∂ v}}{{∂ x}} - \frac{{∂ u}}{{∂ y}}} \right) × Area\)

Circulation= (1 – 0) × 1

∴ Circulation= 1 units