What are the vorticity components at a point (1, 1, 1) for the following flow field ? u = 2x² + 3y, v = – 2xy + 3y² + 3zy, w = – 1·5z² + 2xz – 9y²z
- AΩx = – 21 units, Ωy = – 2 units, Ωz = – 5 unitsCorrect
- BΩx = – 2 units, Ωy = – 12 units, Ωz = – 5 units
- CΩx = – 2 units, Ωy = – 21 units, Ωz = – 5 units
- DΩx = – 5 units, Ωy = – 2 units, Ωz = – 21 units
Solution
Concept
Curl of the velocity field
Formula
Ωx = ∂w/∂y − ∂v/∂z; Ωy = ∂u/∂z − ∂w/∂x; Ωz = ∂v/∂x − ∂u/∂y
Step-by-step solution
1. Ωx = ∂w/∂y − ∂v/∂z = (−18 y z) − (3y) = −18 − 3 = −21. 2. Ωy = ∂u/∂z − ∂w/∂x = 0 − 2z = −2. 3. Ωz = ∂v/∂x − ∂u/∂y = −2y − 3 = −5.
Explanation
The vorticity vector is the curl of the velocity. Each component is obtained by partial differentiation and then evaluated at (1, 1, 1).
Why the other options are wrong
The other options permute or mis-evaluate the three components.
Shortcut
Curl: (∂w/∂y − ∂v/∂z, ∂u/∂z − ∂w/∂x, ∂v/∂x − ∂u/∂y).
Common mistake
Mixing up the order of the partial derivatives.
Correct answer: A) Ω_x = – 21 units, Ω_y = – 2 units, Ω_z = – 5 units