For a two-dimensional flow, if x-component of velocity is u = 6xy – 2x², then y-component of velocity for continuity equation to be satisfied is
- A6y² – 4xy
- B– 6xy + 2x²
- C5x² – 2xy
- D4xy – 3y²Correct
Solution
Concept
Continuity equation for 2D incompressible flow
Formula
∂u/∂x + ∂v/∂y = 0
Step-by-step solution
1. u = 6xy − 2x², so ∂u/∂x = 6y − 4x. 2. ∂v/∂y = −(6y − 4x) = 4x − 6y. 3. Integrate with respect to y: v = 4xy − 3y² (+ f(x), taken as zero).
Explanation
For continuity in two-dimensional incompressible flow, the sum of the velocity gradients is zero. Integrating ∂v/∂y gives v = 4xy − 3y².
Shortcut
Check an option by differentiating it: d(4xy − 3y²)/dy = 4x − 6y ✓.
Common mistake
Forgetting the negative sign between the two gradients.
Correct answer: D) 4xy – 3y²