Which of the following sets of equations represent(s) possible two-dimensional incompressible flows ? 1. u = x + y; v = x – y 2. u = x + 2y; v = x² – y² 3. u = 4x + y; v = x – y² 4. u = xt + 2y; v = x² – yt² 5. u = xt²; v = xyt – y² Select the correct answer using the code given below :
- A1 onlyCorrect
- B1 and 2 only
- C1, 2 and 3 only
- D2, 3, 4 and 5
Solution
Concept
Condition for a possible incompressible 2D flow
Formula
∂u/∂x + ∂v/∂y = 0
Step-by-step solution
1. u = x + y, v = x − y: 1 + (−1) = 0 ✓. 2. u = x + 2y, v = x² − y²: 1 − 2y ≠ 0 ✗. 3. u = 4x + y, v = x − y²: 4 − 2y ≠ 0 ✗. 4. u = xt + 2y, v = x² − yt²: t − t² ≠ 0 ✗. 5. u = xt², v = xyt − y²: t² + xt − 2y ≠ 0 ✗. Only set 1 satisfies continuity.
Explanation
A velocity field is a possible incompressible flow only if its divergence is zero. Only the first set satisfies ∂u/∂x + ∂v/∂y = 0 for all x, y and t.
Why the other options are wrong
Sets 2, 3, 4 and 5 give a non-zero divergence.
Shortcut
Check ∂u/∂x + ∂v/∂y = 0 for each set.
Common mistake
Ignoring the time-dependent terms in sets 4 and 5.
Correct answer: A) 1 only