The flow of incompressible fluid is defined by u = 2, v = 8x. What is the stream function ? (where C = constant)
- Aφ = – 2x² + y + C
- Bφ = – 4x² + 2y + CCorrect
- Cφ = – 4x + 2y² + C
- Dφ = – 4x – 2y² + C
Solution
Concept
Stream function from velocity components
Formula
u = ∂ψ/∂y; v = − ∂ψ/∂x
Step-by-step solution
1. u = ∂ψ/∂y = 2 ⇒ ψ = 2y + f(x). 2. v = − ∂ψ/∂x = 8x ⇒ ∂ψ/∂x = − 8x ⇒ f(x) = − 4x² + C. 3. ψ = − 4x² + 2y + C.
Explanation
Integrating u with respect to y and v with respect to x gives the stream function. Check: ∂ψ/∂y = 2 = u and −∂ψ/∂x = 8x = v.
Why the other options are wrong
The other expressions do not give u = 2 and v = 8x on differentiation.
Shortcut
Check by differentiating: ψy = u, −ψx = v.
Common mistake
Using the wrong sign convention for v.
Correct answer: B) φ = – 4x² + 2y + C