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The velocity potential of a velocity field is given by = x2 – y2 +const. Its stream function will be given by:  
  • a)
    – 2xy + constant
  • b)
    + 2xy + constant
  • c)
    – 2xy + f(x)
  • d)
    – 2xy + f(y)
Correct answer is option 'B'. Can you explain this answer?
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The velocity potential of a velocity field is given by = x2 – y2...
Ans. (b) Use Cauchy – Riemann equation
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The velocity potential of a velocity field is given by = x2 – y2...
Explanation:
Given: Velocity potential (φ) = x2 y2 const
To find: Stream function (ψ)
We know that the velocity potential and stream function are related as follows:
v_x = ∂ψ/∂y
v_y = -∂ψ/∂x

Calculating the stream function:
v_x = ∂φ/∂x = 2xy2
v_y = ∂φ/∂y = 2yx2

Integrating v_x with respect to y, we get:
ψ = ∫v_x dy = xy2 + f(x)

Taking the partial derivative of ψ with respect to x, we get:
∂ψ/∂x = y2 + f'(x)

Comparing this with v_y, we get:
y2 + f'(x) = 2yx2

Solving for f(x), we get:
f(x) = constant - x4/4

Substituting this value of f(x) in ψ, we get:
ψ = xy2 - x4/4 + constant

Therefore, the stream function is given by 2xy - x4/4 + constant (option B).
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The velocity potential of a velocity field is given by = x2 – y2...
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