Velocity from stream function ψ = 2 x y (two-dimensional incompressible flow) Given the stream function ψ(x,y) = 2 x y, find the fluid speed at the point (x, y) = (3, 4).

Introduction / Context:
The stream function ψ(x, y) conveniently describes two-dimensional incompressible flows. Velocity components are obtained by differentiating ψ, and the magnitude (speed) follows from the Euclidean norm of the components. This exercise checks fluency with stream-function relations.

Given Data / Assumptions:

• ψ(x, y) = 2 x y.
• Two-dimensional, incompressible flow in Cartesian coordinates.
• Point of interest: (x, y) = (3, 4).

Concept / Approach:

For 2D incompressible flow in (x, y): u = ∂ψ/∂y and v = −∂ψ/∂x. Compute u and v, then speed V = √(u^2 + v^2).

Step-by-Step Solution:

Verification / Alternative check:

Level curves ψ = constant are orthogonal to velocity potential lines (if defined), and the computed components satisfy the Cauchy–Riemann-type relations for incompressible flow.

Why Other Options Are Wrong:

6 and 8 m/s are component magnitudes, not the resultant; 12 and 15 m/s exceed the correct Pythagorean result.

Common Pitfalls:

Mixing the signs (u, v) definitions or using v = ∂ψ/∂x instead of v = −∂ψ/∂x.

Final Answer:

10 m/s