Stream function properties in 2D incompressible flow — which statements about the stream function ψ are correct?

Introduction / Context:
The 2D stream function ψ(x, y) is a scalar potential whose contours represent streamlines in incompressible flow. It enables automatic satisfaction of continuity and simplifies visualization and analysis of planar flows.

• Two-dimensional, incompressible flow.
• ψ is single-valued and differentiable in the flow domain.
• Velocity components are related to ψ via partial derivatives.

Concept / Approach:

In 2D, velocity components can be obtained from ψ by u = ∂ψ/∂y and v = −∂ψ/∂x (sign convention may vary). Thus, gradients of ψ determine velocity components normal to directions, and constant-ψ lines are streamlines. Circulation around a closed path composed of streamlines is zero for steady, continuous flow without singularities, consistent with the definition.

Step-by-Step Solution:

Verification / Alternative check (if short method exists):

Check continuity: ∂u/∂x + ∂v/∂y = ∂²ψ/∂x∂y − ∂²ψ/∂y∂x = 0, always satisfied, confirming incompressibility.

Why Other Options Are Wrong:

Individually they are each correct; hence the combined correct choice is “all the above.”

Common Pitfalls (misconceptions, mistakes):

Mixing stream function with velocity potential (φ); forgetting sign conventions; assuming ψ exists in 3D the same way (it does not).

Final Answer:

all the above