Stream function properties in 2D incompressible flow — which statements about the stream function ψ are correct?
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Aψ is constant everywhere on any streamline
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Bthe flow around any path in the fluid is zero for continuous flow
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Cthe rate of change of ψ with distance in an arbitrary direction is proportional to the component of velocity normal to that direction
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Dthe velocity vector may be found by differentiating the stream function
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Eall the above
Answer
Correct Answer: all the above
Explanation
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.
Given Data / Assumptions:- 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:
State streamline property: along a streamline, ψ is constant.Relate velocity: u = ∂ψ/∂y, v = −∂ψ/∂x → velocity obtained by differentiating ψ.Directional change: dψ/dn = |∇ψ| along the normal is proportional to normal velocity component, linking spacing of streamlines to speed.For continuous 2D incompressible flow, the net flow across a closed path composed of streamlines is zero, aligning with the stated property.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