Superposition of 2D flows — resultant streamline family from stream functions Two planar, incompressible flow patterns have stream functions ψ1 = x^2 + y^2 and ψ2 = 2xy. If these patterns are superposed, what is the family of streamlines represented by the combined stream function?

Introduction / Context:Stream functions add linearly for incompressible, planar flows, allowing construction of complex patterns by superposing simpler ones. Interpreting the resulting streamline equation helps visualize the flow field and classify it (e.g., uniform flow, source/sink, vortex, shear, stagnation flow).

Given Data / Assumptions:

• ψ1 = x^2 + y^2; ψ2 = 2xy.
• Superposition applies: ψ = ψ1 + ψ2.
• Cartesian coordinates; streamlines are given by ψ = constant.

Concept / Approach:Compute the combined stream function: ψ = x^2 + y^2 + 2xy = (x + y)^2. A streamline is defined by ψ = C, so (x + y)^2 = C. Hence x + y = ±√C, which represents a family of straight lines with slope −1 and different intercepts. Therefore, the combined pattern consists of parallel straight streamlines.

Step-by-Step Solution:Form ψ = ψ1 + ψ2 = (x + y)^2.Set ψ = C → (x + y)^2 = C.Take square root: x + y = constant (±√C) → parallel straight lines.

Verification / Alternative check:Velocity components from ψ are u = ∂ψ/∂y = 2(x + y) and v = −∂ψ/∂x = −2(x + y). Thus v = −u, confirming streamlines of the form x + y = constant.

Why Other Options Are Wrong:Circles, parabolas, hyperbolas, ellipses do not satisfy (x + y)^2 = constant; they arise from different polynomial relations.

Common Pitfalls:

• Confusing stream function with velocity potential; only the former gives streamlines ψ = constant.
• Dropping the cross term and misclassifying the pattern.

Final Answer:A family of parallel straight lines