For an irrotational, incompressible flow field, which governing equation must the velocity potential (or stream function in 2D) satisfy?

Introduction / Context:
Irrotational flow is central to potential-flow theory, used for inviscid approximations around bodies. In such flows, a scalar velocity potential exists and its governing equation provides a powerful mathematical framework for analysis.

Given Data / Assumptions:

• Flow is irrotational (vorticity = 0).
• Incompressible fluid (constant density).
• Smooth, differentiable velocity field with potential function φ.

Concept / Approach:

For incompressible flow, continuity requires divergence of velocity to be zero. With v = grad(φ), divergence-free condition becomes the Laplacian of φ equal to zero. Hence the velocity potential satisfies Laplace’s equation, which is linear and elliptic, enabling superposition and boundary-value methods.

Step-by-Step Solution:

Verification / Alternative check:

In two-dimensional potential flow, the stream function ψ also satisfies ∇^2ψ = 0, consistent with harmonic conjugacy of φ and ψ in simply connected regions.

Why Other Options Are Wrong:

(a) Cauchy–Riemann are compatibility relations between φ and ψ, not the governing PDE. (b) Reynolds transport theorem is an integral balance tool, not a pointwise PDE. (d) Bernoulli provides energy relation along a streamline, not the governing equation for φ.

Common Pitfalls:

Assuming Bernoulli alone defines the flow field; mixing rotational and irrotational assumptions improperly.

Final Answer:

Laplace’s equation