Irrotational flow — the mathematical condition in 2D For a two-dimensional velocity field with components u(x,y) and v(x,y), which relation must hold everywhere for the flow to be irrotational?

Introduction / Context:
Irrotational flow is characterised by zero vorticity. In 2D, the scalar vorticity component normal to the plane (z-component of curl of velocity) must vanish. This condition underpins the existence of a velocity potential and is widely used in potential-flow theory for inviscid analyses around airfoils and bluff bodies (away from boundary layers and wakes).

Given Data / Assumptions:

• Velocity field = (u(x,y), v(x,y)) in the x–y plane.
• Regular differentiability of u and v is assumed.
• Compressibility is not required to define irrotationality (it concerns vorticity, not divergence).

Concept / Approach:
Vorticity vector ω = ∇ × V. In 2D, only ω_z is non-zero in general, with ω_z = ∂v/∂x − ∂u/∂y. Irrotational flow requires ω_z = 0 everywhere, leading to the Cauchy-Riemann-type compatibility that guarantees a scalar velocity potential φ with u = ∂φ/∂x, v = ∂φ/∂y. Note that incompressibility is a separate condition, ∂u/∂x + ∂v/∂y = 0, which may or may not hold simultaneously with irrotationality.

Step-by-Step Solution:
Write ω_z = ∂v/∂x − ∂u/∂y.Set ω_z = 0 for irrotational flow → ∂v/∂x − ∂u/∂y = 0.Recognise that this implies equality of cross-derivatives and the existence of φ.

Verification / Alternative check:
If a potential φ exists such that u = ∂φ/∂x and v = ∂φ/∂y, mixed partial equality gives ∂v/∂x = ∂u/∂y ⇒ ω_z = 0, confirming the condition.

Why Other Options Are Wrong:
∂u/∂x + ∂v/∂y = 0 is the incompressibility (continuity) condition in 2D, not irrotationality.
∂u/∂y + ∂v/∂x = 0 has no general irrotational meaning.
∂u/∂x − ∂v/∂y = 0 is unrelated to ω_z.
u^2 + v^2 = constant is not required for irrotational flow.

Common Pitfalls:

• Conflating zero divergence with zero curl.
• Assuming potential flow implies incompressible; compressible potential flow is also possible (with additional relations).

Final Answer:
∂v/∂x − ∂u/∂y = 0