Governing equation for incompressible, irrotational flow in 3D For steady, incompressible, irrotational flow in three dimensions, the velocity potential φ satisfies which fundamental (Laplacian) equation?

Introduction / Context:
Potential flow theory is frequently applied to seepage and groundwater problems via analogies. In an incompressible, irrotational field, the scalar velocity potential provides a convenient description of the flow, governed by Laplace’s equation.

Given Data / Assumptions:

• Steady flow.
• Incompressible fluid (constant density).
• Irrotational (existence of scalar potential φ such that V = ∇φ).

Concept / Approach:

For incompressible flow, continuity gives ∇·V = 0. If V = ∇φ (irrotational), then ∇·(∇φ) = ∇²φ = 0, the Laplace equation. This harmonic condition underpins flow nets and electrical analogs used in geotechnical seepage analysis.

Step-by-Step Solution:

Verification / Alternative check:

Seepage through earth dams and under sheet piles is solved using Laplace’s equation for hydraulic head or potential with appropriate boundary conditions.

Why Other Options Are Wrong:

(b) is continuity but not the Laplacian for φ; (c) is irrotationality but not the governing equation for φ; (e) is not isotropic Laplace equation; (d) is false because a correct equation exists.

Common Pitfalls:

Confusing conditions on V with the equation for φ; overlooking boundary condition specification importance.

Final Answer:

∇²φ = 0