🎬A Unified Computational Study of Flux Continuity and Vorticity

Across these six demonstrations, we have bridged the gap between vector calculus and physical fluid behavior, focusing on the concepts of flux, density evolution, and vorticity. Our first set of demos validated the Divergence Theorem, illustrating how zero divergence characterizes incompressible helical flow while a positive divergence indicates a mass source where fluid expands outward. The second set applied the Continuity Equation to visualize density as a dynamic variable, showing that fluid "thins out" in source regions ($\nabla \cdot \vec{v} > 0$) and compresses in sink regions ($\nabla \cdot \vec{v} < 0$), causing visible shifts in particle concentration. Finally, the vorticity simulations utilized "paddlewheel" indicators to distinguish between types of circular motion; we proved that rigid body rotation possesses true local "spin" (non-zero curl), whereas an irrotational vortex orbits a center without local rotation because the velocity gradient cancels the orbital curvature. Together, these simulations provide a holistic view of how divergence and curl define the essential properties of a flow field—expansion, mass conservation, and rotation.

🎬Narrated Video

🧵Related Derivation

🧄Verification of the Divergence Theorem for a Rotating Fluid Flow (DT-RFF)

⚒️Compound Page

A Unified Computational Study of Flux Continuity and Vorticity | Resulmationviadean on Notion