0$) and compresses in sink regions ($\nabla \cdot \vec{v} < 0$), causing visible shifts in particle concentration. Finally, the vorticity simulations utilized "paddlewheel" indicators to distinguish between types of circular motion; we proved that rigid body rotation possesses true local "spin" (non-zero curl), whereas an irrotational vortex orbits a center without local rotation because the velocity gradient cancels the orbital curvature. "> 0$) 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. "> 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. ">

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

https://youtu.be/H2ndWQfc-qw

🪜State Diagram: Dynamics and Transitions in Fluid Flow Visualization

The state diagram demonstrations are structured to guide you from basic vector field visualization to complex physical interpretations of divergence, mass conservation, and vorticity.

stateDiagram-v2
    [*] --> Demo1: Baseline Visualization
    
    state "Divergence & Flux Path" as DivergencePath {
        Demo1 --> Demo2: Add Radial Term (k > 0)
        Demo2 --> Demo7: Isolate Flow Components
    }
    
    state "Continuity & Density Path" as DensityPath {
        Demo2 --> Demo3: Apply Continuity Eq (Source)
        Demo3 --> Demo4: Flip Radial Sign (k < 0)
    }
    
    state "Vorticity & Rotation Path" as VorticityPath {
        Demo1 --> Demo5: Shift Focus to Curl
        Demo5 --> Demo6: Invert Velocity Gradient (1/r)
    }

    Demo1: Demo 1 - Helical Flow (∇·v = 0)
    Demo2: Demo 2 - Diverging Helix (Source)
    Demo3: Demo 3 - Density Fading (Source Effect)
    Demo4: Demo 4 - Density Compression (Sink)
    Demo5: Demo 5 - Rigid Body Rotation (∇×v ≠ 0)
    Demo6: Demo 6 - Irrotational Vortex (∇×v = 0)
    Demo7: Demo 7 - Interactive Flux Analysis

⚖️Quadrant 2: Helical & Continuous Flow (P33 Demos)

Helical & Continuous Flow (P33 Demos): A Unified Computational Study of Flux Continuity and Vorticity.

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    quadrant-3 "Structural Mapping & Volumes"
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    "Mass & Density Mapping (P25 Demos)": [0.15, 0.20]
    "Geometric Flux (P27 Demos)": [0.30, 0.85]
    "Surface to Volume (P30 Demos)": [0.75, 0.40]
    "Non-Planar Circulation (P31 Demos)": [0.20, 0.75]
    "Stokes' Theorem Proofs (P32 Demos)": [0.85, 0.90]
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    "Energy Orthogonality (P34 Demos)": [0.70, 0.25]
    "Singularity Management (P35 Demos)": [0.80, 0.15]
    "Generalized Curl Theorem (P37 Demos)": [0.90, 0.80]
 
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