the Ideal Helmholtz Case and the Coupled Boundary Case

The two demos illustrate how the mathematical boundary conditions of a volume dictate the physical independence of different flow types. In Demo 1, the solenoidal field is strictly orthogonal to the surface normal, satisfying the conditions for the Helmholtz decomposition; consequently, the cross-term integral vanishes, and the total kinetic energy is perfectly additive. In contrast, Demo 2 introduces "boundary leakage," where the solenoidal field is forced to align partially with the irrotational gradient. This violation breaks the orthogonality, creating an interaction energy that causes the total energy to deviate from the sum of its parts. Together, these simulations prove that energy conservation between potential and vortex flows is not an inherent property of the fields themselves, but a direct consequence of the boundary constraints.

🎬Narrated Video

https://youtu.be/sETy1at2RjQ

🪜The Energetic Orthogonality of Solenoidal and Irrotational Fields

stateDiagram-v2
    [*] --> OrthogonalSystem : Enforce Boundary Condition ($$w · n = 0$$)
    
    state OrthogonalSystem {
        direction lr
        Condition1: $$w\\ $$ is divergence-free
        Condition2: $$v\\ $$ is curl-free
        Boundary: $$w · n = 0$$ (Tangent to surface)
        Integral: $$I=\\int_V \\vec{v} \\cdot \\vec{w} d V=0$$
        Energy: Total Energy = Sum of Parts
    }

    OrthogonalSystem --> NonOrthogonalSystem : Introduce Boundary Leakage ($$w · n ≠ 0$$)
    
    state NonOrthogonalSystem {
        direction lr
        Violation: $$w\\ $$ crosses surface boundary
        IntegralResult: $$I ≠ 0\\ $$ (Interaction Energy)
        EnergyEffect: Total Energy ≠ Sum of Parts
        Physics: Energy Coupling / Loss of Uniqueness
    }

    NonOrthogonalSystem --> OrthogonalSystem : Restore Boundary Constraints

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    class Condition1,Condition2,Boundary,Integral,Energy,Violation,IntegralResult,EnergyEffect,Physics darkFill

Description

Mathematical Problem: Integral of a Curl-Free Vector Field (CVF)
Example-to-Demo: The Orthogonal Harmony of Helmholtz Decomposition (OH-HD)

🏗️Structural clarification of Poof and Derivation

block-beta
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CC["Criss-Cross"]:6

%% Condensed Notes

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RF["Relevant File"]:6
NV["Narrated Video"]:6
PA("Plotting & Analysis")AA("Animation & Analysis")KT("Summary & Interpretation") ID("Illustration & Demo") VA1("Visual Aid")MG1("Multigraph")

%% Proof and Derivation

PD["Proof and Derivation"]:6
AF("Derivation Sheet"):6
NV2["Narrated Video"]:6
PA2("Plotting & Analysis")AA2("Animation & Analysis")KT2("Summary & Interpretation") ID2("Illustration & Demo")VA2("Visual Aid") MG2("Multigraph")

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%% %% Condensed Notes

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%% Proof and Derivation

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class PD color_5
class AF color_5

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🗒️Downloadable Files - Recursive updates (Feb 10,2026)

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Copyright Notice

All content and images on this page are the property of Sayako Dean, unless otherwise stated. They are protected by United States and international copyright laws. Any unauthorized use, reproduction, or distribution is strictly prohibited. For permission requests, please contact [email protected]

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