The velocity field $v$ represents a steady helical flow, where the fluid rotates around the $z$-axis while simultaneously translating along it. By transforming the Cartesian expression into cylindrical coordinates, we find that the radial velocity component is zero, which is why the mass flux through the cylinder walls is null. The mass flux through the horizontal disc depends purely on the uniform vertical velocity $v_0$, while the flux through the vertical $\phi$-plane depends on the rotational "swirl" component, which increases linearly with the distance from the axis.

🧮Sequence Diagram: Verification and Visualization of Incompressible Helical Flow

This sequence diagram illustrates the step-by-step derivation and verification process for the incompressible helical flow as detailed in the "Derivation sheet".

sequenceDiagram
    participant A as Analyst
    participant FM as Field Model
    participant FC as Flux Calculator
    participant CE as Calculus Engine
    participant V as Visualizer (Demos)

    Note over A, V: Phase 1: Coordinate Transformation
    A->+FM: Input Cartesian Velocity Field (Eq 1)
    FM->-A: Transform to Cylindrical: $$v = (v₀/L)(ρ e_φ + L e_z)$$

    Note over A, V: Phase 2: Flux Analysis
    A->+FC: Calculate Mass Flux (Φ) for $$S_a, S_b, S_c$$
    FC-->>A: $$Φ_a = π ρ₀ v₀ r₀²$$ (Disc)
    FC-->>A: $$Φ_b = 0$$ (Cylinder Wall)
    FC-->>A: $$Φ_c = (ρ₀ v₀ z₀ r₀²) / 2L$$ (φ-Plane)
    A->+V: Trigger Demo 1
    V-->>-A: Render interactive 3D flux across surfaces

    Note over A, V: Phase 3: Divergence & Incompressibility
    A->+CE: Calculate ∇ · v
    CE-->>A: Result = 0 (Incompressible Flow)
    A->+V: Trigger Demo 2
    V-->>-A: Track cube element: Volume change ≈ 0%

    Note over A, V: Phase 4: Vorticity & Rotation
    A->+CE: Calculate ∇ × v
    CE-->>A: Result = $$(2v₀/L) e_z$$ (Rotational Flow)
    A->+V: Trigger Demo 3
    V-->>-A: Render static Vorticity arrow + Rigid-body rotation

🪢Kanban: Visualizing the Dynamics of Helical Fluid Flow

---
config:
 kanban:
  sectionWidth: 260
---
kanban
  ***Derivation Sheet***
   Total Mass Flux Through Cylindrical Surfaces@{ticket: 1st,assigned: Primary,priority: 'Very High'}
   Verification and Visualization of Incompressible Helical Flow@{assigned: SequenceDiagram}
  ***Resulmation***
    Visualizing the Incompressibility and Vorticity of a Steady Helical Flow@{ticket: 2nd, assigned: Demostrate,priority: 'High'}
    compute the mass flux of the fluid through three surfaces@{assigned: Demo1}
    Fluid Element Deformation in Helical Flow (Incompressible)@{assigned: Demo2}
    Vorticity Visualization: Rigid-Body Rotation Flow@{assigned: Demo3}
    Visualizing Fluid Dynamics: Flux, Divergence, and Vorticity Analysis@{assigned: StateDiagram}
  ***IllustraDemo***
    Visualizing Mass Flux in Helical Flows@{ticket: 3rd,priority: 'Low', assigned: Narrademo}
    Anatomy of a Helical Fluid Flow@{assigned: Illustrademo}
    Incompressible Helical Flow A Physical and Mathematical Analysis@{assigned: Illustragram}
    Dynamics and Visual Proofs of Helical Fluid Flow@{assigned: Seqillustrate}
  ***Ex-Demo***
    The Mechanics of Incompressible Helical Flow@{ticket: 4th, assigned: Flowscript,priority: 'Very High'}
    Computational Dynamics and Vector Field Analysis Fluid Flow@{assigned: Flowchart}
    Dynamics and Vector Analysis of Helical Fluid Flow@{assigned: Mindmap}
  ***Narr-graphic***
    The Dynamics and Geometry of Helical Fluid Flow@{ticket: 5th,assigned: Flowstra,priority: 'Very Low'}
    The Geometry of Helical Flow From Calculus to Kinetic Reality@{assigned: Statestra}

Visual and Orchestra