The mass flux out of the closed cylindrical surface is zero, a result confirmed by both the application of the Divergence Theorem and the direct summation of the fluxes through the individual surfaces. The Divergence Theorem revealed that the volume integral of the mass flux is zero because the velocity field $v$ has a zero divergence ( $\nabla \cdot v=0$ ), indicating that the fluid is incompressible and mass-conserving within the volume. This zero net flux was verified by the surface calculations, where the flux out of the top cap ( $\Phi_1= +\rho_0 v_0 \pi r_0^2$) was perfectly balanced by the flux into the bottom cap ($\Phi_2=-\rho_0 v_0 \pi r_0^2$), while the flux through the curved side wall ($\Phi_3$) was also zero due to the specific anti-symmetric nature of the velocity field components on that surface.
The sequence diagram illustrates the pedagogical progression of the demonstrations, moving from basic helical flow visualization to the complex physical analysis of divergence, mass conservation, and vorticity.
sequenceDiagram
participant S as Student / User
participant M as Mathematical Model
participant V as Visualization (Demos)
Note over S, V: Stage 1: Helical Flow & Divergence Theorem
S->>M: Analyze Helical Velocity Field
M->>M: Compute Divergence (∇·v = 0)
S->>V: Load Demo 1
V-->>S: Visualize constant particle spacing (Incompressible)
S->>V: Load Demo 7 (Interactive)
V-->>S: Verify Upward Flux = Downward Flux (Net Flux = 0)
Note over S, V: Stage 2: Sources, Sinks, and Continuity
S->>M: Add Radial Term (vρ = kρ)
M->>M: Compute Positive Divergence (∇·v = 2k)
S->>V: Load Demo 2 (Source)
V-->>S: Particles drift outward (Diverging Helix)
S->>M: Apply Continuity Equation (Dρ/Dt = -ρ∇·v)
S->>V: Load Demo 3 (Source) & Demo 4 (Sink)
V-->>S: Particles fade (thinning) or brighten (compression)
Note over S, V: Stage 3: Vorticity & Local Rotation
S->>M: Calculate Curl (∇×v)
M->>M: Compute Constant Vorticity (2v0/L)
S->>V: Load Demo 5 (Rigid Body Rotation)
V-->>S: Paddlewheels spin locally while orbiting
S->>M: Invert Velocity Gradient (v ∝ 1/r)
M->>M: Compute Zero Curl (∇×v = 0)
S->>V: Load Demo 6 (Irrotational Vortex)
V-->>S: Paddlewheels orbit but maintain orientation (No spin)
<aside> 👏
The document serves as a comprehensive, modular blueprint for a pedagogical tool. It is designed to visually bridge the gap between abstract vector calculus theorems (like Divergence and Curl) and tangible fluid dynamics behaviors (such as vorticity, flux continuity, and helical flow).
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***Derivation Sheet***
Verification of the Divergence Theorem for a Rotating Fluid Flow@{assigned: Primary}
Pedagogical Visualization of Vector Calculus and Fluid Dynamics@{assigned: SequenceDiagram}
***Resulmation***
A Unified Computational Study of Flux Continuity and Vorticity@{assigned: Demostrate}
Helical Fluid Flow@{assigned: Demo1}
Diverging Fluid Flow@{assigned: Demo2}
Continuity Equation-Density fading@{assigned: Demo3}
Continuity Equation-Density increasing@{assigned: Demo4}
Vorticity(Rigid Body Rotation)@{assigned: Demo5}
Irrotational Vortex (No Local Rotation)@{assigned: Demo6}
Divergence Theorem Visualization@{assigned: Demo7}
Dynamics and Transitions in Fluid Flow Visualization@{assigned: StateDiagram}
***GeoMetrics***
Demo 1 Shape Profile@{assigned: Shape1}
Demo 2 Shape Profile@{assigned: Shape2}
Demo 3 Shape Profile@{assigned: Shape3}
Demo 4 Shape Profile@{assigned: Shape4}
Demo 5 Shape Profile@{assigned: Shape5}
Demo 6 Shape Profile@{assigned: Shape6}
Demo 7 Shape Profile@{assigned: Shape7}
Derivation sheet Shape Profile@{assigned: Shape8}
Mindmap Shape Profile@{assigned: Shape9}
State Diagram Shape Profile@{assigned: Shape10}
Sequence Diagram Shape Profile@{assigned: Shape11}
***IllustraDemo***
How Divergence and Curl Define Flow@{assigned: Narrademo}
Visualizing Fluid Dynamics How Vector Calculus Explains Flow@{assigned: Illustrademo}
The Physics of Flow Visualising Vector Calculus in Fluid Dynamics@{assigned: Illustragram}
The Fluid Lens: Mapping Mathematical Flow and Physical Reality@{assigned: Seqillustrate}
***Ex-Demo***
The Mechanics of Helical Flow and Fluid Dynamics@{assigned: Flowscript}
Fluid Dynamics and Divergence Verification@{assigned: Flowchart}
The Mechanics of Fluid Flow and Vector Fields@{assigned: Mindmap}
***Narr-graphic***
Divergence and Curl Analysis@{assigned: Flowstra}
The Mechanics of Fluid Dynamics and Rotational Flow@{assigned: Statestra}
Cylindrical Fluid Flow Flux Verification@{assigned: ChartMeld}
<aside> 👏