The fluid flow demo provides a powerful visual explanation of the Divergence Theorem, demonstrating that for a fluid field with zero divergence, the total outward flux through a closed surface must be zero. The visualization makes it clear that while the fluid has a complex, swirling motion, the amount of fluid flowing out of the top of the cylindrical volume is precisely balanced by the amount of fluid flowing into the bottom, with no flow through the sides, thereby confirming that no fluid is created or destroyed within the volume. This highlights how a single, simple property of the field (zero divergence) can have a significant and easily verifiable consequence for the system as a whole.
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✍️Mathematical Proof
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The total mass flux is zero. Both the direct calculation of the surface integrals and the application of the divergence theorem confirm that no fluid is entering or leaving the closed cylindrical volume.
The divergence theorem simplifies complex problems. By converting a surface integral over a closed surface into a volume integral of the divergence, we can often solve problems more easily. In this case, the divergence was zero, making the volume integral trivial to compute.
Zero divergence means the fluid is incompressible and source-free. The fact that $\nabla. v=0$ indicates that there are no sources or sinks of the fluid within the volume, and the flow is steady-state.
The velocity field has a rotational component. The term $\frac{v_n}{L}\left(y e_x-x e_y\right)$ describes a rotation around the z -axis. The flux through the top and bottom discs is due to the constant $z$-component of the velocity field, $v_z=v_0$. These two fluxes cancel each other out.
Net flux through the side wall is zero. The rotational component of the velocity field is always tangent to the cylindrical surface, so there is no flow through the side wall.
✍️Mathematical Proof
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- Proving the Cross Product Rules with the Levi-Civita Symbol
- Proving the Epsilon-Delta Relation and the Bac-Cab Rule
- Simplifying Levi-Civita and Kronecker Delta Identities
- Dot Cross and Triple Products
- Why a Cube's Diagonal Angle Never Changes
- How the Cross Product Relates to the Sine of an Angle
- Finding the Shortest Distance and Proving Orthogonality for Skew Lines
- A Study of Helical Trajectories and Vector Dynamics
- The Power of Cross Products: A Visual Guide to Precessing Vectors
- Divergence and Curl Analysis of Vector Fields
- Unpacking Vector Identities: How to Apply Divergence and Curl Rules
- Commutativity and Anti-symmetry in Vector Calculus Identities
- Double Curl Identity Proof using the epsilon-delta Relation
- The Orthogonality of the Cross Product Proved by the Levi-Civita Symbol and Index Notation
- Surface Parametrisation and the Verification of the Gradient-Normal Relationship
- Proof and Implications of a Vector Operator Identity
- Conditions for a Scalar Field Identity
- Solution and Proof for a Vector Identity and Divergence Problem
- Kinematics and Vector Calculus of a Rotating Rigid Body
- Work Done by a Non-Conservative Force and Conservative Force
- The Lorentz Force and the Principle of Zero Work Done by a Magnetic Field
- Calculating the Area of a Half-Sphere Using Cylindrical Coordinates
- Divergence Theorem Analysis of a Vector Field with Power-Law Components
- Total Mass in a Cube vs. a Sphere
- Momentum of a Divergence-Free Fluid in a Cubic Domain
- Total Mass Flux Through Cylindrical Surfaces
- Analysis of Forces and Torques on a Current Loop in a Uniform Magnetic Field
- Computing the Integral of a Static Electromagnetic Field
- Surface Integral to Volume Integral Conversion Using the Divergence Theorem
- Circulation Integral vs. Surface Integral
- Using Stokes' Theorem with a Constant Scalar Field
- Verification of the Divergence Theorem for a Rotating Fluid Flow
- Integral of a Curl-Free Vector Field
- Boundary-Driven Cancellation in Vector Field Integrals
- The Vanishing Curl Integral
- Proving the Generalized Curl Theorem
- Computing the Magnetic Field and its Curl from a Dipole Vector Potential
- Proving Contravariant Vector Components Using the Dual Basis
- Verification of Orthogonal Tangent Vector Bases in Cylindrical and Spherical Coordinates
- Vector Field Analysis in Cylindrical Coordinates
- Vector Field Singularities and Stokes' Theorem
- Compute Parabolic coordinates-related properties
- Analyze Flux and Laplacian of The Yukawa Potential
- Verification of Vector Calculus Identities in Different Coordinate Systems
- Analysis of a Divergence-Free Vector Field
- The Uniqueness Theorem for Vector Fields
- Analysis of Electric Dipole Force Field
🧄Proof and Derivation-2
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