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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$\gg$Mathematical Structures Underlying Physical Laws
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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.
The Divergence Theorem states that the total outward flux of a vector field (like our fluid flow) through a closed surface is equal to the integral of the divergence of that field over the volume enclosed by the surface. since no fluid is created or destroyed inside the cylinder, whatever fluid leaves through the top must be replaced by an equal amount of fluid entering from the bottom, resulting in a net flux of zero.
the upward flux is perfectly balanced by the downward flux with zero flux through the sides
the upward flux is perfectly balanced by the downward flux with zero flux through the sides
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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
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
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