🧄Verification of the Divergence Theorem for a Rotating Fluid Flow

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.

🎬the upward flux is perfectly balanced by the downward flux with zero flux through the sides

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.

🖊️Mathematical Proof

Verification of the Divergence Theorem for a Rotating Fluid Flow | Proof and Derivationviadean on Notion