🧄Momentum of a Divergence-Free Fluid in a Cubic Domain

A divergence-free velocity field signifies an incompressible fluid where the density remains constant and mass is conserved, implying no fluid is being created or destroyed within a given volume. However, even in such a flow, total momentum is not necessarily zero. It can result from a non-zero average velocity, such as a constant upward velocity in the example provided. This leads to a net flow of momentum through the volume, demonstrating that divergence-free flows can still exhibit overall movement and momentum.

🎬Visualize both a swirling motion with divergence-free and a spreading motion with divergence under the vector field

Divergence-Free Mode: In this mode, you see particles swirling but staying within the cube, demonstrating that the fluid is neither expanding nor compressing in the x-y plane. The net upward momentum shows how the fluid can still flow without a source or sink. This corresponds to the mathematical concept of a vector field with a zero divergence. Divergent Mode: In this mode, you see particles spreading out from a central point. This visualizes a source of fluid, where fluid is being created and pushed outwards. This corresponds to a vector field with a non-zero divergence.

🖊️Mathematical Proof

Momentum of a Divergence-Free Fluid in a Cubic Domain | Proof and Derivationviadean on Notion