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.

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Divergence-Free Flow

Zero Divergence, Constant Density: The analysis shows that the velocity field is divergence-free ( $\nabla \cdot v=0$ ). This is a crucial property for an incompressible fluid like the one described, where the density ( $\rho_0$ ) is constant.
Conservation of Mass: A zero divergence mathematically confirms that there are no fluid sources or sinks within the volume. In physical terms, this means that fluid is neither being created nor destroyed, and the total mass within any given volume of the fluid remains constant over time. The fluid simply flows through the volume.

Total Momentum

Non-Zero Momentum: Even though the flow is divergence-free, the total momentum of the fluid inside the cube is not zero. This is because the velocity field has a non-zero average velocity.
Component Contributions: The swirling motion of the fluid in the $x^1-x^2$ plane contributes to the $P_1$ and $P_2$ components of the momentum. However, because the velocity is symmetric around the $z$-axis, the integral of these components across the cube results in equal and opposite values, making them non-zero in magnitude but equal and opposite in direction. The flow has a constant upward velocity ( $v_3=v_0$ ), which accumulates across the entire volume of the cube to create a significant total momentum in the $x^3$ direction.
Net Flow: The resulting total momentum vector, $P$, indicates a net flow of momentum through the cube, primarily driven by the constant vertical velocity component.

🎬Demonstration

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.

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