🎬visualize the density fields of Kinetic Energy Momentum and Angular Momentum as a function of time

The visualization highlights a fundamental difference between scalar and vector integrals in fluid dynamics, especially in a symmetric flow: vigorous internal fluid motion does not guarantee net linear momentum. In the case of the symmetric vortex modeled, the Total Kinetic Energy (a scalar integral) and the Total Angular Momentum (a vector quantity measured relative to the center) are large and non-zero. However, the Total Momentum (a vector integral, $\int \rho v d V$ ) is effectively zero. This occurs because the momentum vectors from one side of the rotation are perfectly canceled by the opposing momentum vectors on the other side, illustrating that symmetry in the velocity field leads to a zero net vector sum, even though the energy associated with that motion remains high.

Fluid Mechanics Integrals for Mass and Motion | Proof and Derivationviadean on Notion