🎬From Dust to D-Force-Visualizing the Cauchy Momentum Equation

The internal stress term ( $\nabla \cdot \sigma$ ) is the crucial component that defines cohesive fluid behavior. In the Zero Stress ("Dust") Model ( $\frac{D v}{D t}=g$ ), this term is zero, meaning particles follow simple ballistic trajectories, ignoring their neighbors and passing through intersecting streams. The introduction of simple hydrostatic pressure, leading to the Euler Model ( $\frac{D v}{D t}=-\frac{1}{\rho} \nabla p+g$ ), activates the pressure gradient term ( $-\frac{1}{\rho} \nabla p$ ). This term acts as the internal mechanism for transferring force, causing high-pressure regions to push the fluid into low-pressure regions, which physically manifests as deflection and scattering when streams collide, transforming the independent movement of particles into a true, interacting fluid flow.

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