Category Archives: Fluid

Momentum Equation

If you just have the continuity equation, it is still dead water. Momentum term of the Navier-Stokes equations describes how the fluid reacts to force:

It means the change of velocity is the combined result caused by following effects:

Advection – fluid can transport along its own velocity field.

Pressure – force from pressure gradient. For example, wind blows from high pressure area to where pressure is lower.

External Force applied per unit mass. For example, stirring a cup of tea will cause it to move. Gravity will cause ink to sink in water.

Viscous force due to friction. Sticking glue has more resistance to flow than clean water.

Momentum equation can taken as the fluid version of F = ma.

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Continuity Equation

Time to brush up my math! Today is about the Navier-Stokes (N-S) equations, a set of differential equations describing the behavior of moving fluid. The continuity equation is:

ρ ( rho) is density. u, v, w, is velocity along direction x, y, z. For incompressible fluid density is constant (does not change over time or space), so the equation can be written as


means the divergence (∇) of vector field (velocity in this case) U equals zero (a scalar value). In other words: during the infinitesimal time, through the closed surface of the infinitesimal volume around a given point, inward flow equals outward flow. Anything flows in must flow out, nothing is created, so the mass of the system is preserved. This idea is the very backbone of fluid simulation.

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