Navier-Stokes equations

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Continuity

or, using the substantive derivative:

For an incompressible fluid, is constant, hence the velocity field must be divergence-free:

Momentum

(Also known as the Navier-Stokes equation.)

or, using the substantive derivative:

where is a volumetric force (e.g. for gravity), and is the stress tensor.

The vector quantity is the shear stress. For a Newtonian incompressible fluid,

with being the (dynamic) viscosity.

For an inviscid fluid, the momentum equation becomes Euler's equation for ideal fluids:

References