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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.
Stress
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