While the Euler equations govern ideal “dry” fluids,
the Navier-Stokes equations govern nonideal “wet” fluids,
i.e. fluids with nonzero viscosity.
First of all, we can reuse the incompressibility condition for ideal fluids, without modifications:
Furthermore, from the derivation of the Euler equations,
we know that Newton’s second law can be written as follows,
for an infinitesimal particle of the fluid:
is the material derivative,
is the density, and is the effective force density,
expressed in terms of an external body force (e.g. gravity)
and the Cauchy stress tensor :
From the definition of viscosity,
the stress tensor’s elements are like so for a Newtonian fluid:
Where is the dynamic viscosity.
Inserting this, we calculate in index notation:
Thanks to incompressibility ,
the middle term vanishes, leaving us with:
We assume that the only body force is gravity .
Newton’s second law then becomes:
Dividing by , and replacing
with the kinematic viscosity ,
yields the main equation:
Finally, we can optionally allow incompressible fluids
with an inhomogeneous “lumpy” density ,
by demanding conservation of mass,
just like for the Euler equations:
Putting it all together, the Navier-Stokes equations for an incompressible fluid are given by:
Due to the definition of viscosity as the molecular “stickiness”,
we have boundary conditions for the velocity field :
at any interface, must be continuous.
Likewise, Newton’s third law demands that the normal component
of stress is continuous there.
- B. Lautrup,
Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition,