Derivation of the NavierStokes Equation
 There are three kinds of forces important to fluid mechanics: gravity (body
force), pressure forces, and viscous forces (due to friction). Gravity force,
 Body forces act on the entire element, rather than merely at its
surfaces.
The only body force to be considered here is that due to gravity.
By convention, gravity acts in the negative zdirection, i.e. downward.
 Pressure forces act inward and normal to the surfaces of the element,
and have been discussed previously.
 Finally, there are viscous forces, due to friction acting
on the fluid element because of viscosity in the fluid.
These viscous forces are surface forces, like the forces due to pressure,
but can act in any direction on the surface.
In other words, viscous forces at a surface can have both normal and tangential
(or shear) components.
Viscous forces for incompressible Newtonian fluids

Here, consider only Newtonian fluids.
A Newtonian fluid is one where the stress is linearly proportional to the
strain.
Most common fluids are Newtonian, such as water, air, gasoline, oil, etc.
However, some fluids have a nonlinear relationship between stress and strain.
These fluids are called nonNewtonian. Some examples of
nonNewtonian fluids are cake batter and bread dough.
Blood also has some nonNewtonian properties.
It turns out that the net viscous force
per unit volume for an incompressible
Newtonian fluid is
where the right hand side is the laplacian of the velocity vector.
Note that in hydrostatics, where the velocity is identically zero,
there is no viscous force, regardless of the value of viscosity.
 The sum of all the forces on the element must equal the mass
of the element times its acceleration (Newton's second law). On
a per unit volume basis, the equation of fluid motion is then
The above equation is the famous NavierStokes equation,
valid for incompressible Newtonian flows. Normally, the acceleration
term on the left is expanded as the material acceleration when
writing this equation, i.e.
 Solutions of the full NavierStokes equation will be discussed
in a later module. For now, consider some simplifications of
this equation.
 Hydrostatics (fluid statics)  This is the simplest
possible case, namely when the fluid is either completely at rest
or moving at a constant speed with no acceleration. In such as
case, both acceleration and any velocity derivatives are zero,
and the NavierStokes equation reduces to
 Rigid body motion  The next simplest case is when
there is an acceleration of the fluid, but the entire fluid mass
moves together rigidly as one big chunk. Examples include rigid
body acceleration of a container of fluid in a straight line and
rigid body rotation of a fluid about some axis. In such cases,
although there is viscosity in the fluid, it is not felt since
no fluid particles ever "rub against each other." In
other words, there can be no viscous shear in rigid body motion,
and the NavierStokes equation reduces to
 Inviscid fluid flow  In some practical applications,
the effects of viscosity are negligible. The viscous terms in
the Navier Stokes equation are then neglected, and the equation
reduces to the Euler equation,