Nearly seventy years after Leonhard Euler derived his famous
equations, in 1821, Claude L. M. H. Navier
(1785-1836) added viscous forces and rederived an improved set of
equations
then called the Navier equations. The latter equations
were rederived by Simeon-Denis Poisson in 1831
and George Gabriel Stokes (1819-1903) in 1849. These
equations, which are now called the Navier-Stokes equations,
are widely used in various fluid dynamical
applications, such as computing the flows around
turbine and compressor blades, helicopter rotors, airplanes and missiles.
Specific applications include its use by the National Aeronautics and
Space Administration (NASA) uses it to collect data that complement
those obtained from wind tunnel experiments and to compute the flow
field around a B-747 carrier aircraft transporting the Space Shuttle
Orbiter; by United Technologies Research Center to compute the
multicomponent flowfield patterns inside a centrifugal compressor.

The space shuttle
lifting off from its lauching pad.

The
Navier-Stokes equations are written in
vectorial form as:
The three scalar equations in 3.7
describe the conservation of
momentum laws along the x-, y- and z- directions. Coupling
Equation 3.7 to the conservation of mass and energy
equations yields a system of equations with about 60 partial
derivative terms to be solved at each grid point.
The solution of the Navier-Stokes equations requires
the specification of the conservation of mass equation,
initial and boundary conditions.

The Space Shuttle Columbia.

When the fluid is compressible, the divergence of the
viscous stress tensor is used in place of the viscous term
. The viscous force along
the direction is

where

and

Alternatively, for viscous flows, the Euler's
equations 3.1 can be modified by adding
the viscous terms to it. The x-direction viscous flux
vector is

which is equivalent to

The y-direction viscous flux vector is

+ Philip Emeagwali explaining the derivation of
his partial differential equations.
+

which is equivalent to

The z-direction viscous flux vector is

which is equivalent to

where , the
Reynolds number, is a measure
of the relative effect of the viscous and inertial
forces within the flow. In the term ,
the energy equation describes the work done by the
viscous forces and the heat conductivity of the
fluid. The latter term is defined as

In Equation 3.12, a is the speed of sound,
is the Prandtl number,
k is the conductivity coefficient and

and for ideal fluids

The Prandtl number measures the flow diffusion or convection. For
diffusion, the number is

and for convection the number is

where is the dynamic viscosity, D is the diffusivity, and
is the specific heat at constant pressure. The viscous stress tensor used in
Equations 3.9, 3.10
and 3.11
is defined as