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An excerpt from Philip Emeagwali's
1989 Gordon Bell Prize Report

Emeagwali inside the Space Shuttle.]

+ Philip Emeagwali inside the Space Shuttle. +


Nature's own numbers man
From Oslo, Norway

Making strides in a parallel universe

Inspiration from hard history

Both Gordon Bell Prize winners tackle oil industry problems

Supercomputer produces oil reservoir simulation

IEEE's Gordon Bell Prize Report

Discussions with Gordon Bell Prize Committee

Upstream Interview
From Ghana

You said it ...
Real letters
Real folks

This Man
Is Hot

From London

+ + +



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.

Space Shuttle Columbia
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 x-direction direction is
x-direction viscous force
tau ij

delta ij

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
x-direction viscous flux vector

which is equivalent to

 x-direction equivalent viscous flux

The y-direction viscous flux vector is

y direction viscous flux vector


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

which is equivalent to


The z-direction viscous flux vector is

z-direction viscous flux vector

which is equivalent to

 equivalent z-direction viscous flux vector

where Reynold's number, the Reynolds number, is a measure of the relative effect of the viscous and inertial forces within the flow. In the term Navier-Stokes equations term K, the energy equation describes the work done by the viscous forces and the heat conductivity of the fluid. The latter term is defined as
 definition of navier-stokes equations term K

In Equation 3.12, a is the speed of sound, Prandtl number is the Prandtl number, k is the conductivity coefficient and
definition of the Navier-Stokes
equations term a-squared
and for ideal fluids
Navier-Stokes term
 a -squared for ideal fluids
The Prandtl number measures the flow diffusion or convection. For diffusion, the number is
Prandtl number for diffusion
and for convection the number is
Prandtl number for
where dynamic viscosity is the dynamic viscosity, D is the diffusivity, and Specifi heat under constant pressure is the specific heat at constant pressure. The viscous stress tensor used in Equations 3.9,   3.10 and  3.11 is defined as
viscous stress tensor

where the terms of viscous stress tensor S are

viscous stress tensor

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