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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 where the terms of are Click on emeagwali.com for more information. ©1997. EMEAGWALI.com. All rights reserved.