Airflow is simulated over and past the wing of a high performance aircraft that is using vectored thrust while descending to a few feet above the ground (in ground effect). See here
The Navier-Stokes Equation is a partial differential equation that models fluid flow. Fluids include liquids and gases. Using Navier Stokes we can understand diverse phenomena like airflow over aeroplane wings or rocket bodies, the flow of liquids through pipes and the flow of plasma in stars (magnetohydrodynamics). There are many computer programs available that will divide a fluid flow problem into discrete, finite sized pieces and solve the problem for you (the so-called finite element analysis method). NASA, major defense contractors and car manufacturers all use these programs to solve the sort of problems listed above.
Above: Navier Stocks Equation – Incompressible Liquid with Constant Viscosity (Newtonian Liquid)
It may come as a surprise that even though Navier-Stokes in used so extensively in physics, engineering and industry we know little about the theoretical underpinnings of these equations. Anybody who can answer certain fundamental questions about the Navier-Stokes equations can win a \$1 million award from the Clay Mathematics foundation.
The lack of adequate knowledge about the Navier-Stokes equations represent one of the greatest unsolved problems in 21st century mathematics. The announcement of the \$1 million award certainly attracted my attention but I found Navier-Stokes equations to be so interesting that I taught myself some basics. I would like to share some useful resources:
A good introduction to Navier-Stokes equations can be found on the following links:
(3) A infinitesimal element approach to deriving Navier Stokes
(4) A slightly bizarre but fun link
But the pièce de résistance has got to be the video* of the problem description by the Clay Mathematics Institute. Its an elementary description of the problem that anyone with an understanding of vector calculus can comprehend.
Tip: Read links (1) & (2) of wikipedia before listening to the lecture. You will gain more out of it. In the lecture Cafarelli uses non-standard notation to denote derivatives. For example, 
means 
.
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(*select Navier Stokes existence and smoothness by Luis Caffarelli, University of Texas)
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