[AntonioLao]

One other Millennium Problem worth $1M for its solution is known as the Navier-Stokes equations found in the calculus of fluid motion. It is stated in almost all advanced texts of fluid mechanics that these equations look superficially simple compared to other similar equations. However, with present technology, no one can ever imagine a supercomputer that is capable of solving them in higher than 2 dimensions without specific assumptions and idealizations.

Since the invention of differential calculus by Leibniz and Newton in the middle 17th century and formalized by Cauchy and Weierstrass in the early 19th century using theory of limits its success in different applications are too many to mention here. But let it be known that Modern science and technology cannot function without it. The key idea is the derivative, say dE/dx such that E is a function of x in 1-dimension: E=f(x). In 2-dimension: E=f(x, y). In 3-dimension: E=f(x, y, z). In 4-dimension: E=f(x, y, z, t). In 1738, Bernoulli’s law stated that the force per unit area exerted by a fluid over a surface decreases as the speed of the fluid increases. Using Bernoulli’s principle, Euler successfully derived his equations for incompressible frictionless fluid. Then in 1822 Navier added the viscosity term based on flawed logic yet arrived at the correct equations. Finally, in 1842 Stokes rediscovered Navier’s equations with the correct reasoning. These can be expressed in one vector equation as: ¶V/¶t + (V×Ñ)V + Ñp/r- (a + nѲV) = 0 such that Ñ×V = 0. The missing curl operator (Ñ´) is the reason why no general solutions exist. Note: V is the velocity vector, Ñ is the gradient operator, p is the pressure expressed as energy per unit volume, r is the density of the fluid, a is the acceleration of gravity, n is the viscosity, Ѳ is the Laplacian operator, and Ñ× is the divergence operator.