Wate, Water everywhere ... :)

Water, as my loyal readers know, is an endlessly fascinating subject to yours truly, not only as compound extraordinaire but also as an entity that reacts to its environment in unique ways that boggle the mind, particularly when it comes to turbulence, whether it be driven by wind, gravity or combinations of the two in relationship to the environment in which water resides. With this in mind, water, when impacted by the aforementioned conditions, becomes a chaotic system that can never be fully analysed given the intractable law of initial conditions and the vagaries of an ever changing reality but mathematicians continue to try in spite of the immense barriers nature presents to spite these intrepid researchers. :)

The Navier-Stokes equations capture in a few succinct terms one of the most ubiquitous features of the physical world: the flow of fluids. The equations, which date to the 1820s, are today used to model everything from ocean currents to turbulence in the wake of an airplane to the flow of blood in the heart.

While physicists consider the equations to be as reliable as a hammer, mathematicians eye them warily. To a mathematician, it means little that the equations appear to work. They want proof that the equations are unfailing: that no matter the fluid, and no matter how far into the future you forecast its flow, the mathematics of the equations will still hold. Such a guarantee has proved elusive. The first person (or team) to prove that the Navier-Stokes equations will always work — or to provide an example where they don’t — stands to win one of seven Millennium Prize Problems endowed by the Clay Mathematics Institute, along with the associated $1 million reward.

When reading this, one is simply amazed as to how well math models reality even if it doesn't cover ALL instances of said particular instance of reality being modeled. :)

What Are the Navier-Stokes Equations?

The Navier-Stokes equations govern the motion of fluids and can be seen as Newton's second law of motion for fluids. In the case of a compressible Newtonian fluid, this yields

where u is the fluid velocity, p is the fluid pressure, ρ is the fluid density, and μ is the fluid dynamic viscosity. The different terms correspond to the inertial forces (1), pressure forces (2), viscous forces (3), and the external forces applied to the fluid (4). The Navier-Stokes equations were derived by Navier, Poisson, Saint-Venant, and Stokes between 1827 and 1845.

These equations are always solved together with the continuity equation:

The Navier-Stokes equations represent the conservation of momentum, while the continuity equation represents the conservation of mass.

Any questions?