The Navier Stokes Equations

This week I would like to talk about the Navier-Stokes existence and smoothness problem. I would first like to talk about the history of the problem, then I would like to talk about the problem itself, then I would like to talk about the physics involved in the problem, and finally I would like to talk about changing the problem to allow for its use.

The problem is named after its contributors Claude-Louis Navier and Georg Gabriel Stokes. Navier was french physicist that lived in the late 1700's and early 1800's. He grew up and took interest in engineering and physics. He made a few contributions to math and science, but his main contribution was the Navier-Stokes equations. Stokes, the other contributor, was born in Ireland. Stokes lived in the 1800's roughly the same time period as Navier. Stokes became the head of mathematics at Cambridge, before dying in 1903. His contributions were similar to that of Navier. They both specifically worked on fluid dynamics. They came up with several equations that to me have too many letters from too many alphabets, but some how someone understands them fully.

The problems itself refers to solving the equations created by Navier, and Stokes. As mentioned before the equations deal with fluid dynamics, but there are a few more conditions dealing with the problems. First the fluids are assumed to be non-compressible. Fluids if you are not aware refer to gases and liquids. These fluids are assumed to be incompressible even though fluids are compressible to a small degree, the change in density, income cases, is so small it is negligible. The second condition is the fluid must be viscous. These equations under those conditions then describe fluid mechanics using newton second law of motion, that being force is equal to mass multiplied by acceleration. As you can imagine this is a big deal. Understanding fluids is what gets your airplanes in the air or your water to your house. These equations sometimes have solutions in certain situation, but mathematicians can not seem to prove that these conditions always exist. There is not a certainty that the smoothness condition can always exist in the three dimensions. On top of that if we assume that the smoothness condition always applies then the question remains, if these smoothness solutions exist do the have bounded energy for their mass? Like many of the problems will be talking about, this problem is a millennial problem. That means if you can solve the problem, then you will receive a million dollars. You will also probably be awarded a Field's Medal, and get to be on the front of a newspaper. Society as a whole will still probably care more about what the Kardashians are wearing instead of the solution to an incredibly important explanation of fluid dynamics, but oh well.

At this point it would be good to mention what the equations actually pertain to in fluid dynamics, but the content of the equations would go over a lot of heads and lose people.  Rather then showing all the equations, it would be more appropriate to give a brief description of the general form of the equations. 

This is the general form of the equation. All of the equations can be derived from this equation.

The first "p" represents the density of the fluid. The "v" represent velocity component. The second "p" refers to pressure. The "t" refers to time. The upside down debt refers to gradients or vector derivatives. Now I wish I could fully explain the equation, but I hope it makes more sense to you as a reader than it does to me.

Now these equations are not useless. In fact they are very important. As mentioned there is not a guarantee on our answers. However, there are still ways of getting numerically practical answers out of the equations. For instance, sometime by assuming factors lie change in density or pressure, as negligible you can get a really close answer. So sometimes you do not need to be able to solve the equation straight up, sometimes you can just get close enough for it to work. Hopefully you as an enthusiast will come up with a solution to get rid of these approximations.

Sources http://www.claymath.org/millennium-problems/navier–stokes-equation
              https://www.grc.nasa.gov/www/k-12/airplane/nseqs.html
              https://math.berkeley.edu/~chorin/chorin68.pdf