Navier Stokes: equation to rule them all
A leading Millenium Prize Problem is the Navier-Stokes equation, which, if solved, could model the flow of any fluid - that means how airplanes navigate the skies, how water meanders in a river and how the flow of blood courses through your blood vessels... Understanding these equations in more detail will lead to scientific advances in all of these fields: better aircraft design, improved flood defences, and better drug delivery in the body. Fluids expert and mathematician Keith Moffatt took Tom Crawford down to the deep dark depths of Cambridge's maths lab...
Tom - We've just gone underground and we're stood outside a lab called the Goldstein Lab. It kind of reminds of a secret lair of either a superhero or maybe a super-villain. There are all kinds of complicated looking devices, cameras everywhere, all kinds of equipment and wires coming out of things. So these equations, the Navier-Stokes equations, they are a set of mathematical equations that model the flow of any fluid. That could be air, water, even blood in the body perhaps?
Keith - Yes, that is correct. For the vast majority of fluids particularly air and water, the equations are based on Newton's Laws, so they're very classical. They were first written down in the 19th century and they're highly mathematical in structure.
Tom - So, if we have these equations that model the flow of all of these different kinds of fluids then why is this a millennium problem?
Keith - It is an unsolved problem although many people have tried. There's question of whether solutions of the famous Navier-Stokes equations can or cannot become infinite. You might say it's a problem that you might throw the computer at. We've got extremely powerful computers nowadays, but a computer can never tell you whether a solution actually is going to infinity. A computer programme will always break down before the singularity is reached.
Tom - When I think of singularities, I'm thinking of the Big Bang or a black hole in space. What exactly do you mean by singularity here?
Keith - Well, singularity in general means that you have a system of equations in which one of the variables, any one, may go to infinity.
Tom - Do we have any examples with fluids that exhibit this singularity behaviour?
Keith - The singularity may occur most simply through a consideration of the problem of two tornado-like vortices. If these are forced together, then they go through a process of what's called "Vortex Reconnection". It's a very complex process because each vortex tries to wind around the other. The spatial structure becomes very complex. So the question is, can it become infinitely complex.
Tom - How close are we to actually understanding this problem? How far away is the solution?
Keith - 30 years.
Tom - I'll hold you to that.