A million dollars in cash (£640,000) awaits anyone who can develop a rigorous mathematical model for how fluids flow – this week's Millennium Prize Problem.

Fluids are extremely difficult to analyse because they can flow in such complicated ways. The next time you're bored in the kitchen, take a glass of water and let it stand until it's completely still (which takes longer than you might expect). Then use a straw to release a drop of food colouring from a height into the glass and watch how it disperses. Even better: try imagining how you think it would look.

Most people imagine the dye forming a cloud of colour that gradually disperses. In reality, the colouring will spread out, apparently without mixing, into surprisingly aesthetically pleasing but random patterns. More often than not, the drop will magically form a ring and travel down to the bottom of the glass without spreading out.

That a drop of liquid could form a ring and move undiluted through water goes against all our intuition. But it shouldn't come as a surprise: fluids are always doing this. The same process lies behind the formation of smoke rings.

Our world is awash with fluid. From the blood that courses through our veins to the cytoplasm in every cell, our bodies are dependent on liquids. And we spend our lives at the bottom of the atmospheric ocean that is fluid air. You can't move without stirring this sea of gases, so ubiquitous they almost go unnoticed.

Given how inextricably linked we are to fluids, it is disconcerting to discover that we do not have a precise mathematical understanding of how they move. Well, we have a working model, but there is no guarantee that it will not one day go horribly wrong.

Any mathematical description of how an object moves when you hit it starts with Newton's equation F=ma (the force applied equals the mass of the object multiplied by the resulting acceleration). Fluids, however, are a collection of countless particles all interacting, so we need a variation on F=ma that takes into account the sum effect across all the objects that comprise a fluid. Then we need to consider its "viscosity", which is a measure of how much the particles resist flowing over each other. Water is a low viscosity fluid while honey is high viscosity.

Put all of this together and you get:

Navier-Stokes equation describing how fluids move

Where u is the velocity of the fluid at position x and this changes over time t. The symbol v is the viscosity of the fluid and p represents pressure. Both i and j range from 1 to n, where n is the number of dimensions that the fluid is moving in.

This still isn't the whole story. As you quickly discover while washing up, if you force water down in one place it will burst out somewhere else (normally all over your legs). If you squeeze one part of a water balloon, a different part will expand. If you hit water hard enough, it will not move out of the way and you will hurt yourself – the same resistance that allows a fast-moving stone to bounce across the surface of water.

To incorporate this into our mathematical model, we have a second equation that effectively takes into account fluids' incompressibility:

Navier-Stokes equation 2



These equations were developed simultaneously in the early 1800s by George Stokes in England and Claude-Louis Navier in France. But if they have been around so long, what's the million-dollar prize for? The problem is that for most situations the Navier-Stokes equations are too hard to solve; they tend to result in partial differential equations that are simply too complicated. For the most part, mathematicians have developed numerical workarounds to squeeze out solutions to the equations, but it's a dark art.

Even when the equations can be forced to spit out a solution, in some situations these answers predict that the fluid will accelerate away at infinite speeds, an outcome that is known – rather appropriately – as "blowing up". Predicting fluids with infinite velocity is a sure sign that the mathematical model no longer matches reality.

Worse still, we do not even know whether a solution exists for all fluids. So while we use the Navier-Stokes equations for everything from aeronautical engineering to medical research, there's no guarantee the answers they give will be sensible or even if there will be an answer at all.

To collect the Clay Institute of Mathematics' $1m you need to show mathematically that either the Navier-Stokes equations can always be made to give realistic "not blowing up" answers, or that there is a case where they definitely cannot give such a solution. This has to be done for all fluids in three dimensions – many of these problems are solvable in two dimensions or for low velocities, but fluids get immensely more complicated in three dimensions and when things speed up.

That said, if you succeed in putting fluid dynamics onto solid mathematical foundations, you can use hand-rolled, cognac-infused cigars to conduct all the smoke-ring experiments you want.