Navier-Stokes equations

Motor boat wake

Waves follow our boat as we motor across the lake, and turbulent air currents follow our flight in an airplane. Mathematicians and physicists believe that the breeze and the turbulence can be explained and predicted by understanding the solutions to the Navier-Stokes equations.

  • Everything is Fluid Mechanics and Navier-Stokes equations — Chris Rogers, Tufts Center for Engineering Education and Outreach
  • 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.

    Boat wakes

Fluids move in mysterious ways. Mathematicians aren’t even sure the equations that describe them will work in every situation. So this is a problem for engineers who have to work with fluids every day in designing and creating objects and structures. In addition to boats, cars and planes, engineers must understand and direct the flow of air in electronics like TVs, computers and video game consoles, heating and cooling in homes, stores and schools, and water into and out of buildings, towns and cities.

Bottlenose dolphin surfs on wake from a boat

Basically Navier-Stokes equations say that when a fluid goes through something, like a pipe, or around something, like an airplane wing, the speed and direction of each tiny area of the fluid (liquid or gas) will do its own thing, but there is an equation for that. Actually, lots of pretty complicated equations. What makes it complicated is that everything affects everything else all the time.

Navier-Stokes equations are a set of equations to describe mathematically how velocity, pressure, temperature, and density of a moving fluid are related, and include the effects of viscosity on the flow. It is a math problem, but it describes things like

  • water flowing from a small pipe to a bigger pipe
  • airplane flying faster than the speed of sound
  • air flowing in the lungs of someone with a medical problem
Dispersion – left contains hot water, while the glass on the right contains cold water.

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.

Throughout history, organizations have offered prizes to anyone who could solve a really hard problem, including one for Navier-Stokes Equations. To collect the Clay Institute of Mathematics’ $1million prize 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.

What’s the problem?

  • What do you really know about fluids? What is an example of a “viscous” liquid? How is different from water – a liquid with low viscosity?
  • Have you every seen a liquid do something weird? Flow at different rates? Make interesting patterns?
  • What’s the problem if a fluid doesn’t flow smoothly?

That’s engineering

  • dilute – to make thinner or weaker by the addition of more liquid

Engineering ideas

  • fluids, gas, liquids, equations, random, dilute, disperse, Newton’s equation F=ma, viscosity, resistance, pressure, dimensions, compressibility, partial differential equations, workaround, “blowing up” answers, Computational Fluid Dynamics (CFD)

Do it
Here are some challenges for you to work on…

  • 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 coloring from a height into the glass and watch how it disperses. Even better: try imagining how you think it would look. results – picture, description

Learn more…

  • NASA Navier-Syokes equations – three-dimensional unsteady form of the Navier-Stokes Equations describe how the velocity, pressure, temperature, and density of a moving fluid are related and include the effects of viscosity on the flow.