{"id":3172,"date":"2021-01-10T03:01:19","date_gmt":"2021-01-10T01:01:19","guid":{"rendered":"https:\/\/www.lbscience.org\/en\/2026\/09\/18\/the-equations-worth-a-million-dollars\/"},"modified":"2026-09-18T03:39:41","modified_gmt":"2026-09-18T00:39:41","slug":"the-million-dollar-equations","status":"publish","type":"post","link":"https:\/\/www.lbscience.org\/en\/2021\/01\/10\/the-million-dollar-equations\/","title":{"rendered":"The Million-Dollar Equations"},"content":{"rendered":"

The Navier\u2013Stokes equations describe the behavior of fluids. A fluid is any substance that undergoes continuous deformation under a force acting tangentially to its surface (also called shear stress). If this sounds somewhat confusing, think of a fluid as a material that takes the shape of its container. For example, a gas is (usually) a fluid, and a liquid is of course a fluid, but so are plasma, molten lava, galaxy clusters, and even continental glaciers can be described as fluids (they flow very, very slowly).\u00a0<\/span><\/p>\n

Now that we have explained what falls under the definition of a fluid, we can understand the immense importance of the Navier\u2013Stokes equations. Every fluid in the world can be described by these equations. Whether we want to calculate the trajectories of ballistic missiles in the atmosphere, simulate blood flow through vasculature, or forecast the weather, the Navier\u2013Stokes equations allow us to calculate what the flow field in the system that interests us will look like.<\/span><\/p>\n

The Navier\u2013Stokes equations work so well because they are based on two very fundamental laws of physics: the law of conservation of mass and Newton\u2019s second law of motion. For fluids, the law of conservation of mass leads to an equation called the \u201ccontinuity equation\u201d, which essentially states that if we want to introduce an incompressible fluid (such as water) into a given volume, we must remove an equal amount of fluid from that volume, or \u201cpunch\u201d a hole in it through which the fluid can escape. Newton\u2019s second law states that the rate of change of a fluid\u2019s momentum is equal to the sum of the forces acting on it, and it leads to three additional equations (one for each spatial direction) that describe a competition between the two most important terms in flow: the advection terms and the viscosity terms.<\/span><\/p>\n

Imagine a motorboat traveling across a calm lake. In the region near the boat\u2019s engine, the advection terms dominate, representing the transport of momentum in the fluid (think of them as kinetic energy), and the water churns into vortices of many different sizes. But if we turn off the engine and wait a while, the water\u2019s surface will calm down and all the vortices will disappear. Where did all that kinetic energy go? This is where the viscosity terms come in: they represent the \u201cinternal friction\u201d within the fluid, and they are responsible for converting the system\u2019s kinetic energy into heat. The viscosity terms dominate the flow field in cases of very slow flows, highly viscous liquids (such as honey), and very small system dimensions (such as flow in microchannels or capillaries).<\/span><\/p>\n

Remember the vortices of many different sizes created by the boat\u2019s engine? Their different sizes are the product of a fascinating phenomenon in fluid flow called the energy cascade. In this phenomenon, the kinetic energy of large vortices is transferred to increasingly smaller vortices, until the vortices become so small that viscosity takes over and converts all the kinetic energy into heat. In the video below, you can see a beautiful demonstration of the energy cascade in turbulent flow.<\/span><\/p>\n