Navier–Stokes Equations From Scratch

Fluids like water and air seem too messy to capture with equations because they are made of trillions of tiny bouncing molecules. Instead of tracking every single molecule, we pretend the fluid is a smooth continuous substance. Imagine zooming out until individual molecules blur together into a solid block of liquid that we call a fluid element. Conservation of mass means that fluid cannot magically appear or disappear out of nowhere.

We measure this balance using divergence which calculates whether fluid is spreading out or crowding in. To figure out how our fluid element moves through space, we need to look at all the forces pushing and pulling on it. Pressure, viscosity, and gravity. When we put Newton's law together with our forces, we get the famous Navier Stokes equation balancing inertia against pressure, viscosity, and gravity.

To see this equation in action, let us walk through PCL flow, which is fluid moving steadily between two stationary parallel plates. If we turn viscosity completely off by setting mu to zero, the Navier Stokes equation simplifies into the Oiler equation for invisid fluids. When fluid flows fast enough, the nonlinear convection term causes smooth motion to break down into chaos known as turbulence, remaining an unsolved mathematical puzzle. Fluid motion, mass conservation, force balance.

Navier Stokes. You now understand how the invisible laws of physics govern every drop of water in our world. Generate your own educational videos for free at eduvids. app.

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