We all enjoy watching the waves break and spread their waters swiftly over the shore. But how many of us have ever considered the incredible complexity of water movement? A wave breaks on the sand and separates into hundreds of different currents and bubbles, appearing smooth and regular. These are amazing and unpredictable at the same time. Since fluids can flow in various ways, they are particularly complex to analyse.
We can explain airflow around supersonic planes, highspeed cars, rockets, the flow of incompressible fluids, and many other things with the trickiest equations in the world, known as the Navier-Stokes equation.
It defines everything that flows in the universe, including water, air, smoke, complex weather patterns etc. We witness these things in our everyday life yet remain clueless about the physics behind them. They've been used to simulate the ocean and atmosphere on Earth, Jupiter's Great Red Spot and even how planes fly.
Navier-Stokes equation is the equivalent of Newton's Second law of motion in fluid mechanics. Solving them gives us the potential to change our present understanding of fluid flow and can trigger a new paradigm shift in Fluid mechanics. These equations can accurately predict how fluids will flow in a wide range of scenarios. We can solve them for specific initial conditions and certain restrictive assumptions.
What's appalling is that we haven't even been able to prove that solutions always exist. We know that the Navier-Stokes equation works empirically, but we don't have the analytical solution to the equation. Everything we have achieved so far is only possible by obtaining approximate numeric solutions to the equation. Understanding the Navier-Stokes equation can also help us understand the phenomena of 'Turbulence' – the oldest unsolved problem in physics. The exciting thing is that one can win a million dollars if they find the 'existence and smoothness of the Navier-Stokes equation'. It is one of the 'Millennium Prize Problems' announced by Clay Mathematics Institute in May 2000. They are the collection of seven difficult and important problems in mathematics. Except for the 'Poincaré Conjecture', the remaining six problems haven't been solved to this day.
Due to the complexity of the equation, finding an analytical solution is almost impossible. Instead, people utilise computer simulations to find an approximate discrete solution, which might take days, weeks, or even months. Highly powerful and extremely expensive supercomputers are required to reduce the calculation time. NASA and other multibillion-dollar aerospace corporations are funding research to crack this equation.
Unbeknownst to many, this equation remains in relative obscurity compared to other great equations of nature due to its inherent complexity.
Navier-Stokes equation truly deserves a special place in physics along with the likes of Maxwell's equations, Einstein's mass-energy equivalence etc.