Since 1820 ...

 When two fluids move past each other at different speeds, complex instabilities arise. Mathematicians want to prove that the Navier-Stokes equations can predict what will happen in every possible case.

This exquisite gif showing fluids moving at different speeds is both hypnotic and beguiling at the same time. Seems the very complex but reliable Naiver-Stokes equations are also as described in the Quanta article titled Mathematicians Find Wrinkle in Famed Fluid Equations. 

The Navier-Stokes equations capture in a few succinct terms one of the most ubiquitous features of the physical world: the flow of fluids. The equations, which date to the 1820s, are today used to model everything from ocean currents to turbulence in the wake of an airplane to the flow of blood in the heart.

While physicists consider the equations to be as reliable as a hammer, mathematicians eye them warily. To a mathematician, it means little that the equations appear to work. They want proof that the equations are unfailing: that no matter the fluid, and no matter how far into the future you forecast its flow, the mathematics of the equations will still hold. Such a guarantee has proved elusive. The first person (or team) to prove that the Navier-Stokes equations will always work — or to provide an example where they don’t — stands to win one of seven Millennium Prize Problems endowed by the Clay Mathematics Institute, along with the associated $1 million reward.

George Gabriel Stokes

Claude-Louis Navier