Leonardo da Vinci was famous for his astute observation of the smallest intricacies of nature. His observations were drawn in his infamous notebooks and one in particular, that of water being poured into a river, leads us to another problem: fluid dynamics.
Leonardo da Vinciâs drawing of water being poured into a river.
You can see above that Leonardo mistakenly believed these waves formed Fibonacci spirals, a sequence that we discussed earlier which appears often in nature. Unfortunately, Leonardo da Vinci is wrong -- water formation takes a more chaotic form thatâs difficult to map and even more difficult to project. You can see below that the use of vectors, which is a concept unavailable to da Vinci in the 15th century, is an ideal way of showing the bit-by-bit direction and velocity of water. This is a 2-dimensional representation, so showing this in 3-dimensions and devising a formula that helps us understand fluid dynamics is notoriously difficult.
Vector field of fluid dynamics in two dimensions.
However, another Millenium Problem from the Clay Institute, the Navier Stokes Existence and Smoothness problem is a system of partial differential equations that model the future of any fluid -- not just water! It takes into account several different data points to explain how fluids move and behave in space.
This is how itâs described on the Clay Institutes website: âWaves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations. A fundamental problem in analysis is to decide whether smooth, physically reasonable solutions exist for the NavierâStokes equations. To give reasonable leeway to solvers while retaining the heart of the problem, we ask for a proof of one of the following four statements.â
This system of equations have not yet been proven and though its fundamental components make sense in theory, mathematicians havenât been able to work out all of their details. For example, turbulence is a variable here that still remains unsolved -- its basis in chaos theory solidifies its difficulty in being understood.
Navier-Stokes Equation
It would be helpful to break down this equation by convincing you that this intimidating formula is actually just a compilation of basic statements about how our world works. The first equation states that the mass we are using is conserved -- i.e. that the volume of liquid we will be modeling is consistent and the same throughout.
The second equation can be broken down by whatâs stated on the left and right side. On the left side, this is actually stating Newtonâs Second Law: F = M*A, or force equals mass times acceleration. It looks more complex than what Newton wrote because it is conveying the same statement but through fluid dynamics. Mass is the âPâ in front of the brackets and the other terms in the bracket are expressing acceleration. The mass in this case is not a solid, but the mass of liquid which can be represented as mass per unit volume, or density. Youâll notice the representation of acceleration in the brackets are using vectors since knowing its speed and direction will be instrumental in the modeling of a fluid.
The right side can be broken down to three distinct parts. The first expresses pressure, the second is friction arising from viscosity -- both of which are internal agents in the movement of particles. The third is an external force, which can be anything but more often than not it is gravitational force.
As mentioned before, these equations are useful in modeling fluid dynamics, but theyâre not perfect when trying to reproduce them experimentally. Recently, scientists have been able to leverage artificial intelligence to predict fluid dynamics in certain conditions and it has become nearly identical in its predictions. Some variables, like turbulence, are still chaotic elements that can throw the entire system off -- weâre not sure how we would incorporate turbulence since it is seemingly random.