The physics and math of fluids
I apologize that it has been a while since I last gave an update. As we explored the physics and mathematics of fluids, our trajectory has taken quite a turn for the different: we have been looking at the dynamics of solar systems for the last few sessions. So I think it is a good time to look back on how the sessions have evolved to see how and why we got here, and where I plan to go.
My goal has always been to describe as mathematically as possible the physics of how fluids flow, through this general path:
1. do some hands-on experiments on fluid flow to gather data and intuition
2. analyze the data and describe them quantitatively using formulas
3. start from the theories governing fluid flow and arrive at the same (or similar) formulas
4. understand the theoretical equations governing fluid flow
Steps 1-3 took until about mid January, and I hope we had some fun with the messy and colorful experiments. More importantly, I hope we shared some satisfaction in how the theoretical equations correctly predicted the data that we collected through the experiments. We started with the theoretical equation to derive some formulas, but where does the original theory come from?
What does it take to get to step 4: understand the theoretical equations governing fluid flow? For one, it takes a lot of mathematics that is quite difficult even for graduate students...but I obviously believe that we can get there with careful explanations and demonstrations that rely on a combination of intuition, experience, and new observations.
So why solar systems? The trajectories of heavenly bodies can be described by Newton's law of motion and the universal law of gravitation. I wanted to explore how systems as complex as solar systems can be described by these 2 very simple equations. Newton's law (F = m a) describes motions, that is, relates accelerations (forces), velocities (momenta), and positions. Indeed, Newton's law can be thought of as a simplified version of the laws that describe the motion of fluids, although it is rarely described that way.
We started by understanding the ordinary differential equations of Newton's law using our intuitions about traveling in a car, some algebra, and geometry. When we drive in a straight line and step on the gas in some random pattern, how far away will we end up after a certain time? It is exactly the solution to this problem that will get us to describing the planetary dance of our solar system. We had used our simple "discretized" Newton's law to make a quick program that animated how planets might move if we started them in random locations with random initial speeds, and had some fun watching these movies.
I had always been curious about why objects in space seem to always be spinning and rotating, and why solar systems and galaxies tend to be flat. Having shared that curiosity with the group today, we gave ourselves the challenge to write a program that could predict the formation of the solar system by starting with a "gas cloud" that will clump together and eventually end up as planets that revolve around a sun. I am very happy to say that the group is helping me write a program that I always wanted to try out. Here are some of the notes we have been making and their ideas for algorithm snippets.
We haven't solved all the issues yet.
To go from the solar system to the motion of molecules, we only need to change the forces that the particles feel from gravity to other kinds of forces. Once we can make that jump, we can resume our quest for understanding how fluids flow.