
Airflow is simulated over and past the wing of a high performance aircraft that is using vectored thrust while descending to a few feet above the ground (in ground effect). See here
The Navier-Stokes Equation is a partial differential equation that models fluid flow. Fluids include liquids and gases. Using Navier Stokes we can understand diverse phenomena like airflow over aeroplane wings or rocket bodies, the flow of liquids through pipes and the flow of plasma in stars (magnetohydrodynamics). There are many computer programs available that will divide a fluid flow problem into discrete, finite sized pieces and solve the problem for you (the so-called finite element analysis method). NASA, major defense contractors and car manufacturers all use these programs to solve the sort of problems listed above.

Above: Navier Stocks Equation – Incompressible Liquid with Constant Viscosity (Newtonian Liquid)
It may come as a surprise that even though Navier-Stokes in used so extensively in physics, engineering and industry we know little about the theoretical underpinnings of these equations. Anybody who can answer certain fundamental questions about the Navier-Stokes equations can win a \$1 million award from the Clay Mathematics foundation.
The lack of adequate knowledge about the Navier-Stokes equations represent one of the greatest unsolved problems in 21st century mathematics. The announcement of the \$1 million award certainly attracted my attention but I found Navier-Stokes equations to be so interesting that I taught myself some basics. I would like to share some useful resources:
A good introduction to Navier-Stokes equations can be found on the following links:
(1) Navier Stokes Derivation
(2) Navier Sokes Equations
(3) A infinitesimal element approach to deriving Navier Stokes
(4) A slightly bizarre but fun link
But the pièce de résistance has got to be the video* of the problem description by the Clay Mathematics Institute. Its an elementary description of the problem that anyone with an understanding of vector calculus can comprehend.
Tip: Read links (1) & (2) of wikipedia before listening to the lecture. You will gain more out of it. In the lecture Cafarelli uses non-standard notation to denote derivatives. For example,
means
.
__
(*select Navier Stokes existence and smoothness by Luis Caffarelli, University of Texas)
The upcoming revolution in education
Are you stuck in a not so good university in Kenya or Bihar? Do you not have time for university? Or are you sick of plodding through dry textbooks and bad course materials? Are you not happy with your current instructors?
Don’t worry.
If you have a reasonably high speed internet connection (~256kbps) then the Internet is the answer to all your problems.
Most of the great universities of the world have put all their coursework online. This means that each course has a website. On this website the instructor lists the textbooks he uses, posts weekly reading assignment and problem sets, lecture notes, tests and exams etc. for the world to access. Motivated students not enrolled at that university can study the course themselves through self study.
MIT and the University of California, Berkeley have gone one step further from the above paradigm…They have video taped and put on the web many of their course lectures.
I have been watching video lectures from MIT and Berkeley for the last two years. Just a couple of hours ago I watched the first lecture of an Introduction to Astronomy course (Astro C10 at Berkeley). I plan to “take” the whole course gradually. Then I watched a press conference of the 2006 Nobel Prize winner, George Smoot (see blog entry). After seeing these two lectures I am so charged up and inspired that I just want to share with you the joys of seeing these 50 minute symphonies of reason, insight and humor. Please don’t miss the opportunity to learn from some of the best teachers in the world. For free.
Its easy to understand why putting all course material and video lectures on the Internet is revolutionary:
(1) The best professors in the world can teach as many students as they want, not only those fortunate to get admission into select schools
(2) You can pursue your interest by watching lectures in Astronomy, History, Physics, Math and who-knows-what. Everyone has an urge to learn and grow intellectually but may not have the time to attend night school or university part-time. Videotaped lectures allow flexibility and still allow for high quality inputs to learners.
(3) Lecturers around the world can learn from the best in the business and improve their own skills. Poor universities can show these lectures in classrooms and supplement these lectures with discussions where the local instructor refines and explains what has already been taught. This way you get the best of both worlds: world class lectures and interactive, classroom learning.
(4) There are some other unintended positive benefits. For too long, many universities have gotten away with poor quality instruction because students didn’t know any better. Today you can watch a great lecture from an MIT professor, share it amongst your friends and come to understand what high quality is. Only when you know what high quality is, can you demand quality. Opencourseware video lectures are a positively disruptive technology because they introduce an element of comparison in education. Think back to the days of state controlled TV: Dry, drab and low quality. Then Cable TV arrived and showed us what was happening in the world. Aspirations rose for better quality products, better quality governance and better treatment of citizens. Even drab Indian state TV (Doordarshan) is much better now. Opencourseware promises to do the same for our universities.
Check these links out. Guys please spread the world. People in India are hungry to learn but still don’t know about opencourseware. Some of you may complain that broadband is not easily available in India. Thats true but the situation is fast improving.
(1) MIT Opencourseware site
(2) Video Lectures available at MIT
(3) Berkeley webcast site. Audio is available for most and video for some courses.
(4) A great directory for video lectures
Let our universities spread knowledge far and beyond! Universities should be like lighthouses that spread education far and wide. Let knowledge not be the preserve of the lucky few!
Now this is not the prescription for state control of university tuition. Some might argue: lets keep tuitions very low and just let everyone in college. That would be real open access. Something like that has already been attempted in India with disastrous consequences. For instance, some college tuitions have not been (allowed to be) revised for many decades in the name of open-access and social engineering. Bereft of financial autonomy, many colleges became totally dependent on the government for funding. With government money came corruption, mediocrity and bureaucracy. Universities have become psuedo-government departments where the best and the mediocre are treated at par. Promotions are time bound and salaries are fixed by the government. Every small decision needs to be routed through the government. These are hardly breeding grounds for future Nobel Prize winners. (This is a simplistic view but the topic of this blog is opencourseware so I’ll get back to that soon. A more refined treatment of the decline of Indian university system will have to wait…)
I think the American university system provides a wonderful example of pragmatism. Universities are financially solvent and as a result quite autonomous. Some of them may charge a bomb (e.g. MIT) but they invest that in top quality research and teaching. Loans are provided to needy students and poisonous subsidies avoided. American universities are not elitist as some jealous countries like to point out. Initiatives like opencourseware prove that.
In my view, Opencourseware represents wonderful balance between spreading knowledge in the world at large and maintaining high standards within your institution. Long live Opencourseware!