Figure This Out And Make A Million Bucks: Now With Penguins

First up, the penguins, from Chalkdust, (A Magazine For The Mathematically Curious) May 22:

Penguins: the emperors of fluid dynamics  

Let me guess, you like penguins? I mean, who doesn’t?! And I am guessing you like maths too? Well, what if I told you I could combine the two? Have I captured your interest?

Antarctic penguins experience some of the most extreme weather conditions on Earth, where wind speeds exceed 70mph and temperatures drop below -40°C. However, penguins have adopted an effective survival strategy. The whole colony comes together to form one large huddle to shield the birds from the wind and to conserve each penguin’s heat. While huddling, the birds are constantly waddling around, causing the entire huddle to move. But what are the underlying mechanisms driving this movement? And can we track the huddle’s movement over time?

They’re hot, then they’re cold
To answer these questions, we need to view the problem more mathematically. Intuitively, we know that the penguins are huddling together for warmth, so let’s start by thinking in terms of heat and energy. In other words, let’s consider the thermodynamics of the problem. We identify two key heat transfer processes at work.

First, the penguins are losing heat to the cold air and the fast flow of the wind means that a temperature gradient—a difference in temperature between the air and the penguins—is always sustained. You will have experienced this yourself in everyday (non-Antarctic) life. On a windy day, you feel colder, right? This is because your body loses heat to the air, and the air you just warmed up moves on and is replaced by fresh, cold air for you to lose heat to again.

Second, the penguins are receiving heat from neighbouring birds in the huddle. Once again, think about your everyday life: imagine standing in a crowded queue waiting to buy the new Zelda game. Despite the cool company you are surrounded by, you feel hotter in the crowd as you’re gaining heat from the people around you. It is the same in a penguin huddle (except they can’t play the new Zelda game: their flippers can’t hold the controller). This heating is particularly effective near the huddle centre, where temperatures can reach up to 37.5°C: higher than the average human body temperature!

Unfortunately, this temperature is actually too high and can cause the penguins to overheat. To counteract this, the birds regularly reorganise themselves in the huddle: those in the centre move to the edge and vice versa. In other words, they are all massive Katy Perry fans—by reenacting the song Hot N Cold, all members of the huddle are kept at a nice, cosy average temperature.

We can start making a model of the huddle’s movement based on these two heat transfer processes. First though, let’s add one final condition: we assume that no penguins leave or join the huddle, so the size of the huddle, ie the number of penguins, is conserved. So, there are three things in total affecting the huddle’s movement: cooling by the wind, heating by the penguins and conservation of the huddle size. These three effects become important later, so keep them in mind!

In love with the shape of μ
Our aim is to see how the entire huddle moves over time. Therefore, instead of viewing the huddle as hundreds of individual penguins (a discrete model) we treat it as a continuous ‘blob’ of penguins (a continuum model).

Imagine looking down at the penguin huddle from above and tracing around its edge. The resulting curve is called the huddle boundary and we label it $\mathrm{\mu ....}$

$\mathrm{....}$$\mathrm{MUCH\; MORE}$

And the cash money? The intro to "Fluid Dynamics (and the filth on your phone)" was:

This is one of those fields of study that are so mind-bogglingly complex that, short of having a supercomputer close to hand, we can only approximate as to the details. See also weather, markets, and any other complex/chaotic system you can think of.

So anyone who can get a handle on what is actually going on with this stuff gives a whole 'nother meaning to the concept of smart....

And from the followup post "Think You're Smart Don'tcha: Figure This Out And Make A Million Bucks": 

Turbulence, the oldest unsolved problem in physics

The flow of water through a pipe is still in many ways an unsolved problem.

Werner Heisenberg won the 1932 Nobel Prize for helping to found the field of quantum mechanics and developing foundational ideas like the Copenhagen interpretation and the uncertainty principle. The story goes that he once said that, if he were allowed to ask God two questions, they would be, “Why quantum mechanics? And why turbulence?” Supposedly, he was pretty sure God would be able to answer the first question.

The quote may be apocryphal, and there are different versions floating around. Nevertheless, it is true that Heisenberg banged his head against the turbulence problem for several years.

His thesis advisor, Arnold Sommerfeld, assigned the turbulence problem to Heisenberg simply because he thought none of his other students were up to the challenge—and this list of students included future luminaries like Wolfgang Pauli and Hans Bethe. But Heisenberg’s formidable math skills, which allowed him to make bold strides in quantum mechanics, only afforded him a partial and limited success with turbulence.

Some nearly 90 years later, the effort to understand and predict turbulence remains of immense practical importance. Turbulence factors into the design of much of our technology, from airplanes to pipelines, and it factors into predicting important natural phenomena such as the weather. But because our understanding of turbulence over time has stayed largely ad-hoc and limited, the development of technology that interacts significantly with fluid flows has long been forced to be conservative and incremental. If only we became masters of this ubiquitous phenomenon of nature, these technologies might be free to evolve in more imaginative directions.....

....MUCH MORE at Ars Technica

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At the Max-Planck-Gesellschaft:

April 02, 2013

The American Nobel Prize Laureate for Physics Richard Feynman once described turbulence as “the most important unsolved problem of classical physics”, because a description of the phenomenon from first principles does not exist. This is still regarded as one of the six most important problems in mathematics today.... 

...And The Clay Mathematics Institute. In May 2000 the Institute offered a million dollar prize per problem for solving each of seven problems in math:

Millennium Problems

 

Yang–Mills and Mass Gap
Experiment and computer simulations suggest the existence of a "mass gap" in the solution to the quantum versions of the Yang-Mills equations. But no proof of this property is known.

Riemann Hypothesis
The prime number theorem determines the average distribution of the primes. The Riemann hypothesis tells us about the deviation from the average. Formulated in Riemann's 1859 paper, it asserts that all the 'non-obvious' zeros of the zeta function are complex numbers with real part 1/2.

P vs NP Problem
If it is easy to check that a solution to a problem is correct, is it also easy to solve the problem? This is the essence of the P vs NP question. Typical of the NP problems is that of the Hamiltonian Path Problem: given N cities to visit, how can one do this without visiting a city twice? If you give me a solution, I can easily check that it is correct. But I cannot so easily find a solution.

Navier–Stokes Equation
This is the equation which governs the flow of fluids such as water and air. However, there is no proof for the most basic questions one can ask: do solutions exist, and are they unique? Why ask for a proof? Because a proof gives not only certitude, but also understanding.

Hodge Conjecture
The answer to this conjecture determines how much of the topology of the solution set of a system of algebraic equations can be defined in terms of further algebraic equations. The Hodge conjecture is known in certain special cases, e.g., when the solution set has dimension less than four. But in dimension four it is unknown.

Poincaré Conjecture
In 1904 the French mathematician Henri Poincaré asked if the three dimensional sphere is characterized as the unique simply connected three manifold. This question, the Poincaré conjecture, was a special case of Thurston's geometrization conjecture. Perelman's proof tells us that every three manifold is built from a set of standard pieces, each with one of eight well-understood geometries.

Birch and Swinnerton-Dyer Conjecture
Supported by much experimental evidence, this conjecture relates the number of points on an elliptic curve mod p to the rank of the group of rational points. Elliptic curves, defined by cubic equations in two variables, are fundamental mathematical objects that arise in many areas: Wiles' proof of the Fermat Conjecture, factorization of numbers into primes, and cryptography, to name three.

One of the problems, The Poincaré Conjecture, was solved by Russian mathematician  Grigori Perelman. He turned down the award and the million dollars. He has also turned down The Fields Medal, the highest award in mathematics.

The other six problems are still open, with the Navier–Stokes Equation being the object of our affection.