The Reynolds number: A case-study about a simplified concept

Lately I've been re-reading some older papers on random walks in colloids to prepare for my dissertation proposal. One such paper from a 2007 PRL issue is entitled "Self-Motile Colloidal Particles: From Directed Propulsion to Random Walk." The first sentence reads

"The directed propulsion of small scale objects in water is problematic because of the combination of low Reynolds number and Brownian motion on these length scales."

This struck me as particularly interesting because I had never considered the Reynolds number as useful for describing particle transport within a fluid; rather I always imagined it as a number that somehow answered the yes-or-no question "is this bulk fluid turbulent when flowing?" This conceptualization of the Reynolds number comes from my fluid and thermal systems class in the sophomore engineering curriculum at Rose-Hulman. Since I am so much wiser now than I was then, I decided to re-examine this number and its importance.

After consulting Wikipedia, I now understand the Reynolds number as a dimensionless ratio of the magnitude of the inertial forces transmitted by the fluid to the viscous forces. In other words, higher Reynolds numbers means that the fluid molecules will all move in more-or-less the same direction for longer periods of time and over larger domains. If something perturbs a region of the fluid at very large Reynolds numbers, the perturbation is transmitted by a large-scale, correlated motion of the molecules. This is because the dissipation of the fluid does not suffice to damp this motion.

Going further, this picture explains why scientists who study complexity and emergence love turbulence. If a system can not quickly dampen a fluctuation (in this case a small pocket of correlated motion within the fluid), it grows chaotically into a large-scale turbulence. Often, ordered patterns of fluid motion emerge from this chaos, like in Benard cells. So the Reynolds number leads to much more than a simple answer to a yes-or-no question; the physics behind it describes the chaos and complexity of turbulence itself.

This introspective exercise also demonstrates the short-comings of over-simplifying a concept when teaching it to others. I was blind to the connections between Reynolds number, the microscopic behavior of a turbulent fluid, chaos, and emergence for so long* because I only considered the yes-or-not question above, not the concept behind the number.

*It's been 8 years since I took that class. Yikes, I'm getting old.

A point about negative temperatures

Negative temperatures occur when the derivative of entropy with respect to a system's energy is negative. In other words, the entropy decreases with added energy. As Daniel Schroeder points out in his Introduction to Thermal Physics, this may only occur when the total energy that a system may take is limited, such as a two-state paramagnet. In other, more common systems, such as a gas in a container, the total energy that the system may absorb is practically unlimited. This is why negative temperatures are not observed.

I think a lot of confusion in learning thermodynamics is that the common sense notion of temperature is very different from its thermodynamic definition. Other ideas, such as force or work, do not contradict common sense quite so much and are more readily adopted.

Finally, Schroeder references this article for an experiment in which negative temperature was observed.

I love when new concepts are explained by old ones

I just now learned that photon bunching has an entirely classical analog which corresponds to the fluctuations of the intensity of light with a finite bandwidth. I am well-versed with concepts dealing with the latter as they are tied to optical coherence theory, an important group of concepts in my research. However, I never properly learned what causes photon bunching and always regarded it as somewhat mysterious. Placing it in terms of something I know well is deeply satisfying to me.

I think this type of reaction is true for many scientifically-minded people. When a mysterious phenomenon can be explained by something familiar, we feel satisfied in placing the phenomenon within the logical, mental framework we've built for ourselves. It also explains why creating new concepts and shattering old ones can be so difficult.

Teachers vs. students: when interests don't coincide

I recently had a discussion with a close friend who is not a physicist by training but is very interested in many popular physics ideas. I was attempting to explain several paradoxes and non-intuitive scenarios such as Schroedinger's cat, the twins paradox, and the relativity of simultaneity. However, my friend was not impressed by some of these ideas, like Schroedinger's cat, and proclaimed them as "stupid."

I was of course exasperated by her failure to appreciate these concepts. Out of frustration I refused to explain any further ideas despite her questioning. As you could imagine, my refusal angered her; she interpreted my actions as pompous and arrogant. Fortunately, we are both open-minded to our own faults and quickly apologized to one another.

This bout made me realize that a teacher can't always convey why a topic is interesting. Interest is, after all, a personal attribute that varies between individuals. This mismatch of interests can cause a great deal of friction between a teacher and student and should be recognized by both sides for education to be successful.

Teachers must concede that sometimes students just aren't interested in a topic. A good teacher will be patient, even with difficult students, when they encounter a lack of interest. Eventually, the student will show interest in something that the teacher can help them learn about. Students should acknowledge when they are uninterested but still respect their teachers' enthusiasm. More importantly, they should not interpret their lack of interest as a failure to understand subtleties of a topic.

What's not so clear to me is with whom the greater responsibility for learning should lie. I'm inclined to place the greater burden on the student.

How best to view the internet as a learning tool

I work on some projects that require knowledge of basic biology, such as cell structure, biochemistry, and laboratory technique. However, I was trained as a physicist and engineer, and, as a result, have had an extremely limited education in the biological sciences. For example, my last biology class was anatomy during my junior year in high school.

The internet has been essential in bringing me up to speed in these topics. I've put resources such as MIT's OpenCourseWare and the independent OpenWetWare to good use. Companies such as Invitrogen provide valuable tutorials and explanations on laboratory practices as well. The best part about these resources is that I can find exactly the information that I need to know when I need to know it.

I believe that a very few doubt the usefulness of the web as a learning tool, but how to use it as a tool is certainly a topic of debate. Based on my own experiences, I think that internet learning is best used as an independent collection of bits of knowledge that are accessed as needed.

Let's break this definition down into parts. By independent, I mean that the value of internet resources is determined by the individual who needs to know something. A catalog of optical parameters of semiconductor materials will likely serve little purpose to a field biologist. The downside to this is that the web must contain an exhaustive amount of knowledge to be useful to everyone. If there's a possibility that someone may wish to know something, then it must be contained already on the web [1].

"Bits of knowledge" makes intuitive sense, but a formal definition may not exist. If I wish to know how to stain a cell using immunofluorescence, is each step considered a bit of knowledge, or is the entirety of the process considered one "chunk?" I don't think that this detail is particularly relevant to my discussion, but it is interesting to think about how one may quantify knowledge [2].

Finally, the ability to access knowledge as needed makes it efficient. The human brain can only hold on to a limited amount of data. Some details are best stored on machines; otherwise numerous human specialists would be required to perform complex tasks, each one intimate with one small part of the task. In my graduate work, I can learn about cytoskeletal filaments as needed, or my advisor could hire on a cell biologist to consult me on a small number of issues. The first option is decidedly cheaper. In addition, ease of access is important, and spans topics such as mobile devices, bringing the internet to developing countries, and search algorithms.

So, in my opinion, internet learning is best utilized as a user-valued collection of information that is accessed accordingly. Communications through the internet, such as e-mail correspondence with teachers, is important, and is compatible with my definition since I do not put limits on how knowledge is delivered. Failure to properly use the internet as a learning tool usually comes from poor access (e.g. bad search engine algorithms) or a user improperly identifying what they need to know. In the last case, the success of internet learning cannot be determined by machines; like many things it boils down to the human element.

[1] I can't get the thought of the internet as a causal knowledge database out of my head right now, since it can only contain knowledge that has already been generated. It will never contain knowledge from the future, unless, perhaps, new knowledge can be generated from data it already holds, but that opens the question of the definition of knowledge.

[2] Information theory comes to mind here. The information content of a signal is quantified as a logarithm of the number of symbols in the signal.