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.

Notes from the Chaos Cookbook, Chapters 2 and 3

The remaining portion of chapter 2 in the Chaos Cookbook involves properties of the logistic equation logistic map, which was first used by biologists and ecologists to model population growth. This function displays chaotic behavior as its parameter k is varied between 2.7 and 4, roughly.

There are regions of relative stability in the logistic equation's map's bifurcation plot of the possible output values vs. k, followed subsequently by a period doubling cascade. There is also self-similarity in the plots (I've added one of my own below).

A strange attractor for the logistic equationmap is the set of all possible output values for given k. It is strange because the values do not appear in a predictable or meaningful order.

It's been proved that a period 3 system (three possible output values) indicates the onset of chaos.

Feigenbaum's constant is roughly 4.669. It applies generally to many chaotic systems.

Chapter 3 concerns differential equations, their phase space plots and features, and numerical methods for solving them. I am largely familiar with all of this information, so I only loosely read this chapter.

Of special note in this chapter was a reminder to myself that phase space trajectories for a system of differential equations may not overlap.

Also of note was that, for these systems, a strange attractor is a phase space trajectory that does not settle into some periodic limit cycle and often shoots around to different parts of phase space.

Python code for some of these projects may be found at my personal website: http://quadrule.nfshost.com

Notes from the Chaos Cookbook, Chapter 15

I've skipped ahead to this short chapter in the Chaos Cookbook since I wanted to incorporate some of its ideas into my dissertation proposal. This chapter is entitled "An overview of complexity" and provides a brief and limited definition of what complexity is and several examples to broaden this definition.

Complexity is the study of emergent behavior from systems operating on the verge between stability and chaos. However, chaos is considered a subset of complexity. Complex systems also involve interactions between their individual components. The behavior that emerges from these interactions is often unexpected since the rules of the components don't necessarily predict this behavior.

Examples of complex systems in this book include traffic, autocatalytic systems, sand piles, and economies.

The bunching of cars and subsequent spreading out on highways is an emergent phenomenon that can depend on factors such as driver reaction times, car speeds, and the distances that drivers feel comfortable with when following other cars. I think that the variability in these individual factors leads to the random bunching of cars on the road.

The angle of repose of a sand pile is the angle that the pile makes with the horizontal plane that the pile is on. This angle emerges as the pile grows and may depend on how the pile is formed (dumping, pouring, etc.). Any changes to this angle caused by the addition of more sand leads to small avalanches that "correct" the perturbation so that the angle of repose is maintained. This is known as a self-organized critical state.

Not all sets of system behaviors can give rise to complex behavior.

Economies represent adaptive systems. In these systems, each agent adapts their behavior to the rules of the system to maximize their profits/utility. There is not one best strategy for this; rather, each agent must adapt their strategy according to what the whole system is doing to succeed.

Notes from the Chaos Cookbook, Chapter 2, pp. 37-42

Today I have more notes and insights from the Chaos Cookbook, Chapter 2. I'm trying to understand what characteristics are common to all chaotic systems and what it means for these systems to be chaotic. I'm also trying to find what links the ideas chaos, fractals, emergence, and complexity together. Chapter 2 deals with a specific type of chaotic entity: an iterated function.

"For a function to exhibit chaotic behavior, the function used has to be nonlinear." However, a nonlinear equation does not necessarily display chaotic behavior (example: y = x**2).

An iterated function is one whose value depends on the prior iteration. This is a form of mathematical feedback. Feedback is common in all chaotic systems.

There are two important parameters that describe an iterated function: the initial value and the number of iterations. Chaotic functions may be very sensitive to the initial value, meaning that very small changes to it will produce very drastic differences in the outcome of the iteration process.

Despite the sensitivity of the values of an iterated function to the initial value used, each graph may display common features.

The sequence of numbers obtained by iterating over a function is known as an orbit. Orbits may be stable, unstable, or chaotic.

Notes from The Chaos Cookbook, Chapter 1

I'm reading through a used copy of Joe Pritchard's "The Chaos Cookbook" that I recently obtained from Amazon. This post contains my notes from Chapter 1.

On page 26 he notes that Newton's laws of motion cannot predict the general result of three solid balls colliding simultaneously. At first I thought that this was not true since I remember solving problems like this in general physics. However, after further thought I realized that we can predict which direction a ball will be traveling after the collision only if we already know what happened to the other balls. In other words, we cannot predict the outcome of the collision entirely; we must know something about what happened.

The above example may be considered as a subsequent loss of information that occurs following the collision of all three balls.

Historically speaking, nonlinear systems subjected to analysis were either 1) treated only to first order where higher order effects were negligible (e.g. a simple pendulum displaced slightly from its resting position), or 2) too difficult to analyze thoroughly.

A system may appear periodic in a reduced-dimensional phase space (see Fig. 1.3). A corkscrew trajectory along one axis will appear circular when looking down this axis.

Period doubling leads to splitting in a system's power spectrum. As the number of period doublings occur, does the spectrum fill with peaks and appear as white noise (uniform power spectrum)? Does this form a link to diffusive processes? (The Facebook link to white noise contains some block diagrams. Perhaps these can be shown to be equivalent to some chaotic systems.)

Chaos can emerge both in iterated function systems and from sets of differential equations.

Chaotic systems may eventually settle into seemingly stable states and vice versa.