The Boundary Layer

In Chapter 1 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I introduce a concept from fluid dynamics called the boundary layer.

The behavior of a sphere moving through a fluid illustrates how flow behavior depends on Reynolds number... At very high Reynolds number, viscosity is small but still plays a role because of the no-slip boundary condition at the sphere surface. A thin layer of fluid, called the boundary layer, sticks to the solid surface, causing a large velocity gradient and therefore significant viscous drag.

Life in Moving Fluids,
by Steven Vogel.

For readers who want to know more about the boundary layer, let me quote the start of Chapter 8 of Steven Vogel’s masterpiece Life in Moving Fluids.

At the interface between a stationary solid and a moving fluid, the velocity of the fluid is zero. This, of course, defines the no-slip condition… The immediate corollary of the no-slip condition is that near every such surface is a gradient in the speed of flow. Entirely within the fluid, speed changes from that of the solid to what we call the “free stream” velocity some distance away. Shearing motion is inescapably associated with a gradient in speed, so in these gradients near surfaces, viscosity, fluids’ antipathy to shear, works its mischief, giving rise to skin friction and consequent power consumption. The gradient region is associated with the term “boundary layer”…

The boundary layer… wasn’t so much discovered as it was invented, in the early part of this century, as a great stroke of genius of Ludwig Prandtl. Recognizing the origin of this notion is crucial. In the basic differential equations for moving fluids, the Navier-Stokes equations, some terms result from the inertia of fluids and some from their viscosity… The Reynolds number gives an indication of the relative importance of inertia and viscosity… At Reynolds numbers below unity, inertia can be ignored and nicely predictive rules nonetheless derived—[such as] Stokes’ law for the drag of a sphere… At high Reynolds number, one might expect to get away with neglecting viscosity… It may sound neat, but it all too commonly gets us in trouble—results diverge from physical reality, drag vanishes, and d’Alembert has his paradox.

Prandtl reconciled practical and theoretical fluid mechanics at high Reynolds numbers by recognizing that viscosity could never be totally ignored. What changes with Reynolds number was where it had to be taken into account; initially it mattered everywhere, but as the Reynolds number increased well above unity, viscosity made a difference only in the gradient regions near surfaces. These regions might be small, and they might get ever smaller… as the Reynolds number increased; but as long as the no-slip condition held, a place had to exist where shear rates were high and viscosity was significant. Prandtl called the place in question… the “boundary layer.” In general, a higher Reynolds number implies a thinner boundary layer but a higher shear rate in that boundary layer.

To learn more about the biological significance of the boundary layer see the Chapter 9 in Life in Moving Fluids, which is all about “Life in Velocity Gradients.”

Boundary Layer Theory,
by Schlichting and Gersten

If you want a more rigorous and mathematical analysis of boundary layers, I recommend Boundary Layer Theory by Hermann Schlichting and his student Klaus Gersten. The eighth edition of this book (2000) is cited in IPMB; a revised and updated ninth edition was published in 2017. Schlichting and Gersten write

At the end of the 19th century, fluid mechanics had split into two different directions which hardly had anything more in common. On one side was the science of theoretical hydrodynamics, emanating from Euler’s equations of motion and which had been developed to great perfection. However this had very little practical importance, since the results of this so-called classical hydrodynamics were in glaring contradiction to everyday experience. This was particularly true in the very important case of pressure loss in tubes and channels, as well as that of the drag experienced by a body moved through a fluid. For this reason, engineers, on the other side, confronted by the practical problems of fluid mechanics, developed their own strongly empirical science, hydraulics. This relied upon a large amount of experimental data and differed greatly from theoretical hydrodynamics in both methods and goals.

It is the great achievement of [German scientist] Ludwig Prandtl [1875–1953] which, at the beginning of this century, set forth the way in which these two diverging directions of fluid mechanics could be unified. He achieved a high degree of correlation between theory and experiment, which, in the first half of this century, has led to unimagined successes in modern fluid mechanics. It was already known then that the great discrepancy between the results in classical hydrodynamics and reality was, in many cases, due to neglecting the viscosity effects in the theory. Now the complete equations of motion of viscous flows (the Navier Stokes equations) had been known for some time. However, due to the great mathematical difficulty of these equations, no approach had been found to the mathematical treatment of viscous flows (except in a few special cases). For technically important fluids such as water and air, the viscosity is very small, and thus the resulting viscous forces are small compared to the remaining forces (gravitational force, pressure force). For this reason it took a long time to see why the viscous forces ignored in the classical theory should have an important effect on the motion of the flow.

In his lecture on “Über Flüssigkeitbewegung bei sehr kleiner Reibung” (On Fluid Motion with Very Small Friction) at the Heidelberg mathematical congress in 1904, L. Prandtl... showed how a theoretical treatment could be used on viscous flows in cases of great practical importance. Using theoretical considerations together with some simple experiments, Prandtl showed that the flow past a body can be divided into two regions: a very thin layer close to the body (boundary layer) where the viscosity is important, and the remaining region outside this layer where the viscosity can be neglected. With the help of this concept, not only was a physically convincing explanation of the importance of the viscosity in the drag problem given, but simultaneously, by hugely reducing the mathematical difficulty, a path was set for the theoretical treatment of viscous flows. Prandtl supported his theoretical work by some very simple experiments in a small, self–built water channel, and in doing this reinitiated the lost connection between theory and practice. The theory of the Prandtl boundary layer or the frictional layer has proved to be exceptionally useful and has given considerable stimulation to research into fluid mechanics since the beginning of this century. Under the influence of a thriving flight technology, the new theory developed quickly and soon became, along with other important advances—airfoil theory and gas dynamics—a keystone of modern fluid mechanics.

Introductory Fluid Mechanics L19 p2 — The Boundary Layer Concept.

https://www.youtube.com/watch?v=k37vPSA3E1g

 

E. Bodenschatz — Ludwig Prandtl (1875–1953)

https://www.youtube.com/watch?v=cv952Nhc_vs

Maurice de Broglie and the First Observation of an X-ray Absorption Edge

If you shine x-rays through a material and measure the number absorbed by it, you create an x-ray absorption spectrum. The absorption is related to the cross section; the bigger the cross section, the more the x-rays are absorbed. Figure 15.2 from Intermediate Physics for Medicine and Biology is shown below, where the cross section for carbon is plotted as a function of the x-ray energy. I’ve drawn an oval around what’s the most interesting feature of the plot, the jump in the cross section at an energy of about 0.28 keV. This abrupt rise is known as the K edge, and is an example of an absorption edge. 

A slightly modified version of Figure 15.2 from
Intermediate Physics for Medicine and Biology.

The cross section jumps up when the photon’s energy rises above the binding energy of a K-shell electron [an electron in the innermost energy level]. It’s not a small effect; the cross section increases by more than a factor of ten at the K edge (note that this is a log-log plot). 

When I see such a dramatic effect, I imagine how surprising it must have been for the person who observed it first. Who was the person who discovered the K edge? Maurice de Broglie.

Maurice de Broglie in 1932.

Maurice was the elder brother of the more-famous Louis de Broglie, who Russ Hobbie and I mention when talking about electron waves and the electron microscope. Maurice was born in Paris in 1875. After more than a decade in the French navy, he left the military to study physics. He was interested in x-rays, which were discovered by Wilhelm Röntgen in 1895. In 1913, Maurice published the first observation of an absorption edge (Comptes Rendus, Volume 157, Pages 924–926). When World War I began, he went back to the navy to do research on detecting U-boats (German submarines). After a long career in science, including being awarded the Hughes Medal by the Royal Society of London, he died in 1960 at the age of 85.

Farrel Lytle, an x-ray spectroscopy pioneer, tells Maurice’s story in his review article (Journal of Synchrotron Radiation, Volume 6, Pages 123–134, 1999).

Although Röntgen represents the beginning of X-ray science, the remarkable de Broglie royal family has been significant in both the world of science and the history of France. It has been said that if Maurice did nothing more than convince his younger brother, Louis, to drop his study of history and begin a career in science, he should be memorialized for that alone. But he did considerably more than that. His work in X-ray and atomic physics was innovative and important. Maurice had begun a career as a naval officer, but became interested in the exciting new world of X-rays and physics and resigned his commission. Beginning in the laboratory of Paul Langevin working on the ionization of gases by X-rays, he later built his own laboratory in his personal mansion on rue Châteaubriand. There he became the first in France to work with X-ray diffraction. During these experiments he invented X-ray spectroscopy. The experimental innovation came about when he mounted a single crystal on the cylinder of a recording barometer where the clockwork mechanism rotated it around its vertical axis at 2° h⁻¹. As the crystal rotated, all angles between the incident beam and the diffraction planes (hence, all X-ray energies) were recorded on a photographic plate. In this way he obtained an X-ray line spectrum from the tube with sharp and diffuse lines, bands etc. Two of the absorption bands proved to be the K edges of Ag and Br in the photographic emulsion. This was the first observation of an absorption edge (de Broglie, 1913). It took a few more experiments to reach the correct interpretation of the absorption edges. After the end of the First World War, Maurice gathered a large group of young scientists, all working on X-ray diffraction or X-ray spectroscopy, at the laboratory in his home. Joining him in his work were, among others, Alexandre Dauvillier, Jean Thibaud, Jean-Jacques Trillat, Louis Leprince-Ringuet (all were major contributors to the field of X-ray science) and his young brother, Louis. Maurice’s scientific work and his social position soon made him a major player in the science world.

Apparently it took a while to figure out that the absorption edges belonged to materials in the photographic film and not the x-ray tube or the crystal, but eventually it was all sorted out. A German scientist, Julius Hedwig (1879–1936), independently studied x-ray spectroscopy, and may have observed an x-ray absorption edge before Maurice, but he soon abandoned the work while Maurice pursued it further, becoming the father of x-ray spectroscopy.

Paul Horowitz Discusses The Art of Electronics

The Art of Electronics,
by Horowitz and Hill.

Nine years ago I wrote in this blog about the second edition of Horowtiz and Hill’s textbook The Art of Electronics. At the end of that post I hinted that a new edition of their book was in the works. The third edition of The Art of Electronics appeared in 2015, just in time for Russ Hobbie and me to cite it in the fifth edition of Intermediate Physics for Medicine and Biology.

Recently, I stumbled upon a delightful YouTube video of an interview with Paul Horowitz, explaining how The Art of Electronics began. I’ll keep this post brief, so you’ll have time to watch the video. The host is Limor Fried, who goes by the moniker Ladyada in honor of computer programing pioneer Ada Lovelace. Fried owns the electronics company Adafruit Industries, which is a cross between a business and an educational organization. Notice that during the interview Fried wears a “transistor man” tee shirt; I remember reading about transistor man in The Art of Electronics when I was designing a circuit in John Wikswo’s lab during graduate school.

Enjoy the video, and make The Art of Electronics your go-to book for designing circuits; or, just read it for fun.

Ladyada interview with Paul Horowitz, author of The Art of Electronics.
  https://www.youtube.com/watch?v=iCI3B5eT9NA

 

Meet Limor “Ladyada” Fried at Adafruit Industries.
  https://www.youtube.com/watch?v=SpYMgScKRwk

Thomas Young, Biological Physicist

The Last Man Who Knew Everything,
by Andrew Robinson.

Almost ten years ago in this blog, I speculated about who was the greatest biological physicist of all time, and suggested that it was the German scientist Hermann von Helmholtz. Today, I present another candidate for GOAT: the English physicist and physician Thomas Young. Young’s life is described in Andrew Robinson’s biography The Last Man Who Knew Everything.

Young (1773–1829) went to medical school and was a practicing physician. How did he learn enough math and physics to become a biological physicist? In Young’s case, it was easy. He was a child prodigy and a polymath who learned more through private study than in a classroom. As an adolescent he was studying optics and building telescopes and microscopes. As a teenager he taught himself calculus. By the age of 17 was reading Newton’s Principia. By 21 he was a Fellow of the Royal Society.

Some of his most significant contributions to biological physics were his investigations into physiological optics, including accommodation and astigmatism. In Intermediate Physics for Medicine and Biology, Russ Hobbie and I state that the “ability of the lens to change shape and provide additional converging power is called accommodation.” Robinson describes Young’s experiments that proved the changing shape of the lens of the eye is the mechanism for accommodation. For instance, he was able to rule out a mechanism based on changes in the length of the eyeball by making careful and somewhat gruesome measurements on his own eye as he changed his focus. He showed that patients whose lens had been removed, perhaps because of a cataract, could no longer adjust their focus. He also was one of the first to identify astigmatism, which Russ and I describe as “images of objects oriented at different angles… form at different distances from the lens.”

Young’s name is mentioned in IPMB once, when analyzing the wave nature of light: “Thomas Young performed some interference experiments that could be explained only by assuming that light is a wave.” The Last Man Who Knew Everything describes Young’s initial experiment, where he split a beam of light by letting it pass on each side of a thin card, with the beams recombining to form an interference pattern on a screen. Young presents his famous double-slit experiment in his book A Course of Lectures on Natural Philosophy and the Mechanical Arts. Robinson debates if Young actually performed the double-slit experiment or if for him it was just a thought experiment. In any case, Young’s hypothesis about interference fringes was correct. I’ve performed Young’s double-slit experiment many times in front of introductory physics classes. It establishes that light is a wave and allows students to measure its wavelength. Interference underlies an important technique in medical and biological physics described in IPMB: Optical Coherence Tomography. 

A green laser passing through two slits 0.1 mm apart produces an interference pattern.
Photo by Graham Beards, published in Wikipedia.

Young also studied color vision based on the idea that the retina can detect three primary colors. This work was rediscovered and further developed by Helmholtz fifty years later. Young was also one of the first to suggest that light is a transverse wave and therefore can be polarized.

In Chapter 1 of IPMB, Russ and I define the Young’s modulus, which relates stress to strain in elasticity and plays a key role in biomechanics. Young also studied capillary action and surface tension, two critical phenomena in biology.

Was Young a better biological physicist than Helmholtz? Probably not. Was Young a better scientist? It’s a close call, but I would say yes (Helmholtz had nothing as influential as the double slit experiment). Was Young a better scholar? Almost certainly. In addition to his scientific contributions, he had an extensive knowledge of languages and helped decipher the Rosetta Stone that allowed us to understand Egyptian hieroglyphics. He really was a man who knew everything.