The Viscous Torque on a Rotating Sphere

In Section 9.10 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I write

We saw in Chap. 4 (Stokes' law) that the translational viscous drag on a spherical particle is 6πηav. Similarly, the viscous torque on a rotating sphere is 8πηa³(dθ/dt).

Let’s calculate this torque. We always learn something when we see where such a result comes from.

To begin, we will redo Homework Problem 46 from Chapter 1 that asks you to calculate the translational Stokes’ law by considering a stationary sphere in a moving viscous fluid (equivalent to a sphere moving through a stationary viscous fluid). Below is the analogous problem for a sphere rotating in the same fluid.

Problem 46 ½. Consider a sphere of radius a rotating with angular velocity ω in a fluid of viscosity η. For low Reynolds number flow, the fluid velocity and pressure surrounding the sphere are

          v_φ = ω a³ sinθ/r²

          v_r = v_θ = p = 0.

(a) Show that the no-slip boundary condition is satisfied.

(b) Integrate the shear torque over the sphere surface and find an expression for the net viscous torque on the sphere.

When I first tried to solve part (b), I kept getting an answer that was off by a factor of 2/3. I checked my work several times, but I couldn’t find any mistake. After much fussing, I finally figured out my error. For the shear stress at the sphere surface, I was using η dv_φ/dr. This seemed right at first, but it’s not. The shear stress is actually η (dv_φ/dr – v_φ/r). Why? I could just say that I looked up the expression for the shear strain ε_rφ for spherical coordinates and found it had two terms. But that’s no fair (and no fun). We have to understand what we are doing, not just look things up. Why does the expression for the shear stress have two terms?

Let’s start on page 16 of IPMB, where Russ and I note that the shear stress is the viscosity times the rate of change of the shear strain. We need to see how the shear strain changes with time. There are two cases.

1. The first case will give us the familiar dv/dr expression for the shear stress. Consider an element of fluid with thickness dr, as shown below.

The velocity is in the φ direction, and depends on r. In time T, the top surface of the box moves to the right a distance v_φ(r+dr) T, while the bottom surface moves only v_φ(r) T, forming the dashed box in the figure. The shear strain is the angle θ (see Problem 14 in Chapter 1). Consider the shaded triangle having height dr and angle θ. The length of the bottom side of the triangle is v_φ(r+dr) T – v_φ(r) T. The tangent of θ is therefore

          tanθ = (v_φ(r+dr) T – v_φ(r) T) / dr .

In the limit as dr goes to zero, and for small angles such that tanθ is approximately θ, the shear strain becomes dv_φ/dr T. Therefore, the rate of change of the shear strain is dv_φ/dr, and the contribution to the shear stress is η dv_φ/dr.

This is where I got stuck, until I realized there is a second case we must consider.

2. Even if v_φ does not change with r, we can still get a shear strain because of the curvilinear coordinates. Consider the arc-shaped element of fluid shown below.

Suppose the fluid moves with the same speed, v_φ, on both the top and bottom surfaces. After time T, the fluid element moves to the right and forms the dashed element. The problem is, this dashed shape is no longer an arc aligned with the curvilinear coordinates. It has been sheared! Consider the shaded triangle with angle θ. The top side has a length (r+dr) θ – v_φ T, and the right side has length dr. The ratio of these two sides is tanθ, or for small angles just θ. So

          θ = [(r+dr) θ – v_φ T]/dr

Solving for θ gives

          θ = (v_φ/r) T,

so the shear stress is η v_φ/r.

Notice that in the first case the top side is sheared to the right, whereas in the second case it is sheared to the left. We need a minus sign in case two.

In general, both of these effects act together, so the shear stress is η (dv_φ/dr – v_φ/r).

For a velocity that falls as 1/r², the dv_φ/dr term gives -2/r³, while the -v_φ/r term gives -1/r³, with a sum of -3/r³. I was getting a factor of two when I was supposed to get a factor of three. 

Are you still not convinced about the second term in the stress? Look at it this way. Suppose the velocity were proportional to r. This would imply that the fluid was rotating as if it were a solid body (all the fluid would have the same angular velocity). Such a pure rotation should not result in shear. If we only include the dv_φ/dr term, we would still predict a shear stress. But if we include both terms they cancel, implying no stress.

Let me outline how you do the integral in part (b) of the homework problem above. The torque is the force times distance. The distance from the axis of rotation to the surface where the shear acts (the moment arm) is a sinθ. The force is the shear stress times the area, and the area element is a² sinθ dθ dφ. You end up getting three factors of sinθ: one from the moment arm, one from the shear stress, and one from the area element, so you have to integrate sin³θ.

If you want, you can add a third part to Homework Problem 35 in Chapter 1:

(c) Show that the velocity distribution in Problem 46 ½ is incompressible by verifying that the divergence of the velocity is zero.

Appendix L will help you calculate the divergence in spherical coordinates.

Finally, how did I get the velocity distribution v_φ = ω a³ sinθ/r² that appeared in the homework problem? When the pressure is zero, the velocity during low Reynolds number flow, known as Stokes flow, obeys ∇²v = 0. This is a complicated equation to solve, because v is a vector. In Cartesian coordinates, the Laplacian of a vector is just the Laplacian of its components. In curvilinear coordinates, however, the r, θ, and φ components of the vector mix together in a complicated mess. I will let you try to sort that all out. Don’t say I didn't warn you.

Is Magnetic Resonance Imaging Safe?

In Chapter 18 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss safety issues in magnetic resonance imaging.

Safety issues in MRI include forces on magnetic objects in and around the patient such as aneurysm clips, hairpins, pacemakers, wheel chairs, and gas cylinders (Kanal et al. 2007), absorbed radio-frequency energy (Problem 21), and induced currents from rapidly-changing magnetic field gradients. The rapid changes of magnetic field can stimulate nerves and muscles, cause heating in electrical leads and certain tattoos, and possibly induce ventricular fibrillation. Induced fields are reviewed by Schaefer et al. (2000). Cardiac pacemakers are being designed to be immune to the strong—and rapidly varying—magnetic and rf fields (Santiniet al. 2013).

Recently, two more safety issues have emerged. The first is the possibility of genetic damage caused by MRI. This question is examined in the article “Will an MRI Examination Damage Your Genes?” by Kenneth Foster, John Moulder, and Thomas Budinger (Radiation Research, Volume 187, Pages 1-6, 2017). Foster and Moulder are cited extensively in Chapter 9 of IPMB, when we discuss the risks of low-frequency electric and magnetic fields. They are two of my heroes in the fight against pseudoscience. Budinger has studied MRI safety for years. They write

We conclude that while a few studies raise the possibility that MRI exams can damage a patient’s DNA, they are not sufficient to establish such effects, let alone any health risk to patients. Based on the failure of decades of research on biological effects of static and RF fields to establish genotoxic effects of such fields at levels comparable to those used in clinical MRI, we consider that genotoxic effects of MRI are highly unlikely. The likely increase in risk, if it were present at all, from a one-off MRI exam would surely be very small and possibly nil, but could not be proven to be zero.

In my opinion, the phrase “highly unlikely” is generous.

A second, more serious, safety issue is risks associated with the MRI contrast agent gadolinium. In IPMB, Russ and I explain

Differences in relaxation time are easily detected in an [MRI] image. Different tissues have different relaxation times. A contrast agent containing gadolinium (Gd³⁺), which is strongly paramagnetic, is often used in magnetic resonance imaging. It is combined with many of the same pharmaceuticals used with ^99mTc [an isotope used extensively in nuclear medicine], and it reduces the relaxation time of nearby nuclei. Gadolinium has been used to assess ischemic myocardium.

MRI using gadolinium was recently discussed in a point/counterpoint article (“The Use of Gadolinium-Based Contrast Agents Should be Discontinued Until Proven Safe,” Medical Physics, Volume 44, Pages 3371–3374, 2017). Moderator Colin Orton writes

Gadolinium-based contrast agents (GBCAs) are widely used in MRI to increase the visibility of tissues. Some believe, however, that due to their documented toxicity, clinical use of these agents should be discontinued until proven safe. This is the premise debated in this month’s Point/Counterpoint. Arguing for the Proposition is Stacy Matthews Branch, Ph.D. Dr. Branch is a biomedical consultant, medical writer, and veterinary medical doctor…. Arguing against the Proposition is Michael F. Tweedle, Ph.D. Dr. Tweedle is the Stefanie Spielman Professor of Cancer Imaging and Professor of Radiology at The Ohio State University.

I am a big fan of point/counterpoint articles, and we discuss one every Friday in my Medical Physics class. This debate has more substance than the genetic damage controversy, but I tend to agree with Tweedle when he concludes

But is dissociated Gd a risk factor beyond NSF [Nephrogenic Systemic Fibrosis, a disease shown to be associated with some Gadolinium-based contrast agents]? At what level and for what? Research to better understand the risks of GBCAs should certainly continue. But discontinuation of all GBCAs would result in complete loss of their benefit, probably in loss of human life due to inaccurate or imprecise diagnosis, while we search for an hypothesized chronic toxicity of unknown seriousness that we, at this point, have no reason in evidence to anticipate. The reasonable response to the new findings is further research into chronic tolerance and more discriminating use of the available GBCAs.

A recent article featured in medicalphysicsweb highlighted new MRI contrast agents based on manganese instead of gadolinium, that may be safer.

Manganese-based contrast could allow safer MRI. A team at Massachusetts General Hospital has developed a potential alternative to gadolinium-based MRI contrast agents, which carry significant health risks for some patients and cannot be used in patients with poor renal function. In tests on baboons, the researchers demonstrated that the manganese-based agent Mn-PyC3A produced equivalent contrast enhancement of blood vessels to that of a gadolinium-based agent (Radiology doi:10.1148/radiol.2017170977).

A discussion of significant MRI safety issues can be found here.

So is magnetic resonance imaging safe? For the vast majority of MRIs that do not use any contrast agent, I would say overwhelmingly yes. When gadolinium is used, there is a small risk that in most cases will be far less significant than the benefit of obtaining the image.

#MieToo

Many readers of Intermediate Physics for Medicine and Biology are undergraduate students who may be looking for their first research experience. Now is the time to be applying for summer undergraduate research opportunities; often they have application deadlines in early February. I’ve collected a list of many biomedical research programs on the Oakland University Center for Biomedical Research website. Another place to learn about research programs is Pathways to Science. The National Science Foundation funds a large number of Research Experiences for Undergraduates; find one here. If you are on Twitter, search for tweets with the hashtag #undergraduateresearch.

Be sure to apply to the Summer Internship Program at the National Institutes of Health #NIH. I worked at the NIH intramural program in Bethesda, Maryland during the 1990s, and can think of no better place to do a summer internship. They have a special program aimed at biomedical engineers #Bioengineering, which might be of particular interest to readers of IPMB. The deadline for the general program is March 1, but for the biomedical engineering program it is February 9. Sorry international students, but students eligible for the NIH program must be US citizens or permanent residents.

I had my own career-defining undergraduate research experience at the University of Kansas #KansasU. I worked with physics professor Wes Unruh and his graduate student Robert Bunch studying the scattering of light. We analyzed our data using a theory of light scattering developed by German physicist Gustav Mie (#MieToo...that's supposed to be a joke). My first publication was a 1983 abstract to the March Meeting of the American Physical Society titled “Size distributions of Ni and Co colloids within MgO”. Although in graduate school I switched from condensed matter physics to biological physics, this first exposure to scientific research set me on the path that led ultimately to coauthoring Intermediate Physics for Medicine and Biology.

Most applications for summer research programs require an essay, college transcripts, and letters of recommendation. Gathering all this stuff takes time. So, if you plan to apply for summer research programs, get to work now, now, now!

From Photon to Neuron: Light, Imaging, Vision

From Photon to Neuron,
by Philip Nelson.

The January issue of Physics Today, the magazine of the American Institute of Physics, contains my review of Philip Nelson’s new book From Photon to Neuron: Light, Imaging, Vision. The published review is the result of several iterations with the Physics Today book review editor, which improved it. Below, is my first draft. I think that readers of Intermediate Physics for Medicine and Biology will enjoy Nelson’s books.

From Photon to Neuron: Light, Imaging, Vision
Philip Nelson
Princeton University Press, 2017

Philip Nelson’s book From Photon to Neuron: Light, Imaging, Vision (Princeton University Press, 2017) completes a trilogy begun by Biological Physics (Freeman, 2014) and Physical Models of Living Systems (Freeman, 2015). These works establish Nelson as the preeminent textbook author at the intersection of physics and biology. All three books aim at upper-level undergraduates who already have studied a year of physics and calculus, but the texts are rich enough for the graduate level too.

Is From Photon to Neuron aimed at physicists interested in biology, or biologists interested in physics? Physics students will gain the most from this book. The mathematics (for example, the Fresnel integral) is beyond what most premed students are comfortable with. Biology majors will be challenged, but they need a book like this to improve their quantitative skills. Students with a weak command of calculus and no desire to improve it may find Sonke Johnsen’s excellent The Optics of Life (Princeton University Press, 2011) more palatable. A third-year physics major should be able to handle the math, except for some advanced topics in Part III that seemed out of place in an undergraduate book.

The wave and particle properties of light are both crucial for biology. For instance, diffraction limits your visual acuity, but a rod cell in your retina responds to a single photon. Nelson adopts a light hypothesis like that Richard Feynman presented in QED: The Strange Theory of Light and Matter (Princeton University Press, 1985): photons are governed by a probability amplitude that obeys a stationary-phase principle. Physics students will appreciate this powerful point of view; I am not sure what biology students will make of it. This approach highlights the intimate relationship between quantum mechanics, probability, and vision. For me, it works. Its disadvantage is that you must add a lot of e^iφs to explain simple concepts like reflection and refraction.

Readers who are interested primarily about vision, with little concern for light or imaging, might prefer Robert Rodieck’s masterpiece The First Steps in Seeing (Sinauer, 1998). The books by Rodieck and Nelson share several characteristics: eloquent prose, outstanding artwork (including some beautiful drawings by David Goodsell in From Photon to Neuron), and a quantitative approach that most biology textbooks lack. Nelson’s book, however, is more useful for teaching; it includes homework problems, end-of-chapter summaries, and recommendations for additional reading. Many of the homework exercises require analyzing data that you can download from the author’s website (www.physics.upenn.edu/~pcn). To do these exercises, you must know how to program a computer using MATLAB or similar software (you can download Nelson’s free Student Guide to MATLAB from his website). One critical skill students gain when taking a class using From Neuron to Brain is the ability to write short computer programs to analyze data numerically. Nelson teaches using words, pictures, formulas, and code to construct models and interpret data. His books provide a masterclass in how to integrate these four different approaches into a complete learning experience. Most biology books combine words and pictures, and a few include equations. Nelson’s emphasis on code—or at least his insistence that the students write their own code—sets his books apart. Computerphobes may hesitate initially, but they will gain the most from numerical modeling.

From Photon to Neuron covers topics throughout biological physics. For instance, fluorescence microscopy is a theme Nelson introduces early and revisits often. He devotes one chapter to color vision and another to superresolution microscopy. My favorite chapter begins with Rosalind Franklin’s iconic x-ray diffraction pattern of DNA, and then develops just enough theory to explain how Watson and Crick could, at a glance, obtain the key information they needed to derive their famous structure. Nelson presents enough electrophysiology to describe how absorption of a photon by rhodopsin causes a voltage signal across the neural membrane, and enough physical optics to explain the iridescence of butterfly wings. The network diagrams of signaling cascades seemed a little dry, but that may reflect my own tastes rather than Nelson’s presentation. Other topics include photosynthesis, Fluorescence Resonance Energy Transfer (FRET), and two-photon imaging.

Overall, I found From Photon to Neuron to be an outstanding textbook; a worthy successor to Biological Physics and Physical Models of Living Systems. Philip Nelson has done it again. His books define the field of biological physics.

Brad Roth
Oakland University
Rochester, Michigan

Brad Roth is a professor of physics at Oakland University, and is coauthor with Russell Hobbie of Intermediate Physics for Medicine and Biology (Springer, 2015).

Three books by Philip Nelson.