Friday, March 9, 2012

Glimpses of Creatures in Their Physical Worlds

I have recently finished reading Steven Vogel’s book Glimpses of Creatures in Their Physical Worlds, which is a collection of twelve essays previously published in the Journal of Biosciences. The Preface begins
“The dozen essays herein look at bits of biology, bits that reflect the physical world in which organisms find themselves. Evolution can do wonders, but it cannot escape its earthy context—a certain temperature range, a particular gravitational acceleration, the physical properties of air and water, and so forth. Nor can it tamper with mathematics. Thus the design of organisms—the level of organization at which natural selection acts most directly as well as the focus here—must reflect that physical context. The baseline it provides both imposes constraints and affords opportunities, the co-stars in what follows….”
The first essay is titled “Two Ways to Move Material,” and the two ways it discusses are diffusion and flow. To compare the two quantitatively, Vogel uses the Péclet number, Pe, defined as Pe = VL/D, where V is the flow speed, L the distance, and D the diffusion coefficient. As I read his analysis I suddenly got a sinking feeling: Russ Hobbie and I discussed just such a dimensionless number in Problem 37 of Chapter 4 in the 4th edition of Intermediate Physics for Medicine and Biology, but we called it the Sherwood Number, not the Péclet number. Were we wrong?

Edward Purcell, in his well-known article Life at Low Reynolds Number, introduced the quantity VL/D, which he called simply “S” with no other name. However, in a footnote at the end of the article he wrote “I’ve recently discovered that its official name is the Sherwood number, so S is appropriate after all!” Mark Denny, in his book Air and Water, states that VL/D is the Sherwood number, but in his Encyclopedia of Tide Pools and Rocky Shores he calls it the Péclet number. Vogel, in his earlier book Life in Moving Fluids, introduces VL/D as the Péclet number but adds parenthetically “sometimes known as the Sherwood number”

Some articles report a more complicated relationship between the Péclet and Sherwood number, implying they cannot be the same. For instance, consider the paper Nutrient Uptake by a Self-Propelled Steady Squirmer, by Vanesa Magar, Tomonobu Goto, and T. J. Pedley (Quarterly Journal of Mechanics and Applied Mathematics, Volume 56, Pages 65-91, 2003), in which they write “We find the relationship between the Sherwood number (Sh), a measure of the mass transfer across the surface, and the Péclet number (Pe), which indicates the relative effect of convection versus diffusion”. Similarly, Fumio Takemura and Akira Yabe (Gas Dissolution Process of Spherical Rising Gas Bubbles, Chemical Engineering Science, Volume 53, Pages 2691-2699, 1998) define the Péclet number as VL/D, but define the Sherwood number as αL/D, where α is the mass transfer coefficient at an interface (having, by necessity, the same units as speed, m/s). After reviewing these and other sources, I conclude that Vogel is probably right: VL/D should properly be called the Péclet number and not the Sherwood number, although the distinction is not always clear in the literature.

Now that we have cleared up this Péclet/Sherwood unpleasentness, let’s return to Vogel’s lovely essay about two ways to move material. He calculated the Péclet number for capillaries, using a speed of 0.7 mm/s (close to the 1 mm/s listed in Table 1.4 of Intermediate Physics for Medicine and Biology), a capillary radius of 3 microns (we use 4 microns in Table 1.4), and a oxygen diffusion constant of 1.8 × 10-9 m/s2 (2 × 10-9 m/s2 in Intermediate Physics for Medicine and Biology), and finds a Péclet number of 1.2 (if you use the data in our book, you would get 2). Vogel then argues that the optimum size for capillaries is when the Péclet number is approximately one, so that evolution has created a nearly optimized system. The argument, in my words, is that oxygen transport changes from convection to diffusion in the capillaries. If the Péclet number were much smaller than one, diffusion would dominate and we would be better off with larger capilarries that are farther apart and faster blood flow to improve convection. If the Péclet number were much larger than one, convection would dominate and our circulatory system would be improved by using smaller capilarries closer together, even if that means slower blood flow, to improve diffusion. A Péclet number of about one seems to be the happy medium.

The third essay, “Getting Up to Speed” is also relevant to readers of Intermediate Physics for Medicine and Biology. Our Problem 43 of Chapter 2 is about how high animals can jump.
Problem 43 Let’s examine how high animals can jump [Schmidt-Nielsen (1984), pp. 176-179]. Assume that the energy output of the jumping muscle is proportional to the body mass, M. The gravitational potential energy gained upon jumping to a height h is Mgh (g=9.8 m s-2). If a 3 g locust can jump 60 cm, how high can a 70 kg human jump? Use scaling arguments.
In the next problem, number 44, Russ and I ask the reader to calculate the acceleration of the jumper, which if you solve the problem you will find varies inversely with length.

Vogel analyzes this same topic, but digs a little deeper. Here examines all sorts of jumpers, including seeds and spores that are hurled upward without the help of muscles at all. He finds that the scaling laws from Problems 43 and 44 do indeed hold, but the traditional reasoning behind the law is flawed.
“The diversity of cases for which the scaling rule works ought to raise a flag of suspicion. Why should an argument based on muscle work for systems that do their work with other engines? . . . Something else must be afoot—again, the original argument presumed isometrically built muscle-powered jumpers. In short, the fit of the far more diverse projectiles demands a more general argument for the scaling of projectile performance. . .”
Vogel goes on to show that for small animals, muscles would have to work unrealistically fast in order to produce the accelerations required to jump to a fixed height.
“The old argument has crashed and burned. The work relative to mass of a contracting muscle deteriorates as animals get smaller rather than holding constant—a consequence of the requisite rise in intrinsic speed. Muscle need not and commonly does not power jumps in real time—elastic energy storage in tendons of collagen, in apodemes of chitin, and in pads of resilin provides power amplification. Finally, muscle powers none of those seed and tiny fungal projectiles. Yet acceleration persists in scaling as the classic argument anticipates. . ."
So how does Vogel explain the scaling law?
“A possible alternative emerges if we reexamine the relationship between force and acceleration defined by Newton’s second law. If acceleration indeed scales inversely with length and mass directly with the cube of length, then force should scale with the square of the length. Or, put another way, force divided by the square of the length should remain constant. Force over the square of length corresponds to stress, so we’re saying that stress should be constant. Perhaps our empirical finding that acceleration varies with length tells us that stress in some manner limits the systems.”
Vogel’s book is full of these sorts of physical insights. I recommend it as supplemental reading for those studying from Intermediate Physics for Medicine and Biology.

1 comment:

  1. Wouldn't the potential energy scale as the length cubed in the limit of a small volume of high energy chemical bonds destined for great heights?