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Life in Moving Fluids:
The Physical Biology of Flow (Second edition)
Steven Vogel

Book Description | Reviews | Table of Contents

COPYRIGHT NOTICE: Published by Princeton University Press and copyrighted, © 1994, by Princeton University Press. All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher, except for reading and browsing via the World Wide Web. Users are not permitted to mount this file on any network servers. Follow links for Class Use and other Permissions. For more information, send e-mail to

CHAPTER 1: Remarks at the Start

With the easy confidence of long tradition, the biologist measures the effects of temperature on every parameter of life. Lack of sophistication poses no barrier; heat storage and exchange may be ignored or Arrhenius abused; but temperature is, after time, our favorite abscissa. One doesn't have to be a card-carrying thermodynamicist to wield a thermometer.

By contrast, only a few of us measure the rates at which fluids flow, however potent the possible effects of winds and currents on our particular systems. Fluid mechanics is intimidating, with courses and texts designed for practicing engineers and other masters of vector calculus and similar arcane arts. Besides, we've developed no comfortably familiar and appropriate technology with which to produce and measure the flow of fluid under biologically interesting circumstances. So the effects of flow are too commonly either ignored or else relegated to parentheses, speculation, or anecdote.

A life immersed in a fluid--air or water--is, of course, nothing unusual for an organism. Almost as commonly, the organism and fluid move with respect to each other, either through locomotion, as winds or currents across some sedentary creature, or as fluid passes through internal conduits. Clearly, then, fluid motion is something with which many organisms must contend; as clearly, it ought to be a factor to which the design of organisms reflects adaptation.

It is this particular set of phenomena--the adaptations of organisms to moving fluids--that this book is mainly about. Its intended messages are that such adaptations are of considerable interest and that fluid flow need not be viewed with fear or alarm. Flow may indeed be one of the messier aspects of the physical world, but most of the messiness can be explained in words, simple formulas, and graphs. Quantitative rules are available to bring respectable organization to a wide range of phenomena. With a little experience, one's intuition can develop into a reasonably reliable guide to flows and the forces they generate. Even the technology for experimental work on flows proves less formidable than one might anticipate. Indeed, the underlying complexity of fluid mechanics can be something of a boon, since it greatly restricts the possibilities of exact mathematical solutions or trustworthy simulations. Thus the investigator must often resort to the world of direct experimentation and simple physical models, a world in which the biologist can feel quite at home.

Supplying a comprehensive review of what's known about the interrelationships of the movements of fluids and the design of organisms isn't my intention. I will cite no small number of phenomena and investigations; but they're mainly meant to illustrate the diversity of situations to which flow is relevant and the ways in which such situations can be analyzed. Instead, the main objective is an attempt to imbue the reader with some intuitive feeling for the behavior of fluids under biologically interesting circumstances, to supply some of the comfortable familiarity with fluid motion that most of us have for solids. I take the view that with that familiarity the biologist is likely to notice relationships and phenomena and to put hydrodynamic hypotheses to proper experimental scrutiny. And I feel strongly that such investigations should be unhesitatingly pursued by any biologist and should not constitute the peculiar province of some _au courant_ priesthood.

One might organize a book such as this with either biology or physics as framework. The physical phenomena, though, flow more easily in an orderly and useful sequence; my attempts to make good order of the biological topics inevitably have a more severe air of artificiality. So the physics of flow will provide the skeleton, fleshed out, in turn, by consideration of the bioportentousness of each item. Where relevance to a particularly large segment of biology wants examination, as when considering velocity gradients or drag, wholly biological chapters will be interjected. While I hope that the arrangement will be effective for both the biologist seeking an easy entry into fluids and the fluid mechanist dazzled by biological diversity, I've opted for the biologist where hard choices had to be made. The reader may be a little vague on the distinctions between work and power or stress and strain but is assumed to be quite sound on vertebrates and invertebrates as well as cucumbers and sea cucumbers. For one thing, I'm very much a biologist myself; for another, the relevant biological details are easy to obtain from textbooks or other references. Since the framework is physical rather than biological, topics such as seed dispersal and suspension feeding will wander in and out; since specific biological topics sometimes involve several applications of fluid mechanics, a little redundancy has been inevitable.

Biological examples will vary from well-established through half-baked to wildly speculative; I'll try to indicate the degree of confidence with which each should be vested. Speculation is the crucial raw material of science, and it seems especially important at this stage of this subject. If some assertion generates either enough antipathy or enthusiasm to provoke a decent investigation, then we'll all be a bit farther ahead. In any case, I claim no proprietary rights to any hypothesis here, whether it's explicit or implied. (But neither does any idea come warranted against post facto silliness.) Incidentally, it's an entirely practical procedure to fix on some physical phenomenon and then go looking to see how organisms have responded to it. While this may sound like shooting at a wall and then drawing targets around the bullet holes, science is, after all, an opportunistic affair rather than a sporting proposition. But whatever the respectability of the approach, it does at least take some advantage of the way things are organized here.

It's probably not the best idea to use this book solely as a reference, with the index for intromission. The reader coming from biology ought to move in sequence through at least the basic material on viscosity, the principle of continuity, Bernoulli's equation, the Reynolds number, and the characteristics of velocity gradients. The book certainly can be used out of order or with parts omitted, but certain pitfalls lurk. Biologists have been all too willing to use equations with no more than a guileless glance to check that the right variables were represented. As a result, much mischief has been perpetrated. Bernoulli's equation doesn't work well in boundary layers and has little direct applicability to circulatory systems. The Hagen-Poiseuille equation presumes that flow is laminar and that one is far from the entrance to a pipe. Stokes' law applies (in general) to small spheres, not large ones, and it usually needs a correction for spheres of gas moving in a liquid medium. Equations certainly abound in these pages, but they can be found as easily elsewhere. More important are the discussions of how and where they apply--when to rush in and when to fear to tread. Especially in this latter matter, the specific sequential treatment of topics is of significance.

Inclusions, Exclusions, General Sources

At least for physical phenomena, some idea of what's covered in this book is apparent from the table of contents. Naturally, the omissions are less evident, but it's important that they be mentioned. Physical fluid mechanics ramifies in many directions; much of it carries the unmistakable odor of our technological concerns and has little relevance to biology. Thus we can safely ignore such things as high speed (compressible) flows and the flows of rarefied gases. For reasons that flatter neither author nor most readers, the mathematical niceties of fluid mechanics will not loom large here. More attention will be given to phenomena that I judged simple or widespread than to ones that I regarded as more complex or specialized. The focus here is on flow in and around organisms, decent-sized chunks of organisms, and small assemblages of organisms, and this focus has necessitated some omissions. Only a little will be said about the statics of fluids, biometeorology, and the flow of substances that are incompletely fluid, such as the contents of cells and life's various slurries and slimes. Thus I'll largely eschew the treacherous quagmires of mitosis and cyclosis, of surface waves and ocean currents, of the hydraulics of streams and rivers, of atmospheric winds and wind-driven circulation in lakes, and of flow through porous media. In addition, I've made some essentially arbitrary exclusions and will have little to say about the sensory side of responses to flows and the effects of flow on chemical communication in either air or water. Flows either driven by or involving temperature differences--convective heat transfer--are a major omission rationalized only by a lack of space and steam. Most of these topics are treated well elsewhere, and sources of enlightenment will be mentioned at appropriate points.

More extensive and detailed information on fluid dynamics may be obtained from conventional engineering textbooks, such as those by Streeter and Wylie (1985) or Massey (1989). The basic processes that will be of concern here were about as well understood fifty or sixty years ago as they are today, so the age of the source is usually immaterial. Indeed, the earlier generations of fluid dynamicists may have worried more than their successors about low-speed phenomena and other items that turn out to have biological relevance; and inexpensive reprints of several quite useful old texts are currently available. Prandtl and Tietjens' (1934) _Applied Hydro- and Aeromechanics_ is particularly good on boundary layers and drag. _Fluid Mechanics for Hydraulic Engineers_ (1938) contains a fine treatment of dimensional analysis and the origin of the common dimensionless indices, while _Elementary Mechanics of Fluids_ (1946) covers an unusually diverse range of topics; both are by Hunter Rouse. Goldstein's (1938) two volumes, _Modern Developments in Fluid Dynamics_, have nice verbal descriptions of phenomena in between the equations. Mises (1945) gives excellent explanations of how wings and propellers work in _Theory of Flight_. With several of these books at hand one can usually find an understandable and intuitively satisfying explanation of a puzzling aspect of flow. Finally, both aesthetically and technically pleasing, there's _An Album of Fluid Motion_ (1982), by Van Dyke.

Trustworthy popular accounts are surprisingly scarce, with no treatment of fluid mechanics coming anywhere near the breadth and elegance that Gordon's (1978) _Structures_ brings to solid mechanics. My favorite three are Von K rm n's (1954) _Aerodynamics_, Sutton's (1955) _The Science of Flight_, and Shapiro's (1961) _Shape and Flow_. The first is particularly witty, the second is especially clear and honest, and the third, _mirabile dictu_, focuses on fluid phenomena that biologists should encounter.

Several other general sources deserve mention. _Fluid Behavior in Biological Systems_, by Leyton (1975), is nearest to this book in intent. It gives less attention to drag, boundary layers, and propulsion but more to flow in porous media, heat transfer, non-Newtonian fluids, thermodynamics, and micrometeorology. Ward-Smith's _Biophysical Aerodynamics_ (1984) focuses on seed dispersal and animal flight. _Waves and Beaches _(1980) by Bascom, _Biology and the Mechanics of the Wave-Swept Environment_ by Denny (1988), and _Air and Water_, also by Denny (1993), are invaluable for anyone working along the edge of the ocean; in effect the latter picks up in both elegance and focus where the former leaves off. And the privately published compendium of Hoerner (1965), _Fluid-Dynamic Drag_, is filled with information about the behavior of simple objects to which analogous data for organisms can be compared.


The how-to-do-it aspects of biological fluid mechanics presents special snares. High technology is certainly no stranger to the microscopist, molecular biologist, or neurophysiologist. But few of us are facile at making and measuring moving fluids, and there's rarely a lab-down-the- hall where folks are already, so to speak, well immersed in flow. Faced with some upcoming investigation, one's first impulse is to seek out a friendly engineer, who then prescribes a hot-wire or laser anemometer; the problem is thereby reduced to a search for kilobucks. There is, though, a second and less ordinary problem. The technology used by the engineers is a product of, by, and for engineers. The range of magnitude of the flows we usually need to produce and of the forces we typically want to measure is rather different, and engineering technology is often as inappropriate as it is expensive.

In over thirty years of facing problems of flow, I've found that the devices I've had to use were, compared to those of my colleagues in more established fields of biology, rather cheap. On the other hand, they have rarely been available prepackaged and precalibrated, with a factory to phone when all else fails. Flow tanks, wind tunnels, flow meters, anemometers, and force meters have simply been built in my laboratory as needed. I've developed a fair contempt for fancy commercial gear except for items of very general applicability--digital voltmeters, power supplies, potentiometric chart recorders, variable speed motors, gears and pulleys, electronic stroboscopes, analog-to-digital converters, video cameras and recorders, and so forth. The consistently most valuable tools have been lathe, milling machine, and drill press--but I'm a quite unreconstructed primitivist with the perverse passion of the impecunious and impatient. The first edition of this book included appendices on making and measuring flows. I've not retained them, since the earlier edition is still available as a reference and since I'm now convinced that it would take a whole book to do the job properly.

Dimensions and Units

Fluid mechanics makes use of a wide array of different variables, some (density, viscosity, lift, drag) familiar and others (circulation, friction factor, pressure coefficient) out of the biologist's normal menagerie of terms. A list of symbols and quantities used in this book precedes the index; it might be worth putting a protruding label on that page. Matters will be somewhat simplified if the reader pays a little attention to the dimensions that attach to each quantity. By dimensions I don't mean units. Thus velocity always has dimensions of length per unit time, whether data are given in units of meters per second, miles per hour, or furlongs per fortnight. Dimensions take on rather special significance in fluid mechanics not just as a result of the general messiness of the subject but because of a condition that may sound trivial but proves surprisingly potent. _For an equation to have any applicability to the real world, not only must the two sides be numerically equal, but they must be dimensionally equal as well_. The general statement asserts that proper equations must be dimensionally homogeneous--each term must have the same dimensions. When theory, memory, or intuition fails, this injunction can go a long way toward indicating the form of an appropriate equation.

An example should clarify the matter. Assume you want an equation relating the tension in the wall of a sphere or a cylinder to the pressure inside--perhaps you know the surface tension of water and want to know what this does to the internal pressure of a gas bubble (in connection with Chapter 15 or 17). Tension has dimensions of force per unit length (as you can tell, if need be, from the units in which surface tension is quoted). Pressure has dimensions of force per unit area (as in pounds of force per square inch of area). To relate pressure to tension, clearly pressure must be multiplied by some linear dimension of the sphere or cylinder. Thus the equation will be of the form

tension = constant x radius x pressure,

where we know nothing about the constant except that it's dimensionless. Evidently a given tension generates a larger pressure when the radius is small than when the radius is large. Even without further information about the constant we're no longer surprised that (surface tension being constant) tiny bubbles have high internal pressures. And it ought to take a higher pressure to generate a given tension when the radius is small, so we're much less mystified about why it's relatively hard to _start_ blowing up a balloon despite the obvious flaccidity of the rubber. We're no longer surprised that a small plant cell can withstand pressure differences of many atmospheres across its thin walls, nor are we puzzled at how arterioles can get by with much thinner walls than that of the aorta when both are subjected to similar internal pressures.

The easiest way to compare the dimensions of different variables is to reduce them to combinations of a few so-called fundamental dimensions. We will require only three such dimensions here: _length_, _mass_, and _time_ (_temperature_ is a frequent fourth). This reduction is accomplished by use of definitions or previously memorized functions. For instance, force is mass times acceleration, and therefore force has fundamental dimensions of MLT-2. Pressure or stress is force per area, so both have dimensions of ML-1T-2. Incidentally, this latter example emphasizes the fact that just because two quantities have the same dimensions doesn't mean that they are synonymous or equivalent. With such simple manipulations, the fundamental dimensions of each term in almost any equation can be obtained. Considerable use will be made of this sort of dimensional reasoning in forthcoming chapters. More extensive and formal introductions to the subject of dimensional analysis can be found in books by Bridgman (1931) and Langhaar (1951); biological contexts are supplied by McMahon and Bonner (1983) and Pennycuick (1992).

Not only constants but also variables may be dimensionless and still be rich in relevance to the real world. Any number that is the ratio of two quantities measured in the same dimensions will be dimensionless. A fairly simple example is _strain_, as used in solid mechanics. Strain is the ratio of the extension in length of a stressed object to its original, unstressed length. Unstressed length is simple and commonly fixed; things get even more interesting when several of the quantities in a dimensionless index decide to vary. Such indices turn out to be scales that achieve quite useful simplifications of otherwise complex situations and that can lead to remarkable insights into what really matters beneath a confusion of varying quantities. Thus surface-to-volume ratio depends on size as well as shape; it has a dimension of inverse length (_L_-1). By contrast, surface cubed over volume squared is dimensionless and quite indifferent to the size of an object per se; so, for instance, a cube has the same value whatever its size. It's therefore much more useful as an index of shape. Dimensionless numbers, usually named after their first promulgators, find wide use in fluid mechanics; while initially they seem odd, one rapidly achieves proper contemptuous familiarity even with graphs of one dimensionless number plotted against another. I hardly need mention that dimensionless numbers are automatically unitless and therefore quite indifferent to which system of units is in use.

Units, though, can't be completely ignored. All variables in a dimensionally proper equation ought to be given in a consistent set of units. An earlier generation of biologists, when they meant metric units, usually used something approximating the physicist's centimeter-gram- second (CGS) system, along with a few oddities such as heat as calories and pressure as millimeters of mercury of pressure. We're now enjoined to adopt a specific version of the metric system common to all of science, the SI or "SystSme Internationale," which will be used here. Fundamental units (for the fundamental dimensions) are kilograms (mass), meters (length), and seconds (time). SI allows the common prefixes going up or down by factors of 1000 (mega-, milli-, micro-, and so forth) to be attached to any fundamental or derived unit. Only a single prefix, though, may be used with a single unit, and the prefix must attach to the numerator. Thus meganewtons per square meter is legitimate but newtons per square millimeter is not. I'll only stoop to such frowned-upon units as centimeters, liters, and kilometers as a kind of vernacular where no calculations are contemplated. SI units often get mildly ludicrous in the context of organisms. Thus the drag of a cruising fruit fly (Vogel 1966) is about three micronewtons. And Wainwright et al. (1976) give the strength of spider silk as about 10 to the 9th newtons per square meter of cross section; it would take one hundred billion (U.S.) strands to get that combined area. But consistency is really a sufficiently compensatory blessing.

Table 1.1 gives a list of quantities with their fundamental dimensions and SI units. For further introduction to SI units and conventions, see _Quantities, Units, and Symbols_ by the Symbols Committee of the Royal Society (1975) or _The International System of Units_ by Goldman and Bell (1986). For the inevitable nuisance of dealing with different systems of units, Pennycuick's (1988) booklet, _Conversion Factors_, is an absolute godsend. It's cheap enough so one might scatter a few copies around home, office, and laboratory.

If you're decently scrupulous about consistency of dimensions and use of units then you never have to specify units when giving equations, a considerable convenience. The main place where the system falters at all is when equations with variable exponents are fitted to empirical data, as in any statement such as "metabolic rate is proportional to body mass to the 0.75 power." The exponent of proportionality comes out the same in any set of units, but the constant with which it forms an equation does not. Moreover, the rule about dimensional homogeneity is violated unless one tacitly heaps all the unpleasantness on the constant of proportionality. If the expression is written as an equation rather than a proportionality, the usual practice is merely to specify the set of units that must be used. Here the practice among fluid mechanists and biologists is about the same, both being practical people who have scattered data, imperfect theories, and the like.

Measurements and Accuracy

First, a few words about what's meant by accuracy. As Eisenhart (1968) has pointed out, lack of accuracy reflects two distinct disabilities in data. First, there's imprecision, or lack of repeatability of determinations. And second, there's systematic error, or bias, the gap between the measured value and some actual, "true" value--the tendency to measure something other than what was really intended. In the kinds of problems we'll discuss, unavoidable imprecision is usually pretty horrid, at least by the standards of physical rather than biological sciences. Thus it's rarely worth great attention to fine standards for calibration where these just drive systematic errors well below what proves to be the more intractable problem of imprecision.

Quantities such as density can be measured very precisely. But the inevitable irregularities in flows, the nature of vortices and turbulence, and quite a few other phenomena severely limit the precision with which the behavior and effects of moving fluids can be usefully measured. The drag of an object measured in one wind tunnel will often differ considerably from that measured in another, while a third datum will result from towing the object through otherwise still air. If a figure of, say, 1 m s-1 is cited as the transition point from laminar to turbulent flow in some pipe, that figure should not be interpreted as 1 ñ 0.01 or often even 1 ñ 0.1 m s-1. With extreme care it may be possible to postpone the transition to 10 m s-1 or more. Some empirical formulas given in standard works, especially those for convective flows, use constants with three significant figures. My own experience suggests that such numbers should be viewed with enormous skepticism for anything except, perhaps, the very specific experimental conditions under which they were determined. And citing calculations for Reynolds numbers (Chapter 5), for instance, without minimally rounding off to the nearest part in a hundred is at the least a bit self-deceptive.

The development of electronic calculators has had a pernicious effect on our notions of precision. The art estimating just how much precision is truly necessary to resolve the question at hand has suffered from the demise of the slide rule. As a practical matter, the flow of fluids, even without organisms, is a subject that enjoys barely a slide rule level of precision--rarely better than one percent and often much worse.

Relative Motion

Another matter ought to be set straight at the start, an item that arises with some frequency among biologists taking their first look at fluid flow. Frames of reference can be chosen for convenience, and the surface of the earth upon which we walk has no absolute claim as a "correct" reference frame. One might imagine that a seed carried beneath a parachute of fluffy fibers will trail behind the fibers as the unit is blown across the landscape by a steady wind. In fact the image is quite wrong for anything beyond the initial events of detachment from the plant--when the surface of the earth is still an active participant--as can be seen in Figure 1.1. Farther along (if the wind is steady) neither any longer "knows" anything about what the ground's doing so the seed hangs below the fluff--the surface of the earth now constitutes a reference frame that's both misleading and unnecessarily complex. People who've traveled in balloons commonly comment on the silence and windlessness they experience and its incongruence with clear visual evidence that the ground is moving beneath.

A more drastic if less commonly observed case is that of a "ballooning" spider. (For a general account of the phenomenon, see Bishop 1990.) A young spider spins a long silk strand that extends downwind from it. Eventually it lets go and drifts along. One might expect that the spider is thereafter pulled behind the silken line until the whole system settles to the earth. What will happen in the absence of gusts, vortices, and gradients is that the spider will fall downward but at a speed that's much reduced by the drag of the line. The line will gradually shift from running horizontally to running vertically, forming a relatively high drag, low weight element that extends _downstream_ (here, of course, _upward_) from the falling spider. The line lags behind, slowing descent as a result of its high drag relative to its weight. (Humphrey 1987 estimates that the line has over 75% of the drag of the system while contributing less than a tenth of a percent of the weight.)

Failing to keep in mind a proper reference frame can generate odd misconceptions as well as obscure some real problems. From time to time statements appear in the avian literature about a problem facing (or, perhaps we should say chasing) a bird flying downwind--it must somehow keep from getting its tail feathers ruffled. But consider: once launched, a bird simply does not (and, indeed, cannot) fly downwind with respect to the local flow around it. If the wind with respect to the ground goes in the same direction as the bird, then the bird just flies that much faster with respect to the ground. The problem is a really nasty one for slow fliers such as migrating monarch butterflies that can't make progress (once again with respect to the ground) against even modest winds (Gibo and Pallett 1979).

Worse, you might really get stung by confusion about relative motion and frames of reference. Imagine being chased by a swarm of mayhem- minded killer bees, as in Figure 1.2. A decent breeze is blowing, so to get a little more speed you run downwind. Bad move--you're quickly bee-set. Honeybees can fly (despite a lot of lore to the contrary) only about 7.5 m s-1 (Nachtigall and Hanauer-Thieser 1992), but that's equivalent to about a four-minute mile. With a 4.5 m s-1 breeze from behind, they'll go all of 12 m s-1 _with respect to the ground_. You may gain a little from the tail wind, but the bees will automatically get full value. What if you run upwind? You may be slowed down slightly, but the bees will be dramatically retarded--down to 3 m s-1 _with respect to the ground_. Only about a nine-minute mile is needed for you to stay ahead.

The ability to shift reference frames for our convenience can effect more than conceptual simplifications. No special justification is needed for using an experimental system in which the organism is stationary and the fluid environment moves as a substitute for a reality in which the organism swims, flies, or falls through a fluid stationary with respect to the surface of the earth. That's the main reason for the popularity of such devices as wind tunnels and flow tanks for working on flying and swimming. I've used a vertical wind tunnel to provide upward breezes just equal to the falling rates of seeds. Shrewd choice of reference frame is an old tradition--assuming (not proving!) a sun-centered system permitted Copernicus to simplify enormously the complex cosmology of Ptolemy.

Balancing Forces

When considering solid bodies moving through fluids, Newton's first law has to be put into a somewhat more distant perspective than might have been the practice in one's first physics course. Bodies may continue in steady, rectilinear motion unless imposed upon by external forces; but imposition of external forces is just what fluids do to solid bodies that have the temerity to force passage. Chief among the external forces is, of course, drag.1 [See footnote at bottom of file] In practice, then, a body traveling steadily with respect to a fluid (ah--frames of reference) must be exerting a force on the fluid exactly sufficient to balance the fluid's force on the body. That force might be provided by the action of wings or tail acting as thrust generators. Or it might be supplied by the action of gravity on the body, as when a body sinks steadily, when the downward force of gravity balances the combined upward forces of buoyancy and drag. Or it might be provided by some other solid structure that transmits forces--a mounting strut supporting an object in a flow tank or wind tunnel or the branches and trunk supporting the leaves of a tree.

In combination, force balance and reference frame can make some subtle but substantial trouble, and the utility of quite a lot of literature is compromised by insufficient attention to them. Say you persuade an insect to beat its wings as hard as it can while attached to a fine wire, and you even manage to attach the fine wire to a device that will measure the force the insect produces. What will this tell you about how fast the insect can fly or about how much force it can generate at top speed? Almost nothing! Even if you (separately) measure how fast the insect can fly, you are on very unsafe ground if you multiply that speed and the force you've measured to get the insect's power output. Similarly, if you want to know the drag of the insect at its maximum speed, you can't just put the thing in a wind tunnel set to that speed and measure the force it exerts on a mount. If the insect is doing exactly what it ought to, then the mount will feel no force at all because the drag of the insect's body will exactly balance the thrust produced by the wings. If it's not beating its wings then you get a force where normally no net force would be exerted. Fortunately, there are ways around such difficulties; what's important at this point is that they be recognized.

I do regret a little the admonitory tone of much of this initial chapter. A lot of it reflects some scar tissue induced by abrasions in the form of written material that I've been expected to evaluate. Even though I quite obviously mean what I've said here, its general character contrasts a bit with my first rule of fluid mechanics--you have to be breezy if you expect to make waves.


1 Perhaps someone (me, for instance) ought to say a word against the common usage of that needlessly redundant and pretentious term, "drag force." Drag is a force, is always a force, and is nothing but a force.

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