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Physics of Life Reviews ••• (••••) •••–•••www.elsevier.com/locate/plrev

Review

The constructal unification of biological and geophysical design

Adrian Bejan a,∗, James H. Marden b

a Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 2708-0300, USAb Department of Biology, 208 Mueller Lab, Pennsylvania State University, University Park, PA 16802, USA

Received 23 July 2008; received in revised form 6 November 2008; accepted 6 December 2008

Communicated by L. Peliti

Abstract

Here we show that the emergence of scaling laws in inanimate (geophysical) flow systems is analogous to the emergence ofallometric laws in animate (biological) flow systems, and that features of evolutionary “design” in nature can be predicted basedon a principle of physics (the constructal law): “For a finite-size flow system to persist in time (to live) it must evolve in such away that it provides easier and easier access to its currents”, meaning that the configuration and function of flow systems changeover time in a predictable way that improves function, distributes imperfection, and creates geometries that best arrange high andlow resistance areas or volumes. This theoretical unification of the phenomena of animate and inanimate flow design generation isillustrated with examples from biology (lung design, animal locomotion) and the physics of fluid flow (river basins, turbulent flowstructure, self-lubrication). The place of this design-generation principle as a self-standing law in thermodynamics is discussed.Natural flow systems evolve by acquiring flow configuration in a definite direction in time: existing configurations are replaced byeasier flowing configurations.© 2008 Elsevier B.V. All rights reserved.

Keywords: Design in nature; Constructal law; Scaling laws; Animal locomotion; Lung design; River basins; Organ sizes

Contents

1. The phenomenon of generation of configuration in nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1. The physics phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2. Constructal law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2. Animate flow architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1. Tree-shaped flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2. Lungs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3. Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4. Organ sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5. Rhythm: flying, running and swimming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3. Inanimate flow architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

* Corresponding author.E-mail address: [email protected] (A. Bejan).

Please cite this article in press as: A. Bejan, J.H. Marden. The constructal unification of biological and geophysical design. Physics of LifeReviews (2008), doi:10.1016/j.plrev.2008.12.002

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2 A. Bejan, J.H. Marden / Physics of Life Reviews ••• (••••) •••–•••

3.1. River basins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2. Drawings, as properties of flow systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3. Channel cross-sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4. Turbulent structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.5. Zipf distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4. A unifying perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.1. Constructal law and thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2. Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.3. First principles versus ad-hoc optimality postulates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

“Motion (flow) is the cause of every life”

(Leonardo da Vinci)

“Design (drawing) . . . is the root of all science”

(Michelangelo Buonarroti)

1. The phenomenon of generation of configuration in nature

1.1. The physics phenomenon

“Design in nature” is a topic that is gaining increasing attention over the entire range of science (e.g., Refs. [1–5]).Interest is spreading from biology—from the classical debate between Darwinism and the Argument from Design—tothe center of scientific theory, which is physics (physis in Greek, the being of everything; natura in Latin, she whogives birth to everything).

A physics-based examination of design in nature runs counter to prevailing tendencies. For example, Dawkins [6]keeps the discussion limited to biology. He wants “to know how (complicated things) came into existence and whythey are so complicated”. But, he argues that “the explanation (. . . ) is likely to be broadly the same for complicatedthings everywhere in the universe; the same for us, for chimpanzees, worms, oak trees and monsters from outer space.On the other hand, it will not be the same for ‘simple’ things, such as rocks, clouds, rivers, galaxies and quarks. Theseare the stuff of physics. Chimps and dogs and bacteria are the stuff of biology”.

A broader, more unifying point of view has begun to emerge from biology, physics and engineering science.The ‘stuff of physics’ is not necessarily ‘simple’. Think of the turbulent flow architecture of the atmosphere, nextto the flow by diffusion over and through Dawkins’ complicated bacteria. The physics question is key: why does“designedness” (complexity, configuration, rhythm) happen everywhere, in animate and inanimate systems alike?Why does the design generation phenomenon persist in time? This question is no different than other broad questionsin the history of physics (e.g., why does everything, rocks and animals, have “weight”?). Indeed, what is the physicsprinciple that governs the generation of design-like features and behavior throughout nature?

The most recent such questioner comes from biology. Turner [7–9] argues that biologists must come to grips withthe fact that “tinkering” alone does not explain design. In Turner’s book, the “tinkerer” is Darwinian process, whilethe “accomplice” is the something in addition, which is absolutely necessary if we are to make sense of what puzzlesus. As the accomplice, Turner relies on the concept of “intentionality”, defined as “actions that are implicitly gearedtoward creating some future or other abstract state”. He does not say what universal characteristics the future stateshould have. Why does the tinkerer keep on tinkering, and what is doing the tinkering? [To preempt any confusion,this paper is not headed toward a creationist argument and here we declare explicitly that taking any excerpt fromthis paper and presenting it as if we are arguing for a spiritual source for “designedness” will be an intentional act ofdishonesty.]

In this article we review the recent literature that shows that the occurrence of design features in biological systemsis similar to and can be reasoned based on the same principle as the occurrence of design features in geophysicalsystems. This paper is not a complete review, because the field is vast and book size reviews of separate domains existalready. The paper is even less a criticism of research that has illuminated the generation of design in fields that have



A. Bejan, J.H. Marden / Physics of Life Reviews ••• (••••) •••–••• 3

Nomenclature

A area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m2

Ai area of river basin of order i . . . . . . . . . . m2

CD drag coefficientD thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . mDwi drainage density, LT i/Ai

Fdrag drag force . . . . . . . . . . . . . . . . . . . . . . . . . . . NFsi stream frequency, Nsi/Ai

g gravitational acceleration. . . . . . . . . . . m/s2

i order of construction, complexity levelL distance traveled . . . . . . . . . . . . . . . . . . . . . mLT i total length of all streams on Ai

m mass flow rate . . . . . . . . . . . . . . . . . . . . . kg/sM body mass . . . . . . . . . . . . . . . . . . . . . . . . . . kgMo organ mass . . . . . . . . . . . . . . . . . . . . . . . . . . kgN number of bifurcation levels

Ni number of streams present on Ai

RBi number of largest streams on Ai−1 dividedby the number of largest streams on Ai

RLi length of the largest stream on Ai divided bythe length of the largest stream on Ai−1

S entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . J/KSgen entropy generation rate . . . . . . . . . . . . W/Kt time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sv speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m/sW1 vertical destruction of useful energy . . . . . JW2 horizontal destruction of useful energy . . J�z altitude difference . . . . . . . . . . . . . . . . . . . . mν kinematic viscosity . . . . . . . . . . . . . . . . m/s2

ρa air density . . . . . . . . . . . . . . . . . . . . . . . kg/m3

ρb animal body density . . . . . . . . . . . . . . kg/m3

been investigated separately. A lot has been achieved, and it is documented in a voluminous and fast growing literaturein biology [1,10–12], geophysics [13–15], social organization [16], and bioinspired engineering [17,18]. These booksand articles are themselves reviews of various sectors of natural sciences, and their massive size and combined breadthargue in favor of the importance and timeliness of a unifying theory.

For example, West and Brown [19] reviewed a growing body of work that puts physics principles behind theobserved allometric scaling laws of living bodies and organs that can be modeled as tree-shaped flow systems (fordetails see Ref. [20] and Section 4.3). They start by recognizing design, or configuration (they speak of “evolvedbranching networks that transport a variety of resources”), and end with one question, which is a question of physics:“Does some fixed point or deep basin of attraction in the dynamics of natural selection ensure that all life is organizedby a few fundamental principles and that energy is a prime determinant of biological structure”? The answer is yes,as Banavar et al. [21] demonstrated by showing that the evolution of and the optimality principle working for rivernetworks can explain the quarter power scaling law that characterizes the time scale of metabolism in living organisms.

We argue that the answer is yes on a much broader scale, in view of the physics results that are appearing across theboard, from biology to geophysics and social dynamics. The objective of our article is to explore this broader view,the world of animate and inanimate flows together as physics, and to show that the progress that physics has made inexplaining design in biology is in fact fundamental and applicable across the board. Our physics objective is similarto Perlovsky’s [22] in his unification of the modeling of the mind based on first principles of physics.

1.2. Constructal law

To make our point, we review a collection of developments from biology, geophysics, technology evolution andsocial dynamics. These examples are complementary to the tree-shaped biological examples reviewed by West andBrown [19] and river systems of Banavar et al. [21]: they are complementary because they cannot all be reasoned onthe basis of those author’s models. All these examples are brought together by constructal theory, which is the viewthat the generation of flow configuration is a universal physics phenomenon, the many manifestations of which areaccounted by one principle (the constructal law; [23]):

“For a finite-size flow system to persist in time (to live) it must evolve in such a way that it provides easier andeasier access to its currents”.

To understand the meaning of this general and compact statement, in the next two sections we illustrate its im-plications in a variety of settings, which are grouped into two previously immiscible classes: animate and inanimate




systems. The constructal law recognizes the time direction in which the evolution of the flow configuration progresses.Existing flow configurations are replaced by configurations that flow more easily. The constructal law is simply aboutthe physics meaning of the time direction of configuration evolution.

As noted by Perlovsky [22], physics is the search for basic laws, a few universal “first principles” describing awealth of observed phenomena. How the constructal law accounts for large numbers of empirical observations of thesame kind was demonstrated in much more comprehensive reviews [4,23–26] that we are now trying to bring to theattention of the physics and biology research communities. The constructal law is a first principle because it cannot bederived from other first principles (if it could be derived, then it would be a theorem). How the constructal law relatesto the other laws of thermodynamics is outlined in Section 4.1.

Jordan and Scheuring [27] wrote that natural scientists always look for characteristic patterns, rules and laws innature—laws for physicists, rules for biologists, and patterns for ecologists. With the constructal law we view all theseimages as a single phenomenon of flow configuration generation in time, and place the law and the phenomenon inphysics.

2. Animate flow architectures

2.1. Tree-shaped flows

Prominent in the surge of interest in design in nature are the tree-shaped networks with and without loops, whichdominate practically everything that flows, from lungs and vascularized tissues to river basins. Descriptions andexplanations have been offered on a case by case basis (as ad hoc models, albeit with optimization and scaling featuresthat suggest broader applicability, e.g., Section 4.3), in physiology, geophysics, ecology and social dynamics [16,27–37]. These are important advances, and their place in our language and textbooks has been based on the concept offractal algorithms [38–40]. Description is not prediction (theory, principle). The downside of fractal descriptions isthat they suggest that pattern is “complex” and that complexity expands infinitely in nature. This is not the case, infact, pattern is “pattern” (easy enough for us to discern) because it is simple, not complicated.

Here we seek to connect this visible body of work with other bodies of work (less visible, and seemingly unrelated)by showing that a simple statement of physics summarizes the occurrence of design-like features across the board,from biology to geophysics and social dynamics. Fig. 1 places tree-like networks in the context of other classesof natural design whose geometries and functional characteristics have been predicted based on the constructal law[4,23]. For example, tree-shaped water flow networks are predictable from a balance between the resistances of twoflow mechanisms, low-resistivity flow (streams) along channels, and high-resistivity flow (seepage, diffusion) acrossthe interstices between channels. One flow mechanism sustains the other, diffusion at the smallest (interstitial) scale,and channel flow at scales larger than the diffusion scale. Not the other way around. Both mechanisms are necessary(and, consequently, present) when the flow system is large enough. On the other hand, when the system is smallenough one mechanism (diffusion) is more effective than two.

2.2. Lungs

Both the fractal and constructal approaches have been applied to analyzing the geometry of human lungs, withthe aim of determining the geometry that offers greater access to flow. Because fractal networks diverge into smallerbranches, infinitely, predictions of circulatory systems based on fractal geometry must include an additional postulatethat the smallest dimension is finite and of a particular dimension. Critically, this approach cannot address the rela-tionship and transition point between channel-free diffusion and its interface with low-resistivity channels, althoughby using an empirical end point to the fractal geometry it manages to incorporate nature’s solution to the problem.

Constructal theory accounts for the lung configuration based on one general principle, stated at the end of the firstsection. The constructal approach begins with a recognition of purpose and constraint, which for lungs is to deliveroxygen to the blood within the volume constrained by chest cavity dimensions. Lung geometry could be as simple asa single large sac or a duct system ending in many small sacs. The former has a large diffusive resistance whereas thelatter features more resistance due to friction as air passes at higher velocity through the ducts. To provide the easiestaccess for blood to oxygen, the global resistance (sum of diffusive resistance and friction) needs to be minimized.




Fig. 1. Commonly observed phenomena of generation of flow configuration in nature. Top row: river drainage basins, bronchial trees, round ductcross-sections, and open channel cross-sections. Middle row: cracks in shrinking solids, dendritic solidification, and splat vs. splash when a liquiddroplet hits a wall. Bottom row: laminar vs. eddy shear flow, and animal locomotion (flying, running and swimming).

Using a constructal approach, it is fairly simple to show [41] that a bifurcating series of tubes, with N bifurcationsconnected to 2N alveoli minimizes total resistance for human dimensions when N = 23 (Fig. 2).

Designs featuring fewer or more bifurcations have a larger sum of diffusive-resistance and duct-resistance losses:N = 23 is the optimal distribution of imperfection for the dimensions assumed in that study. The prediction of 23 bi-furcations matches closely the actual number of bifurcations in the human lung [42], a concordance which is startlinggiven that the three authors of the demonstration [41] are geophysicists, not respiratory physiologists. General princi-ples of lung design also emerged, for example the prediction that the ratio of the square of the airway diameter to itslength should be constant within a species and is related to the characteristics of the space allocated to the respiratoryprocess [41]. Other research approaches to lung biomechanics and design have provided fruitful perspectives [43,44],most or all of which incorporate principles that we show here to be general.

2.3. Evolution

Design optimization of lungs presumably occurred by the familiar process of natural selection and Darwinianevolution: small heritable variations in lung design that improved blood access to oxygen led to higher reproductivesuccess, and the accumulation of these changes over time resulted ultimately in the current design. Optimality ap-




Fig. 2. The effect of the number of bifurcations on the global resistance to air flow in mammalian airways and lungs [41].

proaches have a long history not only in geophysics [45,46] but also in biology [3] (cf. Section 4.3), however, exceptfor the emergence of a fairly general theory for animal foraging behavior [47] these have been approached in a caseby case fashion. A more unified approach is possible because, as the examples presented here illustrate, biologicaloptimization models can be derived from the constructal law in the same fashion as models for optimization of designof inanimate flow structures.

In this context it is interesting to add the statement from Rinaldo et al. [15] in their analysis of river networks that“nature works through imperfect searches of dynamically accessible optimal configurations”. This is fully concordantwith the way biologists understand Darwinian evolution. Organic evolution is a dynamic process that continuouslysearches the pool of random mutations (accessible features, i.e. only those mutations that arise can be tested and incor-porated if successful) for better designs, and there is a large amount of imperfection involved (genetic drift, selectionon linked alleles, extinction, dispersal limitation, environmental heterogeneity in space and time, etc.). Constructaltheory addresses the general problem of the time-evolution of design by showing how and why all flow systems (inan-imate and animate) develop flow configuration by a process analogous to the way natural selection works in biologyand the way river networks acquire their geometry. Thus, we propose that work on dynamic processes and features ofinanimate entities such as river systems can be united via constructal theory with the dynamic processes and featuresof living systems. The unification occurs by understanding how and why configurations that provide better access fortheir currents (i.e. less impediment to flow) are more likely to be present in the future.

Design-like features summarized as scaling laws are receiving increasing attention (for reviews, see [1]). For ex-ample, the optimized geometry of the dichotomous branching in blood flow and pulmonary airflow (Murray’s rule)means that the tree-shaped flow distributes the wall shear stress uniformly throughout the flow space (e.g., [48,49]).Allometric arguments show that aortic flow velocity should be constant across species, and that aortic shear stressesshould be proportional to the body mass raised to the power −3/8 [50]. Majumdar et al. [51] have shown that ap-parent deviations from Murray’s rule are explained by applying the same principle to asymmetric bifurcations. As ageneral mindset for discovery, we suggest that the existence of a scaling relationship is a hint that there is an underly-ing constructal evolutionary process that is ripe for discovery, theoretical understanding, and opportunity for unifyingperspectives, and that this holds for both animate and inanimate systems.

2.4. Organ sizes

The relations between organ sizes and body mass are another class of scaling laws that continue to draw atten-tion [52]. For these, constructal theory states that the existence of a characteristic organ size is demanded by a trade-offbetween the flow enhancement (reduced loss of useful energy) with increasing size of the organ and the loss of usefulenergy associated with sustaining and carrying the organ as a component of the whole animal design [53,54]. Thepredicted existence of characteristic organ size does not mean that one organ should have the same relative size in




all animals regardless of environment. Consider, for example, the size of the eye in a nocturnal vs. diurnal mammal(much larger in nocturnal), or the size of the leg bones in a burrowing vs. a running animal. Clearly the cost-benefitrelationship for organ function will depend on the environment, and ultimately it is the fitness of the whole organismthat dictates the evolution of design features of its components. The point we seek to make is that there are universalprinciples that drive the evolution of central tendencies of design even though the outcome of that evolution will varytremendously among species and environments. An example is provided in Miguel [55], where constructal theory wasused to show that the contact surface of stony corals should be larger (and tree shaped) when the flowing ambient isless rich in nutrients.

Here is how the prediction of the existence of organ size leads to more predictions. If the fluid flow through theorgan is in the laminar regime, then the pressure difference (�P ) that drives the flow is proportional to mL/D4,where m, L and D are the scales of the mass flow rate, duct lengths and duct diameters. The required pumping poweris proportional to m�P , or m2L/D4, cf. [56]. Next, the organ length scales (L,D) are proportional to the organvolume raised to the power 1/3, or to M

1/3o , where Mo is the mass of the organ. The mass flowrate m is proportional

to the metabolic rate, which is proportional to M3/4 (cf. [1,20,57]), where M is the mass of the animal body.In sum, we expect the pumping power to be proportional to M3/2/Mo, which confirms that larger organs should

require less pumping power. On the other hand, the constructal theory of animal locomotion (Section 2.5) shows thatthe power required to transport the organ is of order MogV , where the body speed V is proportional to M1/6.

All together, these scaling rules indicate that the total power required by the organ must scale as the sumAM3/2/Mo + MogM1/6, where A is a constant. This sum becomes smaller and smaller as the organ size Mo ap-proaches g−1/2M2/3. This M exponent agrees with the organ size scaling noted empirically for organs permeated bylaminar flow, for example the brain, Mo ∼ M0.7 [17,52]. In addition, this theoretical step also teaches that on a planetwith a stronger gravitational field the relative organ size should be smaller than on earth.

The heart is an organ in which the flow is turbulent. The pressure difference across it varies as m2L/D5 (cf.[56]), and the pumping power as m3L/D5 ∼ M9/4/M

4/3o . The power required to carry the heart has the same scale

as before, MogM1/6. The total power requirement reaches its lowest value when the organ mass Mo is proportionalto g−3/7M25/28. The exponent 25/28 = 0.89 agrees well with the allometric exponent (0.97) noted for the hearts ofall animals [17,52]. The scaling rules derived here also show that the empirical exponent for the heart size must belarger than for other organ sizes if the flow is turbulent, not laminar. Because all these theoretical developments areconsequences of the constructal theory of animal locomotion, they cannot be derived from the tree model of Westet al. [20].

2.5. Rhythm: flying, running and swimming

Generation of flow configuration occurs not only spatially but also temporally. Examples treated based on theconstructal law are respiration rate and heart beating [4]: both frequencies must decrease as M−1/4 as the body massincreases. Another time pattern example is animal locomotion: larger animals travel faster, are stronger, and oscillatetheir bodies and limbs less frequently. This is true for all fliers, runners and swimmers [58]. Animal mass moves onthe surface of the earth in the same way as the rivers, the winds and the oceanic currents, by finding geometries, flowcharacteristics and rhythms that allow them to move their mass the greatest distance per expenditure of useful energy(exergy or, approximately, food, fuel).

Note that this is a statement of central tendency and does not deny the presence of variation (idiosyncratic individ-ual species and/or departures by entire phylogenetic lineages) around the scaling pattern. Phylogeny creates geneticarchitectures that, like boulders in rivers, are slow to erode. Furthermore, because of ecological variation, not allspecies are selected to maximize distance per cost. This is an important point, because it acknowledges that there aremultiple factors that shape species and create variation, just as for example constant exposure to a high wind and inputof surrounding sediment would cause a different form of river basin to evolve by processes that we ordinarily addresswithout reference to wind (e.g. [15]) or an exhaustive list of other factors that create idiosyncratic variation.

A simple analysis predicts the magnitude and scaling of velocity (Fig. 3), stride or stroke frequency, and net forceoutput for all of these types of animal locomotion. The details of this theoretical work are available in the literature[58–61]. Here we review the main steps as a way to illustrate how constructal theory can be applied to a biologicaldesign problem. The feature that is common to all types of locomotion is global minimization of loss of usefulenergy—to friction (cost of horizontal motion) and gravity (cost of vertical motion)—per distance traveled. Neither




Fig. 3. Mass scaling of velocity, stride or stroke frequency, and force output for animals that run, swim, and fly. Lines show the relationshipspredicted from the theory. In plots A, B, these predictions ignore constants of order 1, and therefore are expected to be accurate to within an orderof magnitude [58].




of these costs can be avoided completely. They can be balanced so that their sum is minimum, at which point theirmass flow over the earth is thermodynamically equivalent to the mass flow of a river or any other inanimate flowconfiguration (which we propose evolve by analogous mechanisms governed by a common law of physics).

For example, a bird in flight destroys useful energy in two ways. One is the vertical loss: the body has weight, it fallsincrementally, and the bird performs work to lift itself back to the cruising altitude,W1 ∼ MgLb . In this scale-analysisestimate, M is the body mass and Lb the body length scale. The competing “imperfection” is the horizontal loss. Thebird performs work in order to advance horizontally against air friction, W2 ∼ FdragL, where Fdrag ∼ ρaV

2L2bCD and

Fdrag, ρa , V and CD ∼ 1 are the drag force, air density, cruising speed and drag coefficient. The total loss per unit ofdistance traveled is

(1)W1 + W2

L∼ MgLb

L+ ρaV

2L2b

where L ∼ V t , and t is the Galilean time of free fall during one flapping time, t ∼ (Lb/g)1/2. The right side of Eq. (1)is a sum of two terms combined as A/V + BV 2, where A and B are known factors. This sum can be minimized byselecting the cruising speed V , and the result is

(2)V ∼(

ρb

ρa

)1/3

g1/2ρ−1/6b M1/6

where ρb is the body density, ρb ∼ M/L3b . At this speed, the bird reaches optimal distribution of imperfection, that is

W1 ∼ W2 in an order of magnitude sense, and (W1 + W2)/L ∼ Mg.Cruising flight is an optimized rhythm in which the work of repositioning the body on the vertical is matched by

the work of advancing it horizontally. The balance is required by two competing trends: the vertical loss decreases andthe horizontal loss increases as the flying speed increases. Balance is achieved by flapping such that the flying speedis just right. Constructal theory predicts that flying speeds should be distributed in proportion to the body mass raisedto the power 1/6. Flapping frequencies should be proportional to the body mass raised to the power −1/6. Predictionsagree well with observations over the entire range of flying bodies (Fig. 3).

Previous work on bird flight predicted the same velocity scaling [62,63] based on specific features of bird designrather than generalizable principles. The utility of a physics law approach is revealed when we treat running in thesame way as flying to predict the speeds and stride frequencies of runners. The saw-toothed trajectory of a runninganimal is more familiar to us (we also run). Running is a succession of cycles involving the same two losses. Oneloss is the lifting of the body weight to a height proportional to the body length (the vertical deviation dictated bylimb length). This work is the vertical loss, because when the body lands its gravitational potential energy is destroyedin the legs and the ground. The second is the horizontal loss: the work performed to overcome friction against theground, the surrounding air and internal body parts. The vertical and horizontal losses compete, and when they arein balance their sum is minimal. This optimized intermittence is characterized by a speed proportional to M1/6 and astride frequency proportional to M−1/6.

Another prediction is that the average force exerted over the stride or stroke cycle should be on the same orderof magnitude and scale isometrically with the body weight (Mg), even though force output from individual musclesscales with cross sectional area or ∼M2/3. This agrees with the force-weight measurements across all body sizes, forall animals that fly, run and swim [64,65], Fig. 3. Furthermore, the useful energy needed per distance traveled scalesas the body weight (Mg), for all types of locomotion.

As an optimized intermittence, running is similar to flying. But, is it also like swimming? The seemingly obviousanswer is no, because the common view of neutrally buoyant bodies (fish) is that gravity does not matter. The reasonwhy running and swimming are no different than flying (this in spite of the fact that flyers do not touch the ground)is that the ground supports and feels every body that moves relative to it. The same ground serves as reference—themuch bigger body—against which all moving bodies push.

To advance horizontally by one body length, the swimming body must do work equivalent to lifting a body of waterof its own size to a height equal to its body length scale. This body of water must be moved out of the way so that theswimmer may take its place. Aside from compressibility of water and bodies within it, the only way for the displacedwater to move is up, because the free surface is deformable and the bottom is not.

The water lifted by the swimming fish induces a temporary local elevation of the free surface (albeit spread over alarge water surface area and therefore usually imperceptible to us), and a greater local pressure on the lake bottom. The




Table 1Theoretical structure of constructal river basins [54].

i Ai/A0 LT i/A1/20 Ni RLi RBi Dωi/A

1/20 FsiA0 Fsi/D

1/2ωi

LMi/A1/2i

0 1 1/2 1 – – 1/2 1 4 1/21 4 7/2 5 3 4 7/8 5/4 1.63 3/42 42 35/2 21 2 4 35/22 21/16 1.10 3/43 43 76 85 2 4 76/64 85/64 0.94 3/44 44 316 341 2 4 316/256 341/256 0.087 3/4

River 1.5–3.5 3–5 0.7 ∼1.4basins Horton Horton Melton Hack

ground resists the force variations created by every thing that moves: the runner, the flier and the swimmer. Havingrecognized that displacement of a body-sized parcel of water in the vertical plane is physically equivalent to verticaldisplacement of the entire body during running or flying, we were able explain the previously known but unexplainedfact that swimming speeds, stroke frequencies, and force outputs scale in the same fashion as they do for other formsof locomotion.

The ability to predict and describe complex features of animal design from a physics law differs markedly fromone of the major viewpoints of evolutionary biology, which maintains that evolution involves a large degree of chanceand historical contingency, to the extent that if the evolutionary process was reset to its beginning, a very differentset of creatures would evolve [66]. Without denying the role of contingency in macroevolution, the work reviewedhere on lung design, organ size and animal locomotion shows that an evolutionary process should consistently andpredictably produce creatures with certain lung configurations and organ masses that, according to their body mass,move at predictable speeds, stroke/stride frequencies and force outputs. The species and their features would bedifferent, but would have predictable characteristics if they evolved again on this planet or on another planet with adifferent gravitational force and density of the gaseous and liquid environment. An ability to understand and evenpredict complex features of organismal design from physics theory is a long sought and exciting development forbiology.

3. Inanimate flow architectures

3.1. River basins

Throughout the world, river basins evolve in such a way that they are united through statistical correlations (e.g.,Horton’s laws) that indicate similarity, for example the fact that there tend to be approximately 4 tributary streamchannels for each next-order downstream channel (Table 1). These correlations are analogous to allometric correla-tions in physiology. To predict the laws of river basin design without any recourse to empiricism (i.e. simplest form,yielding the least specific but most generalizable result, including transitions from diffusion to channel flow), onecan rely on the constructal law [54,67]. An illustration of this is shown in Fig. 4, where four square areas A0 collectrainwater and channel it away as a stream of length L0/2, width D0 and depth d0. For the flow resistance along thechannel with free surface to be minimal, there must be a universal proportionality between D1 and d1 (this geometricfeature is a consequence of the constructal law, cf. [23]). This means that the optimized river channel has only onetransversal dimension, which is called D1. The volume occupied by the eroded river bed (i.e., the volume of all thechannels) is a measure of its age, and at a particular point in time, it is fixed.

The sizes of the river channels in Fig. 4 a and b will evolve such that the ensuing flow structure poses less and lessflow resistance for a given volume rate of water input to the flow system (drainage basin). The flow along each channelis turbulent, which means that the drop in altitude along the channel (�zi ) is roughly proportional to m2

i Li/D5i , where

Li is the channel length and mi is the mass flow rate through the channel. The construct of Fig. 4a is obtained by twosuccessive pairings, and the result is a square. The overall �z for the 4A0 construct of Fig. 4 is minimum whenD1/D2 = 2−3/7.

Performed for Fig. 4b, the same analysis yields D0/D1 = 2−4/7, and D1/D2 = 2−2/7. The overall flow resistancesof the two 4A0 constructs is such that �za/�zb = 1.964. This shows that the design of Fig. 4b is significantly better:




Fig. 4. Square river basins with four A0 area elements: (a) square obtained by two successive airings (A0 → 2A0,2A0 → 4A0); (b) square obtainedby one quadrupling move (A0 → 4A0) [54].

Fig. 5. River basin with eight A0 area elements: (a) construct obtained by pairing two square constructs of size 4A0; (b) eight A0 elementsassembled in one move [54].

its global resistance is half of what it is in Fig. 4a. In other words, to assemble four elements is much better than topair two A0 elements and, later, pair two 2A0 constructs.

Fig. 5 is a comparison of two river basins of size 8A0 to see whether quadrupling (Fig. 5a) is also better thanassembling eight A0 elements in one move (Fig. 5b). The optimization of the distribution of channel flow volume inFig. 5a yields D0/D1 = 2−4/7, D1/D2 = 2−2/7, and D2/D3 = 2−3/7. In the case of Fig. 5b, the optimal distributionof channel sizes is D0/D1 = 2−5/7, D1/D2 = 2−2/7, D2/D3 = (2/3)2/7, and D3/D4 = (3/4)2/7. The resulting globalflow resistances (�z/m2) are proportional to each other, but not equal, �za/�zb = 0.815.

In sum, quadrupling is best, because it fits between pairing (Fig. 4a) and assemblies of eight constituents (Fig. 5b).Admittedly, this is a rough identification of the integer 4 as a rule of construction, but it is sufficient to view constructalriver basins as area constructs with 4 constituents at each new level of assembly. Also, like the animal locomotiontheory above, this uses no fitting constants or modeling of any type. Unlike numerical modeling, the result is simpleto the point of appearing naïve or primitive compared to complex numerical models, but the reader must keep in mindthe difference between predictive results from pure theory vs. estimates from fitted empirical data.




Fig. 6. The evolution of the cross-sectional configuration of a stream composed of two liquids, low viscosity and high viscosity. In time, the lowviscosity liquid coats all the walls, and the high viscosity liquid migrates toward the center. This tendency of “self-lubrication” is the action of theconstructal law of the generation of flow configuration in geophysics (e.g., volcanic discharges) and in many biological systems [26].

If we neglect the size and fixed shape of the smallest area element (A0 = L20) over which there is a balance between

slow flow (seepage through wet ground on the hill slope) and fast flow (the smallest rivulet [note that constructal theorycould be used to predict the point at which there is a balance between seepage and channel flow (cf. [4]), i.e. where thelocation at which the 4-fold branching should stop, whereas the fractal sequence is infinite and has no inherent way topredict end points, i.e. cutoffs]), we can use the quadrupling rule to predict the morphological features of increasinglycomplex river basins.

Table 1 shows the start of the construction [54], where i is the order of the construct, A0 is the smallest squareconstruct, Ni is the number of streams of all sizes present on Ai , LT i is the total length of all the streams presenton Ai,RLi is the ratio of the length of the largest stream on Ai divided by the length of the largest stream onAi−1,RBi is the number of the largest streams on Ai−1 divided by the number of the largest streams on Ai , Dωi iscalled drainage density and is equal to LT i/Ai , Fsi is the stream frequency (Fsi = Ni/Ai ), and Nsi .

In Table 1 the empirical correlations do not account for the size of the basin, but the theory does. The agreementbetween theory and empirical correlations is excellent across the board, and it improves as the size and complexity ofthe basin (i) increases. Horton’s empirical correlation of stream lengths indicates that RL is a constant with a valuebetween 1.5 and 3.5, while the theory yields 2. Horton’s empirical correlation of stream numbers shows that FB isa constant with a value between 3 and 5, while the constructal value is 4. Melton’s correlation states the Fs/D

2ω is a

constant approximately equal to 0.7, which agrees with the theoretical values. Finally, Hack’s correlation indicates thatthe length of the mainstream LMi is proportional to Ab

i , where Ai is the basin area and b is an exponent between 0.5and 0.56. In the last column of Table 1 we used b = 0.5 to show in dimensionless terms that the constructal architecturealso anticipates the trend correlated by Hack.

3.2. Drawings, as properties of flow systems

The prediction of the scaling laws of river basins illustrates an essential feature of all flow systems that is tradition-ally overlooked. That feature is the drawing, the actual layout of what moves on the background (e.g., Figs. 1, 2, 4–6and Refs. [4,23]). “Flow” represents the movement of one entity relative to another (the background). To describe aflow, we speak of what the flow carries (fluid, heat, mass), how much it carries (mass flow rate, [animate or inanimate],heat current), and where the stream is located in the available space. The “where” is the drawing, i.e. the design. Thisobservation is essential, because there is a vast literature on flow design in which the drawings and maps of how thingsflow are not a concept. The tree networks assumed in the model of West et al. [20] are said to be “space filling”, yet,how the tree flow fills the space is not considered at all. As such the assumed tree model is not a design, and, thisreduces the breadth and applicability of the model.

Another such example is provided by Jordan and Scheuring [27] who reviewed a significant body of researchshowing that flow networks of many kinds have been used in ecology. They defined carefully the properties of theflow networks, for example, the directness, the sign, and the weight (engineering translation: the sense of the flow, theflow resistance, and the flow rate). Absent from this descriptive vocabulary is the drawing, i.e. the actual placement ofthe streams in space. Clearly this presents a challenge for systems involving mobile animals that are not spatially fixeddesign features, but we suggest that there may be new ways to understand both the features of the mobile elementsthemselves (as we have done for locomotion of runners, swimmers, and fliers) and the way they are distributed inspace and time.

Flow systems have boundaries that constrain the flow, and these must be specified in a design. This is in line withcurrent doctrine (fluid mechanics, heat and mass transfer, etc.), where “prediction” begins with defining the system




by stating the boundary conditions, and then solving the conservation laws (mass, momentum, energy, species). Theresult of the analysis is the flow field that resides within the postulated boundaries, e.g., the laminar flow inside a rigidpipe with a pressure difference end to end. In the present context it means that to state the boundary conditions isto postulate and fix the drawing. Such a flow system has configuration (the postulated drawing) but it is not free tomorph. According to the constructal law, the ability of this flow system to provide easier access to its currents will notchange, and the physics of the rigid flow drawing is covered fully by the conservation laws.

In contrast to such rigid systems, natural flow systems morph more or less freely (depending on a host of factors),and their evolution is a rich sequence of flow problems. In this evolution, the drawing itself constitutes an additionalunknown, and the constructal law is the additional principle that (along with the conservation laws) completes thephysics of the natural flow system.

If one specifies how the flow will interact with and alter the boundaries (e.g., the flow through an elastic tube), isthe problem then fully stated and the flow and its boundaries are deducible from the conservation laws without needof additional principles? The answer is yes if the design is pre-specified, but that begs the question of where the originof that design and how design emerges from the most rudimentary flow in evolving systems. A natural flow systemdoes not have a drawing, tube or anything else. The natural system has freedom to morph. The natural system is astream that originates from one point and must find easier access to another point, area or volume. The natural flow(the stream) fills a certain volume, but this volume does not have configuration. It generates configuration—better andbetter for flowing—as time passes. Along the way, the stream draws for itself the tube (if a blood vessel, for example).It is this aspect of natural flow system physics that the constructal law covers. The difference between this aspect andthe classical treatment of flows (with drawings that are assumed, not generated) is treated in detail in Refs. [53,54,68],and is the domain covered by the constructal law in physics.

3.3. Channel cross-sections

The scaling laws of river basins are the geophysical (inanimate) counterparts of the allometric laws of biology,and they arise because, in time, easier flowing configurations tend to replace lesser functioning configurations. Onceestablished, global flow characteristics are similar to evolutionarily stable strategies in biology (e.g., behavioral orlife history tactics that cannot be bettered by alternative tactics once they are present in a certain frequency in apopulation [69]). However, even after the flow system arrives at or near its constructal configuration (also called“equilibrium flow architecture” [68]) it is free to morph. This is because the configuration is not rigid; in fact, itis quite dynamic in mature river basins. At “equilibrium” the flow architecture acquires global features that do notchange, such as the global flow resistance, and the scaling laws that describe hierarchy in the tree design of the riverbasin (Table 1), but the local details remain subject to chance and perturbation. At equilibrium, the flow architecture isthe most free to morph [68], because it is through progressive increases in freedom to morph (i.e. erosion of the barriersthat imposed the most resistance and most constrained the geometry) that the design rose to this high level of globalflow performance. The unpredictable appearance and disappearance of Oxbox lakes on relatively flat floodplains ofmature rivers exemplifies this freedom to morph.

Familiar tree-shaped networks like lungs and rivers are not the only flow configurations of interest. Fig. 1 shows aselection of the most common classes of natural design generation phenomena. The tree-shaped flow networks are abig part of the picture, but no more so than all the other examples. Each represents a class of configurations that occurin enormous numbers. Each class unites the animate with the inanimate. The round cross-section (the circle) occursnaturally in blood vessels, bronchial passages, intestines, underground rivers, and earth worms. The circle is muchsimpler than the tree, and, like the tree, it occurs everywhere.

The biological evolution of duct cross-section cannot be witnessed in blood vessels and bronchial passages becauseour observation time scale (lifetime) is too short in comparison with the time scale of the evolution of species traits.In contrast, the morphing of a round gallery can be observed during erosion processes in soil, following a suddenrainfall, or in the vicinity of a ruptured water main. It can be observed in the evolution of a volcanic lava conduit,where lava with lower viscosity coats the wall of the conduit, and lava with higher viscosity positions itself near thecentral part of the cross-section [70,71]. To have it the other way—high viscosity on the periphery and low viscosityin the center—would create greater resistance to flow and evolution thereof would be a violation of the constructallaw.




Additional information about the evolution of flow configuration is provided by laboratory simulations [70,71] oflava flow with high-viscosity intrusions (Fig. 6). Initially, the intrusion has a flat cross-section, and is positioned nearthe wall of the conduit. In time, i.e. downstream, the intrusion not only migrates toward the center of the cross-sectionbut also develops a round cross-section of its own. This natural tendency of self-lubrication matches what is alwaysobserved when a jet (laminar or turbulent) is injected into a fluid reservoir. If the jet has a flat cross-section, thenfurther downstream it develops into one or more thicker jets with round cross-sections. The opposite trend is notobserved: a round jet does not evolve into a flat jet.

Pattern is evident in the shapes of the cross-sections of streams with a free surface: large rivers are wider anddeeper than smaller rivers. A universal proportionality exists between width and depth in rivers of all sizes. Cross-sections differ from river to river, and from place to place along the same river (primarily in response to variations inthe presence and geometry of erosion-resistant rocks), but in rough scaling terms all cross-sections are geometricallysimilar [72].

3.4. Turbulent structure

A shear flow transmits momentum in the direction perpendicular to the fluid motion. The faster fluid packet entrainsits neighbor to move faster. The transmission of momentum can occur in two ways, by packet to packet friction(viscous diffusion, laminar flow) and by organized motion (streams, rolls, eddies, turbulence). The two mechanismsare seen everywhere: no two eddies are alike, even in the same shear flow. Yet, the transition from one mechanism tothe other is sharp, reproducible and firmly established [73]. Without this empirical and highly dependable knowledge,no hydraulic, aerodynamic and biomedical engineering would be possible.

The origin of “turbulence” lies in the existence of two mechanisms for the flow of momentum: diffusion andstreams (convection). We commented already that this two-mechanism competition is prevalent in nature (Section 2.2),for example, in the generation of pattern in river basins, where diffusion is the seepage down the hill slopes, andconvection is the organized flow along channels. At small times, in a shear flow the thickness of the shear zone (themixing region, length scale D) grows very fast, as t1/2. At longer times, the faster growth is by organized motion(rolling, eddies), when D is proportional to t . The intersection of the t1/2 and t curves dictates the time scale andlength scale of the first eddy. This event corresponds to a “local Reynolds number” Rel = V D/ν of order 102, whichagrees with all known observations of transition to turbulence [73–78].

The smallest eddies in a turbulent flow field must have V D/ν ∼ 102, where V is the local turning (peripheral)speed, and D is the eddy transversal length scale. Here is how this constructal-design feature manifests itself through-out much more complex flow architectures such as the lung. Consider one branching, from a tube with mean airvelocity V1 and diameter D1, into two tubes with V2 and D2. Because D2/D1 = 2−1/3 and mass conservation, therespective local Reynolds numbers (Rel,1 = V1D1/ν, Rel,2 = V2D2/ν) form the ratio Rel,1/Rel,2 = 22/3 > 1. Thismeans that turbulence, if it occurs, is a phenomenon in the larger tube. It also explains why during peak inspiration(increasing V1,2) the flow turbulence spreads to several generations of smaller tubes below the trachea (because theRel values of all the tubes increase).

The main theoretical result contributed by constructal theory is that the occurrence of eddies is predicted, notassumed. The eddy is not the result of an assumed “disturbance”, or of an ad-hoc “instability” postulate that thefastest growing deformation is selected [79]. Each eddy is an expression of the balance between two momentumtransport mechanisms, in the same way that every rivulet is in balance with the seepage across the area allocated tothe rivulet and moving animals are optimally balancing two forms of energy dissipation. The eddy is the necessary(better) configuration, under the right circumstances.

Pattern is also evident in the behavior of liquid droplets when they hit a solid surface. The smashed droplet is asplat (disc) or a splash (crown, with fingers and dendrites), Fig. 1. Smaller and slower droplets come to rest as splats.Larger and faster droplets come to rest as splashes. The naturally occurring transition between one flow configurationand the other is firmly established in empirical correlations. Ink-jet printing, high-tech metal deposition (film coating)and forensic science of blood splatter would not be possible without this natural pattern. Splats and splashes selectthemselves always in the same way so that the flow comes to rest faster (i.e. along rapidly evolved geometries thatcreated less flow resistance during the time that energy was being dissipated), and the surviving external configuration(peripheral shape) can be predicted [80].




3.5. Zipf distributions

Unexpected theoretical predictions emerge when one invokes the constructal law in classical fields where “design”has been studied in great depth, but only descriptively. One example is the prediction of the Zipf distribution of cityranks versus city sizes (Ref. [54], pp. 774–779), which was made possible by recognizing the optimal allocation offlow paths to areas, in the sense of Figs. 4 and 5 for river scaling in this paper.

The most recent constructal theory development is the prediction of the Zipf distribution of tree ranks versus treesizes in forests [81], which is a consequence of the tendency of the forest (and each tree) to facilitate the flow of water,from ground to air. The theory also predicts the basic features of root, trunk and canopy architecture, including therules of Leonardo and Fibonacci.

To predict the Zipf distribution had been impossible, as demonstrated by Fontanari and Perlovsky [82]. The predic-tion of the Zipf distribution of city sizes suggested [54] that all the domains in which Zipf distributions are observed(information, news, language, brain, computing) are homes to tree-shaped flow systems with architectures optimallyallocated in space. These domains are constructal flow systems: river basins and deltas akin to those presented in thisreview. Geometry, geography, tree architectures, freedom to morph, persistent hierarchy (e.g., Ref. [83]), and optimalfinite complexity (visible pattern) are constructal properties of the flow of information.

4. A unifying perspective

4.1. Constructal law and thermodynamics

It is important to recognize that the constructal law is different than (i.e. complementary to) the other laws ofthermodynamics. The constructal law was proposed as an addition to the laws of thermodynamics [23,84]. To seewhy, consider an isolated thermodynamic system (no work, heat and mass flow across the system boundary) that isinitially in a state of internal non-uniformity (e.g., regions of higher and lower pressures or temperature, separated byinternal partitions that suddenly break). The first law states that the energy of the system stays constant. The secondlaw accounts for observations that describe a tendency in time: in time, if changes occur inside the system all flowsproceed one way, from high to low. Analytically, this trend is noted by writing that the entropy of the system increasesduring each change [23].

The first and second laws speak of a black box, without geometry. They say nothing about the configurations (thedrawings, maps, pathways) of the things that flow.

The generation of flow configuration is an entirely different phenomenon of physics, which is not accounted forby the first law and the second law. The time sequence of drawings that the flow system exhibits as it evolves is thephenomenon covered by the constructal law. Not the drawings per se, but the time direction in which they morph ifgiven freedom. No drawing in nature is “predetermined” or “destined” to be or to become a particular image. Theactual evolution or lack of evolution (the rigidity) of the drawing depends on many factors, which are mostly random.The constructal law statement is not about maximizing, minimizing, optimizing or destiny.

In this brief review of thermodynamics, we used the isolated system formulation because of its extreme simplic-ity, not because it is prevalent in nature. The system is a “non-equilibrium system”, which evolves in time towardequilibrium (no internal flows), and which exhibits the phenomenon of generation of flow configurations during thisevolution.

Natural flow systems, biological or not, are best modeled as open systems. The simplest are those that can be mod-eled to function in the steady state, i.e. in time-independent fashion. These too are non-equilibrium thermodynamicsystems, because differences and gradients of their properties are present inside their volumes. For any such system,the first law states that energy is conserved as it flows through the system. The second law states that the entropyoutflow must be greater than the entropy inflow. This difference between the outflow and the inflow is the rate ofentropy generation Sgen, which is constant because of the steady state. The entropy (S) and all the other inventories(properties) of the system (energy, volume, mass, enthalpy, exergy, etc.) are constant in the steady state [23].

To the thermodynamics of open systems, the constructal law adds that the steady state will continue if the flowsystem is not free to change its configuration. The constructal law also says that if the system morphs freely, then itwill evolve (for example, as a sequence of steady states) such that its configuration develops and its currents flow moreand more easily. What changes in the properties and fluxes of the system from one configuration to the next depends




on what is fixed, for example, what drives the flows through the system. In any case, easier and easier flowing meansthat globally the system configuration poses less and less resistance, and that the rate of entropy generation tendsin one direction: down or up, depending on what kind of forcing is fixed (for details, see [85]). The constructal lawaccounts for the unique direction of changes in Sgen. In this way, the constructal law unifies two contradictory fields,which claim the minimization of Sgen or the maximization of Sgen as working ad-hoc principles of optimality (see alsoSection 4.3 and [85]).

4.2. Time

The arrow of time is aligned with two phenomena, not one. The classical phenomenon is captured by the secondlaw of thermodynamics, i.e. the arrow of irreversibility (every thing flows one way, from high to low), which wasexpressed analytically by Clausius, by defining the new property “entropy” and stating that all systems have a tendencyto generate entropy [86]. The new phenomenon is the time direction of how every thing acquires architecture: existingconfigurations survive when they change in time toward easier flowing configurations (the constructal law).

4.3. First principles versus ad-hoc optimality postulates

A new physical theory must show its relation to first principles. This we have done in this unified presentationof animate and inanimate design phenomena. All the examples, from animal allometry to geophysical scaling laws,obey the known and accepted principles of physics. The fact that they all exhibit “designedness” is the something “inaddition”, which is not covered by the known and accepted first principles.

Science, especially in the past two decades, has been very active in this direction: to identify the principle or prin-ciples that can be used to account for the occurrence of design-like features across the board. Many ad-hoc principleshave been proposed: minimization of entropy generation, maximization (sic) of entropy generation, minimization offlow resistance, maximization (sic) of flow resistance (e.g., animal hair insulation), minimization of travel time, theaxiom of uniform stresses, the three assumptions made in West et al.’s [20] model, etc.

These are optimality (min, max, end objective) statements. Each principle works in its particular domain. None isuniversal. For example, West et al.’s [20] model is based not on one ad-hoc postulate but on at least three: the flowarchitecture is space filling (read: tree), the size of the smallest element in this structure is fixed, and the pumpingpower is minimum. The constructal law, which was stated one year earlier [84], covers these three postulates [87]. Inthis article we showed that the constructal law covers a wide diversity of natural design phenomena that lie outsidethe domain covered by the West et al. model, for example: inanimate flows, flows that are not dendritic, turbulence,river basins, animal locomotion, organ sizes, self-lubrication, splash-splat droplet behavior, Zipf distributions, etc.

A first principle has unifying power. One more example is Murray’s rule of dichotomous branching (Section 2.3),which is a consequence of the constructal law. This rule also means that wall shear stresses must be distributeduniformly through the flow volume, which applies so well that we now speak of an “axiom” of uniform shear stressesin vascular design. The constructal law expands the reach of this statement to the entire solid body, because theconstructal organ size (Section 2.4) is better when all its stressed elements are stressed to the same, allowable level.Bone and muscle sizes and shapes come from this tendency, to support more with a fixed amount of solid. The sizes,shapes and numbers of branches in botanical trees are deducible in the same manner [81]. Flow systems and solidstructures owe their designs to the same tendency, under the same principle. According to constructal theory, solidmechanical structures are conduits for the flow of stresses [56].

Ultimately, the constructal law is not about optimality and end objective, even though it predicts optimized featuresof particular phenomena such as animal locomotion speeds. More fundamentally it is simply about one overlookedphysics phenomenon (the generation of configuration) and the fact that this phenomenon evolves in a particular direc-tion in time. Our proposal is to judge this law on its merits, across the board, and in relation to (i.e. in addition to) theother laws. This is why the comparative discussion with the laws of thermodynamics (above) is important.

This is an exciting development for physics, but it should also resonate in biology. For example, the discovery thatfundamental features of animal design can be predicted from physics theory provides substantial support for the view[88] that “evolution shows an eerie predictability, leading to the direct contradiction of the currently accepted wisdomthat insists on evolution being governed by the contingencies of circumstance (most famously expressed in Gould’smetaphor of “re-running the tape of life”)”. Furthermore, the idea that organic evolution is analogous to the way form




evolves in inanimate flow systems is a novel concept that has the potential to unite perspectives and approaches acrossdisparate disciplines. We suggest that the constructal law provides a powerful tool for examining and understandingvariation in both the animate and inanimate compartments of nature.

Acknowledgement

Prof. Bejan’s research was sponsored by grants from the US Air Force Office of Scientific Research and theNational Science Foundation. Prof. J.H. Marden’s research was supported by NSF ER-0412651.

References

[1] Hoppeler H, Weibel ER, editors. Scaling functions to body size: Theories and facts. J Exp Biol 2005;208(9) [special issue].[2] Weibel ER. Symmorphosis. Cambridge: Harvard University Press; 2000.[3] Alexander M. Optima for animals. Princeton: Princeton University Press; 1996.[4] Bejan A. Shape and structure, from engineering to nature. Cambridge, UK: Cambridge University Press; 2000.[5] Whitfield J. Order out of chaos. Nature 2005;436:905–7.[6] Dawkins R. The blind watchmaker. New York: Norton; 1996.[7] Turner JS. The Tinkerer’s accomplice. Cambridge: Harvard University Press; 2000.[8] Turner S. Homeostasis, Complexity, and the problem of biological design. Emergence: Complexity and Organization 2008;10(2):76–89.[9] Bejan A. Design in nature: tinkering and the constructal law. The Quarterly Review of Biology 2008;83:91–4.

[10] Niklas KJ. Plant allometry: The scaling of form and process. Chicago: University of Chicago Press; 1994.[11] Brown JH, West GB. Scaling in biology. Oxford: Oxford University Press; 2000.[12] Hubbell SP. The unified neutral theory of biodiversity and biogeography. Princeton: Princeton University Press; 2001.[13] Rodriguez-Iturbe I, Rinaldo A. Fractal river basins. Cambridge: Cambridge University Press; 1997.[14] Meakin P. Fractals, scaling and growth far from equilibrium. Cambridge: Cambridge University Press; 1998.[15] Rinaldo A, Banavar JR, Maritan A. Trees, networks, and hydrology. Water Resour Res 2006;42:W06D07.[16] Bejan A, Merkx GW, editors. Constructal theory of social dynamics. New York: Springer; 2007.[17] Ahlborn BK. Zoological physics. Berlin: Springer; 2004.[18] Vogel S. Life’s devices. Princeton: Princeton University Press; 1988.[19] West GB, Brown JH. Life’s universal scaling laws. Phys Today 2004:36–42.[20] West GB, Brown JH, Enquist BJ. A general model for the origin of allometric scaling laws in biology. Science 1997;276:122–6.[21] Banavar JR, Maritan A, Rinaldo A. Size and form in efficient transportation networks. Nature 1999;399:130–2.[22] Perlovsky LI. Toward physics of the mind: concepts, emotions, consciousness, and symbols. Phys Life Rev 2006;3:23–55.[23] Bejan A. Advanced engineering thermodynamics. 2nd edition. New York: Wiley; 1997.[24] Poirier H. Une théorie explique l’intelligence de la nature. Science & Vie 2003;1034:44–63.[25] Reis AH. Constructal theory: from engineering to physics, and how flow systems develop shape and structure. Appl Mech Rev 2006;59:269–

82.[26] Bejan A, Lorente S. Constructal theory of generation of configuration in nature and engineering. J Appl Phys 2006;100:041301.[27] Jordan F, Scheuring I. Network ecology: topological constraints on ecosystem dynamics. Phys Life Rev 2004;1:139–72.[28] Levin SA. The problem of pattern and scale in ecology: the R.H. MacArthur award lecture. Ecology 1992;73:1943–67.[29] Levy M, Solomon S. Power laws are logarithmic Boltzmann laws. Int J Modern Phys C 1996;7:595–601.[30] Sneppen K, Newman MEJ. Coherent noise, scale invariance and intermittency in large systems. Physica D, Nonlinear Phenomena

1997;110:209–22.[31] Sornette D, Cont R. Convergent multiplicative processes repelled from zero: Power laws and truncated power laws. J Physique I 1997;7:431–

44.[32] Bak P. How nature works: The science of self-organized criticality. Berlin: Copernicus-Springer; 1997.[33] Colaiori F, Flammini A, Banavar JR, Maritan A. Analytical and numerical study of optimal channel networks. Phys Rev E 1997;55:1298–310.[34] Marsili M, Zhang YC. Interacting individuals leading to Zipf’s law. Phys Rev Lett 1998;80:2741–4.[35] Dodds PS, Rothman DH, Weitz JA. Re-examination of the 3/4 law of metabolism. J Theor Biol 2001;209:9–27.[36] Newman MEJ. The structure and function of complex networks. SIAM Rev 2003;45(2):157–256.[37] Park J, Newman MEJ. Origin of degree correlations in the Internet and other networks. Phys Rev E 2003;68(2):026112.[38] Mandelbrot BB. The fractal geometry of nature. New York: Freeman; 1982.[39] Avnir D, Biham O, Lidar D, Malcai O. Is the geometry of nature fractal? Nature 1998;279:39–40.[40] Nottale L. Fractal space–time and microphysics. Singapore: World Scientific; 1993.[41] Reis AH, Miguel AF, Aydin M. Constructal theory of flow architectures of the lungs. Medical Physics 2004;31:1135–40.[42] Horsfield K, Dart G, Olson DE, Filley GF, Cumming G. Models of the human bronchial tree. J Appl Physiol 1971;31:207–17.[43] Kitaoka H, Takaki R, Suki B. A three-dimensional model of the human airway tree. J Appl Physiol 1999;87:2207–17.[44] Tawhai MH, Burrowes KS. Developing integrative computational models of pulmonary structure. The Anatomical Record (Part B: New Anat)

2003;275B:207–18.[45] Malkus WVR. The heat transport and spectrum of turbulence. Proc Roy Soc A 1954;225:196–212.




[46] Paltridge GW. Global dynamics and climate – a system of minimum entropy exchange. Q J R Meteorol Soc 1975;101:475–84.[47] Stephens DW, Krebs JR. Foraging theory. Princeton: Princeton University Press; 1986.[48] Stroev PV, Hoskins PR, Easson WJ. Distribution of wall shear rate throughout the arterial tree: a case study. Atherosclerosis 2007;191(2):276–

80.[49] Karau KL, Krenz GS, Dawson CA. Branching exponent heterogeneity and wall shear stress distribution in vascular trees. Am J Physiol Heart

Circ Physiol 2001;280:1256–63.[50] Weinberg PD, Ethier CR. Twenty-fold difference in hemodynamic wall shear stress between murine and human aortas. J Biomechanics

2007;40(7):1594–8.[51] Majumdar A, Alencar AM, Buldyrev SV, Hantos Z, Lutchen KR, Stanley HE, et al. Relating airway diameter distributions to regular branching

asymmetry in the lung. Phys Rev Lett 2005;95:168101.[52] Schmidt-Nielsen K. Scaling – why is animal size so important? Cambridge: Cambridge University Press; 1984.[53] Bejan A, Lorente S. La loi constructale. Paris: L’Harmattan; 2005.[54] Bejan A. Advanced engineering thermodynamics. 3rd edition. Hoboken: Wiley; 2006.[55] Miguel AF. Constructal pattern formation in stony corals, bacterial colonies and plant roots under different hydrodynamics conditions. J Theor

Biol 2006;242:954–61.[56] Bejan A, Lorente S. Design with constructal theory. Hoboken: Wiley; 2008.[57] Bejan A. The tree of convective streams: its thermal insulation function and the predicted 3/4-power relation between body heat loss and body

size. Int J Heat Mass Transfer 2001;44:699–704.[58] Bejan A, Marden JH. Unifying constructal theory for scale effects in running, swimming and flying. J Exp Biol 2006;209:238–48.[59] Bejan A, Marden JH. Constructing animal locomotion from new thermodynamics theory. Am Scientist 2006;94:342–9.[60] Bejan A, Marden JH. Locomotion: une même loi pour tous. Pour la Science 2006;346.[61] Bejan A, Marden JH. Termodinamica de la locomotion animal. Investigacion y Ciencia 2006:6–14.[62] Lighthill J. Aerodynamic aspects of animal flight. 5th fluid science lecture, British Hydrodynamics Research Association, June 1974.[63] Greenewalt CH. The flight of birds. The significant dimensions, their departure from the requirements of dimensional similarity, and the effect

on flight aerodynamics of that departure. Trans Amer Phil Soc 1975;65:1–67.[64] Marden JH, Allen LR. Molecules, muscles and machines: universal characteristics of motors. Proc Nat Acad Sci 2002;99:4161–6.[65] Marden JH. Scaling of maximum net force output by motors locomotion. J Exp Biol 2005;208:1653–64.[66] Gould SJ. Wonderful life: The burgess shale and the nature of history. New York: Norton; 1990.[67] Reis AH. Constructal view of scaling laws of river basins. Geomorphology 2006;78:201–6.[68] Bejan A, Lorente S. The constructal law and the thermodynamics of flow systems with configuration. Int J Heat Mass Transfer 2004;47:3203–

14.[69] Smith JM. Evolution and the theory of games. Cambridge: Cambridge University Press; 1982.[70] Carrigan CR, Eichelberger JC. Zoning of magmas by viscosity in volcanic conduits. Nature 1990;343:248–51.[71] Carrigan CR. Two-component magma transport and the origins of composite intrusions and lava flows. In: Ryan MP, editor. Magmatic

systems. New York: Academic Press; 1994.[72] Scheidegger A. Theoretical geomorphology. 2nd edition. Berlin: Springer; 1970.[73] Bejan A. Convection heat transfer. 3rd edition. Hoboken: Wiley; 2004.[74] Bejan A. Entropy generation through heat and fluid flow. New York: Wiley; 1982.[75] Levi E. A universal Strouhal law. ASCE J Eng Mech 1983;109:718–27.[76] Mikic BB, Vujisic B, Kapat J. Turbulent transition and maintenance of turbulence; implications to heat transfer augmentation. Int J Heat Mass

Transfer 1994;37(Suppl. 1):425–31.[77] Cervantes JG, Solorio FJ, Mendez F. On the buckling and the Strouhal law of fluid columns: the case of turbulent jets and wakes. J Fluids

Struct 2003;17:1203–11.[78] Childress S, Dudley R. Transition from ciliary to flapping mode in swimming mollusk: flapping flight as a bifurcation in Rew . J Fluid Mech

2004;498:257–88.[79] Rayleigh JWS. On the stability, or instability of certain fluid motions. Proc London Math Soc 1880;XI:57–70.[80] Bejan A, Gobin D. Constructal theory of droplet impact geometry. Int J Heat Mass Transfer 2006;49:2412–9.[81] Bejan A, Lorente S, Lee J. Unifying constructal theory of tree roots, canopies and forests. J Theor Biol 2008;254:529–40.[82] Fontanari JF, Perlovsky LI. Solvable null model for the distribution of word frequencies. Phys Rev E 2004;70:042901.[83] Bejan A. Why university rankings do not change: education as a natural hierarchical flow architecture. Int J Design & Nature 2007;2(4):319–

27.[84] Bejan A. Constructal-theory network of conducting paths for cooling a heat generating volume. Int J Heat Mass Transfer 1997;40:799–816.

[published 1 Nov. 1996].[85] Reis AH, Bejan A. Constructal theory of global circulation and climate. Int J Heat Mass Transfer 2006;49:1857–75.[86] Clausius R. Mechanical theory of heat. London: John Van Voorst; 1867.[87] Bejan A. Constructing a theory of scaling and more. Phys Today 2005:20.[88] Conway Morris S. University of Edinburgh Gifford Lectures, 2007.

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