In
biophysical fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) a ...
, Murray's law is a potential relationship between
radii at
junctions in a network of fluid-carrying
tubular pipes
Pipe(s), PIPE(S) or piping may refer to:
Objects
* Pipe (fluid conveyance), a hollow cylinder following certain dimension rules
** Piping, the use of pipes in industry
* Smoking pipe
** Tobacco pipe
* Half-pipe and quarter pipe, semi-circula ...
. Its simplest version proposes that whenever a branch of
radius
In classical geometry, a radius (plural, : radii) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', ...
splits into two branches of radii
and
, then all three radii should obey the equation
If network flow is
smooth and
leak-free, then systems that obey Murray's law minimize the resistance to flow through the network. For
turbulent
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
networks, the law takes the same form but with a different
characteristic exponent .
Murray's law is observed in the
vascular
The blood vessels are the components of the circulatory system that transport blood throughout the human body. These vessels transport blood cells, nutrients, and oxygen to the tissues of the body. They also take waste and carbon dioxide away f ...
and
respiratory systems of animals,
xylem
Xylem is one of the two types of transport tissue in vascular plants, the other being phloem. The basic function of xylem is to transport water from roots to stems and leaves, but it also transports nutrients. The word ''xylem'' is derived from ...
in plants, and the respiratory system of insects. In principle, Murray's law also applies to
biomimetic engineering, but human designs rarely exploit the law.
Murray's law is named after
Cecil D. Murray, a
physiologist
Physiology (; ) is the scientific study of functions and mechanisms in a living system. As a sub-discipline of biology, physiology focuses on how organisms, organ systems, individual organs, cells, and biomolecules carry out the chemica ...
at
Bryn Mawr College
Bryn Mawr College ( ; Welsh: ) is a women's liberal arts college in Bryn Mawr, Pennsylvania. Founded as a Quaker institution in 1885, Bryn Mawr is one of the Seven Sister colleges, a group of elite, historically women's colleges in the United ...
, who first argued that efficient transport might determine the structure of the human
vascular system
The blood circulatory system is a system of organs that includes the heart, blood vessels, and blood which is circulated throughout the entire body of a human or other vertebrate. It includes the cardiovascular system, or vascular system, t ...
.
Assumptions
Murray's law assumes material is passively transported by the flow of fluid in a network of
tubular pipes
Pipe(s), PIPE(S) or piping may refer to:
Objects
* Pipe (fluid conveyance), a hollow cylinder following certain dimension rules
** Piping, the use of pipes in industry
* Smoking pipe
** Tobacco pipe
* Half-pipe and quarter pipe, semi-circula ...
,
[: "Poisseuille's equation for the law of flow in cylindrical tubes may be taken as a very fair approximation giving the relationships required."] and that said network requires energy both to maintain flow and structural integrity. Variation in the fluid
viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the int ...
across scales will affect the Murray's law exponent, but is usually too small to matter.
At least two different conditions are known in which the cube exponent is optimal.
In the first, organisms have free (variable) circulatory volume. Also, maintenance energy is not proportional to the pipe material, but instead the quantity of working fluid. The latter assumption is justified in metabolically active biological fluids, such as
blood
Blood is a body fluid in the circulatory system of humans and other vertebrates that delivers necessary substances such as nutrients and oxygen to the cells, and transports metabolic waste products away from those same cells. Blood in the cir ...
. It is also justified for metabolically ''inactive'' fluids, such as air, as long as the energetic "cost" of the infrastructure scales with the
cross-sectional area
In geometry and science, a cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional spaces. Cutting an object into slices creates many parallel cross-sections. The ...
of each tube; such is the case for all known biological tubules.
In the second, organisms have fixed circulatory volume and pressure, but wish to minimize the resistance to flow through the system. Equivalently, maintenance is negligible and organisms with to maximize the
volumetric flow rate
In physics and engineering, in particular fluid dynamics, the volumetric flow rate (also known as volume flow rate, or volume velocity) is the volume of fluid which passes per unit time; usually it is represented by the symbol (sometimes ). I ...
.
Although most derivations of Murray's law assume a
steady state
In systems theory, a system or a process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those properties ''p' ...
flow field
Flow may refer to:
Science and technology
* Fluid flow, the motion of a gas or liquid
* Flow (geomorphology), a type of mass wasting or slope movement in geomorphology
* Flow (mathematics), a group action of the real numbers on a set
* Flow (psyc ...
, the same results apply for flow in tubes that have a width moderate-to-small relative to the flow
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
.
Derivation in laminar, mass-conservative networks with non-negligible upkeep
Murray's original derivation uses the first set of assumptions described above. She begins with the
Hagen–Poiseuille equation
In nonideal fluid dynamics, the Hagen–Poiseuille equation, also known as the Hagen–Poiseuille law, Poiseuille law or Poiseuille equation, is a physical law that gives the pressure drop in an incompressible and Newtonian fluid in laminar flow ...
, which states that for fluid of
dynamic viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the int ...
, flowing
laminarly through a
cylindrical
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an in ...
pipe of radius and length , the
volumetric flow rate
In physics and engineering, in particular fluid dynamics, the volumetric flow rate (also known as volume flow rate, or volume velocity) is the volume of fluid which passes per unit time; usually it is represented by the symbol (sometimes ). I ...
associated with a pressure drop is
and the corresponding
power consumed is
Said pipe contains volume . If the maintenance power density is , then the total power consumed (from both flow and upkeep) is
Minimizing this quantity depends on precisely which variables the organism is free to manipulate, but the minimum invariably occurs when the two terms are proportional to each other. In that minimal case, the proportionality determines a relationship between and . Canceling common factors and taking a square root,
That is, when using as little energy as possible, the
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
flowing through the pipe must be proportional to the cube of the pipe's radius. Since flow is leakless, the total flow rate into a junction
must
Must (from the Latin ''vinum mustum'', "young wine") is freshly crushed fruit juice (usually grape juice) that contains the skins, seeds, and stems of the fruit. The solid portion of the must is called pomace and typically makes up 7–23% of th ...
be the total flow rate out:
Substituting () then gives Murray's law with .
Exponents for other networks
If the network does not rely on transported material getting "swept up in the flow", but instead expects it to passively diffuse, then resistance to transport is minimized when : that is,
The same law would apply to a
direct-current electrical grid
An electrical grid is an interconnected network for electricity delivery from producers to consumers. Electrical grids vary in size and can cover whole countries or continents. It consists of:Kaplan, S. M. (2009). Smart Grid. Electrical Power ...
composed of wires of only one material, but varying
diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid f ...
.
For
turbulent flow
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between t ...
, transport resistance is minimized when ; that is:
In general, networks intermediate between diffusion and laminar flow are expected to have characteristic exponents between 2 and 3, at least approximately.
Experimental tests
Murray's law has been verified in chicks; dog intestines and lungs; cat
mesentery
The mesentery is an organ that attaches the intestines to the posterior abdominal wall in humans and is formed by the double fold of peritoneum. It helps in storing fat and allowing blood vessels, lymphatics, and nerves to supply the intesti ...
; and human intestines and lung capillaries. Mice
genetically engineered to lack the blood-vessel-wall protein
elastin
Elastin is a protein that in humans is encoded by the ''ELN'' gene. Elastin is a key component of the extracellular matrix in gnathostomes (jawed vertebrates). It is highly elastic and present in connective tissue allowing many tissues in the bo ...
have smaller and thinner blood vessels, but still obey Murray's law.
In humans, large vessels, such as the
aorta
The aorta ( ) is the main and largest artery in the human body, originating from the left ventricle of the heart and extending down to the abdomen, where it splits into two smaller arteries (the common iliac arteries). The aorta distributes o ...
or
trachea
The trachea, also known as the windpipe, is a cartilaginous tube that connects the larynx to the bronchi of the lungs, allowing the passage of air, and so is present in almost all air- breathing animals with lungs. The trachea extends from t ...
, do not appear to obey Murray's law, instead obeying a Murray's law with exponent close to 2. But flow in those vessels is also partially turbulent, and so should exhibit an exponent nearer to than to 3.Insects do not have a fully-fledged circulatory system, instead relying on passive
diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical ...
through the
haemocoel. For those networks, Murray's law predicts constant cross-sectional area, which is observed.
The same arguments that imply Murray's law also imply that the distribution of tubules should exhibit a specific
power law
In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one q ...
scaling with size. Plant
xylem
Xylem is one of the two types of transport tissue in vascular plants, the other being phloem. The basic function of xylem is to transport water from roots to stems and leaves, but it also transports nutrients. The word ''xylem'' is derived from ...
is known to exhibit that scaling except in scenarios where the passages double as
structural support A structural support is a part of a building or structure that provides the necessary stiffness and strength in order to resist the internal forces (vertical forces of gravity and lateral forces due to wind and earthquakes) and guide them safely to ...
s.
History
The first phenomenon now recognized as Murray's law is
Young's rule for circulatory systems, which states that two identical subcapillaries should combine to form a capillary with radius about larger, and dates to the
early 19th century. Bryn Mawr physiologist
Cecil D. Murray published the law's modern, general formulation in 1926, but it languished in a disciplinary no-man's-land for the next fifty years: too trivial for physicists and too complicated for biologists. Interest in the law revived in the 1970s.
Applications
In circulatory system governed by Murray's law with ,
shear stress
Shear stress, often denoted by ( Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. '' Normal stress'', on ...
on vessel walls is roughly constant. Consequently, variations in shear stress are a sign of deviation from Murray's law; Rodbard and Zamir suggest that such variations stimulate
homeostatic
In biology, homeostasis (British also homoeostasis) (/hɒmɪə(ʊ)ˈsteɪsɪs/) is the state of steady internal, physical, and chemical conditions maintained by living systems. This is the condition of optimal functioning for the organism an ...
growth or contraction.
Murray's law rarely applies to engineered materials, because man-made transport routes attempt to reduce flow resistance by minimizing branching and maximizing diameter.
Murray materials
Materials that obey Murray's law at the microscale, known as Murray materials, are expected to have favorable flow characteristics, but their construction is difficult, because it requires tight control over
pore size typically over a wide range of scales.
Lim ''et al'' propose designing
microfluidic "
labs on a chip" in accord with Murray's law to minimize flow resistance during analysis. Conventional lithography does not support such construction, because it cannot produce channels of varying depth.
Seeking long-lived
lithium battery
Lithium battery may refer to:
* Lithium metal battery, a non-rechargeable battery with lithium as an anode
** Rechargeable lithium metal battery, a rechargeable counterpart to the lithium metal battery
* Lithium-ion battery, a rechargeable batte ...
electrode
An electrode is an electrical conductor used to make contact with a nonmetallic part of a circuit (e.g. a semiconductor, an electrolyte, a vacuum or air). Electrodes are essential parts of batteries that can consist of a variety of materials ...
s, Zheng ''et al'' constructed Murray materials out of layers of
sintered
Clinker nodules produced by sintering
Sintering or frittage is the process of compacting and forming a solid mass of material by pressure or heat without melting it to the point of liquefaction.
Sintering happens as part of a manufacturing ...
zinc oxide
Zinc oxide is an inorganic compound with the formula . It is a white powder that is insoluble in water. ZnO is used as an additive in numerous materials and products including cosmetics, food supplements, rubbers, plastics, ceramics, glass, cement ...
nanoparticle
A nanoparticle or ultrafine particle is usually defined as a particle of matter that is between 1 and 100 nanometres (nm) in diameter. The term is sometimes used for larger particles, up to 500 nm, or fibers and tubes that are less than 10 ...
s. The evaporation rate of the dissolved zinc oxide
solvent
A solvent (s) (from the Latin '' solvō'', "loosen, untie, solve") is a substance that dissolves a solute, resulting in a solution. A solvent is usually a liquid but can also be a solid, a gas, or a supercritical fluid. Water is a solvent for ...
controlled the size of the pores in each layer; the network was then just layers of ZnO with different pore sizes placed atop each other.
Because
power plant
A power station, also referred to as a power plant and sometimes generating station or generating plant, is an industrial facility for the generation of electric power. Power stations are generally connected to an electrical grid.
Many ...
working fluid
For fluid power, a working fluid is a gas or liquid that primarily transfers force, motion, or mechanical energy. In hydraulics, water or hydraulic fluid transfers force between hydraulic components such as hydraulic pumps, hydraulic cylinders, a ...
s typically funnel into many small tubules for efficient
heat transfer
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy ( heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conducti ...
, Murray's law may be appropriate for
nuclear reactor
A nuclear reactor is a device used to initiate and control a fission nuclear chain reaction or nuclear fusion reactions. Nuclear reactors are used at nuclear power plants for electricity generation and in nuclear marine propulsion. Heat fr ...
design.
References
Footnotes
Further reading
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* {{Cite journal , last1=Zheng , first1=Xianfeng , last2=Shen , first2=Guofang , last3=Wang , first3=Chao , last4=Li , first4=Yu , last5=Dunphy , first5=Darren , last6=Hasan , first6=Tawfique , author-link6=Tawfique Hasan , last7=Brinker , first7=C. Jeffrey , last8=Su , first8=Bao-Lian , date=2017-04-06 , title=Bio-inspired Murray materials for mass transfer and activity , journal=Nature Communications , language=en , volume=8
, pages=14921 , bibcode=2017NatCo...814921Z , doi=10.1038/ncomms14921 , issn=2041-1723 , pmc=5384213 , pmid=28382972
Botany
Equations
Developmental biology