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Duration of urination does not change with body sizePatricia J. Yanga, Jonathan Phama, Jerome Chooa, and David L. Hua,b,1
Schools of aMechanical Engineering and bBiology, Georgia Institute of Technology, Atlanta, GA 30332
Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved May 14, 2014 (received for review February 6, 2014)
Many urological studies rely on models of animals, such as rats and
generate jets. Instead, they urinate using a series of drops, which
pigs, but their relation to the human urinary system is poorly
is shown by the 0.03-kg lesser dog-faced fruit bat and the 0.3-kg
understood. Here, we elucidate the hydrodynamics of urination
rat in Fig. 2 A–C, respectively.

across five orders of magnitude in body mass. Using high-speed
Fig. 1H shows the urination time for 32 animals across six
videography and flow-rate measurement obtained at Zoo Atlanta,
orders of magnitude of body mass from 0.03 to 8,000 kg. De-
we discover that all mammals above 3 kg in weight empty their
spite this wide range in mass, urination time remains constant,
bladders over nearly constant duration of 21 ± 13 s. This feat is
T = 21 ± 13 s (n = 32), for all animals heavier than 3 kg. This
possible, because larger animals have longer urethras and thus,
invariance is noteworthy, considering that an elephant's blad-
higher gravitational force and higher flow speed. Smaller mam-
der, at 18 L, is nearly 3,600 times larger in volume than a cat's
mals are challenged during urination by high viscous and capillary
bladder at 5 mL. Using the method of least squares, we fit the
forces that limit their urine to single drops. Our findings reveal
data to a clear scaling law shown by the dashed line: T ∼ M0.13(Fig. 1H).

that the urethra is a flow-enhancing device, enabling the urinary
For small animals, urination is a high-speed event of 0.01- to
system to be scaled up by a factor of 3,600 in volume without
2-s duration and therefore, quite different from the behavior of
compromising its function. This study may help to diagnose uri-
the large animals observed that urinate for 21 s. Fig. 1H shows
nary problems in animals as well as inspire the design of scalable
urination time across 11 small animals, including one bat, five
hydrodynamic systems based on those in nature.

rats, and five mice. Their body masses ranged from 0.03 to 0.3 kg.

The large error bar for the rats is caused by bladder fullness
urology allometry scaling Bernoulli's principle
varying across individuals. Fig. 2D shows the time course of theurine drop's radius measured by image analysis of high-speed
Medical and veterinary urology often relies on simple, non- video of a rat. To rationalize the striking differences between
invasive methods to characterize the health of the urinary
large and small animals, we turn to mathematical modeling of
system (1, 2). One of the most easily measured characteristics of
the urinary system.

the urinary system is its flow rate (3), changes in which may beused to diagnose urinary problems. The expanding prostates of
Modeling Assumptions. Urination may be simply described math-
aging males may constrict the urethra, decreasing urine flow
ematically. Fig. 1E shows a schematic of the urinary system,
rate (4). Obesity may increase abdominal pressure, causing
consisting of a bladder of volume V and the urethra, which is
incontinence (5). Studies of these ailments and others often
assumed to be a straight vertical pipe of length L and diameter
involve animal subjects of a range of sizes (6). A study of
D. We assume that the urethra has such a thin wall that its in-ternal and external diameters are equal. Urination begins when
urination in zero gravity involved a rat suspended on two legs for
the smooth muscles of the bladder pressurize the urine to P
long periods of time (7), whereas other studies involve mice (8),
measured relative to atmospheric pressure. After an initial tran-
dogs (1), and pigs (9). Despite the wide range of animals used in
sient of duration that depends on the system size, a steady flow of
urological studies, the consequences of body size on urination
speed u is generated.

remain poorly understood.

Previous medical and veterinary studies, particularly cystome-
The bladder serves a number of functions, as reviewed by
trography and ultrasonography, report substantial data on the
Bentley (10). In desert animals, the bladder stores water to be
anatomy, pressure, and flow rate of the urinary system. Fig. 3
retrieved at a time of need. In mammals, the bladder acts as
shows urethral length (8, 15–25) and diameter (15, 24–34), flow
a waterproof reservoir to be emptied at a time of convenience.

rate (35–42), bladder capacity (25, 43–49), and bladder pressure
This control of urine enables animals to keep their homes sanitary
(1, 35, 39, 40, 43, 46, 50) for over 100 individuals across 13 species.

and themselves inconspicuous to predators. Stored urine may evenbe used in defense, which one knows from handling rodents
Several misconceptions in urology have important repercus-
sions in the interpretation of healthy bladder function. For in-
Animals eject fluids for waste elimination, communication, and
stance, several investigators state that urinary flow is driven
defense from predators. These diverse systems all rely on the
entirely by bladder pressure. Consequently, their modeling of the
fundamental principles of fluid mechanics, which we use to
bladder neglects gravitational forces (11–13). Others, such as
predict urination duration across a wide range of mammals. In
Martin and Hillman (14), contend that urinary flow is driven by
this study, we report a mathematical model that clarifies mis-
a combination of both gravity and bladder pressure. In this study,
conceptions in urology and unifies the results from 41 in-
we elucidate the hydrodynamics of urination across animal size,
dependent urological and anatomical studies. The theoretical
showing the effects of gravity increase with increasing body size.

framework presented may be extended to study fluid ejectionfrom animals, a universal phenomenon that has received little
In Vivo Experiments. We filmed the urination of 16 animals andobtained 28 videos of urination from YouTube, listed in
Author contributions: P.J.Y. and D.L.H. designed research; J.P. and J.C. performed re-
. show that urination style is strongly de-
search; P.J.Y. and D.L.H. analyzed data; and P.J.Y. and D.L.H. wrote the paper.

pendent on animal size. Here, we define an animal as large if it is
The authors declare no conflict of interest.

heavier than 3 kg and an animal as small if it is lighter than 1 kg.

This article is a PNAS Direct Submission.

Large animals, from dogs to elephants, produce jets and sheets
1To whom correspondence should be addressed. Email:
of urine, which are shown in Fig. 1 A–D. Small animals, including
This article contains supporting information online at
rodents, bats, and juveniles of many mammalian species, cannot
11932–11937 PNAS August 19, 2014 vol. 111 no. 33

Jetting urination by large animals, including (A) elephant, (B) cow, (C) goat, and (D) dog. Inset of cow is reprinted from the public domain and cited in
. (E) Schematic of the urinary system. (F) Ultrasound image of the bladder and urethra of a female human. The straight arrow indicates theurethra, and the curved arrow indicates the bladder. Reproduced with permission from ref. 20, (Copyright 2005, Radiological Society of North America). (G)Transverse histological sections of the urethra from a female pig. Reproduced with permission from ref. 9, (Copyright 2001, Elsevier). (H) The relationshipbetween body mass and urination time.

Table 1 shows the corresponding allometric relationships to be
This shape factor is nearly constant across species and body
used in numerical predictions for flow rate and urination time.

mass and consistent with the value of 0.17 found by Wheeler
We begin by showing that the urinary system is isometric (i.e.,
et al. (55).

it has constant proportions across animal size). Fig. 3A shows
Peak bladder pressure is difficult to measure in vivo, and in-
the relation between body mass M and urethral dimensions
stead, it is estimated using pressure transducers placed within the
(length L and diameter D). As shown by the nearly parallel
bladders of anesthetized animals. Pressure is measured when the
trends for L and D (L = 35M0.43 and D = 2M0.39), the aspect
bladder is filled to capacity by the injection of fluid. This tech-
ratio of the urethra is 18. Moreover, the exponents are close to
nique yields a nearly constant bladder pressure across animal
the expected isometric scaling of M1/3. Fig. 3B shows the re-
size: Pbladder = 5.2 ± 0.86 kPa (n = 8), which is shown in Fig. 3D.

lationship between body mass and bladder capacity. The bladder's
The constancy of bladder pressure at 5.2 kPa is consistent with
capacity is V ∼ M0.97, and the exponent of near unity indicates
other systems in the body. One prominent example is the re-
spiratory system, which generates pressures of 10 kPa for animals
In ultrasonic imaging (Fig. 1F), the urethra appears circular
spanning from a mosquito to an elephant (56).

(20). However, in histology (Fig. 1G), the urethra is clearlycorrugated, which decreases its cross-sectional area (9). The
Steady-State Equation of Urine Flow. We model the flow as steady
presence of such corrugation has been verified in studies in
state and the urine as an incompressible fluid of density ρ,
which flow is driven through the urethra (51, 52), although the
viscosity μ, and surface tension σ. The energy equation re-
precise shape has been too difficult to measure. We proceed
lates the pressures involved, each of which has units of energy
by using image analysis to measure cross-sectional area A
per cross-sectional area of the urethra per unit length downthe urethra:
from urethral histological diagrams of dead animals in theabsence of flow (9, 53, 54). We define a shape factor α = 4A/
πD2, which relates the urethral cross-sectional area with re-
Pbladder + Pgravity = Pinertia + Pviscosity + Pcapillary:
spect to that of a cylinder of diameter D. Fig. 3C shows theshape factor α = 0.2 ± 0.05 (n = 5) for which the corrected
Each term in Eq. 1 may be written simply by considering the
cross-sectional area is 20% of the original area considered.

pressure difference between the entrance and exit of the
PNAS August 19, 2014 vol. 111 no. 33 11933

Dripping urination by small animals. (A) A
rat's excreted urine drop. (B) A urine drop releasedby the lesser dog-faced fruit bat Cynopterus bra-chyotis. Courtesy of Kenny Breuer and SharonSwartz. (C) A rat's urine drop grows with time. Insetis reprinted from the public domain and cited in . (D) Time course of the drop radii of therat (triangles) along with prediction from our model(blue dotted line, α = 0.5; green solid line, α = 1; reddashed line, α = 0.2).

urethra. The combination of bladder and hydrostatic pressure
our derivations here, however, we assume that the transient
drives urine flow. Bladder pressure Pbladder is a constant given
phase is negligible.

in Fig. 3D. We do not model the time-varying height in thebladder, because bladders vary greatly in shape (57). Thus,
Large Animals Urinate for Constant Duration. Bladder pressure,
hydrostatic pressure scales with urethral length: P
gravity, and inertia are dominant for large animals, which can be
gravity ∼ ρgL,
where g is the acceleration of gravity. Dynamic pressure Pinertia
shown by consideration of the dimensionless groups in
scales as ρu2/2 and is associated with the inertia of the flow.

The viscous pressure drop in a long cylindrical pipe is given by
the Darcy–Weisbach equation (58): Pviscosity = fD(Re)ρLu2/2αD.

αD as the effective diameter of the pipe to keep the cross-
sectional area of the pipe consistent with experiments. TheDarcy friction factor f
The urination time T, the time to completely empty the bladder,
D is a function of the Reynolds number
Re = ρuD/μ, such that f
may be written as the ratio of bladder capacity to time-averaged
D(Re) = 64/Re for laminar flow and
flow rate, T = V/Q. We define the flow rate as Q = uA, where A =
D(Re) = 0.316/Re1/4 for turbulent flow (104 < Re < 105). Drops
generated from an orifice of effective diameter
απD2/4 is the cross-sectional area of the urethra. Using Eq. 4 to
a capillary force (59) of P
substitute for flow speed yields
αD. Substituting these terms
into Eq. 1, we arrive at
By isometry, bladder capacity V ∼ M and urethral length and
The relative magnitudes of the five pressures enumerated in
diameter both scale with M1/3; substitution of these scalings into
Eq. 2 are prescribed by six dimensionless groups, including the
Eq. 5 yields urination time T ∼ M1/6 ≈ M0.16 in the limit of in-
aforementioned Reynolds number and Darcy friction factor
creasing body mass. Agreement between predicted and mea-
and well-known Froude Fr = u= gL and Bond Bo = ρgD2/σ
sured scaling exponents is very good (0.13 compared with 0.16).

numbers (60) as well as dimensionless groups pertaining to the
We, thus, conclude that our scaling has captured the observed
urinary system, the urethra aspect ratio As = D/L, and pressure
invariance in urination time.

ratio Pb = Pbladder/ρgL. Using these definitions, we nondimen-
We go beyond a simple scaling by substituting the measured
sionalize Eq. 2 to arrive at
allometric relationships from Table 1 for L, D, α, V, and Pbladderinto Eq. 5, yielding a numerical prediction for urination time.

This prediction (Fig. 1H, solid line) is shown compared with ex-
perimental values (Fig. 1H, dashed line). The general trend is
captured quite well. We note that numerical values are under-predicted by a factor of three across animal masses, likely because
In the following sections, we solve Eq. 3 in the limits of large and
of the angle and cross-section of the urethra in vivo.

small animals.

How can an elephant empty its bladder as quickly as a cat?
In we apply a variation of the Washburn law
Larger animals have longer urethras and therefore, greater hy-
(61) to show that the steady-state model given in Eq. 2 is
drostatic pressure driving flow. Greater pressures lead to higher
accurate for most animals. Animals lighter than 100 kg ach-
flow rates, enabling the substantial bladders of larger animals to
ieve 90% of their flow velocity within 4 s; however, for animals
be emptied in the same duration as those of their much smaller
such as elephants, the transient phase can be substantial. For

0.01 0.1 1 10 100 1000
0.01 0.1 1 10 100 1000
0.01 0.1 1 10 100 1000
0.01 0.1 1 10 100 1000
10000 0.1 1 10 100
The relation between body mass M and properties of the urinary system. (A) Urethral length L (green triangles) and diameter D (blue circles). (B)
Bladder capacity V. (C) Shape factor α associated with the urethral cross-section. (D) Bladder pressure Pbladder. (E) Flow rate of males. (F) Flow rate of females.

Symbols represent experimental measurements, dashed lines represent best fits to the data, and solid lines represent predictions from our model.

Our model provides a consistent physical picture on consid-
pressures and urethral anatomy (15, 16, 50) (Pbladder = 6.03 kPa,
eration of flow rate. Combining Eq. 4 and the definition of flow
L = 20 mm, D = 0.8 mm).

rate (Q = uA) yields
To determine urination time, we turn to the dynamics of drop
filling. A spherical drop is filled by the influx of urine, Q = απD2u/4.

By conservation of mass, dV
drop/dt = Q, a first-order differential
equation that may be easily integrated to obtain the drop volumeV(t). We assume that the initial drop corresponds to a sphere of the
Our model gives insight into the distinct flow-rate scalings
same diameter as the urethra,
αD. Thus, the radius of the
observed for both male and female mammals. Male mammals
spherical urine drop may be written
generally stand on four legs and have a penis that extends
downward, enabling them to urinate vertically. Assuming isometry
(D ∼ M1/3 and L ∼ M1/3), flow rate scales as Q ∼ M5/6 ≈ M0.83
in the limit of large body mass. This predicted exponent is with-in 10% of the observed scaling for males: QM ∼ M0.92. By sub-stituting the allometric relations from Table 1 into Eq. 6, we
Combining Eq. 8 and the numerical solution for Eq. 7, we com-
compute a numerical prediction for flow rate (Fig. 3E, solid line)
pute the time course of the drop radius. This prediction is com-
that is five times higher than experimental flow rates (Fig. 3E,
pared with experimental values in Fig. 2D. We find that the
dashed line). This overprediction is roughly consistent with our
prediction is highly sensitive to the value of α. Without consid-
underprediction for urination time.

eration of the corrugated cross-section, a prediction of α = 1
Female mammals have anatomy such that the urethral exit is
(Fig. 2D, green solid line) yields a flow rate that is too high.

near the anus: thus, many female animals urinate horizontally.

Using the shape factor α = 0.2 (Fig. 2D, red dashed line), our
The scaling of Eq. 6 without the gravitational term is Q ∼ M2/3 ≈
model predicts a flow speed of u = 1.2 m/s, which fits the data
M0.67, and the exponent is in correspondence to that found in our
fairly well. Using nonlinear least-squares fitting in Matlab, the
experiments for females: Q
best fit to the experimental data yields an intermediate value of
F ∼ M0.66. Substituting allometric
relations from Table 1 yields a numerical prediction (Fig. 3F,
α = 0.5 (Fig. 2D, blue dotted line).

solid line) that remains in good agreement with experiments.

The drop does not grow without limit but falls when its gravi-
tational force, scaling as 4πR3ρg=3, overcomes its attaching cap-
Small Animals Urinate Quickly and for Constant Duration. Bladder
illary force to the urethra, scaling as π αDσ. Equating these two
pressure, viscous pressure, and capillary pressure are dominant
forces yields the final drop radius before detachment,
for small animals, which is shown by the associated dimension-less groups in In this limit, Eq. 2 reduces to
which does a fair job of predicting the drop size. We predict drop
which we solve numerically for flow speed u. To predict the flow
radii for rats and mice of 1.3 and 1.1 mm, respectively, which are
speed of a rat, inputs to this equation include the rat's bladder
two times as large as experimental values of 2.0 ± 0.1 (n = 5) and
PNAS August 19, 2014 vol. 111 no. 33 11935
Measured allometric relationships for the urinary
urethra. In the medical literature, the function of the urethra was
system of animals
previously unknown. It was simply defined as a conduit betweenbladder and genitals. In this study, we find that the urethra is
analogous to Pascal's Barrel: by providing a water-tight pipe to
Duration of urination
direct urine downward, the urethra increases the gravitational
force acting on urine and therefore, the rate at which urine is
Urethral diameter
expelled from the body. Thus, the urethra is critical to the
bladder's ability to empty quickly as the system is scaled up.

Engineers may apply this result to design a system of pipes and
reservoirs for which the drainage time does not depend on system
Flow rate for females
size. This concept of a scalable hydrodynamic system may be used
Flow rate for males
in the design of portable reservoirs, such as water towers, waterbackpacks, and storage tanks.

Body mass M given in kilograms. Duration of urination corresponds to
Why is urination time 21 s, and why is this time constant across
animals heavier than 3 kg. Urethral length and diameter, shape factor, blad-
animal sizes? The numerical value of 21 s arises from the un-
der capacity, bladder pressure, and flow rates correspond to animals heavierthan 0.02 kg.

derlying physics involving the physical properties of urine as wellas the dimensions of the urinary system. Our model shows thatdifferences in bladder capacity are offset by differences in flow
2.2 ± 0.4 mm (n = 5), respectively. We suspect this difference is
rate, resulting in a bladder emptying time that does not change
caused by our underestimation of urethral perimeter at the exit.

with system size. Such invariance has been observed in a number
For such a large drop to remain attached, we require the attach-
of other systems. Examples include the height of a jump (64) and
ment diameter to be larger by a factor of two, which is quite
the number of heartbeats in a lifetime (65). Many of these examples
possible, because the urethral exit is elliptical.

arise from some aspect of isometry, such as with our system.

Substituting Eq. 9 into Eq. 8, the time to eject one drop may
From a biological perspective, the invariance of urination time
suggests its low functional significance. Because bladder volume
is 4.6 mL/kg body mass and daily urine voided is 26 mL/kg body
mass (66), mammals urinate 5.6 times/d. Because the time to
− 1 ≈ 4D cos θ
urinate once is 21 s, the daily urination time is 2 min, which canbe translated to 0.2% of an animal's day, a negligible portion
The predictions of maximum drop size and time to fall are in
compared with other daily activities, such as eating and sleeping,
excellent correspondence with observed values for rats and mice.

for which most animals take care to avoid predation. Thus, uri-
Using Eq. 10, we predict drop falling times of 0.05 and 0.15 s for rats
nation time likely does not influence animal fitness. The geometry
and mice, respectively, which are nearly identical to experimental
of the urethra, however, may play a role in other activities of
values of 0.06 ± 0.05 (n = 5) and 0.14 ± 0.1 s (n = 14), respectively.

high functional significance, such as ejaculation.

A scaling for urine duration for small animals is not straight-
In our study, we found that urination time is highly sensitive
forward because of the nonlinearity of Eq. 7. We conduct a scaling
to urethral cross-section. This dependency is particularly high for
analysis in the limit of decreasing animal size for which the Rey-
small animals for which urine flow is resisted by capillary and
nolds number approaches zero. Because of isometry, V ∼ M and D ∼
viscous forces, which scale with the perimeter of the urethra. More
M1/3. Rewriting Eq. 7 at low Reynolds number, we have u ∼ D,
accurate predictions for small animals require measurements of
and therefore, the time to eject one drop from Eq. 10 scales as
the urethral exit perimeter and the urethral cross-section, which is
Tdrop ∼ Bo−1 ∼ M−2/3. Using Eq. 9, the final drop size is Rf ∼ D ∼ M1/3.

known to vary with distance down the urethra (67). Current
By conservation of mass, a full bladder of volume V can produce
models of noncircular pipe flow are not applicable, because they
N spherical drops, where N = 3V =4πR3 ∼ M2=3. Thus, the urina-
only account for infinitesimal corrugations (68). Additional math-
tion time for small animals T = NTdrop is constant and there-
ematical techniques as well as accurate urethral measurements
fore, independent of animal size. This prediction indicates
are needed to increase correspondence with experiments.

that small animals urinate for different durations than largeanimals, which is in correspondence with experiments. Our
Materials and Methods
experiments indicate that mammals of mass 0.03–0.3 kg uri-
We filmed urination of animals using both Sony HDR-XR200 and high-speed
nate for durations of 0.1–2 s. We have insufficient range of
cameras (Vision Research v210 and Miro 4). The masses of animals are taken
masses for small animals to conclude our prediction that
from annual veterinary procedures or measured using an analytical balance.

urination time is constant in this regime.

Flow rate Q is measured by simultaneous high-speed videography and
The model yields insight into the challenges faced by small
manual urine collection using containers of appropriate size and shape. We
animals. In Eq. 7, flow speed is positive only if Pbladder αD ≥ 4σ,
use the open-source software Tracker to measure the time course of the
where σ is the surface tension of urine, which for humans is com-
radius of urine drops produced by rats and mice.

parable with the surface tension of water (62). Thus, we predictthat the smallest urethra to expel urine has a diameter of
ACKNOWLEDGMENTS. We acknowledge photographer C. Hobbs and our
4σ= αPbladder ∼ 0:1 mm. According to our allometric trends, the
hosts at Zoo Atlanta (R. Snyder), the University of Georgia (L. Elly), the
smallest animal that can urinate independently corresponds to
Atlanta Humane Society (A. Lopez), and the animal facilities at Georgia Tech
a body mass of 0.8 g and urethral length of 1.7 mm. This mass
(L. O'Farrell). We thank YouTube contributors, including Alex Cobb, Cole
corresponds to that of altricial mice (0.5–3 g), which are de-
Onyx, demondragon115, drakar2835, ElMachoPrieto83, Ilze Darguze, Joe
pendent on their mother's licking of excreted urine drops (63).

BERGMANN, Joey Ponticello, krazyboy35, laupuihang, MegaTobi89, Mixetc,mpwhat, MrTitanReign, relacsed, RGarrido121, ronshausen63, Sandro Puelles,
Silvia Lugli, and Tom Holloway. Our funding sources were National ScienceFoundation Faculty Early Career Development Program (Division of Physics)
The urinary system works effectively across a range of length scales.

Grant 1255127 for the modeling and Georgia Tech President's Undergraduate
This robustness is caused by the hydrodynamic contribution of the
Research Awards for the experiments.

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23. Balke JM, Boever WJ, Ellersieck MR, Seal US, Smith DA (1988) Anatomy of the
54. Praud C, Sebe P, Mondet F, Sebille A (2003) The striated urethral sphincter in female
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reference to development of techniques for artificial breeding. J Reprod Fertil
55. Wheeler AP, Morad S, Buchholz N, Knight MM (2012) The shape of the urine stream—
from biophysics to diagnostics. PLoS ONE 7(10):e47133.

24. Hildebrandt TB, et al. (2000) Ultrasonography of the urogenital tract in elephants
56. Kim W, Bush JWM (2012) Natural drinking strategies. J Fluid Mech 705:7–25.

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57. Damaser MS, Lehman SL (1995) The effect of urinary bladder shape on its mechanics
reproductive function. Zoo Biol 19(5):321–332.

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25. Fowler ME, Mikota SK (2006) Biology, Medicine, and Surgery of Elephants (Wiley-
58. Ghiaasiaan S (2011) Convective Heat and Mass Transfer (Cambridge Univ Press,
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PNAS August 19, 2014 vol. 111 no. 33 11937
Supporting Videos
Video S1. Urination of a rat, mass of 0.24 kg. Time slowed by 33 X.

Video S2. Urination of a goat, mass of 70 kg. Time slowed by 17 X.

Video S3. Urination of a cow, mass of 640 kg. Time slowed by 33 X.

Video S4. Urination of an elephant, mass of 3540 kg. Time slowed by 33 X.

Unsteady hydrodynamic urination model for large and small animals
In this section, we estimate the time for the urine flow to achieve steady state. The column of urinedescends due to gravitational and bladder pressure forces. This descent is resisted by viscosity,fluid inertia and capillary pressure.

At time t = 0 the urethra is empty. We parametrize the height of urine in the urethra with a
height z(t), measured from the bladder. Consider Figure 1E in the main text. We consider a con-trol volume including the mass of urine with the bladder and urethra. Conservation of momentumfor this control volume may be written
z = A (Pbladder + Pgravity − Pinertia − Pviscosity − Pcapillary) .

Each terms in Equation (1) has units of force and may be written simply. The mass of urine in theurethra is m = ρπαD2z/4. Using techniques presented by Bush∗, the added mass associated withacceleration of fluid in the bladder may be written ma = 7ρπα3/2D3/48 . Bladder pressure Pbladderis constant. Hydrostatic pressure Pgravity scales as ρgz where g is the gravitational acceleration.

Dynamic pressure Pinertia scales as ρ ˙z2/2, and is associated with inertia of the flow. Assuminglaminar flow, the pressure drop due to viscosity in a long cylindrical pipe is Pviscosity = 32µz ˙z/αD2.

The capillary force of drops generated from an orifice of effective diameter
αD is Pcapillary =
4σ/ αD. Substituting these terms into Equation (1), we arrive at
We compute velocity ˙
z in Equation (2) using a Runge-Kutta single-step solver (ode45) in Matlab.

Inputs to this equation include allometric relationships of bladder pressure, urethral diameter, andshape factor, given in Table 1. Figure S1 shows the time course of urine velocity. For animallighter than 100 kg, the flow reaches 90% of its final velocity in 4 seconds, which is 15% of the thetime to empty the bladder, 21 seconds. We thus conclude our steady state model reported in themain text is accurate for animals lighter than 100 kg. For larger animals such as elephants, thetransition can be substantial.

∗Bush JWM (2010) 18.357 Interfacial Phenomena, Fall 2010. Avaliable at http://ocw.mit.edu Accessed 2 May,
Figure S1. Time course of urine velocity.

**less G **100

**Mass (kg) **

Figure S2. Values of dimensionless groups including the Froude Fr, Bond Bo, Reynolds Re, aspect ratioAs, and ratio of bladder and hydrostatic pressure Pb.

Table S1. Duration of urination.
**Mass (kg) **

**Duration (s) **

Experiment at Georgia Tech
K. Breuer and S. Swartz, Brown University
Experiment at Georgia Tech
Applehead Chihuahua
Experiment at local park
Youtube, Mattern (2000)
Experiment at Zoo Atlanta
Experiment at Zoo Atlanta
Youtube, Wilcox (1997)
Youtube, Wilcox (1997)
Youtube, Wilcox (1997)
Youtube, Wilson (2001)
Youtube, Mil er (1997)
Youtube, Brown (1996)
Youtube, Starkey (1992)
Youtube, Nowak (1999)
Youtube, Lynette (2013)
Youtube, Lynette (2013)
Youtube, Linnaeus (1758)
Youtube, Kingdon (1988)
Youtube, Marvin (1992)
Youtube, Marvin (1992)
Youtube, Marvin (1992)
Youtube, Bongianni (1988)
Youtube, Potts (1997)
Indian Rhinoceros
Youtube, Toon (2002)
Youtube, Toon (2002)
Youtube, Toon (2002)
Youtube, Toon (2002)
Experiment at Zoo Atlanta
Youtube, Shoshani (1982)
Youtube, Shoshani (1982)
Youtube, Shoshani (1982)
Youtube, Shoshani (1982)
Youtube, Shoshani (1982)
Youtube, Shoshani (1982)
Table S2. Urethral length.
**Mass (kg) **

**Length (mm) **

Wister Rat (N=20)
Sprague-Dawley Rat (N=61)
Dunkin Hartley Guinea Pig
Normal Adult Cat
Normal Adult Cat
Normal Adult Cat
Normal Adult Cat
Normal Adult Cat
Normal Adult Cat
Normal Adult Cat
Normal Adult Cat
Normal Adult Cat
Normal Adult Cat
Normal Adult Cat
Normal Adult Cat
Normal Adult Dog
Normal Adult Dog
Normal Adult Dog
Normal Adult Dog
Normal Adult Dog
Mongrel Dog (N=10)
Normal Adult Dog
Normal Adult Dog
Normal Adult Dog
Prasad (2005), Ogden (2004)
African Lion (N=7)
Lueders (2012), Nowak (1999)
African Elephant (5 years)
Balke (1988), Krumrey (1968)
African Elephant (5 years)
Balke (1988), Krumrey (1968)
African Elephant (6 years)
Balke (1988), Krumrey (1968)
African Elephant (6 years)
Balke (1988), Krumrey (1968)
African Elephant (6 years)
Balke (1988), Krumrey (1968)
African Elephant (7 years)
Balke (1988), Krumrey (1968)
African Elephant (9 years)
Balke (1988), Krumrey (1968)
African Elephant (9 years)
Balke (1988), Krumrey (1968)
African Elephant (9 years)
Balke (1988), Krumrey (1968)
African Elephant (10 years)
Balke (1988), Krumrey (1968)
African Elephant (10 years)
Balke (1988), Krumrey (1968)
African Elephant (10 years)
Balke (1988), Krumrey (1968)
African Elephant (11 years)
Balke (1988), Krumrey (1968)
African Elephant (14 years)
Balke (1988), Krumrey (1968)
Hildebrandt (2000), Nowak (1999)
African Elephant (23 years)
Balke (1988), Krumrey (1968)
African Elephant (25 years)
Balke (1988), Krumrey (1968)
Fowler (2006), Nowak (1999)
Table S3. Urethral diameter.
**Mass (kg) **

**Diameter (mm) **

Wister Rat (N=10)
Hybrid Rat (12 months) (N=5)
Russel (1996), Tasaki (2009)
Hybrid Rat (32 months) (N=5)
Russel (1996), Tasaki (2009)
Adult Rat (N=176)
Kunstyvr (1982), Perrin (2003)
Short Hair Cat (7 weeks) (N=6)
Root (1996), Sturman (1985)
Short Hair Cat (7 weeks) (N=6)
Root (1996), Sturman (1985)
Short Hair Cat (7 months) (N=5)
Root (1996), Scott (1970)
Short Hair Cat (7 months) (N=5)
Root (1996), Lein (1983)
Gray (1918), Ogden (2004)
Gray (1918), Ogden (2004)
Man (71.7 years) (N=32)
Tsujimoto (2003), Ogden (2004)
Miniature Horse (N=7)
Hereford X Angus Bul (N=96)
Light Horse (N=53)
Heavy Horse (N=15)
Hildebrandt (2000), Nowak (1999)
African Elephant (N=6)
Hildebrandt (1998)
Asian Elephant (N=2)
Hildebrandt (1998)
Hildebrandt (1998)
Table S4. Shape factor, bladder capacity, and bladder pressure.
**Mass (kg) **

**Shape factor α **

Caceci, Johnston (1985)
Skarva, Ogden (2004)
**Mass (kg) **

**Bladder capacity (mL) **

Sprague Dawley Rat
Applehead Chihuahua
Experiment at local park
Mongrel Dog (N=14)
Fowler (2006), Shoshani (1982)
**Mass (kg) **

**Bladder pressure (kPa) **

Sprague Dawley Rat (N=7)
Wister Rat (N=3)
Sprague Dawley Rat (N=18)
Table S5. Urine flow rate.
**Mass (kg) **

**Flow rate (mL/s) **

Experiment at Georgia Tech
Experiment at Georgia Tech
Experiment at Georgia Tech
Experiment at Georgia Tech
Experiment at Georgia Tech
Woman (3-4 years)
Segura (1997), Ogden (2004)
Woman (5-6 years)
Segura (1997), Ogden (2004)
Woman (7-8 years)
Segura (1997), Ogden (2004)
Woman (9-11 years)
Segura (1997), Ogden (2004)
Woman (12-14 years)
Segura (1997), Ogden (2004)
Experiment at local park
Woman (59 years) (N = 183)
Madersbacher (1998) , Ogden (2004)
Woman (55 years) (N = 185)
Nitti (1999) , Ogden (2004)
L. Ely, University of Georgia
L. Ely, University of Georgia
Wister Rat (N=2)
Sprague Dawley Rat (N=18)
Dunkin Hartley Guinea Pig (N=4)
Applehead Chihuahua
Experiment at local park
Segura (1997), Ogden (2004)
Segura (1997), Ogden (2004)
Segura (1997), Ogden (2004)
Man (9-11 years)
Segura (1997), Ogden (2004)
Man (12-14 years)
Segura (1997), Ogden (2004)
Nigerian Dwarf Goat
Experiment at Zoo Atlanta
Experiment at Zoo Atlanta
Japanese Man (N=271)
Schmidt (2003), Schmidt (2002)
American Man (N=467)
Man (53.8 years) (N=58)
Folkestad (2004), Ogden (2004)
Table S6. Bibliography for images and animal masses.
**Online Image **

Man vyi. (2005) *Jersey cattle in Jersey*, Wikimedia Commons
Inset in **Figure 1(c) **http://commons.wikimedia.org/wiki/File:Jersey_cattle_in_Jersey.jpg

Stephens, J. (1992) *Lobund-Wistar Rat *
Inset in **Figure 2(b) **https://visualsonline.cancer.gov/details.cfm?imageid=2568

Animal silhouettes Public Domain Pictures
in **Figure 1(h) **

http://www.publicdomainpictures.net/ Caceci, T. (2008) *Canine penis; H&E stain, paraffin section (decalcified), 20x*, VM8054: Veterinary Histology
Skarva, F. *Cross-Section of a Normal Human Penis Showing the Urethra and Corpora Spongiosum, H&E Stain, LM X12*
**Body masses of animals considered **

Nowak RM, Paradiso JL (1999) *Walker's mammals of the world* (The Johns Hopkins University Press, Baltimore, MD) Vol. 1, 6th Ed.
Mattern MY, McLennan DA (2000) Phylogeny and speciation of felids. *Cladistics* 16(2):232–253.
Wilcox C (1997) *The Great Dane *(Capstone, North Mankato, MN).
Wilson D, Burnie D (2001) *Animal: the definitive visual guide to the world's wildlife* (DK Publishing, New York).
Mil er-Schroeder P (1997) *Goril as* (Raintree Steck-Vaughn, Austin, TX).
Brown G (1996) *The Great Bear Almanac* (Globe Pequot, Guilford, CT).
Starkey P, Mwenya E, Stares J (1994) *Improving animal traction technology. *Proceedings of the First Workshop of the Animal Traction Network for Eastern and Southern Africa held 18-23 January 1992, Lusaka, Zambia* *(Technical Centre for Agricultural and Rural Cooperation, Wageningen, The Netherlands).
Lynette R (2013) *South American Tapirs* (Bearport Publishing, New York).
Linnaeus C (1758) *Systema naturae per regna tria naturae, secundum classes, ordines, genera, species, cum characteribus, differenti s, synonymis, locis.* (Laurenti Salvii, Stockholm), 10th Ed.
Kingdon J (1988) *East African Mammals: An Atlas of Evolution in Africa, Part A: Carnivores* (University of Chicago Press, Chicago) Vol. 3.
Hal MH, Comerford PM (1992) Pasture and hay for horses. *Cooperative Extension, The Pennsylvania State University *32:1–4.
Bongianni M (1988) *Simon and Schuster's guide to horses and ponies of the world *(Simon and Schuster, New York).
Potts S (1997) *The American Bison* (Capstone, Mankato, MN).
Toon A, Toon S (2002) *Rhinos* (Voyageur Press, Minneapolis, MN).
Shoshani J, Eisenberg JF (1982) Elephas maximus. *Mammalian Species *182:1–8.
Ogden CL, Fryar CD, Carrol MD, Flegal KM (2004) *Mean Body Weight, Height, and Body Mass Index: United States 1960-2002 *(Department of Health and Human Services, Centers for Disease Control and Prevention, National Center for Health Statistics). Krumrey WA, Buss IO (1968) Age estimation, growth, and relationships between body dimensions of the female African elephant. *Journal of Mammalogy* 49(1):22–31. Tasaki M, et al. (2009) Simultaneous induction of non-neoplastic and neoplastic lesions with highly proliferative hepatocytes fol owing dietary exposure of rats to tocotrienol for 2 years. *Archives of Toxicology *83(11):1021–1030. Perrin D, Soulage C, Pequignot J, Geloen A (2003) Resistance to obesity in Lou/C rats prevents ageing-associated metabolic alterations. *Diabetologia *46(11):1489–1496. Sturman J, Moretz R, French J, Wisniewski H (1985) Postnatal taurine deficiency in the kitten results in a persistence of the cerebel ar external granule cel layer: correction by taurine feeding. *Journal of Neuroscience Research *13(4):521–528.
Scott P, Hafez E (1970) *Reproduction and Breeding Techniques for Laboratory Animals*. (Lea and Febiger, Philadelphia).
Lein D, Concannon P (1983) Infertility and fertility treatments and management in the queen and tomcat. *Current Therapy VIII. Kirk, R.(ed). *(Elsevier Saunders, Philadelphia)* *936–987.
Schmidt F, Shin P, Jorgensen TM, Djurhuus JC, Constantinou CE (2002) Urodynamic patterns of normal male micturition: Influence of water consumption on urine production and detrusor function. *The Journal of Urology *168(4 Pt 1):1458–1463.
Supporting Information
Yang et al. 10.1073/pnas.1402289111
Urination of a rat (mass of 0.24 kg). Time slowed by 33 times.

Urination of a goat (mass of 70 kg). Time slowed by 17 times.

Urination of a cow (mass of 640 kg). Time slowed by 33 times.

Urination of an elephant (mass of 3,540 kg). Time slowed by 33 times.

Other Supporting Information Files

Source: http://blacklight.fi/~jok/yang.pdf

The Canadian Consensus Document on ME/CFS Our comparison study examined differences between patients meeting the Canadian clinicaland the Fukuda et al. criteria for ME/CFS, with people who had chronically fatiguing illnessexplained by a psychiatric condition. The Canadian Clinical Criteria selected patients with morephysical functional impairment, more fatigue/weakness, neurocognitive and neurologicalsymptoms and had more variables that significantly differentiated them from the psychiatriccomparison group than did the Fukuda et al. criteria. The findings do suggest that the Canadiancriteria point to the potential utility in designating post-exertional malaise and fatigue, sleepdysfunction, pain, clinical neurocognitive, and clinical autonomic/neuroimmunoendocrinesymptoms as major criteria.

ISRAEL FÜR „ANFÄNGER UND KENNER" UMWELT – NATUR – GESCHICHTE – NACHHALTIGKEIT Die Reise führt Sie zu ökologischen, sozialen, landwirtschaftlichen Projekten des modernen Israel, ohne dabei die geschichtlichen Orte und Aspekte zu vergessen. 1. Tag, Samstag: Anreise Ankunft Flughafen Tel Aviv. Treffen im Hotel mit unserer Reiseleitung. Je nach Ankunftszeit Rundgang in Tel Aviv oder Yaffo. Ü/HP Tel Aviv.