Jurin's law
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Jurin's law, or capillary rise, is the simplest analysis of
capillary action Capillary action (sometimes called capillarity, capillary motion, capillary rise, capillary effect, or wicking) is the process of a liquid flowing in a narrow space without the assistance of, or even in opposition to, any external forces li ...
—the induced motion of liquids in small channels—and states that the maximum height of a liquid in a capillary tube is
inversely proportional In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constan ...
to the tube's
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 fo ...
. Capillary action is one of the most common fluid mechanical effects explored in the field of
microfluidics Microfluidics refers to the behavior, precise control, and manipulation of fluids that are geometrically constrained to a small scale (typically sub-millimeter) at which surface forces dominate volumetric forces. It is a multidisciplinary field th ...
. Jurin's law is named after
James Jurin James Jurin FRS FRCP (baptised 15 December 168429 March 1750) was an English scientist and physician, particularly remembered for his early work in capillary action and in the epidemiology of smallpox vaccination. He was a staunch proponent o ...
, who discovered it between 1718 and 1719.See: * James Jurin (1718
"An account of some experiments shown before the Royal Society; with an enquiry into the cause of some of the ascent and suspension of water in capillary tubes,"
''Philosophical Transactions of the Royal Society of London'', 30 : 739–747. * James Jurin (1719
"An account of some new experiments, relating to the action of glass tubes upon water and quicksilver,"
''Philosophical Transactions of the Royal Society of London'', 30 : 1083–1096.
His quantitative law suggests that the maximum height of liquid in a capillary tube is inversely proportional to the tube's diameter. The difference in height between the surroundings of the tube and the inside, as well as the shape of the meniscus, are caused by
capillary action Capillary action (sometimes called capillarity, capillary motion, capillary rise, capillary effect, or wicking) is the process of a liquid flowing in a narrow space without the assistance of, or even in opposition to, any external forces li ...
. The mathematical expression of this law can be derived directly from
hydrostatic Fluid statics or hydrostatics is the branch of fluid mechanics that studies the condition of the equilibrium of a floating body and submerged body "fluids at hydrostatic equilibrium and the pressure in a fluid, or exerted by a fluid, on an imme ...
principles and the
Young–Laplace equation In physics, the Young–Laplace equation () is an algebraic equation that describes the capillary pressure difference sustained across the interface between two static fluids, such as water and air, due to the phenomenon of surface tension or w ...
. Jurin's law allows the measurement of the surface tension of a liquid and can be used to derive the
capillary length The capillary length or capillary constant, is a length scaling factor that relates gravity and surface tension. It is a fundamental physical property that governs the behavior of menisci, and is found when body forces (gravity) and surface forces ...
.


Formulation

The law is expressed as : \qquad h = \frac , where * ''h'' is the liquid height; * '' \gamma '' is the surface tension; * ''θ'' is the
contact angle The contact angle is the angle, conventionally measured through the liquid, where a liquid–vapor interface meets a solid surface. It quantifies the wettability of a solid surface by a liquid via the Young equation. A given system of solid, liq ...
of the liquid on the tube wall; * ''ρ'' is the mass
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
(mass per unit volume); * ''r''0 is the tube radius; * ''g'' is the
gravitational acceleration In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodi ...
. It is only valid if the tube is cylindrical and has a radius (''r''0) smaller than the
capillary length The capillary length or capillary constant, is a length scaling factor that relates gravity and surface tension. It is a fundamental physical property that governs the behavior of menisci, and is found when body forces (gravity) and surface forces ...
(\lambda_^2=\gamma/\rho g). In terms of the capillary length, the law can be written as : \lambda_^2=\frac .


Examples

For a water-filled glass tube in air at standard conditions for temperature and pressure, at 20 °C, , and . Because water spreads on clean glass, the effective equilibrium contact angle is approximately zero. For these values, the height of the water column is :h\approx \ \mbox. Thus for a radius glass tube in lab conditions given above, the water would rise an unnoticeable . However, for a radius tube, the water would rise , and for a radius tube, the water would rise . Capillary action is used by many plants to bring up water from the soil. For tall trees (larger than ~10 m (32 ft)), other processes like osmotic pressure and negative pressures are also important.


History

During the 15th century,
Leonardo da Vinci Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, Drawing, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially res ...
was one of the first to propose that mountain streams could result from the rise of water through small capillary cracks. It is later, in the 17th century, that the theories about the origin of capillary action begin to appear.
Jacques Rohault Jacques Rohault (; 1618 – 27 December 1672) was a French philosopher, physicist and mathematician, and a follower of Cartesianism. Life Rohault was born in Amiens, the son of a wealthy wine merchant, and educated in Paris. Having grown up with ...
erroneously supposed that the rise of the liquid in a capillary could be due to the suppression of air inside and the creation of a vacuum. The astronomer
Geminiano Montanari Geminiano Montanari (1 June 1633 – 13 October 1687) was an Italian astronomer, lens-maker, and proponent of the experimental approach to science. He was a member of various learned academies, notably the Accademia dei Gelati. Montanari's famous ...
was one of the first to compare the capillary action to the circulation of
sap Sap is a fluid transported in xylem cells (vessel elements or tracheids) or phloem sieve tube elements of a plant. These cells transport water and nutrients throughout the plant. Sap is distinct from latex, resin, or cell sap; it is a separ ...
in plants. Additionally, the experiments of
Giovanni Alfonso Borelli Giovanni Alfonso Borelli (; 28 January 1608 – 31 December 1679) was a Renaissance Italian physiologist, physicist, and mathematician. He contributed to the modern principle of scientific investigation by continuing Galileo's practice of testin ...
determined in 1670 that the height of the rise was inversely proportional to the radius of the tube.
Francis Hauksbee Francis Hauksbee the Elder FRS (1660–1713), also known as Francis Hawksbee, was an 18th-century English scientist best known for his work on electricity and electrostatic repulsion. Biography Francis Hauksbee was the son of draper and common c ...
, in 1713, refuted the theory of Rohault through a series of experiments on capillary action, a phenomenon that was observable in air as well as in vacuum. Hauksbee also demonstrated that the liquid rise appeared on different geometries (not only circular cross sections), and on different liquids and tube materials, and showed that there was no dependence on the thickness of the tube walls.
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...
reported the experiments of Hauskbee in his work ''
Opticks ''Opticks: or, A Treatise of the Reflexions, Refractions, Inflexions and Colours of Light'' is a book by English natural philosopher Isaac Newton that was published in English in 1704 (a scholarly Latin translation appeared in 1706). (''Opti ...
'' but without attribution. It was the English physiologist
James Jurin James Jurin FRS FRCP (baptised 15 December 168429 March 1750) was an English scientist and physician, particularly remembered for his early work in capillary action and in the epidemiology of smallpox vaccination. He was a staunch proponent o ...
, who finally in 1718 confirmed the experiments of Borelli and the law was named in his honour.


Derivation

The height h of the liquid column in the tube is constrained by the
hydrostatic pressure Fluid statics or hydrostatics is the branch of fluid mechanics that studies the condition of the equilibrium of a floating body and submerged body " fluids at hydrostatic equilibrium and the pressure in a fluid, or exerted by a fluid, on an imm ...
and by the surface tension. The following derivation is for a liquid that rises in the tube; for the opposite case when the liquid is below the reference level, the derivation is analogous but pressure differences may change sign.


Laplace pressure

Above the interface between the liquid and the surface, the pressure is equal to the atmospheric pressure p_. At the meniscus interface, due to the surface tension, there is a pressure difference of \Delta p=p_-p_, where p_is the pressure on the convex side; and \Delta p is known as Laplace pressure. If the tube has a circular section of radius r_0, and the meniscus has a spherical shape, the radius of curvature is r=r_0/\cos\theta, where \theta is the
contact angle The contact angle is the angle, conventionally measured through the liquid, where a liquid–vapor interface meets a solid surface. It quantifies the wettability of a solid surface by a liquid via the Young equation. A given system of solid, liq ...
. The Laplace pressure is then calculated according to the Young-Laplace equation:\Delta p=\frac, where \gamma is the surface tension.


Hydrostatic pressure

Outside and far from the tube, the liquid reaches a ground level in contact with the atmosphere. Liquids in communicating vessels have the same pressures at the same heights, so a point \rm w, inside the tube, at the same liquid level as outside, would have the same pressure p_=p_. Yet the pressure at this point follows a vertical pressure variation as :p_=p_+\rho gh, where g is the
gravitational acceleration In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodi ...
and \rho the density of the liquid. This equation means that the pressure at point \rm w is the pressure at the interface plus the pressure due to the weight of the liquid column of height h. In this way, we can calculate the pressure at the convex interface p_=p_-\rho g h=p_-\rho g h.


Result at equlibrium

The hydrostatic analysis shows that \Delta p=\rho g h, combining this with the Laplace pressure calculation we have:\rho g h =\frac{r_0}, solving for h returns Jurin's law.


References

Fluid dynamics Hydrology