Hydrostatics

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Table of Hydraulics and Hydrostatics, from the 1728 Cyclopaedia Table of Hydraulics and Hydrostatics, Cyclopaedia, Volume 1.jpg
Table of Hydraulics and Hydrostatics, from the 1728 Cyclopædia

Fluid statics or hydrostatics is the branch of fluid mechanics that studies "fluids at rest and the pressure in a fluid or exerted by a fluid on an immersed body". [1]

Fluid mechanics is the branch of physics concerned with the mechanics of fluids and the forces on them. It has applications in a wide range of disciplines, including mechanical, civil, chemical and biomedical engineering, geophysics, astrophysics, and biology.

Contents

It encompasses the study of the conditions under which fluids are at rest in stable equilibrium as opposed to fluid dynamics, the study of fluids in motion. Hydrostatics are categorized as a part of the fluid statics, which is the study of all fluids, incompressible or not, at rest.

Mechanical equilibrium (in classical mechanics) a particle is in mechanical equilibrium if the net force on that particle is zero

In classical mechanics, a particle is in mechanical equilibrium if the net force on that particle is zero. By extension, a physical system made up of many parts is in mechanical equilibrium if the net force on each of its individual parts is zero.

In fluid mechanics, a fluid is said to be in hydrostatic equilibrium or hydrostatic balance when it is at rest, or when the flow velocity at each point is constant over time. This occurs when external forces such as gravity are balanced by a pressure-gradient force. For instance, the pressure-gradient force prevents gravity from collapsing Earth's atmosphere into a thin, dense shell, whereas gravity prevents the pressure gradient force from diffusing the atmosphere into space.

Fluid dynamics subdiscipline of fluid mechanics that deals with fluid flow—the natural science of fluids (liquids and gases) in motion

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 and hydrodynamics. Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation,

Hydrostatics is fundamental to hydraulics, the engineering of equipment for storing, transporting and using fluids. It is also relevant to geophysics and astrophysics (for example, in understanding plate tectonics and the anomalies of the Earth's gravitational field), to meteorology, to medicine (in the context of blood pressure), and many other fields.

Hydraulics liquid engineering

Hydraulics is a technology and applied science using engineering, chemistry, and other sciences involving the mechanical properties and use of liquids. At a very basic level, hydraulics is the liquid counterpart of pneumatics, which concerns gases. Fluid mechanics provides the theoretical foundation for hydraulics, which focuses on the applied engineering using the properties of fluids. In its fluid power applications, hydraulics is used for the generation, control, and transmission of power by the use of pressurized liquids. Hydraulic topics range through some parts of science and most of engineering modules, and cover concepts such as pipe flow, dam design, fluidics and fluid control circuitry. The principles of hydraulics are in use naturally in the human body within the vascular system and erectile tissue. Free surface hydraulics is the branch of hydraulics dealing with free surface flow, such as occurring in rivers, canals, lakes, estuaries and seas. Its sub-field open-channel flow studies the flow in open channels.

Engineering applied science

Engineering is the application of knowledge in the form of science, mathematics, and empirical evidence, to the innovation, design, construction, operation and maintenance of structures, machines, materials, devices, systems, processes, and organizations. The discipline of engineering encompasses a broad range of more specialized fields of engineering, each with a more specific emphasis on particular areas of applied mathematics, applied science, and types of application. See glossary of engineering.

Geophysics Physics of the Earth and its vicinity

Geophysics is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term geophysics sometimes refers only to the geological applications: Earth's shape; its gravitational and magnetic fields; its internal structure and composition; its dynamics and their surface expression in plate tectonics, the generation of magmas, volcanism and rock formation. However, modern geophysics organizations use a broader definition that includes the water cycle including snow and ice; fluid dynamics of the oceans and the atmosphere; electricity and magnetism in the ionosphere and magnetosphere and solar-terrestrial relations; and analogous problems associated with the Moon and other planets.

Hydrostatics offers physical explanations for many phenomena of everyday life, such as why atmospheric pressure changes with altitude, why wood and oil float on water, and why the surface of still water is always flat and horizontal whatever the shape of its container.

Atmospheric pressure, sometimes also called barometric pressure, is the pressure within the atmosphere of Earth. The standard atmosphere is a unit of pressure defined as 1013.25 mbar (101325 Pa), equivalent to 760 mmHg (torr), 29.9212 inches Hg, or 14.696 psi. The atm unit is roughly equivalent to the mean sea-level atmospheric pressure on Earth, that is, the Earth's atmospheric pressure at sea level is approximately 1 atm.

Altitude or height is defined based on the context in which it is used. As a general definition, altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The reference datum also often varies according to the context. Although the term altitude is commonly used to mean the height above sea level of a location, in geography the term elevation is often preferred for this usage.

History

Some principles of hydrostatics have been known in an empirical and intuitive sense since antiquity, by the builders of boats, cisterns, aqueducts and fountains. Archimedes is credited with the discovery of Archimedes' Principle, which relates the buoyancy force on an object that is submerged in a fluid to the weight of fluid displaced by the object. The Roman engineer Vitruvius warned readers about lead pipes bursting under hydrostatic pressure. [2]

Cistern Waterproof receptacle for holding liquids, usually water

A cistern is a waterproof receptacle for holding liquids, usually water. Cisterns are often built to catch and store rainwater. Cisterns are distinguished from wells by their waterproof linings. Modern cisterns range in capacity from a few litres to thousands of cubic metres, effectively forming covered reservoirs.

Aqueduct (water supply) any system of pipes, ditches, canals, tunnels, and other structures used to convey water supply

An aqueduct is a watercourse constructed to carry water from a source to a distribution point far away. In modern engineering, the term aqueduct is used for any system of pipes, ditches, canals, tunnels, and other structures used for this purpose. The term aqueduct also often refers specifically to a bridge on an artificial watercourse. The word is derived from the Latin aqua ("water") and ducere. Aqueducts were used in ancient Greece, ancient Egypt, and ancient Rome. In modern times, the largest aqueducts of all have been built in the United States to supply the country's biggest cities. The simplest aqueducts are small ditches cut into the earth. Much larger channels may be used in modern aqueducts. Aqueducts sometimes run for some or all of their path through tunnels constructed underground. Modern aqueducts may also use pipelines. Historically, agricultural societies have constructed aqueducts to irrigate crops and supply large cities with drinking water.

Fountain piece of architecture which ejects water

A fountain is a piece of architecture which pours water into a basin or jets it into the air to supply swimming water and/or for a decorative or dramatic effect.

The concept of pressure and the way it is transmitted by fluids was formulated by the French mathematician and philosopher Blaise Pascal in 1647.

France Republic with mainland in Europe and numerous oversea territories

France, officially the French Republic, is a country whose territory consists of metropolitan France in Western Europe and several overseas regions and territories. The metropolitan area of France extends from the Mediterranean Sea to the English Channel and the North Sea, and from the Rhine to the Atlantic Ocean. It is bordered by Belgium, Luxembourg and Germany to the northeast, Switzerland and Italy to the east, and Andorra and Spain to the south. The overseas territories include French Guiana in South America and several islands in the Atlantic, Pacific and Indian oceans. The country's 18 integral regions span a combined area of 643,801 square kilometres (248,573 sq mi) and a total population of 67.3 million. France, a sovereign state, is a unitary semi-presidential republic with its capital in Paris, the country's largest city and main cultural and commercial centre. Other major urban areas include Lyon, Marseille, Toulouse, Bordeaux, Lille and Nice.

Mathematician person with an extensive knowledge of mathematics

A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.

Philosopher person with an extensive knowledge of philosophy

A philosopher is someone who practices philosophy. The term "philosopher" comes from the Ancient Greek, φιλόσοφος (philosophos), meaning "lover of wisdom". The coining of the term has been attributed to the Greek thinker Pythagoras.

Hydrostatics in ancient Greece and Rome

Pythagorean Cup

The "fair cup" or Pythagorean cup, which dates from about the 6th century BC, is a hydraulic technology whose invention is credited to the Greek mathematician and geometer Pythagoras. It was used as a learning tool.

The cup consists of a line carved into the interior of the cup, and a small vertical pipe in the center of the cup that leads to the bottom. The height of this pipe is the same as the line carved into the interior of the cup. The cup may be filled to the line without any fluid passing into the pipe in the center of the cup. However, when the amount of fluid exceeds this fill line, fluid will overflow into the pipe in the center of the cup. Due to the drag that molecules exert on one another, the cup will be emptied.

Heron's fountain

Heron's fountain is a device invented by Heron of Alexandria that consists of a jet of fluid being fed by a reservoir of fluid. The fountain is constructed in such a way that the height of the jet exceeds the height of the fluid in the reservoir, apparently in violation of principles of hydrostatic pressure. The device consisted of an opening and two containers arranged one above the other. The intermediate pot, which was sealed, was filled with fluid, and several cannula (a small tube for transferring fluid between vessels) connecting the various vessels. Trapped air inside the vessels induces a jet of water out of a nozzle, emptying all water from the intermediate reservoir.

Pascal's contribution in Hydrostatics

Pascal made contributions to developments in both hydrostatics and hydrodynamics. Pascal's Law is a fundamental principle of fluid mechanics that states that any pressure applied to the surface of a fluid is transmitted uniformly throughout the fluid in all directions, in such a way that initial variations in pressure are not changed.

Pressure in fluids at rest

Due to the fundamental nature of fluids, a fluid cannot remain at rest under the presence of a shear stress. However, fluids can exert pressure normal to any contacting surface. If a point in the fluid is thought of as an infinitesimally small cube, then it follows from the principles of equilibrium that the pressure on every side of this unit of fluid must be equal. If this were not the case, the fluid would move in the direction of the resulting force. Thus, the pressure on a fluid at rest is isotropic; i.e., it acts with equal magnitude in all directions. This characteristic allows fluids to transmit force through the length of pipes or tubes; i.e., a force applied to a fluid in a pipe is transmitted, via the fluid, to the other end of the pipe. This principle was first formulated, in a slightly extended form, by Blaise Pascal, and is now called Pascal's law.

Hydrostatic pressure

In a fluid at rest, all frictional and inertial stresses vanish and the state of stress of the system is called hydrostatic. When this condition of V = 0 is applied to the Navier-Stokes equation, the gradient of pressure becomes a function of body forces only. For a barotropic fluid in a conservative force field like a gravitational force field, pressure exerted by a fluid at equilibrium becomes a function of force exerted by gravity.

The hydrostatic pressure can be determined from a control volume analysis of an infinitesimally small cube of fluid. Since pressure is defined as the force exerted on a test area (p = F/A, with p: pressure, F: force normal to area A, A: area), and the only force acting on any such small cube of fluid is the weight of the fluid column above it, hydrostatic pressure can be calculated according to the following formula:

where:

For water and other liquids, this integral can be simplified significantly for many practical applications, based on the following two assumptions: Since many liquids can be considered incompressible, a reasonable good estimation can be made from assuming a constant density throughout the liquid. (The same assumption cannot be made within a gaseous environment.) Also, since the height h of the fluid column between z and z0 is often reasonably small compared to the radius of the Earth, one can neglect the variation of g. Under these circumstances, the integral is simplified into the formula:

where h is the height zz0 of the liquid column between the test volume and the zero reference point of the pressure. This formula is often called Stevin's law. [3] [4] Note that this reference point should lie at or below the surface of the liquid. Otherwise, one has to split the integral into two (or more) terms with the constant ρliquid and ρ(z′)above. For example, the absolute pressure compared to vacuum is:

where H is the total height of the liquid column above the test area to the surface, and patm is the atmospheric pressure, i.e., the pressure calculated from the remaining integral over the air column from the liquid surface to infinity. This can easily be visualized using a pressure prism.

Hydrostatic pressure has been used in the preservation of foods in a process called pascalization. [5]

Medicine

In medicine, hydrostatic pressure in blood vessels is the pressure of the blood against the wall. It is the opposing force to oncotic pressure.

Atmospheric pressure

Statistical mechanics shows that, for a gas of constant temperature, T, its pressure, p will vary with height, h, as:

where:

This is known as the barometric formula, and may be derived from assuming the pressure is hydrostatic.

If there are multiple types of molecules in the gas, the partial pressure of each type will be given by this equation. Under most conditions, the distribution of each species of gas is independent of the other species.

Buoyancy

Any body of arbitrary shape which is immersed, partly or fully, in a fluid will experience the action of a net force in the opposite direction of the local pressure gradient. If this pressure gradient arises from gravity, the net force is in the vertical direction opposite that of the gravitational force. This vertical force is termed buoyancy or buoyant force and is equal in magnitude, but opposite in direction, to the weight of the displaced fluid. Mathematically,

where ρ is the density of the fluid, g is the acceleration due to gravity, and V is the volume of fluid directly above the curved surface. [6] In the case of a ship, for instance, its weight is balanced by pressure forces from the surrounding water, allowing it to float. If more cargo is loaded onto the ship, it would sink more into the water – displacing more water and thus receive a higher buoyant force to balance the increased weight.

Discovery of the principle of buoyancy is attributed to Archimedes.

Hydrostatic force on submerged surfaces

The horizontal and vertical components of the hydrostatic force acting on a submerged surface are given by the following: [6]

where:

Liquids (fluids with free surfaces)

Liquids can have free surfaces at which they interface with gases, or with a vacuum. In general, the lack of the ability to sustain a shear stress entails that free surfaces rapidly adjust towards an equilibrium. However, on small length scales, there is an important balancing force from surface tension.

Capillary action

When liquids are constrained in vessels whose dimensions are small, compared to the relevant length scales, surface tension effects become important leading to the formation of a meniscus through capillary action. This capillary action has profound consequences for biological systems as it is part of one of the two driving mechanisms of the flow of water in plant xylem, the transpirational pull.

Hanging drops

Without surface tension, drops would not be able to form. The dimensions and stability of drops are determined by surface tension. The drop's surface tension is directly proportional to the cohesion property of the fluid.

See also

Related Research Articles

Pressure Force distributed continuously over an area

Pressure is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure is the pressure relative to the ambient pressure.

Relative density, or specific gravity, is the ratio of the density of a substance to the density of a given reference material. Specific gravity usually means relative density with respect to water. The term "relative density" is often preferred in scientific usage. It is defined as a ratio of density of particular substance with that of water.

Buoyancy An upward force that opposes the weight of an object immersed in fluid

Buoyancy or upthrust, is an upward force exerted by a fluid that opposes the weight of an immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the pressure at the bottom of a column of fluid is greater than at the top of the column. Similarly, the pressure at the bottom of an object submerged in a fluid is greater than at the top of the object. The pressure difference results in a net upward force on the object. The magnitude of the force is proportional to the pressure difference, and is equivalent to the weight of the fluid that would otherwise occupy the volume of the object, i.e. the displaced fluid.

Settling The process by which particulates settle to the bottom of a liquid and form a sediment

Settling is the process by which particulates settle to the bottom of a liquid and form a sediment. Particles that experience a force, either due to gravity or due to centrifugal motion will tend to move in a uniform manner in the direction exerted by that force. For gravity settling, this means that the particles will tend to fall to the bottom of the vessel, forming a slurry at the vessel base.

Specific gravity Relative density compared to water

Specific gravity is the ratio of the density of a substance to the density of a reference substance; equivalently, it is the ratio of the mass of a substance to the mass of a reference substance for the same given volume. Apparent specific gravity is the ratio of the weight of a volume of the substance to the weight of an equal volume of the reference substance. The reference substance for liquids is nearly always water at its densest ; for gases it is air at room temperature. Nonetheless, the temperature and pressure must be specified for both the sample and the reference. Pressure is nearly always 1 atm (101.325 kPa).

Siphon device

The word siphon is used to refer to a wide variety of devices that involve the flow of liquids through tubes. In a narrower sense, the word refers particularly to a tube in an inverted 'U' shape, which causes a liquid to flow upward, above the surface of a reservoir, with no pump, but powered by the fall of the liquid as it flows down the tube under the pull of gravity, then discharging at a level lower than the surface of the reservoir from which it came.

Internal wave Gravity waves that oscillate within a fluid medium with density variation with depth, rather than on the surface

Internal waves are gravity waves that oscillate within a fluid medium, rather than on its surface. To exist, the fluid must be stratified: the density must decrease continuously or discontinuously with depth/height due to changes, for example, in temperature and/or salinity. If the density changes over a small vertical distance, the waves propagate horizontally like surface waves, but do so at slower speeds as determined by the density difference of the fluid below and above the interface. If the density changes continuously, the waves can propagate vertically as well as horizontally through the fluid.

Hydraulic head specific measurement of liquid pressure above a geodetic datum

Hydraulic head or piezometric head is a specific measurement of liquid pressure above a vertical datum.

Torricellis law

Torricelli's law, also known as Torricelli's theorem, is a theorem in fluid dynamics relating the speed of fluid flowing out of an orifice to the height of fluid above the opening. The law states that the speed of efflux, v, of a fluid through a sharp-edged hole at the bottom of a tank filled to a depth h is the same as the speed that a body would acquire in falling freely from a height h, i.e. , where g is the acceleration due to gravity. This last expression comes from equating the kinetic energy gained, , with the potential energy lost, mgh, and solving for v. The law was discovered by the Italian scientist Evangelista Torricelli, in 1643. It was later shown to be a particular case of Bernoulli's principle.

A reference atmospheric model describe how the ideal gas properties of an atmosphere change, primarily as a function of altitude, and sometimes also as a function of latitude, day of year, etc. A static atmospheric model has a more limited domain, excluding time. A standard atmosphere is defined by the World Meteorological Organization as "a hypothetical vertical distribution of atmospheric temperature, pressure and density which, by international agreement, is roughly representative of year-round, midlatitude conditions."

In fluid mechanics, pressure head is the height of a liquid column that corresponds to a particular pressure exerted by the liquid column on the base of its container. It may also be called static pressure head or simply static head. It is mathematically expressed as:

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 behaviour of menisci, and is found when body forces (gravity) and surface forces are in equilibrium.

Free surface surface of a fluid that is subject to constant perpendicular normal stress and zero parallel shear stress; surface not created by the container

In physics, a free surface is the surface of a fluid that is subject to zero parallel shear stress, such as the interface between two homogeneous fluids, for example liquid water and the air in the Earth's atmosphere. Unlike liquids, gases cannot form a free surface on their own. Fluidized/liquified solids, including slurries, granular materials, and powders may form a free surface.

The depth–slope product is used to calculate the shear stress at the bed of an open channel containing fluid that is undergoing steady, uniform flow. It is widely used in river engineering, stream restoration, sedimentology, and fluvial geomorphology. It is the product of the water depth and the mean bed slope, along with the acceleration due to gravity and density of the fluid.

Pascals law principle in fluid mechanics

Pascal's law is a principle in fluid mechanics that states that pressure at a point has infinite direction, and thus a pressure change at any point in a confined incompressible fluid is transmitted throughout the fluid such that the same change occurs everywhere. The law was established by French mathematician Blaise Pascal in 1647–48.

Radiation stress The depth-integrated excess momentum flux caused by the presence of the surface gravity waves, which is exerted on the mean flow

In fluid dynamics, the radiation stress is the depth-integrated – and thereafter phase-averaged – excess momentum flux caused by the presence of the surface gravity waves, which is exerted on the mean flow. The radiation stresses behave as a second-order tensor.

Jurins law

One of the most common fluid mechanical effects that is often explored in micro fluids is capillarity i.e the induced motion of liquids in small channels of which the most simplest case is capillary rise or Jurin's law, named after James Jurin who discovered between 1718 and 1719. His quantitative law suggests that the maximum height of a capillary tube is inversely proportional to the diameter. The difference in height between the surroundings of the tube and the inside and the shape of the meniscus, are caused by capillary action. The mathematical expression of this law can be derived directly from hydrostatic principles and from Young–Laplace equation. Jurin's law allows the measurement of the surface tension of a liquid and can be used to derive the capillary length.

Vertical pressure variation is the variation in pressure as a function of elevation. Depending on the fluid in question and the context being referred to, it may also vary significantly in dimensions perpendicular to elevation as well, and these variations have relevance in the context of pressure gradient force and its effects. However, the vertical variation is especially significant, as it results from the pull of gravity on the fluid; namely, for the same given fluid, a decrease in elevation within it corresponds to a taller column of fluid weighing down on that point.

A level probe is a special pressure transmitter for level measurement of liquids in open vessels and tanks. Level probes are submerged directly into the liquid and remain permanently floating above the tank bottom. The measurement is carried out according to the hydrostatic principle. The gravity pressure of the liquid column causes an expansion of the pressure-sensitive sensor element, which converts the measured pressure into an electrical standard signal. The connecting cable of level probes has several tasks to fulfil. In addition to the power supply and signal forwarding, the level sensor is held in place by the cable. The cable also includes a thin air tube that directs the ambient air pressure to the level probe. Level probes are therefore usually designed as relative pressure sensors, which use the current ambient pressure as the zero point of their measuring range.

References

  1. "Hydrostatics". Merriam-Webster. Retrieved 11 September 2018.
  2. Marcus Vitruvius Pollio (ca. 15 BCE), "The Ten Books of Architecture", Book VIII, Chapter 6. At the University of Chicago's Penelope site. Accessed on 2013-02-25.
  3. Bettini, Alessandro (2016). A Course in Classical Physics 2—Fluids and Thermodynamics. Springer. p. 8. ISBN   978-3-319-30685-8.
  4. Mauri, Roberto. Transport Phenomena in Multiphase Flow. Springer. p. 24. ISBN   978-3-319-15792-4 . Retrieved 3 February 2017.
  5. Brown, Amy Christian (2007). Understanding Food: Principles and Preparation (3 ed.). Cengage Learning. p. 546. ISBN   978-0-495-10745-3.
  6. 1 2 Fox, Robert; McDonald, Alan; Pritchard, Philip (2012). Fluid Mechanics (8 ed.). John Wiley & Sons. pp. 76–83. ISBN   978-1-118-02641-0.

Further reading