Hydrostatics

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Table of Hydraulics and Hydrostatics, from the 1728 Cyclopædia

Fluid statics or hydrostatics is the branch of fluid mechanics that studies incompressible fluids at rest. 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.

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.

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 water is always flat and horizontal whatever the shape of its container.

History[edit]

Some principles of hydrostatics have been known[to whom?] 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[1]

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

Hydrostatics in ancient Greece and Rome[edit]

Pythagorean Cup[edit]

Main article: 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 Pythagorus. 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[edit]

Main article: 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[edit]

Main article: Pascal's Law

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[edit]

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[edit]

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 reasonably 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 z − z0 of the liquid column between the test volume and the zero reference point of the pressure. 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.[2]

Medicine[edit]

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[edit]

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[edit]

Main article: 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.[3] 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.

The Principle of Archimedes proved experimentally in the following manner: Taking a body hanging in the small dynamometer read the indication of weight. Then keeping the body hanging on the dynamometer immersed in a glass completely overflowing with water, but have previously put in a deep dish. Immersing the body in water, the indication of the dynamometer will be smaller than the previous (outside water). At the same time, we see that some of the water from the overflowing glass will be poured on the plate. If you weigh the quantity of water that overflowed will see that this will be equal to the difference in body weight outside and inside the water. This test is more accurate if successfully used special "weir tank." After the above experiment simplifying the definition of the Principle of Archimedes saying that: Each body is immersed in a liquid loses both by weight, as the weight of the liquid that displaces.

Accordingly, when a body is found inside a liquid will be observed two main forces (resultants) each. The body weight and the force applied to this buoyancy. Depending to the value are taking these resultants each time a further three cases are distinguished:

1st case: The body weight is greater that the buoyancy. In this case the body is immersed.

2nd case: The body weight is equal to the buoyancy. In this case the body is suspended in the liquid, i.e. standstill wherever found in the liquid, and

3rd case: The body weight is less than pursued in this buoyancy. In this case the body is not immersed, when floating.

It is obvious that all three cases depending on the specific weight of the body (solid or liquid) which can be respectively greater or less than the specific gravity of the liquid. For example, wood, cork, the oil floating on the water, while the iron, aluminum, mercury sink.

The Principle of Archimedes finds very wide application in daily life mainly in engineering. Anything that floats, such as ships, all lighter water bodies, the human body, the floats, amphibious vehicles etc. obey the principle. But more interested in the Principle is Shipbuilding, the science which deals in construction of ships. There the principle of Archimedes studied, analyzed and implemented in all its details. Maximum implementation of the Principle of Archimedes observed in Underwater and Water Tanks constantly change their buoyancy values (in corresponding cases to negative, zero and positive). The Principle of Archimedes also apply in aerostatic e.g. in balloons

Hydrostatic force on submerged surfaces[edit]

The horizontal and vertical components of the hydrostatic force acting on a submerged surface are given by the following:[3]

where:

  • pc is the pressure at the centroid of the vertical projection of the submerged surface
  • A is the area of the same vertical projection of the surface
  • ρ is the density of the fluid
  • g is the acceleration due to gravity
  • V the volume of fluid directly above the curved surface

Liquids (fluids with free surfaces)[edit]

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[edit]

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[edit]

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[edit]

References[edit]

  1. ^ 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.
  2. ^ Brown, Amy Christian (2007). Understanding Food: Principles and Preparation (3 ed.). Cengage Learning. p. 546. ISBN 978-0-495-10745-3. 
  3. ^ a b 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[edit]

  • Batchelor, George K. (1967), An Introduction to Fluid Dynamics, Cambridge University Press, ISBN 0-521-66396-2
  • Falkovich, Gregory (2011), Fluid Mechanics (A short course for physicists), Cambridge University Press, ISBN 978-1-107-00575-4
  • Kundu, Pijush K.; Cohen, Ira M. (2008), Fluid Mechanics (4th revised ed.), Academic Press, ISBN 978-0-12-373735-9
  • Currie, I. G. (1974), Fundamental Mechanics of Fluids, McGraw-Hill, Inc., ISBN 0-07-015000-1
  • Massey, B.; Ward-Smith, J. (2005), Mechanics of Fluids (8th ed.), Taylor & Francis, ISBN 978-0-415-36206-1
  • White, Frank M. (2003), Fluid Mechanics, McGraw–Hill, ISBN 0-07-240217-2

External links[edit]