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Hydrostatics

From Wikipedia, the free encyclopedia
(Redirected fromFluid statics)
Branch of fluid mechanics that studies fluids at rest
This article is about the discipline branch. For the concept, seeHydrostatic equilibrium.
Table of Hydraulics and Hydrostatics, from the 1728Cyclopædia
Part of a series on
Continuum mechanics
J=−Ddφdx{\displaystyle J=-D{\frac {d\varphi }{dx}}}

Hydrostatics is the branch offluid mechanics that studiesfluids athydrostatic equilibrium[1] and "the pressure in a fluid or exerted by a fluid on an immersed body".[2] The word "hydrostatics" is sometimes used to refer specifically to water and other liquids, but more often it includes both gases and liquids, whethercompressible orincompressible.

It encompasses the study of the conditions under which fluids are at rest instable equilibrium. It is opposed tofluid dynamics, the study of fluids in motion.

Hydrostatics is fundamental tohydraulics, theengineering of equipment for storing, transporting and using fluids. It is also relevant togeophysics andastrophysics (for example, in understandingplate tectonics and the anomalies of theEarth's gravitational field), tometeorology, tomedicine (in the context ofblood pressure), and many other fields.

Hydrostatics offers physical explanations for many phenomena of everyday life, such as whyatmospheric pressure changes withaltitude, why wood and oil float on water, and why the surface of still water is alwayslevel according to thecurvature of the earth.

History

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Some principles of hydrostatics have been known in an empirical and intuitive sense since antiquity, by the builders of boats,cisterns,aqueducts andfountains.Archimedes is credited with the discovery ofArchimedes' Principle, which relates thebuoyancy force on an object that is submerged in a fluid to the weight of fluid displaced by the object. TheRoman engineerVitruvius warned readers aboutlead pipes bursting under hydrostatic pressure.[3]

The concept of pressure and the way it is transmitted by fluids was formulated by theFrenchmathematician andphilosopherBlaise Pascal in 1647.[citation needed]

Hydrostatics in ancient Greece and Rome

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Pythagorean Cup

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Main article:Pythagorean cup

The "fair cup" orPythagorean 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.[citation needed]

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

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Main article:Heron's fountain

Heron's fountain is a device invented byHeron 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 severalcannula (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.[citation needed]

Pascal's contribution in hydrostatics

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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

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Due to the fundamental nature of fluids, a fluid cannot remain at rest under the presence of ashear stress. However, fluids can exertpressurenormal 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, thepressure on a fluid at rest isisotropic; 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 calledPascal's law.[citation needed]

Hydrostatic pressure

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See also:Vertical pressure variation

In a fluid at rest, all frictional and inertial stresses vanish and the state of stress of the system is calledhydrostatic. When this condition ofV = 0 is applied to theNavier–Stokes equations for viscous fluids orEuler equations (fluid dynamics) for ideal inviscid fluid, the gradient of pressure becomes a function of body forces only. The Navier-Stokes momentum equations are:

Navier–Stokes momentum equation (convective form)

ρDuDt=−∇[p−ζ(∇⋅u)]+∇⋅{μ[∇u+(∇u)T−23(∇⋅u)I]}+ρg.{\displaystyle \rho {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}=-\nabla [p-\zeta (\nabla \cdot \mathbf {u} )]+\nabla \cdot \left\{\mu \left[\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }-{\tfrac {2}{3}}(\nabla \cdot \mathbf {u} )\mathbf {I} \right]\right\}+\rho \mathbf {g} .}

By setting theflow velocityu=0{\displaystyle \mathbf {u} =\mathbf {0} }, they become simply:

0=−∇p+ρg{\displaystyle \mathbf {0} =-\nabla p+\rho \mathbf {g} }

or:

∇p=ρg{\displaystyle \nabla p=\rho \mathbf {g} }

This is the general form of Stevin's law: thepressure gradient equals thebody forceforce density field.

Let us now consider two particular cases of this law. In case of aconservative body force withscalar potentialϕ{\displaystyle \phi }:

ρg=−∇ϕ{\displaystyle \rho \mathbf {g} =-\nabla \phi }

the Stevin equation becomes:∇p=−∇ϕ{\displaystyle \nabla p=-\nabla \phi }

That can be integrated to give:

Δp=−Δϕ{\displaystyle \Delta p=-\Delta \phi }

So in this case the pressure difference is the opposite of the difference of the scalar potential associated to the body force.In the other particular case of a body force of constant direction along z:

g=−g(x,y,z)k^{\displaystyle \mathbf {g} =-g(x,y,z){\hat {k}}}

the generalised Stevin's law above becomes:

∂p∂z=−ρ(x,y,z)g(x,y,z){\displaystyle {\frac {\partial p}{\partial z}}=-\rho (x,y,z)g(x,y,z)}

That can be integrated to give another (less-) generalised Stevin's law:

p(x,y,z)−p0(x,y)=−∫0zρ(x,y,z′)g(x,y,z′)dz′{\displaystyle p(x,y,z)-p_{0}(x,y)=-\int _{0}^{z}\rho (x,y,z')g(x,y,z')dz'}

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 consideredincompressible, 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 heightΔz{\displaystyle \Delta z} of the fluid column betweenz andz0 is often reasonably small compared to the radius of the Earth, one can neglect the variation ofg. Under these circumstances, one can transport out of the integral the density and the gravity acceleration and the law is simplified into the formula

Δp(z)=ρgΔz,{\displaystyle \Delta p(z)=\rho g\Delta z,}

whereΔz{\displaystyle \Delta z} is the heightz −z0 of the liquid column between the test volume and the zero reference point of the pressure. This formula is often calledStevin's law.[4][5] One could arrive to the above formula also by considering the first particular case of the equation for a conservative body force field: in fact the body force field of uniform intensity and direction:

ρg(x,y,z)=−ρgk^{\displaystyle \rho \mathbf {g} (x,y,z)=-\rho g{\hat {k}}}

is conservative, so one can write the body force density as:

ρg=∇(−ρgz){\displaystyle \rho \mathbf {g} =\nabla (-\rho gz)}

Then the body force density has a simple scalar potential:

ϕ(z)=−ρgz{\displaystyle \phi (z)=-\rho gz}

And the pressure difference follows another time the Stevin's law:

Δp=−Δϕ=ρgΔz{\displaystyle \Delta p=-\Delta \phi =\rho g\Delta z}

The 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 constantρliquid andρ(z′)above. For example, theabsolute pressure compared to vacuum is

p=ρgΔz+p0,{\displaystyle p=\rho g\Delta z+p_{\mathrm {0} },}

whereΔz{\displaystyle \Delta z} is the total height of the liquid column above the test area to the surface, andp0 is theatmospheric 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 apressure prism.

Hydrostatic pressure has been used in the preservation of foods in a process calledpascalization.[6]

Medicine

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In medicine, hydrostatic pressure inblood vessels is the pressure of the blood against the wall. It is the opposing force tooncotic pressure. In capillaries, hydrostatic pressure (also known as capillary blood pressure) is higher than the opposing “colloid osmotic pressure” in blood—a “constant” pressure primarily produced by circulating albumin—at the arteriolar end of the capillary. This pressure forces plasma and nutrients out of the capillaries and into surrounding tissues. Fluid and the cellular wastes in the tissues enter the capillaries at the venule end, where the hydrostatic pressure is less than the osmotic pressure in the vessel.[7]

Atmospheric pressure

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Statistical mechanics shows that, for a pureideal gas of constant temperatureT in the earth gravitational field, its pressure,p will vary with height,h, as

p(h)=p(0)e−MghkT{\displaystyle p(h)=p(0)e^{-{\frac {Mgh}{kT}}}}

where

This is known as thebarometric formula, and may be derived from assuming the pressure ishydrostatic.

If there are multiple types of molecules in the gas, thepartial 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

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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,

F=ρgV{\displaystyle F=\rho gV}

whereρ is the density of the fluid,g is the acceleration due to gravity, andV is the volume of fluid directly above the curved surface.[8] In the case of aship, 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.[citation needed]

Discovery of the principle of buoyancy is attributed toArchimedes.

Hydrostatic force on submerged surfaces

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The horizontal and vertical components of the hydrostatic force acting on a submerged surface are given by the following formula:[8]

Fh=pcAFv=ρgV{\displaystyle {\begin{aligned}F_{\mathrm {h} }&=p_{\mathrm {c} }A\\F_{\mathrm {v} }&=\rho gV\end{aligned}}}

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 is the volume of fluid directly above the curved surface

Liquids (fluids with free surfaces)

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

Capillary action

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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 ameniscus throughcapillary 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 inplantxylem, thetranspirational pull.

Hanging drops

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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

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  • Communicating vessels – Set of internally connected containers containing a homogeneous fluid
  • D-DIA – Apparatus used for high pressure and high temperature deformation experiments
  • Hydrostatic test – Non-destructive test of pressure vessels
  • Triaxial shear test – Stress test with different levels of stress in each perpendicular direction

References

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  1. ^"Fluid Mechanics/Fluid Statics/Fundamentals of Fluid Statics - Wikibooks, open books for an open world".en.wikibooks.org. Retrieved2021-04-01.
  2. ^"Hydrostatics".Merriam-Webster. Retrieved11 September 2018.
  3. ^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.
  4. ^Bettini, Alessandro (2016).A Course in Classical Physics 2—Fluids and Thermodynamics. Springer. p. 8.ISBN 978-3-319-30685-8.
  5. ^Mauri, Roberto (8 April 2015).Transport Phenomena in Multiphase Flow. Springer. p. 24.ISBN 978-3-319-15792-4. Retrieved3 February 2017.
  6. ^Brown, Amy Christian (2007).Understanding Food: Principles and Preparation (3 ed.). Cengage Learning. p. 546.ISBN 978-0-495-10745-3.
  7. ^ This article incorporatestext available under theCC BY 4.0 license.Betts, J Gordon; Desaix, Peter; Johnson, Eddie; Johnson, Jody E; Korol, Oksana; Kruse, Dean; Poe, Brandon; Wise, James; Womble, Mark D; Young, Kelly A (September 16, 2023).Anatomy & Physiology. Houston: OpenStax CNX. 26.1 Body fluids and fluid compartments.ISBN 978-1-947172-04-3.
  8. ^abFox, Robert; McDonald, Alan; Pritchard, Philip (2012).Fluid Mechanics (8 ed.).John Wiley & Sons. pp. 76–83.ISBN 978-1-118-02641-0.

Further reading

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  • 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 rev. ed.). Academic Press.ISBN 978-0-12-373735-9.
  • Currie, I. G. (1974).Fundamental Mechanics of Fluids. McGraw-Hill.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

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Look uphydrostatics in Wiktionary, the free dictionary.
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