Fluid mechanics is the branch of
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
concerned with the
mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objec ...
of
fluids (
liquids,
gases, and
plasmas) and the
forces on them.
It has applications in a wide range of disciplines, including
mechanical
Mechanical may refer to:
Machine
* Machine (mechanical), a system of mechanisms that shape the actuator input to achieve a specific application of output forces and movement
* Mechanical calculator, a device used to perform the basic operations ...
,
aerospace
Aerospace is a term used to collectively refer to the atmosphere and outer space. Aerospace activity is very diverse, with a multitude of commercial, industrial and military applications. Aerospace engineering consists of aeronautics and ast ...
,
civil,
chemical and
biomedical engineering
Biomedical engineering (BME) or medical engineering is the application of engineering principles and design concepts to medicine and biology for healthcare purposes (e.g., diagnostic or therapeutic). BME is also traditionally logical sciences ...
,
geophysics
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'' so ...
,
oceanography,
meteorology
Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did no ...
,
astrophysics, and
biology
Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary ...
.
It can be divided into
fluid statics
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 im ...
, the study of fluids at rest; and
fluid dynamics, the study of the effect of forces on fluid motion.
It is a branch of
continuum mechanics, a subject which models matter without using the information that it is made out of atoms; that is, it models matter from a ''macroscopic'' viewpoint rather than from ''microscopic''. Fluid mechanics, especially fluid dynamics, is an active field of research, typically mathematically complex. Many problems are partly or wholly unsolved and are best addressed by
numerical methods, typically using computers. A modern discipline, called
computational fluid dynamics
Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate ...
(CFD), is devoted to this approach.
Particle image velocimetry, an experimental method for visualizing and analyzing fluid flow, also takes advantage of the highly visual nature of fluid flow.
Brief history
The study of fluid mechanics goes back at least to the days of
ancient Greece
Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cu ...
, when
Archimedes investigated fluid statics and
buoyancy
Buoyancy (), or upthrust, is an upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the ...
and formulated his famous law known now as the
Archimedes' principle, which was published in his work ''
On Floating Bodies
''On Floating Bodies'' ( el, Περὶ τῶν ἐπιπλεόντων σωμάτων) is a Greek-language work consisting of two books written by Archimedes of Syracuse (287 – c. 212 BC), one of the most important mathematicians, physicis ...
''—generally considered to be the first major work on fluid mechanics. Rapid advancement in fluid mechanics began with
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, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially rested on ...
(observations and experiments),
Evangelista Torricelli
Evangelista Torricelli ( , also , ; 15 October 160825 October 1647) was an Italian physicist and mathematician, and a student of Galileo. He is best known for his invention of the barometer, but is also known for his advances in optics and wo ...
(invented the
barometer),
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
(investigated
viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the int ...
) and
Blaise Pascal (researched
hydrostatics, formulated
Pascal's law), and was continued by
Daniel Bernoulli with the introduction of mathematical fluid dynamics in ''Hydrodynamica'' (1739).
Inviscid flow was further analyzed by various mathematicians (
Jean le Rond d'Alembert,
Joseph Louis Lagrange,
Pierre-Simon Laplace,
Siméon Denis Poisson) and viscous flow was explored by a multitude of
engineers
Engineers, as practitioners of engineering, are professionals who invent, design, analyze, build and test machines, complex systems, structures, gadgets and materials to fulfill functional objectives and requirements while considering the li ...
including
Jean Léonard Marie Poiseuille
Jean Léonard Marie Poiseuille (; 22 April 1797 – 26 December 1869) was a French physicist and physiologist.
Poiseuille was born in Paris, France, and he died there on 26 December 1869.
Fluid flow
From 1815 to 1816 he studied at the École ...
and
Gotthilf Hagen. Further mathematical justification was provided by
Claude-Louis Navier and
George Gabriel Stokes in the
Navier–Stokes equations, and
boundary layers
In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary condi ...
were investigated (
Ludwig Prandtl,
Theodore von Kármán), while various scientists such as
Osborne Reynolds,
Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
, and
Geoffrey Ingram Taylor advanced the understanding of fluid viscosity and
turbulence.
Main branches
Fluid statics
Fluid statics
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 im ...
or hydrostatics is the branch of fluid mechanics that studies
fluids at rest. It embraces the study of the conditions under which fluids are at rest in
stable
A stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the ...
equilibrium; and is contrasted with
fluid dynamics, the study of fluids in motion. 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 level whatever the shape of its container. Hydrostatics is fundamental to
hydraulics, the
engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
of equipment for storing, transporting and using
fluids. It is also relevant to some aspects of
geophysics
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'' so ...
and
astrophysics (for example, in understanding
plate tectonics
Plate tectonics (from the la, label= Late Latin, tectonicus, from the grc, τεκτονικός, lit=pertaining to building) is the generally accepted scientific theory that considers the Earth's lithosphere to comprise a number of larg ...
and anomalies in the
Earth's gravitational field), to
meteorology
Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did no ...
, to
medicine
Medicine is the science and practice of caring for a patient, managing the diagnosis, prognosis, prevention, treatment, palliation of their injury or disease, and promoting their health. Medicine encompasses a variety of health care pr ...
(in the context of
blood pressure), and many other fields.
Fluid dynamics
''
Fluid dynamics'' is a subdiscipline of fluid mechanics that deals with ''fluid flow''—the science of liquids and gases in motion. Fluid dynamics offers a systematic structure—which underlies these
practical disciplines—that embraces empirical and semi-empirical laws derived from
flow measurement
Flow measurement is the quantification of bulk fluid movement. Flow can be measured in a variety of ways. The common types of flowmeters with industrial applications are listed below:
* a) Obstruction type (differential pressure or variable area ...
and used to solve practical problems. The solution to a
fluid dynamics problem typically involves calculating various properties of the fluid, such as
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
,
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
,
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. Mathematicall ...
, and
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
, as functions of space and time. It has several subdisciplines itself, including ''
aerodynamics
Aerodynamics, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dy ...
'' (the study of air and other gases in motion) and ''hydrodynamics'' (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating
forces and
movements on
aircraft
An aircraft is a vehicle that is able to flight, fly by gaining support from the Atmosphere of Earth, air. It counters the force of gravity by using either Buoyancy, static lift or by using the Lift (force), dynamic lift of an airfoil, or in ...
, determining the
mass flow rate of
petroleum
Petroleum, also known as crude oil, or simply oil, is a naturally occurring yellowish-black liquid mixture of mainly hydrocarbons, and is found in geological formations. The name ''petroleum'' covers both naturally occurring unprocessed crud ...
through pipelines, predicting evolving
weather patterns, understanding
nebulae in
interstellar space and modeling
explosions. Some fluid-dynamical principles are used in
traffic engineering and crowd dynamics.
Relationship to continuum mechanics
Fluid mechanics is a subdiscipline of
continuum mechanics, as illustrated in the following table.
In a mechanical view, a fluid is a substance that does not support
shear stress
Shear stress, often denoted by ( Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. '' Normal stress'', on ...
; that is why a fluid at rest has the shape of its containing vessel. A fluid at rest has no shear stress.
Assumptions
The assumptions inherent to a fluid mechanical treatment of a physical system can be expressed in terms of mathematical equations. Fundamentally, every fluid mechanical system is assumed to obey:
*
Conservation of mass
*
Conservation of energy
*
Conservation of momentum
*
The continuum assumption
For example, the assumption that mass is conserved means that for any fixed
control volume (for example, a spherical volume)—enclosed by a
control surface—the
rate of change of the mass contained in that volume is equal to the rate at which mass is passing through the surface from ''outside'' to ''inside'', minus the rate at which mass is passing from ''inside'' to ''outside''. This can be expressed as an
equation in integral form over the control volume.
The is an idealization of
continuum mechanics under which fluids can be treated as
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
, even though, on a microscopic scale, they are composed of
molecules. Under the continuum assumption, macroscopic (observed/measurable) properties such as density, pressure, temperature, and bulk velocity are taken to be well-defined at "infinitesimal" volume elements—small in comparison to the characteristic length scale of the system, but large in comparison to molecular length scale. Fluid properties can vary continuously from one volume element to another and are average values of the molecular properties. The continuum hypothesis can lead to inaccurate results in applications like supersonic speed flows, or molecular flows on nano scale.
Those problems for which the continuum hypothesis fails can be solved using
statistical mechanics. To determine whether or not the continuum hypothesis applies, the
Knudsen number, defined as the ratio of the molecular
mean free path
In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as ...
to the characteristic length
scale, is evaluated. Problems with Knudsen numbers below 0.1 can be evaluated using the continuum hypothesis, but molecular approach (statistical mechanics) can be applied to find the fluid motion for larger Knudsen numbers.
Navier–Stokes equations
The Navier–Stokes equations (named after
Claude-Louis Navier and
George Gabriel Stokes) are
differential equations that describe the force balance at a given point within a fluid. For an
incompressible fluid with vector velocity field
, the Navier–Stokes equations are
:
.
These differential equations are the analogues for deformable materials to Newton's equations of motion for particles – the Navier–Stokes equations describe changes in
momentum (
force) in response to
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
and viscosity, parameterized by the
kinematic viscosity here. Occasionally,
body force
In physics, a body force is a force that acts throughout the volume of a body.
Springer site - Book 'Solid mechanics'preview paragraph 'Body forces'./ref>
Forces due to gravity, electric fields and magnetic fields are examples of body forces. ...
s, such as the gravitational force or Lorentz force are added to the equations.
Solutions of the Navier–Stokes equations for a given physical problem must be sought with the help of
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
. In practical terms, only the simplest cases can be solved exactly in this way. These cases generally involve non-turbulent, steady flow in which the
Reynolds number is small. For more complex cases, especially those involving
turbulence, such as global weather systems, aerodynamics, hydrodynamics and many more, solutions of the Navier–Stokes equations can currently only be found with the help of computers. This branch of science is called
computational fluid dynamics
Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate ...
.
Inviscid and viscous fluids
An inviscid fluid has no
viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the int ...
,
. In practice, an inviscid flow is an
idealization, one that facilitates mathematical treatment. In fact, purely inviscid flows are only known to be realized in the case of
superfluidity. Otherwise, fluids are generally viscous, a property that is often most important within a
boundary layer near a solid surface,
where the flow must match onto the
no-slip condition at the solid. In some cases, the mathematics of a fluid mechanical system can be treated by assuming that the fluid outside of boundary layers is inviscid, and then
matching its solution onto that for a thin
laminar boundary layer.
For fluid flow over a porous boundary, the fluid velocity can be discontinuous between the free fluid and the fluid in the porous media (this is related to the Beavers and Joseph condition). Further, it is useful at low
subsonic speeds to assume that gas is
incompressible—that is, the density of the gas does not change even though the speed and
static pressure change.
Newtonian versus non-Newtonian fluids
A Newtonian fluid (named after
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
) is defined to be a
fluid whose
shear stress
Shear stress, often denoted by ( Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. '' Normal stress'', on ...
is linearly proportional to the
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
in the direction
perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
to the plane of shear. This definition means regardless of the forces acting on a fluid, it ''continues to flow''. For example, water is a Newtonian fluid, because it continues to display fluid properties no matter how much it is stirred or mixed. A slightly less rigorous definition is that the
drag of a small object being moved slowly through the fluid is proportional to the force applied to the object. (Compare
friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative lateral motion of ...
). Important fluids, like water as well as most gasses, behave—to good approximation—as a Newtonian fluid under normal conditions on Earth.
By contrast, stirring a
non-Newtonian fluid can leave a "hole" behind. This will gradually fill up over time—this behavior is seen in materials such as pudding,
oobleck, or
sand
Sand is a granular material composed of finely divided mineral particles. Sand has various compositions but is defined by its grain size. Sand grains are smaller than gravel and coarser than silt. Sand can also refer to a textural class ...
(although sand isn't strictly a fluid). Alternatively, stirring a non-Newtonian fluid can cause the viscosity to decrease, so the fluid appears "thinner" (this is seen in non-drip
paints). There are many types of non-Newtonian fluids, as they are defined to be something that fails to obey a particular property—for example, most fluids with long molecular chains can react in a non-Newtonian manner.
Equations for a Newtonian fluid
The constant of proportionality between the viscous stress tensor and the velocity gradient is known as the
viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the int ...
. A simple equation to describe incompressible Newtonian fluid behavior is
:
where
:
is the shear stress exerted by the fluid ("
drag")
:
is the fluid viscosity—a constant of proportionality
:
is the velocity gradient perpendicular to the direction of shear.
For a Newtonian fluid, the viscosity, by definition, depends only on
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
, not on the forces acting upon it. If the fluid is
incompressible the equation governing the viscous stress (in
Cartesian coordinates) is
:
where
:
is the shear stress on the
face of a fluid element in the
direction
:
is the velocity in the
direction
:
is the
direction coordinate.
If the fluid is not incompressible the general form for the viscous stress in a Newtonian fluid is
:
where
is the second viscosity coefficient (or bulk viscosity). If a fluid does not obey this relation, it is termed a
non-Newtonian fluid, of which there are several types. Non-Newtonian fluids can be either plastic, Bingham plastic, pseudoplastic, dilatant, thixotropic, rheopectic, viscoelastic.
In some applications, another rough broad division among fluids is made: ideal and non-ideal fluids. An ideal fluid is non-viscous and offers no resistance whatsoever to a shearing force. An ideal fluid really does not exist, but in some calculations, the assumption is justifiable. One example of this is the flow far from solid surfaces. In many cases, the viscous effects are concentrated near the solid boundaries (such as in boundary layers) while in regions of the flow field far away from the boundaries the viscous effects can be neglected and the fluid there is treated as it were inviscid (ideal flow). When the viscosity is neglected, the term containing the viscous stress tensor
in the Navier–Stokes equation vanishes. The equation reduced in this form is called the
Euler equation.
See also
*
Transport phenomena
*
Aerodynamics
Aerodynamics, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dy ...
*
Applied mechanics
Applied mechanics is the branch of science concerned with the motion of any substance that can be experienced or perceived by humans without the help of instruments. In short, when mechanics concepts surpass being theoretical and are applied and e ...
*
Bernoulli's principle
*
Communicating vessels
*
Computational fluid dynamics
Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate ...
*
Compressor map A compressor map is a chart which shows the performance of a turbomachinery compressor. This type of compressor is used in gas turbine engines, for supercharging reciprocating engines and for industrial processes, where it is known as a dynamic com ...
*
Secondary flow
*
Different types of boundary conditions in fluid dynamics
Boundary conditions in fluid dynamics are the set of constraints to boundary value problems in computational fluid dynamics. These boundary conditions include inlet boundary conditions, outlet boundary conditions, wall boundary conditions, constant ...
References
Further reading
*
*
*
*
*
External links
Free Fluid Mechanics booksAnnual Review of Fluid MechanicsCFDWiki– the Computational Fluid Dynamics reference wiki.
Educational Particle Image Velocimetry – resources and demonstrations
{{DEFAULTSORT:Fluid Mechanics
Civil engineering