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 r ...
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 objects r ...
of
fluid
In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
s (
liquid
A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter (the others being solid, gas, a ...
s,
gas
Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma).
A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...
es, and
plasma
Plasma or plasm may refer to:
Science
* Plasma (physics), one of the four fundamental states of matter
* Plasma (mineral), a green translucent silica mineral
* Quark–gluon plasma, a state of matter in quantum chromodynamics
Biology
* Blood pla ...
s) and the
force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
s 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 of ...
,
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 astrona ...
,
civil
Civil may refer to:
*Civic virtue, or civility
*Civil action, or lawsuit
* Civil affairs
*Civil and political rights
*Civil disobedience
*Civil engineering
*Civil (journalism), a platform for independent journalism
*Civilian, someone not a membe ...
,
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'' som ...
,
oceanography
Oceanography (), also known as oceanology and ocean science, is the scientific study of the oceans. It is an Earth science, which covers a wide range of topics, including ecosystem dynamics; ocean currents, waves, and geophysical fluid dynamic ...
,
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 not ...
,
astrophysics
Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline said, Astrophysics "seeks to ascertain the nature of the h ...
, 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 i ...
.
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 imme ...
, the study of fluids at rest; and
fluid dynamics
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'' (the study of air and other gases in motion) an ...
, the study of the effect of forces on fluid motion.
It is a branch of
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
, 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
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
, 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 th ...
(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 cult ...
, when
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
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 p ...
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, Drawing, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially res ...
(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 work o ...
(invented the
barometer),
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 ...
(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 inte ...
) and
Blaise Pascal
Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic Church, Catholic writer.
He was a child prodigy who was educated by his father, a tax collector in Rouen. Pa ...
(researched
hydrostatics, formulated
Pascal's law
Pascal's law (also Pascal's principle or the principle of transmission of fluid-pressure) is a principle in fluid mechanics given by Blaise Pascal that states that a pressure change at any point in a confined incompressible fluid is transmitted ...
), and was continued by
Daniel Bernoulli
Daniel Bernoulli FRS (; – 27 March 1782) was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for his applications of mathematics to mechan ...
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
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
,
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 limit ...
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 Po ...
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
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
.
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 imme ...
or hydrostatics is the branch of fluid mechanics that studies
fluid
In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
s 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
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'' (the study of air and other gases in motion) an ...
, the study of fluids in motion. Hydrostatics offers physical explanations for many phenomena of everyday life, such as why
atmospheric pressure changes with
altitude
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
, 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 method, 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 rang ...
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'' som ...
and
astrophysics
Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline said, Astrophysics "seeks to ascertain the nature of the h ...
(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 large ...
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 not ...
, 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 pract ...
(in the context of
blood pressure
Blood pressure (BP) is the pressure of circulating blood against the walls of blood vessels. Most of this pressure results from the heart pumping blood through the circulatory system. When used without qualification, the term "blood pressure" r ...
), and many other fields.
Fluid dynamics
''
Fluid dynamics
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'' (the study of air and other gases in motion) an ...
'' 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
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'' (the study of air and other gases in motion) an ...
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 is a ...
,
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 and e ...
,
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 ...
, 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 dyn ...
'' (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
force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
s and
movement
Movement may refer to:
Common uses
* Movement (clockwork), the internal mechanism of a timepiece
* Motion, commonly referred to as movement
Arts, entertainment, and media
Literature
* "Movement" (short story), a short story by Nancy Fu ...
s on
aircraft
An aircraft is a vehicle that is able to fly by gaining support from the air. It counters the force of gravity by using either static lift or by using the dynamic lift of an airfoil, or in a few cases the downward thrust from jet engines ...
, 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
Weather is the state of the atmosphere, describing for example the degree to which it is hot or cold, wet or dry, calm or stormy, clear or cloudy. On Earth, most weather phenomena occur in the lowest layer of the planet's atmosphere, the ...
patterns, understanding
nebula
A nebula ('cloud' or 'fog' in Latin; pl. nebulae, nebulæ or nebulas) is a distinct luminescent part of interstellar medium, which can consist of ionized, neutral or molecular hydrogen and also cosmic dust. Nebulae are often star-forming regio ...
e in
interstellar space and modeling
explosions
An explosion is a rapid expansion in volume associated with an extreme outward release of energy, usually with the generation of high temperatures and release of high-pressure gases. Supersonic explosions created by high explosives are known ...
. Some fluid-dynamical principles are used in
traffic engineering and crowd dynamics.
Relationship to continuum mechanics
Fluid mechanics is a subdiscipline of
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
, 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 the ot ...
; 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
In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means th ...
*
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
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
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 ...
, even though, on a microscopic scale, they are composed of
molecules
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioche ...
. 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
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
. 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 a ...
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
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
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
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
(
force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
) 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 and e ...
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. Bo ...
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 mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
. 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
In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...
is small. For more complex cases, especially those involving
turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
, 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 th ...
.
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 inte ...
,
. 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
Laminar means "flat". Laminar may refer to:
Terms in science and engineering:
* Laminar electronics or organic electronics, a branch of material sciences dealing with electrically conductive polymers and small molecules
* Laminar armour or "band ...
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
Subsonic may refer to:
Motion through a medium
* Any speed lower than the speed of sound within a sound-propagating medium
* Subsonic aircraft, a flying machine that flies at air speeds lower than the speed of sound
* Subsonic ammunition, a type o ...
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, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the grea ...
) is defined to be a
fluid
In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
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 the ot ...
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 is a ...
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 gradi ...
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 can ...
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 t ...
). 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 of s ...
(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
paint
Paint is any pigmented liquid, liquefiable, or solid mastic composition that, after application to a substrate in a thin layer, converts to a solid film. It is most commonly used to protect, color, or provide texture. Paint can be made in many ...
s). 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 inte ...
. 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
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
) 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
200px, Leonhard Euler (1707–1783)
In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include ...
.
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 dyn ...
*
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
Communicating vessels or vasesMario Bunge, ''Philosophy of Science: From Explanation to Justification'', 1998, , p. 369 are a set of containers containing a homogeneous fluid and connected sufficiently far below the top of the liquid: when the l ...
*
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 th ...
*
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 comp ...
*
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, constan ...
References
Further reading
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*
*
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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