In

Encyclopedia: Fluid dynamics

National Committee for Fluid Mechanics Films (NCFMF)

containing films on several subjects in fluid dynamics (in RealMedia format)

Gallery of fluid motion

"a visual record of the aesthetic and science of contemporary fluid mechanics," from the

List of Fluid Dynamics books

{{Authority control Piping Aerodynamics Continuum mechanics

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 ...

and 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 ...

, fluid dynamics is a subdiscipline of fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...

that describes the flow of fluids— liquids and gases. It has several subdisciplines, 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 moment
Moment or Moments may refer to:
* Present time
Music
* The Moments, American R&B vocal group Albums
* ''Moment'' (Dark Tranquillity album), 2020
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* ''Moments'' (Christine Guldbrand ...

s 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 weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation.
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 the calculation of various properties of the fluid, such as flow velocity, 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.
Before the twentieth century, ''hydrodynamics'' was synonymous with fluid dynamics. This is still reflected in names of some fluid dynamics topics, like magnetohydrodynamics and hydrodynamic stability, both of which can also be applied to gases.
Equations

The foundational axioms of fluid dynamics are the conservation laws, specifically, conservation of mass,conservation of linear 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 ...

, and conservation of energy (also known as the First Law of Thermodynamics
The first law of thermodynamics is a formulation of the law of conservation of energy, adapted for thermodynamic processes. It distinguishes in principle two forms of energy transfer, heat and thermodynamic work for a system of a constant am ...

). These are based on classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...

and are modified in quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

and general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...

. They are expressed using the Reynolds transport theorem.
In addition to the above, fluids are assumed to obey the continuum assumption
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical an ...

. Fluids are composed of molecules that collide with one another and solid objects. However, the continuum assumption assumes that fluids are continuous, rather than discrete. Consequently, it is assumed that properties such as density, pressure, temperature, and flow velocity are well-defined at infinitesimally small points in space and vary continuously from one point to another. The fact that the fluid is made up of discrete molecules is ignored.
For fluids that are sufficiently dense to be a continuum, do not contain ionized species, and have flow velocities that are small in relation to the speed of light, the momentum equations for Newtonian fluids are the Navier–Stokes equations—which is a non-linear set of differential equations that describes the flow of a fluid whose stress depends linearly on flow velocity gradients and pressure. The unsimplified equations do not have a general closed-form solution, so they are primarily of use in 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 ...

. The equations can be simplified in several ways, all of which make them easier to solve. Some of the simplifications allow some simple fluid dynamics problems to be solved in closed form.
In addition to the mass, momentum, and energy conservation equations, a thermodynamic
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of ...

equation of state that gives the pressure as a function of other thermodynamic variables is required to completely describe the problem. An example of this would be the perfect gas equation of state:
:$p=\; \backslash frac$
where is 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 ...

, is 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 is the absolute temperature
Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.
Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic ...

, while is the gas constant and is molar mass for a particular gas. A constitutive relation may also be useful.
Conservation laws

Three conservation laws are used to solve fluid dynamics problems, and may be written inintegral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...

or differential form. The conservation laws may be applied to a region of the flow called a ''control volume''. A control volume is a discrete volume in space through which fluid is assumed to flow. The integral formulations of the conservation laws are used to describe the change of mass, momentum, or energy within the control volume. Differential formulations of the conservation laws apply Stokes' theorem to yield an expression that may be interpreted as the integral form of the law applied to an infinitesimally small volume (at a point) within the flow.
Classifications

Compressible versus incompressible flow

All fluids are compressible to an extent; that is, changes in pressure or temperature cause changes in density. However, in many situations the changes in pressure and temperature are sufficiently small that the changes in density are negligible. In this case the flow can be modelled as anincompressible flow
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An ...

. Otherwise the more general compressible flow equations must be used.
Mathematically, incompressibility is expressed by saying that the density of a fluid parcel In fluid dynamics, within the framework of continuum mechanics, a fluid parcel is a very small amount of fluid, identifiable throughout its dynamic history while moving with the fluid flow. As it moves, the mass of a fluid parcel remains constant, ...

does not change as it moves in the flow field, that is,
: $\backslash frac\; =\; 0\; \backslash ,\; ,$
where is the material derivative, which is the sum of local and convective derivatives. This additional constraint simplifies the governing equations, especially in the case when the fluid has a uniform density.
For flow of gases, to determine whether to use compressible or incompressible fluid dynamics, the Mach number of the flow is evaluated. As a rough guide, compressible effects can be ignored at Mach numbers below approximately 0.3. For liquids, whether the incompressible assumption is valid depends on the fluid properties (specifically the critical pressure and temperature of the fluid) and the flow conditions (how close to the critical pressure the actual flow pressure becomes). Acoustic problems always require allowing compressibility, since sound waves are compression waves involving changes in pressure and density of the medium through which they propagate.
Newtonian versus non-Newtonian fluids

All fluids, except superfluids, are viscous, meaning that they exert some resistance to deformation: neighbouring parcels of fluid moving at different velocities exert viscous forces on each other. The velocity gradient is referred to as a strain rate; it has dimensions .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 ...

showed that for many familiar fluids such as water
Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as ...

and air, the stress due to these viscous forces is linearly related to the strain rate. Such fluids are called Newtonian fluids. The coefficient of proportionality is called the fluid's viscosity; for Newtonian fluids, it is a fluid property that is independent of the strain rate.
Non-Newtonian fluids have a more complicated, non-linear stress-strain behaviour. The sub-discipline of rheology
Rheology (; ) is the study of the flow of matter, primarily in a fluid ( liquid or gas) state, but also as "soft solids" or solids under conditions in which they respond with plastic flow rather than deforming elastically in response to an ap ...

describes the stress-strain behaviours of such fluids, which include emulsion
An emulsion is a mixture of two or more liquids that are normally immiscible (unmixable or unblendable) owing to liquid-liquid phase separation. Emulsions are part of a more general class of two-phase systems of matter called colloids. Alth ...

s and slurries, some viscoelastic materials such as blood
Blood is a body fluid in the circulatory system of humans and other vertebrates that delivers necessary substances such as nutrients and oxygen to the cells, and transports metabolic waste products away from those same cells. Blood in the cir ...

and some polymer
A polymer (; Greek '' poly-'', "many" + '' -mer'', "part")
is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic a ...

s, and ''sticky liquids'' such as latex
Latex is an emulsion (stable dispersion) of polymer microparticles in water. Latexes are found in nature, but synthetic latexes are common as well.
In nature, latex is found as a milky fluid found in 10% of all flowering plants (angiosperms ...

, honey
Honey is a sweet and viscous substance made by several bees, the best-known of which are honey bees. Honey is made and stored to nourish bee colonies. Bees produce honey by gathering and then refining the sugary secretions of plants (primar ...

and lubricants.
Inviscid versus viscous versus Stokes flow

The dynamic of fluid parcels is described with the help ofNewton's second law
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in mo ...

. An accelerating parcel of fluid is subject to inertial effects.
The Reynolds number is a dimensionless quantity which characterises the magnitude of inertial effects compared to the magnitude of viscous effects. A low Reynolds number () indicates that viscous forces are very strong compared to inertial forces. In such cases, inertial forces are sometimes neglected; this flow regime is called Stokes or creeping flow.
In contrast, high Reynolds numbers () indicate that the inertial effects have more effect on the velocity field than the viscous (friction) effects. In high Reynolds number flows, the flow is often modeled as an inviscid flow
In fluid dynamics, inviscid flow is the flow of an inviscid (zero-viscosity) fluid, also known as a superfluid. The Reynolds number of inviscid flow approaches infinity as the viscosity approaches zero. When viscous forces are neglected, suc ...

, an approximation in which viscosity is completely neglected. Eliminating viscosity allows the Navier–Stokes equations to be simplified into the Euler equations. The integration of the Euler equations along a streamline in an inviscid flow yields Bernoulli's equation. When, in addition to being inviscid, the flow is irrotational everywhere, Bernoulli's equation can completely describe the flow everywhere. Such flows are called potential flows, because the velocity field may be expressed as the 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 ...

of a potential energy expression.
This idea can work fairly well when the Reynolds number is high. However, problems such as those involving solid boundaries may require that the viscosity be included. Viscosity cannot be neglected near solid boundaries because the no-slip condition generates a thin region of large strain rate, the boundary layer, in which 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 ...

effects dominate and which thus generates vorticity. Therefore, to calculate net forces on bodies (such as wings), viscous flow equations must be used: inviscid flow theory fails to predict drag forces, a limitation known as the d'Alembert's paradox.
A commonly used model, especially in 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 ...

, is to use two flow models: the Euler equations away from the body, and boundary layer equations in a region close to the body. The two solutions can then be matched with each other, using the method of matched asymptotic expansions.
Steady versus unsteady flow

A flow that is not a function of time is called steady flow. Steady-state flow refers to the condition where the fluid properties at a point in the system do not change over time. Time dependent flow is known as unsteady (also called transient). Whether a particular flow is steady or unsteady, can depend on the chosen frame of reference. For instance, laminar flow over asphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...

is steady in the frame of reference that is stationary with respect to the sphere. In a frame of reference that is stationary with respect to a background flow, the flow is unsteady.
Turbulent flows are unsteady by definition. A turbulent flow can, however, be statistically stationary. The random velocity field is statistically stationary if all statistics are invariant under a shift in time. This roughly means that all statistical properties are constant in time. Often, the mean field is the object of interest, and this is constant too in a statistically stationary flow.
Steady flows are often more tractable than otherwise similar unsteady flows. The governing equations of a steady problem have one dimension fewer (time) than the governing equations of the same problem without taking advantage of the steadiness of the flow field.
Laminar versus turbulent flow

Turbulence is flow characterized by recirculation,eddies
In fluid dynamics, an eddy is the swirling of a fluid and the reverse current created when the fluid is in a turbulent flow regime. The moving fluid creates a space devoid of downstream-flowing fluid on the downstream side of the object. Fluid ...

, and apparent random
In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual ran ...

ness. Flow in which turbulence is not exhibited is called laminar. The presence of eddies or recirculation alone does not necessarily indicate turbulent flow—these phenomena may be present in laminar flow as well. Mathematically, turbulent flow is often represented via a Reynolds decomposition
In fluid dynamics and turbulence theory, Reynolds decomposition is a mathematical technique used to separate the expectation value of a quantity from its fluctuations.
Decomposition
For example, for a quantity u the decomposition would be
u(x,y,z ...

, in which the flow is broken down into the sum of an average
In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7 ...

component and a perturbation component.
It is believed that turbulent flows can be described well through the use of the Navier–Stokes equations. Direct numerical simulation (DNS), based on the Navier–Stokes equations, makes it possible to simulate turbulent flows at moderate Reynolds numbers. Restrictions depend on the power of the computer used and the efficiency of the solution algorithm. The results of DNS have been found to agree well with experimental data for some flows.
Most flows of interest have Reynolds numbers much too high for DNS to be a viable option, given the state of computational power for the next few decades. Any flight vehicle large enough to carry a human ( > 3 m), moving faster than is well beyond the limit of DNS simulation ( = 4 million). Transport aircraft wings (such as on an Airbus A300 or Boeing 747) have Reynolds numbers of 40 million (based on the wing chord dimension). Solving these real-life flow problems requires turbulence models for the foreseeable future. Reynolds-averaged Navier–Stokes equations (RANS) combined with turbulence modelling provides a model of the effects of the turbulent flow. Such a modelling mainly provides the additional momentum transfer by the Reynolds stresses
In fluid dynamics, the Reynolds stress is the component of the total stress tensor in a fluid obtained from the averaging operation over the Navier–Stokes equations to account for turbulent fluctuations in fluid momentum.
Definition
The veloci ...

, although the turbulence also enhances the heat
In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is ...

and mass transfer. Another promising methodology is large eddy simulation (LES), especially in the guise of detached eddy simulation Detached eddy simulation (DES) is a modification of a Reynolds-averaged Navier–Stokes equations (RANS) model in which the model switches to a subgrid scale formulation in regions fine enough for large eddy simulation (LES) calculations.
Details
...

(DES)—which is a combination of RANS turbulence modelling and large eddy simulation.
Other approximations

There are a large number of other possible approximations to fluid dynamic problems. Some of the more commonly used are listed below. * The '' Boussinesq approximation'' neglects variations in density except to calculatebuoyancy
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 ...

forces. It is often used in free convection
Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the c ...

problems where density changes are small.
* ''Lubrication theory
In fluid dynamics, lubrication theory describes the flow of fluids (liquids or gases) in a geometry in which one dimension is significantly smaller than the others. An example is the flow above air hockey tables, where the thickness of the air l ...

'' and '' Hele–Shaw flow'' exploits the large aspect ratio of the domain to show that certain terms in the equations are small and so can be neglected.
* '' Slender-body theory'' is a methodology used in Stokes flow problems to estimate the force on, or flow field around, a long slender object in a viscous fluid.
* The ''shallow-water equations
The shallow-water equations (SWE) are a set of hyperbolic partial differential equations (or parabolic if viscous shear is considered) that describe the flow below a pressure surface in a fluid (sometimes, but not necessarily, a free surface). ...

'' can be used to describe a layer of relatively inviscid fluid with a free surface, in which surface gradients are small.
* '' Darcy's law'' is used for flow in porous media, and works with variables averaged over several pore-widths.
* In rotating systems, the '' quasi-geostrophic equations'' assume an almost perfect balance between pressure gradients and the Coriolis force. It is useful in the study of atmospheric dynamics
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 ...

.
Multidisciplinary types

Flows according to Mach regimes

While many flows (such as flow of water through a pipe) occur at low Mach numbers ( subsonic flows), many flows of practical interest in aerodynamics or inturbomachines
Turbomachinery, in mechanical engineering, describes machines that transfer energy between a rotor and a fluid, including both turbines and compressors. While a turbine transfers energy from a fluid to a rotor, a compressor transfers energy ...

occur at high fractions of ( transonic flows) or in excess of it ( supersonic or even hypersonic flows). New phenomena occur at these regimes such as instabilities in transonic flow, shock waves for supersonic flow, or non-equilibrium chemical behaviour due to ionization in hypersonic flows. In practice, each of those flow regimes is treated separately.
Reactive versus non-reactive flows

Reactive flows are flows that are chemically reactive, which finds its applications in many areas, includingcombustion
Combustion, or burning, is a high-temperature exothermic redox chemical reaction between a fuel (the reductant) and an oxidant, usually atmospheric oxygen, that produces oxidized, often gaseous products, in a mixture termed as smoke. Combus ...

( IC engine), propulsion devices ( rockets, jet engines, and so on), detonations
Detonation () is a type of combustion involving a supersonic exothermic front accelerating through a medium that eventually drives a shock front propagating directly in front of it. Detonations propagate supersonically through shock waves with ...

, fire and safety hazards, and astrophysics. In addition to conservation of mass, momentum and energy, conservation of individual species (for example, mass fraction of methane
Methane ( , ) is a chemical compound with the chemical formula (one carbon atom bonded to four hydrogen atoms). It is a group-14 hydride, the simplest alkane, and the main constituent of natural gas. The relative abundance of methane ...

in methane combustion) need to be derived, where the production/depletion rate of any species are obtained by simultaneously solving the equations of chemical kinetics
Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is to be contrasted with chemical thermodynamics, which deals with the direction in ...

.
Magnetohydrodynamics

Magnetohydrodynamics is the multidisciplinary study of the flow ofelectrically conducting
Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...

fluids in electromagnetic fields. Examples of such fluids include plasmas, liquid metals, and salt water
Saline water (more commonly known as salt water) is water that contains a high concentration of dissolved salts (mainly sodium chloride). On the United States Geological Survey (USGS) salinity scale, saline water is saltier than brackish w ...

. The fluid flow equations are solved simultaneously with Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits ...

of electromagnetism.
Relativistic fluid dynamics

Relativistic fluid dynamics studies the macroscopic and microscopic fluid motion at large velocities comparable to the velocity of light. This branch of fluid dynamics accounts for the relativistic effects both from thespecial theory of relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The law ...

and the general theory of relativity. The governing equations are derived in Riemannian geometry for Minkowski spacetime
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the ...

.
Fluctuating hydrodynamics

This branch of fluid dynamics augments the standard hydrodynamic equations with stochastic fluxes that model thermal fluctuations. As formulated byLandau
Landau ( pfl, Landach), officially Landau in der Pfalz, is an autonomous (''kreisfrei'') town surrounded by the Südliche Weinstraße ("Southern Wine Route") district of southern Rhineland-Palatinate, Germany. It is a university town (since 1990 ...

and Lifshitz,
a white noise contribution obtained from the fluctuation-dissipation theorem of statistical mechanics
is added to the viscous stress tensor and heat flux.
Terminology

The concept of pressure is central to the study of both fluid statics and fluid dynamics. A pressure can be identified for every point in a body of fluid, regardless of whether the fluid is in motion or not. Pressure can be measured using an aneroid, Bourdon tube, mercury column, or various other methods. Some of the terminology that is necessary in the study of fluid dynamics is not found in other similar areas of study. In particular, some of the terminology used in fluid dynamics is not used influid 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 ...

.
Terminology in incompressible fluid dynamics

The concepts of total pressure and dynamic pressure arise from Bernoulli's equation and are significant in the study of all fluid flows. (These two pressures are not pressures in the usual sense—they cannot be measured using an aneroid, Bourdon tube or mercury column.) To avoid potential ambiguity when referring to pressure in fluid dynamics, many authors use the term static pressure to distinguish it from total pressure and dynamic pressure. Static pressure is identical to pressure and can be identified for every point in a fluid flow field. A point in a fluid flow where the flow has come to rest (that is to say, speed is equal to zero adjacent to some solid body immersed in the fluid flow) is of special significance. It is of such importance that it is given a special name—a stagnation point. The static pressure at the stagnation point is of special significance and is given its own name— stagnation pressure. In incompressible flows, the stagnation pressure at a stagnation point is equal to the total pressure throughout the flow field.Terminology in compressible fluid dynamics

In a compressible fluid, it is convenient to define the total conditions (also called stagnation conditions) for all thermodynamic state properties (such as total temperature, total enthalpy, total speed of sound). These total flow conditions are a function of the fluid velocity and have different values in frames of reference with different motion. To avoid potential ambiguity when referring to the properties of the fluid associated with the state of the fluid rather than its motion, the prefix "static" is commonly used (such as static temperature and static enthalpy). Where there is no prefix, the fluid property is the static condition (so "density" and "static density" mean the same thing). The static conditions are independent of the frame of reference. Because the total flow conditions are defined by isentropically bringing the fluid to rest, there is no need to distinguish between total entropy and static entropy as they are always equal by definition. As such, entropy is most commonly referred to as simply "entropy".About

Fields of study

*Acoustic theory Acoustic theory is a scientific field that relates to the description of sound waves. It derives from fluid dynamics. See acoustics for the engineering approach.
For sound waves of any magnitude of a disturbance in velocity, pressure, and density w ...

*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 ...

* Aeroelasticity
* Aeronautics
*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 ...

*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 ...

* Geophysical fluid dynamics
* Hemodynamics
* Hydraulics
*Hydrology
Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and environmental watershed sustainability. A practitioner of hydrology is call ...

* Hydrostatics
* Electrohydrodynamics
* Magnetohydrodynamics
*Quantum hydrodynamics
In condensed matter physics, quantum hydrodynamics is most generally the study of hydrodynamic-like systems which demonstrate quantum mechanical behavior. They arise in semiclassical mechanics in the study of metal and semiconductor devices, in wh ...

Mathematical equations and concepts

* Airy wave theory *Benjamin–Bona–Mahony equation
The Benjamin–Bona–Mahony equation (BBM equation, also regularized long-wave equation; RLWE) is the partial differential equation
:u_t+u_x+uu_x-u_=0.\,
This equation was studied in as an improvement of the Korteweg–de Vries equation (KdV e ...

*Boussinesq approximation (water waves)
In fluid dynamics, the Boussinesq approximation for water waves is an approximation valid for weakly non-linear and fairly long waves. The approximation is named after Joseph Boussinesq, who first derived them in response to the observation by ...

*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 ...

* Elementary flow
* Helmholtz's theorems
*Kirchhoff equations
In fluid dynamics, the Kirchhoff equations, named after Gustav Kirchhoff, describe the motion of a rigid body in an ideal fluid.
:
\begin
& = \times \vec \omega + \times \vec v + \vec Q_h + \vec Q, \\0pt
& = \times \vec \omega + \vec F_h ...

*Knudsen equation
In fluid dynamics, the Knudsen equation is used to describe how gas flows through a tube in free molecular flow. When the mean free path of the molecule
A molecule is a group of two or more atoms held together by attractive forces known ...

* Manning equation
*Mild-slope equation
In fluid dynamics, the mild-slope equation describes the combined effects of diffraction and refraction for water waves propagating over bathymetry and due to lateral boundaries—like breakwaters and coastlines. It is an approximate model, deriv ...

* Morison equation
* Navier–Stokes equations
*Oseen flow
In fluid dynamics, the Oseen equations (or Oseen flow) describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compare ...

* Poiseuille's law
* Pressure head
* Relativistic Euler equations
* Stokes stream function
* Stream function
* Streamlines, streaklines and pathlines
* Torricelli's Law
Types of fluid flow

* Aerodynamic force *Convection
Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the c ...

* Cavitation
* Compressible flow
*Couette flow
In fluid dynamics, Couette flow is the flow of a viscous fluid in the space between two surfaces, one of which is moving tangentially relative to the other. The relative motion of the surfaces imposes a shear stress on the fluid and induces flow. ...

* Effusive limit
* Free molecular flow
*Incompressible flow
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An ...

*Inviscid flow
In fluid dynamics, inviscid flow is the flow of an inviscid (zero-viscosity) fluid, also known as a superfluid. The Reynolds number of inviscid flow approaches infinity as the viscosity approaches zero. When viscous forces are neglected, suc ...

*Isothermal flow
Isothermal flow is a model of compressible fluid flow whereby the flow remains at the same temperature while flowing in a conduit.Shapiro, A.H., ''The Dynamics and Thermodynamics of Compressible Fluid Flow, Volume 1'', Ronald Press, 1953. In th ...

* Open channel flow
* Pipe flow
* Pressure-driven flow
* Secondary flow
* Stream thrust averaging
* Superfluidity
* Transient flow
*Two-phase flow
In fluid mechanics, two-phase flow is a flow of gas and liquid — a particular example of multiphase flow. Two-phase flow can occur in various forms, such as flows transitioning from pure liquid to vapor as a result of external heating, sep ...

Fluid properties

*List of hydrodynamic instabilities
This is a list of hydrodynamic and plasma instabilities named after people (eponymous instabilities).
{, class="wikitable"
! Instability !! Field !! Named for
, -
, Benjamin–Feir instability , , Surface gravity waves , , T. Brooke Benjamin an ...

* Newtonian fluid
* Non-Newtonian fluid
*Surface tension
Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) t ...

* Vapour pressure
Fluid phenomena

*Balanced flow
In atmospheric science, balanced flow is an idealisation of atmospheric motion. The idealisation consists in considering the behaviour of one isolated parcel of air having constant density, its motion on a horizontal plane subject to selected for ...

* Boundary layer
* Coanda effect
*Convection cell
In the field of fluid dynamics, a convection cell is the phenomenon that occurs when density differences exist within a body of liquid or gas. These density differences result in rising and/or falling currents, which are the key characteristics ...

* Convergence/Bifurcation
* Darwin drift
* Drag (force)
*Droplet vaporization
The vaporizing droplet (droplet vaporization) problem is a challenging issue in fluid dynamics. It is part of many engineering situations involving the transport and computation of sprays: fuel injection, spray painting, aerosol spray, flashing re ...

* Hydrodynamic stability
*Kaye effect
The Kaye effect is a property of complex liquids which was first described by the British engineer Alan Kaye in 1963.
While pouring one viscous mixture of an organic liquid onto a surface, the surface suddenly spouted an upcoming jet of liquid wh ...

* Lift (force)
* Magnus effect
*Ocean current
An ocean current is a continuous, directed movement of sea water generated by a number of forces acting upon the water, including wind, the Coriolis effect, breaking waves, cabbeling, and temperature and salinity differences. Depth conto ...

* Ocean surface waves
* Rossby wave
*Shock wave
In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a me ...

*Soliton
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the me ...

* Stokes drift
*Teapot effect
The teapot effect, also known as dribbling, is a fluid dynamics phenomenon that occurs when a liquid being poured from a container runs down the spout or the body of the vessel instead of flowing out in an arc.
Markus Reiner coined the term "te ...

* Thread breakup
*Turbulent jet breakup
Turbulent jet breakup is the phenomena of the disintegration of a liquid/gas jet due to turbulent forces acting either on the surface of the jet or present within the jet itself. Turbulent jet breakup is mainly caused by an interplay of aerodynami ...

* Upstream contamination
* Venturi effect
*Vortex
In fluid dynamics, a vortex ( : vortices or vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in ...

* Water hammer
*Wave drag
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...

*Wind
Wind is the natural movement of air or other gases relative to a planet's surface. Winds occur on a range of scales, from thunderstorm flows lasting tens of minutes, to local breezes generated by heating of land surfaces and lasting a few ...

Applications

* Acoustics *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 ...

* Cryosphere science
*EFDC Explorer
EFDC_Explorer (EE) is a Windows-based GUI for pre- and post processing of the Environmental Fluid Dynamics Code (EFDC). The program is developed and supported by the engineering company DSI. EFDC_Explorer is designed to support model set-up, grid ...

* Fluidics
* Fluid power
* Geodynamics
*Hydraulic machinery
Hydraulic machines use liquid fluid power to perform work. Heavy construction vehicles are a common example. In this type of machine, hydraulic fluid is pumped to various hydraulic motors and hydraulic cylinders throughout the machine ...

*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 ...

*Naval architecture
Naval architecture, or naval engineering, is an engineering discipline incorporating elements of mechanical, electrical, electronic, software and safety engineering as applied to the engineering design process, shipbuilding, maintenance, and ...

* Oceanography
* Plasma physics
* Pneumatics
*3D computer graphics
3D computer graphics, or “3D graphics,” sometimes called CGI, 3D-CGI or three-dimensional computer graphics are graphics that use a three-dimensional representation of geometric data (often Cartesian) that is stored in the computer for th ...

Fluid dynamics journals

* '' Annual Review of Fluid Mechanics'' * ''Journal of Fluid Mechanics
The ''Journal of Fluid Mechanics'' is a peer-reviewed scientific journal in the field of fluid mechanics. It publishes original work on theoretical, computational, and experimental aspects of the subject.
The journal is published by Cambridge U ...

''
* '' Physics of Fluids''
* '' Physical Review Fluids''
* ''Experiments in Fluids
''Experiments in Fluids'' is a scientific, peer-reviewed scientific journal published monthly by Springer Science+Business Media. The journal presents contributions that employ existing experimental techniques to gain an understanding of the under ...

''
* ''European Journal of Mechanics B: Fluids''
* ''Theoretical and Computational Fluid Dynamics''
* ''Computers and Fluids''
* '' International Journal for Numerical Methods in Fluids''
* ''Flow, Turbulence and Combustion
''Flow, Turbulence and Combustion'' is a peer-reviewed scientific journal on fluid mechanics. It covers original research on fluid mechanics and combustion, with the areas of interest including industrial, geophysical, and environmental application ...

''
Miscellaneous

* Important publications in fluid dynamics * Isosurface * Keulegan–Carpenter number * Rotating tank * Sound barrier * Beta plane * Immersed boundary method *Bridge scour
Bridge scour is the removal of sediment such as sand and gravel from around bridge abutments or piers. Hydrodynamic scour, caused by fast flowing water, can carve out ''scour holes'', compromising the integrity of a structure.
In the United Sta ...

*Finite volume method for unsteady flow Unsteady flows are characterized as flows in which the properties of the fluid are time dependent. It gets reflected in the governing equations as the time derivative of the properties are absent.
For Studying Finite-volume method for unsteady flow ...

* Flow visualization
See also

* * * * * * * * * * * * * * * * * * * * * * (hydrodynamic) * * * * * * * * * * * * * (aerodynamics) * * * * * * * * *References

Further reading

* * * * * Originally published in 1879, the 6th extended edition appeared first in 1932. * Originally published in 1938. * *Encyclopedia: Fluid dynamics

Scholarpedia
''Scholarpedia'' is an English-language wiki-based online encyclopedia with features commonly associated with open-access online academic journals, which aims to have quality content in science and medicine.
''Scholarpedia'' articles are writ ...

External links

National Committee for Fluid Mechanics Films (NCFMF)

containing films on several subjects in fluid dynamics (in RealMedia format)

Gallery of fluid motion

"a visual record of the aesthetic and science of contemporary fluid mechanics," from the

American Physical Society
The American Physical Society (APS) is a not-for-profit membership organization of professionals in physics and related disciplines, comprising nearly fifty divisions, sections, and other units. Its mission is the advancement and diffusion of k ...

List of Fluid Dynamics books

{{Authority control Piping Aerodynamics Continuum mechanics