## Fluid dynamics |

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In **fluid dynamics** is a subdiscipline of **hydrodynamics** (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating

Fluid dynamics offers a systematic structure—which underlies these

Before the twentieth century, *hydrodynamics* was synonymous with fluid dynamics. This is still reflected in names of some fluid dynamics topics, like ^{[1]}

The foundational axioms of fluid dynamics are the

In addition to the above, fluids are assumed to obey the **continuum assumption**. 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

For fluids that are sufficiently dense to be a continuum, do not contain ionized species, and have flow velocities small in relation to the speed of light, the momentum equations for ^{[citation needed]}

In addition to the mass, momentum, and energy conservation equations, a

where *p* is pressure, ρ is density, *T* the absolute temperature, while *R _{u}* is the

Three conservation laws are used to solve fluid dynamics problems, and may be written in *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

Mass continuity (conservation of mass): The rate of change of fluid mass inside a control volume must be equal to the net rate of fluid flow into the volume. Physically, this statement requires that mass is neither created nor destroyed in the control volume,^{[2]}and can be translated into the integral form of the continuity equation:

- Above, is the fluid density,
**u**is theflow velocity vector, and*t*is time. The left-hand side of the above expression is the rate of increase of mass within the volume and contains a triple integral over the control volume, whereas the right-hand side contains an integration over the surface of the control volume of mass convected into the system. Mass flow into the system is accounted as positive, and since the normal vector to the surface is opposite the sense of flow into the system the term is negated. The differential form of the continuity equation is, by thedivergence theorem :

Conservation of momentum :Newton's second law of motion applied to a control volume, is a statement that any change in momentum of the fluid within that control volume will be due to the net flow of momentum into the volume and the action of external forces acting on the fluid within the volume.

- In the above integral formulation of this equation, the term on the left is the net change of momentum within the volume. The first term on the right is the net rate at which momentum is convected into the volume. The second term on the right is the force due to pressure on the volume's surfaces. The first two terms on the right are negated since momentum entering the system is accounted as positive, and the normal is opposite the direction of the velocity and pressure forces. The third term on the right is the net acceleration of the mass within the volume due to any
body forces (here represented by*f*_{body}).Surface forces , such as viscous forces, are represented by**, the net force due to**shear forces acting on the volume surface. The momentum balance can also be written for a*moving*control volume.^{[3]}

- The following is the differential form of the momentum conservation equation. Here, the volume is reduced to an infinitesimally small point, and both surface and body forces are accounted for in one total force,
*F*. For example,*F*may be expanded into an expression for the frictional and gravitational forces acting at a point in a flow.

- In aerodynamics, air is assumed to be a
Newtonian fluid , which posits a linear relationship between the shear stress (due to internal friction forces) and the rate of strain of the fluid. The equation above is a vector equation in a three-dimensional flow, but it can be expressed as three scalar equations in three coordinate directions. The conservation of momentum equations for the compressible, viscous flow case are called the Navier–Stokes equations.^{[2]}

Conservation of energy : Althoughenergy can be converted from one form to another, the totalenergy in a closed system remains constant.

- Above,
*h*isenthalpy ,*k*is thethermal conductivity of the fluid,*T*is temperature, and is the viscous dissipation function. The viscous dissipation function governs the rate at which mechanical energy of the flow is converted to heat. Thesecond law of thermodynamics requires that the dissipation term is always positive: viscosity cannot create energy within the control volume.^{[4]}The expression on the left side is amaterial derivative .

All fluids are

Mathematically, incompressibility is expressed by saying that the density ρ of a

where *D*/*Dt* is the

For flow of gases, to determine whether to use compressible or incompressible fluid dynamics, the

All fluids 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

*sticky liquids* such as ^{[citation needed]}

The dynamic of fluid parcels is described with the help of

The *Re*<<1) indicates that viscous forces are very strong compared to inertial forces. In such cases, inertial forces are sometimes neglected; this flow regime is called

In contrast, high Reynolds numbers (*Re*>>1) 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

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

A commonly used^{[citation needed]} model, especially in

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^{[6]}). Whether a particular flow is steady or unsteady, can depend on the chosen frame of reference. For instance, laminar flow over a

^{[7]}

The random field

U(x,t) 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

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.

Turbulence is flow characterized by recirculation,

It is believed that turbulent flows can be described well through the use of the ^{[8]}

Most flows of interest have Reynolds numbers much too high for DNS to be a viable option,^{[9]} given the state of computational power for the next few decades. Any flight vehicle large enough to carry a human (L > 3 m), moving faster than 20 m/s (72 km/h) is well beyond the limit of DNS simulation (Re = 4 million). Transport aircraft wings (such as on an

While many flows (e.g. flow of water through a pipe) occur at low

Reactive flows are flows that are chemically reactive, which finds its applications in many areas such as

Relativistic fluid dynamics studies the macroscopic and microscopic fluid motion at large velocities comparable to the ^{[10]} This branch of fluid dynamics accounts the relativistic effects both from the

There are a large number of other possible approximations to fluid dynamic problems. Some of the more commonly used are listed below.

- The
neglects variations in density except to calculateBoussinesq approximation buoyancy forces. It is often used in freeconvection problems where density changes are small. andLubrication theory exploits the largeHele–Shaw flow aspect ratio of the domain to show that certain terms in the equations are small and so can be neglected.is a methodology used inSlender-body theory Stokes flow problems to estimate the force on, or flow field around, a long slender object in a viscous fluid.- The
can be used to describe a layer of relatively inviscid fluid with ashallow-water equations free surface , in which surfacegradients are small. is used for flow inDarcy's law porous media , and works with variables averaged over several pore-widths.- In rotating systems, the
assume an almostquasi-geostrophic equations perfect balance betweenpressure gradients and theCoriolis force . It is useful in the study ofatmospheric dynamics .

Other Languages

العربية: جريان الموائع

asturianu: Fluidodinámica

bosanski: Dinamika fluida

català: Dinàmica de fluids

čeština: Proudění

Deutsch: Fluiddynamik

español: Fluidodinámica

Esperanto: Fluidodinamiko

euskara: Fluidoen dinamika

فارسی: دینامیک شارهها

français: Dynamique des fluides

galego: Dinámica de fluídos

客家語/Hak-kâ-ngî: Liù-thí thûng-li̍t-ho̍k

한국어: 유체동역학

हिन्दी: तरल गतिकी

Ido: Fluido dinamiko

Bahasa Indonesia: Dinamika fluida

italiano: Fluidodinamica

lietuvių: Hidrodinamika

magyar: Hidrodinamika

македонски: Основни својства на флуидите

Bahasa Melayu: Dinamik bendalir

Nederlands: Vloeistofmechanica

日本語: 流体力学

norsk: Fluiddynamikk

norsk nynorsk: Væskedynamikk

Oromoo: Fluid dynamics

polski: Dynamika płynów

română: Dinamica fluidelor

русский: Гидроаэродинамика

Scots: Fluid deenamics

shqip: Fluidodinamika

සිංහල: තරල ගති විද්යාව

Simple English: Fluid dynamics

slovenščina: Dinamika fluidov

српски / srpski: Динамика флуида

srpskohrvatski / српскохрватски: Dinamika fluida

suomi: Virtausdynamiikka

svenska: Fluidmekanik

Tagalog: Dinamika ng likido

தமிழ்: பாய்ம இயக்கவியல்

ไทย: พลศาสตร์ของไหล

Türkçe: Akışkanlar dinamiği

українська: Гідроаеродинаміка

اردو: سيالی حرکيات

Tiếng Việt: Động lực học chất lưu

Winaray: Pluwidodinamika

中文: 流體動力學