Inphysics,physical chemistry andengineering,fluid dynamics is a subdiscipline offluid mechanics that describes the flow offluids –liquids andgases. It has several subdisciplines, includingaerodynamics (the study of air and other gases in motion) andhydrodynamics (the study of water and other liquids in motion). Fluid dynamics has a wide range of applications, including calculatingforces andmoments onaircraft, determining themass flow rate ofpetroleum throughpipelines,predicting weather patterns, understandingnebulae ininterstellar space, understanding large scalegeophysical flows involving oceans/atmosphere andmodelling fission weapon detonation.

Fluid dynamics offers a systematic structure—which underlies thesepractical disciplines—that embraces empirical and semi-empirical laws derived fromflow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves the calculation of various properties of the fluid, such asflow velocity,pressure,density, andtemperature, 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, likemagnetohydrodynamics andhydrodynamic stability, both of which can also be applied to gases.^[1]

The foundational axioms of fluid dynamics are theconservation laws, specifically,conservation of mass,conservation of linear momentum, andconservation of energy (also known as thefirst law of thermodynamics). These are based onclassical mechanics and are modified inquantum mechanics andgeneral relativity. They are expressed using theReynolds transport theorem.

In addition to the above, fluids are assumed to obey thecontinuum assumption. At small scale, all 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 atinfinitesimally 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 forNewtonian fluids are theNavier–Stokes equations—which is anon-linear set ofdifferential 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 generalclosed-form solution, so they are primarily of use incomputational fluid dynamics. 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.^{[citation needed]}

In addition to the mass, momentum, and energy conservation equations, athermodynamic 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 theperfect gas equation of state:

${\displaystyle p=\frac{\rho R_{u}T}{M}}$ 

wherep ispressure,ρ isdensity, andT is theabsolute temperature, whileR_u is thegas constant andM ismolar mass for a particular gas. Aconstitutive relation may also be useful.

Three conservation laws are used to solve fluid dynamics problems, and may be written inintegral ordifferential form. The conservation laws may be applied to a region of the flow called acontrol 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 applyStokes' 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.

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:

${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}{\iiint }_V\rho dV=-}$  ${\displaystyle S}$ ${\displaystyle \rho \mathbf{u}\cdot d\mathbf{S}}$ 

Above,ρ is the fluid density,u is theflow velocity vector, andt 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 to the sense of flow into the system the term is negated. The differential form of the continuity equation is, by thedivergence theorem:$${\displaystyle \frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}+\mathrm{\nabla }\cdot (\rho \mathbf{u})=0}$$ 

Conservation of momentum

See also:Cauchy momentum equation

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.

${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}{\iiint }_V\rho \mathbf{u}dV=-}$  ${\displaystyle {}_S}$ ${\displaystyle (\rho \mathbf{u}\cdot d\mathbf{S})\mathbf{u}-}$  ${\displaystyle S}$ ${\displaystyle pd\mathbf{S}}$ ${\displaystyle {\displaystyle +{\iiint }_V\rho {\mathbf{f}}_{\text{body}}dV+{\mathbf{F}}_{\text{surf}}}}$ 

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 velocityu and pressure forces. The third term on the right is the net acceleration of the mass within the volume due to anybody forces (here represented byf_body).Surface forces, such as viscous forces, are represented byF_surf, the net force due toshear forces acting on the volume surface. The momentum balance can also be written for amoving 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.$${\displaystyle \frac{D\mathbf{u}}{Dt}=\mathbf{F}-\frac{\mathrm{\nabla }p}{\rho }}$$ 

In aerodynamics, air is assumed to be aNewtonian 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 is called the Navier–Stokes equations.^[2]

Conservation of energy

See also:First law of thermodynamics (fluid mechanics)

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

$${\displaystyle \rho \frac{Dh}{Dt}=\frac{Dp}{Dt}+\mathrm{\nabla }\cdot \left(k\mathrm{\nabla }T\right)+\mathrm{\Phi }}$$ 

Above,h is the specificenthalpy,k is thethermal conductivity of the fluid,T is temperature, andΦ is the viscous dissipation function. The viscous dissipation function governs the rate at which the 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 arecompressible 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. Otherwise the more generalcompressible flow equations must be used.

Mathematically, incompressibility is expressed by saying that the densityρ of afluid parcel does not change as it moves in the flow field, that is,$${\displaystyle \frac{\mathrm{D}\rho }{\mathrm{D}t}=0,}$$ where⁠D/Dt⁠ is thematerial derivative, which is the sum oflocal andconvective 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, theMach 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, sincesound waves are compression waves involving changes in pressure and density of the medium through which they propagate.

All fluids, exceptsuperfluids, 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 astrain rate; it has dimensionsT⁻¹.Isaac Newton showed that for many familiar fluids such aswater andair, thestress due to these viscous forces is linearly related to the strain rate. Such fluids are calledNewtonian 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 ofrheology describes the stress-strain behaviours of such fluids, which includeemulsions andslurries, someviscoelastic materials such asblood and somepolymers, andsticky liquids such aslatex,honey andlubricants.^[5]

The dynamic of fluid parcels is described with the help ofNewton's second law. An accelerating parcel of fluid is subject to inertial effects.

TheReynolds number is adimensionless quantity which characterises the magnitude of inertial effects compared to the magnitude of viscous effects. A low Reynolds number (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 calledStokes or creeping flow.

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 aninviscid flow, an approximation in which viscosity is completely neglected. Eliminating viscosity allows theNavier–Stokes equations to be simplified into theEuler equations. The integration of the Euler equations along a streamline in an inviscid flow yieldsBernoulli's equation. When, in addition to being inviscid, the flow isirrotational everywhere, Bernoulli's equation can completely describe the flow everywhere. Such flows are calledpotential flows, because the velocity field may be expressed as thegradient 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 theno-slip condition generates a thin region of large strain rate, theboundary layer, in whichviscosity effects dominate and which thus generatesvorticity. Therefore, to calculate net forces on bodies (such as wings), viscous flow equations must be used: inviscid flow theory fails to predictdrag forces, a limitation known as thed'Alembert's paradox.

A commonly used^[6] model, especially incomputational fluid dynamics, is to use two flow models: the Euler equations away from the body, andboundary layer equations in a region close to the body. The two solutions can then be matched with each other, using themethod of matched asymptotic expansions.

A flow that is not a function of time is calledsteady 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^[8]). Whether a particular flow is steady or unsteady, can depend on the chosen frame of reference. For instance, laminar flow over asphere 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, bestatistically stationary. The random velocity fieldU(x,t) is statistically stationary if all statistics are invariant under a shift in time.^[9]: 75 This roughly means that all statistical properties are constant in time. Often, the meanfield 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.

Turbulence is flow characterized by recirculation,eddies, and apparentrandomness. Flow in which turbulence is not exhibited is calledlaminar. 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 aReynolds decomposition, in which the flow is broken down into the sum of anaverage component and a perturbation component.

It is believed that turbulent flows can be described well through the use of theNavier–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.^[10]

Most flows of interest have Reynolds numbers much too high for DNS to be a viable option,^[9]: 344 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; 45 mph) is well beyond the limit of DNS simulation (Re = 4 million). Transport aircraft wings (such as on anAirbus A300 orBoeing 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 withturbulence modelling provides a model of the effects of the turbulent flow. Such a modelling mainly provides the additional momentum transfer by theReynolds stresses, although the turbulence also enhances theheat andmass transfer. Another promising methodology islarge eddy simulation (LES), especially in the form ofdetached eddy simulation (DES) — a combination of LES and RANS turbulence modelling.

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

• TheBoussinesq approximation neglects variations in density except to calculatebuoyancy forces. It is often used in freeconvection problems where density changes are small.
• Lubrication theory andHele–Shaw flow exploits the largeaspect 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 inStokes flow problems to estimate the force on, or flow field around, a long slender object in a viscous fluid.
• Theshallow-water equations can be used to describe a layer of relatively inviscid fluid with afree surface, in which surfacegradients are small.
• Darcy's law is used for flow inporous media, and works with variables averaged over several pore-widths.
• In rotating systems, thequasi-geostrophic equations assume an almostperfect balance betweenpressure gradients and theCoriolis force. It is useful in the study ofatmospheric dynamics.

While many flows (such as flow of water through a pipe) occur at lowMach numbers (subsonic flows), many flows of practical interest in aerodynamics or inturbomachines occur at high fractions ofM = 1 (transonic flows) or in excess of it (supersonic or evenhypersonic 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 flows are flows that are chemically reactive, which finds its applications in many areas, includingcombustion (IC engine),propulsion devices (rockets,jet engines, and so on),detonations, fire and safety hazards, and astrophysics. In addition to conservation of mass, momentum and energy, conservation of individual species (for example, mass fraction ofmethane in methane combustion) need to be derived, where the production/depletion rate of any species are obtained by simultaneously solving the equations ofchemical kinetics.

Magnetohydrodynamics is the multidisciplinary study of the flow ofelectrically conducting fluids inelectromagnetic fields. Examples of such fluids includeplasmas, liquid metals, andsalt water. The fluid flow equations are solved simultaneously withMaxwell's equations of electromagnetism.

Relativistic fluid dynamics studies the macroscopic and microscopic fluid motion at large velocities comparable to thevelocity of light.^[11] This branch of fluid dynamics accounts for the relativistic effects both from thespecial theory of relativity and thegeneral theory of relativity. The governing equations are derived inRiemannian geometry forMinkowski spacetime.

This branch of fluid dynamics augments the standard hydrodynamic equations with stochastic fluxes that model thermal fluctuations.^[12]As formulated byLandau andLifshitz,^[13]awhite noise contribution obtained from thefluctuation-dissipation theorem ofstatistical mechanicsis added to theviscous stress tensor andheat flux.

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 bemeasured 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.

The concepts of total pressure anddynamic pressure arise fromBernoulli'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 termstatic 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—astagnation 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.

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 byisentropically 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".

• List of publications in fluid dynamics
• List of fluid dynamicists

• Acheson, D. J. (1990).Elementary Fluid Dynamics. Clarendon Press.ISBN 0-19-859679-0.
• Batchelor, G. K. (1967).An Introduction to Fluid Dynamics. Cambridge University Press.ISBN 0-521-66396-2.
• Chanson, H. (2009).Applied Hydrodynamics: An Introduction to Ideal and Real Fluid Flows. CRC Press, Taylor & Francis Group, Leiden, The Netherlands, 478 pages.ISBN 978-0-415-49271-3.
• Clancy, L. J. (1975).Aerodynamics. London: Pitman Publishing Limited.ISBN 0-273-01120-0.
• Lamb, Horace (1994).Hydrodynamics (6th ed.). Cambridge University Press.ISBN 0-521-45868-4. Originally published in 1879, the 6th extended edition appeared first in 1932.
• Milne-Thompson, L. M. (1968).Theoretical Hydrodynamics (5th ed.). Macmillan. Originally published in 1938.
• Shinbrot, M. (1973).Lectures on Fluid Mechanics. Gordon and Breach.ISBN 0-677-01710-3.
• Nazarenko, Sergey (2014),Fluid Dynamics via Examples and Solutions, CRC Press (Taylor & Francis group),ISBN 978-1-43-988882-7
• Encyclopedia: Fluid dynamicsScholarpedia

• National Committee for Fluid Mechanics Films (NCFMF), containing films on several subjects in fluid dynamics (inRealMedia format)
• Gallery of fluid motion, "a visual record of the aesthetic and science of contemporary fluid mechanics," from theAmerican Physical Society
• List of Fluid Dynamics books