physics Physics is the scientific study of matter, its Elementary particle, 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 whi ...

, a perfect fluid or ideal fluid is a

fluid In physics, a fluid is a liquid, gas, or other material that may continuously motion, move and Deformation (physics), deform (''flow'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are M ...

that can be completely characterized by its rest frame

mass density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek language, Greek letter rho), although the Latin letter ''D'' (or ''d'') ...

\rho_m

and ''isotropic''

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 eve ...

. Real fluids are viscous ("sticky") and contain (and conduct) heat. Perfect fluids are idealized models in which these possibilities are ignored. Specifically, perfect fluids have no

shear stress Shear stress (often denoted by , Greek alphabet, Greek: tau) is the component of stress (physics), stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross secti ...

es,

viscosity Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...

, or

heat conduction Thermal conduction is the diffusion of thermal energy (heat) within one material or between materials in contact. The higher temperature object has molecules with more kinetic energy; collisions between molecules distributes this kinetic energy u ...

. A

quark–gluon plasma Quark–gluon plasma (QGP or quark soup) is an interacting localized assembly of quarks and gluons at Thermodynamic equilibrium#Local and global equilibrium, thermal (local kinetic) and (close to) chemical (abundance) equilibrium. The word ''plasm ...

and

graphene Graphene () is a carbon allotrope consisting of a Single-layer materials, single layer of atoms arranged in a hexagonal lattice, honeycomb planar nanostructure. The name "graphene" is derived from "graphite" and the suffix -ene, indicating ...

are examples of nearly perfect fluids that could be studied in a laboratory.

Non-relativistic fluid mechanics

classical mechanics Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...

, ideal fluids are described by

Euler equations In mathematics and physics, many topics are eponym, 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 their own unique function, e ...

. Ideal fluids produce no drag according to d'Alembert's paradox. If a fluid produced drag, then work would be needed to move an object through the fluid and that work would produce heat or fluid motion. However, a perfect fluid can not dissipate energy and it can't transmit energy infinitely far from the object. A flock of birds in the medium of air is an example of a perfect fluid; an electron gas is also modeled as a perfect fluid.

Superfluidity

Superfluids are fluids with zero viscosity, however in practice superfluids cannot be accurately described as a perfect fluid. In the two-fluid model, superfluids are macroscopically considered as having two-coexisting phases, a mixture between a normal fluid and a perfect fluid.

Cosmology and astrophysics

Perfect fluids are a

fluid solution In general relativity, a fluid solution is an exact solutions in general relativity, exact solution of the Einstein field equation in which the gravitational field is produced entirely by the mass, momentum, and stress density of a fluid. In ast ...

used in

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

to model idealized distributions of

matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic pa ...

, such as the interior of a star or an isotropic universe. In the latter case, the symmetry of the cosmological principle and the

equation of state In physics and chemistry, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or internal energy. Most mo ...

of the perfect fluid lead to Friedmann equation for the

expansion of the universe The expansion of the universe is the increase in proper length, distance between Gravitational binding energy, gravitationally unbound parts of the observable universe with time. It is an intrinsic and extrinsic properties (philosophy), intrins ...

. 22.1.3 The Friedmann equations of motion

Formulation

In space-positive

metric signature In mathematics, the signature of a metric tensor ''g'' (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and z ...

tensor notation, the

stress–energy tensor The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress ...

of a perfect fluid can be written in the form :

T^ = \left( \rho_m + \frac \right) \, U^\mu U^\nu + p \, \eta^ ,

where ''U'' is the 4-velocity

vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...

of the fluid and where

\eta_ = \operatorname(-1,1,1,1)

is the metric tensor of Minkowski spacetime. The case where p=0 describes a dust solution. When

p=\rho_m c^2/3

, it describes a

photon gas In physics, a photon gas is a gas-like collection of photons, which has many of the same properties of a conventional gas like hydrogen or neon – including pressure, temperature, and entropy. The most common example of a photon gas in equilibrium ...

(radiation). In time-positive

tensor notation, the

of a perfect fluid can be written in the form :

T^ = \left( \rho_\text + \frac \right) \, U^\mu U^\nu - p \, \eta^ ,

where

U

is the 4-velocity of the fluid and where

\eta_ = \operatorname(1,-1,-1,-1)

is the metric tensor of Minkowski spacetime. This takes on a particularly simple form in the rest frame :

\left \begin \rho_e &0&0&0\\0&p&0&0\\0&0&p&0\\0&0&0&p\end \right

where

\rho_\text = \rho_\text c^2

is the ''

energy density In physics, energy density is the quotient between the amount of energy stored in a given system or contained in a given region of space and the volume of the system or region considered. Often only the ''useful'' or extractable energy is measure ...

'' and

p

is the ''pressure'' of the fluid. Perfect fluids admit a Lagrangian formulation, which allows the techniques used in field theory, in particular, quantization, to be applied to fluids. Relativistic Euler equations read :

\partial_\nu T^=0

in the non relativistic limit, these equations reduce to the usual Euler equations.

Non-relativistic fluid mechanics

Superfluidity

Cosmology and astrophysics

Formulation

See also

References

Further reading