Fluid Dynamics / Hydrodynamics |

**In General** > s.a. history of physics;
physics teaching; sound [including
differential geometry viewpoint]; symmetry breaking [history].

* __History__: It is the
field in which people have been working for the longest time with the most
meagre results; The problem is that at the basic level it involves an infinite
number of ordinary differential equations, and we know that even a finite number
of ordinary differential equations have a peculiar behaviour, like strange
attractors; We can understand this mathematically from the fact that the Euler
equation came from an approximation of the fluid by small fluid elements.

* __And fundamental physics__: In principle one would
start from the Boltzmann equation and derive from it the Navier-Stokes equation.

* __Formalism__: The motion
of a Newtonian incompressible fluid is described by the Navier-Stokes equations
of momentum conservation and the continuity equation,
in the absence of density variations, magnetic fields and heat sources; To
solve them, usually one assumes a finite volume *V*, and prescribes
the velocity vector *u* on ∂*V*.

* __Status__: 1987, In the
compressible fluid case, not even the 1D problem is understood.

* __Decomposition__: Motion
of a continuous fluid can be decomposed into an "incompressible" rearrangement
which preserves volumes (described by the Euler equation), and a gradient map
that transfers fluid elements in a way unaffected by any pressure or elasticity
(described by the Zel'dovich approximation, used to model the motion of a self-gravitating
fluid in cosmology).

* __Hydrodynamical approach__:
It works for Δ*t* \(\gg\) collision time and lengths Δ*x* \(\gg\) collision length.

@ __Books__: Goldstein 60;
Von Mises & Friedrichs 71;
Marchioro & Pulvirenti 94;
Massey 06;
Kambe 07; Buresti 12; in Thorne & Blandford 15; Bernard 15; Regev et al 16 [in physics and astrophysics].

@ __Geometric__: de Montigny JPA(03); Kambe 09 [and dynamical systems]; Gawlik et al PhyD(11)-a1010 [variational discretizations of complex-fluid dynamics].

@ __General references__: issue JMP(07)#6
[mathematical aspects]; García-Colín et al PRP(08) [beyond the Navier-Stokes equation, Burnett hydrodynamics];
López-Arias EJP(12)-a1103 [Thomas Young and the behavior of air streams];
Sasa PRL(14) [hydrodynamics from the Hamiltonian description of particle systems]; Dostoglou et al JMS(15)-a1406 [in the limit of infinitely-many particles]; > s.a. computational physics.

@ __Equation of state__: Friedman et al PRL(89)
[and astrophysics]; Eliezer et al 02; Silbergleit ap/02 [Klein-Gordon
field in cosmology].

> __Other general topics__: see Continuous Media;
Emergence [vs molecular dynamics]; fluctuations;
Navier-Stokes Equation and Euler Equations.

> __Online resources__:
see Wikipedia page.

**Relativistic Hydrodynamics**

@ __General references__: in Dixon 78;
Geroch et al JMP(01)gq [Lagrange formulation];
Sklarz & Horwitz FP(01) [continuous media, including viscosity];
Ivanov a0905-conf [geometrical modeling];
Chen & Spiegel CQG(11)-a1107 [causal];
Kovtun JPA(12) [hydrodynamic fluctuations];
Rezzolla & Zanotti 13;
Disconzi Nonlin(14)-a1310 [viscous];
García-Perciante et al JSP(15)-a1406 [stability];
Christodoulou & Lisibach a1411 [self-gravitating, phase transition];
Jensen et al a1701 [effective field theory, superspace formalism];
> s.a. computational physics; gravitating matter
[fluid spheres]; Maxwell-Lorentz Equations; solution methods
for einstein's equation [fluid-gravity correspondence]; time in gravity.

@ __Dissipative__: in Dixon 78;
Geroch & Lindblom PRD(90),
AP(91);
Geroch JMP(95);
Kreiss et al JMP(97)gq;
Anile et al gq/98;
Calzetta & Thibeault PRD(01) [interacting with scalar field];
Geroch gq/01 [re hyperbolic theories of dissipation];
Silva et al GRG(02)gq [evolution];
Molnár et al EPJC(10)-a0907 [numerical methods];
Andersson & Comer CQG(15)-a1306 [covariant action principle];
Disconzi et al IJMPD-a1510 [first-order formulation, and cosmology];
Crossley et al a1511 [in curved spacetime, effective field theory];
Pimentel et al GRG(16)-a1604 [energy-momentum tensor].

@ __In curved spacetimes__: Duggal JMP(89);
Krisch & Glass JMP(02)gq/01,
PRD(09)-a0908 [anisotropic];
Love & Cianci PTRS(11)-a1208 [using the Chapman-Enskog procedure];
Bemfica et al a1708 [viscous, coupled to gravity];
Van den Bergh a1710 [rotating and twisting];
> s.a. FLRW spacetimes.

**Other Types** > s.a. condensed matter [liquids]; gas; membranes; molecular physics [polymer fluids]; perfect fluid; superfluids; Viscoelasticity.

* __Incompressible__: A fluid with equation of state *ρ* = constant.

* __Non-perfect fluids__: There are heat-conducting, viscous, particle-creating,
and/or anisotropic ones.

* __ Complex fluids__: Binary mixtures in which two phases coexist;
Examples are solid–liquid (suspensions or solutions of macromolecules such as polymers), solid-gas (granular),
liquid-gas (foams) and liquid-liquid (emulsions); They exhibit unusual mechanical responses to applied stress or strain,
including transitions between solid-like and fluid-like behavior, due to the geometrical constraints that the phase
coexistence imposes and characteristics such as high disorder, caging, and clustering on multiple length scales;
> s.a. Wikipedia page.

* __Lattice gas models__: Computer simulations (notably 2D hexagonal lattice).

@ __Dissipative__: Rajeev JPCS(13)-a1004 [geometric
formulation]; Andersson & Comer CQG(06) [and superfluid neutron stars]; Glorioso et al a1701 [effective field theory].

@ __Hyperfluids__: Obukhov & Tresguerres PLA(93)gq/00; Obukhov PLA(96)gq/00.

@ __Complex fluids__: Gast & Russel PT(98)dec; Shen & Cheung PT(10)sep.

@ __Quantum fluids__: Tsubota et al PRP(13) [rev]; Gripaios & Sutherland PRL(15)-a1406; Suto JMP(15)-a1504 [probability distribution of the total momentum]; > s.a. bose-einstein condensation; condensed matter; gas; ideal gas; superfluids.

@ __Related topics__: Roberts CEJP(11)ht/04 [fluid-like generalization of membranes]; Rajeev IJMPA(08)-a0705 [with
short-distance cutoff, non-commutative]; Doering et al JMP(12)#11 [incompressible, turbulence and mixing]; news pw(18)jan [active fluids, and quantum mechanics].

**Related Concepts and Phenomenology** > s.a. Bernoulli Equation;
Continuity Equation; Circulation Theorem;
critical phenomena; Equation of State.

* __Plateau-Rayleigh instability__:
A fluid cylinder longer than its circumference in energetically unstable to breakup.

@ __Stability__: Plateau 1873, Rayleigh PLMS(1878);
Chandrasekhar PRS(64) [liquid drops];
Joseph 76; > s.a. Instabilities.

@ __Ordinary physics__: Burgess et al PRL(01)
+ pn(00)dec [dripping];
Lohse PT(03)feb [bubbles].

@ __Microscale description__: Celani et al PRL(12) [failure of the overdamped approximation and entropic anomaly].

@ __Smooth Particle Hydrodynamics__: Inutsuka JCP(02)ap;
Cossins PhD(10)-a1007 [rev];
Price JCP(11)-a1012 [and magnetohydrodynamics]; Springel ARAA(10)-a1109 [in astrophysics]; Chiaki & Yoshida MNRAS(15)-a1504 [particle splitting based on Voronoi diagrams]; Price et al a1702 [in astrophysics].

@ __Cosmology, structure formation__: Bouchet ap/96-ln [perturbations];
Gibson JFE(00)ap [turbulence, viscosity, etc];
Mohayaee & Sobolevskii PhyD(08)-a0712; Cervantes-Cota & Klapp a1306-ch [rev].

@ __Astrophysics emphasis__: Thompson 06; Ogilvie JPP(16)-a1604-ln [and magnetohydrodynamics].

@ __Astrophysics, instabilities__: Hartle & Sharp ApJ(67);
Friedman & Schutz ApJ(75); Bardeen et al ApJ(77);
Friedman CMP(78);
Hiscock & Lindblom AP(83), PRD(85);
Semelin et al PRD(01)ap/99.

> __Other phenomenology__:
see chaos; dark energy;
electromagnetism with matter; Floating;
Flux [flow rate]; Froude Number;
gravitational collapse; magnetism [magnetohydrodynamics];
meta-materials [suspensions]; phase transitions;
Pressure; relativistic cosmology;
Rheology; thermodynamics;
turbulence [including Magnus, Reynolds Number, examples];
viscosity [including bound]; wave phenomena.

> __Other related topics__:
see Adiabatic Transformation; bianchi I models [effects]; energy-momentum tensor; Enstrophy; knots; Knudsen Number.

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