Fluid Dynamics  

In General > s.a. Continuous Media; Emergence [vs molecular dynamics].
* History: It is the field in which people have been working for the longest time with the most meagre results; The problem is that it involves an infinite number of ode's, and we know that even a finite number of ode's 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.
* Status: 1987, In the compressible fluid case, not even the 1D problem is understood.
* Formalism: The motion of a Newtonian incompressibe 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.
* 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: Works for t collision time and lengths x collision length.
* Lattice gas models: Computer simulations (notably 2D hexagonal lattice).

Related Concepts > s.a. Bernoulli Equation; Continuity Equation; Circulation Theorem; Equation of State; fluctuations.
$ Navier-Stokes equations: The non-linear pde's describing the time evolution of the fluid velocity and pressure of an incompressible viscous homogeneous Newtonian fluid in terms of given initial velocity and external body forces; They express momentum conservation, and are obtained applying Newton's laws to the flow of the fluid, and adding a term that accounts for energy lost through viscosity

t u + (u · ) u = ––1 p+ 2u ,

where u satisfies · u = 0, p = pressure, = density (constant), and = (kinematic molecular) viscosity; > s.a. mathematics.
@ Equation of state: Friedman et al PRL(89) [and astrophysics]; Eliezer et al 02; Silbergleit ap/02 [Klein-Gordon field in cosmology].
@ Euler equation: Euler NCASP(1761)-a0804; Nachtergaele & Yau CMP(03) [from quantum dynamics]; Frauendiener CQG(03) [relativistic].
@ Navier-Stokes equation: Succi 01 [lattice Boltzmann equation; r PT(02)dec]; Streater mp/01 [corrections]; Brüger et al JCP(04) [high-order numerical solution]; Gill & Zachary mp/07, mp/07 [initial data for global solutions]; > s.a. Boltzmann Equation.
@ Smooth Particle Hydrodynamics: Inutsuka JCP(02)ap.

Related Phenomena and Applications > s.a. chaos; collapse; relativistic cosmology; thermodynamics.
* 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].
@ Cosmology, structure formation: Bouchet ap/96-in [perturbations]; Gibson JFE(00)ap [turbulence, viscosity, etc]; Mohayaee & Sobolevskii PhyD(08)-a0712.
@ Instabilities in astrophysics: 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 topics: see Adiabatic Transformation; dark energy; Froude Number; magnetism [magnetohydrodynamics]; molecular physics [polymer fluids]; phase transition; Rheology; turbulence [including Magnus, Reynolds Number]; Viscosity [including bound]; wave phenomena.

Types > s.a. bianchi I models [effects]; condensed matter [liquids]; gas; membranes; perfect fluid; Superfluids.
* Incompressible: A fluid with equation of state = constant.
* Non-perfect fluids: Heat-conducting, viscous, particle-creating, and/or anisotropic ones.
@ Non-perfect: in Dixon 78; Duggal JMP(89); Geroch & Lindblom PRD(90), AP(91); Kreiss et al JMP(97)gq; Anile et al gq/98; Geroch gq/01 [re hyperbolic theories of dissipation]; Krisch & Glass JMP(02)gq/01, PRD-a0908 [anisotropic]; Silva et al GRG(02)gq [evolution]; > s.a. FRW spacetimes.
@ Hyperfluids: Obukhov & Tresguerres PLA(93)gq/00; Obukhov PLA(96)gq/00.
@ Other types: de Montigny JPA(03) [non-relativistic, geometric]; Rajeev IJMPA(08)-a0705 [with short-distance cutoff, non-commutative].

References > s.a. history of physics; physics teaching; sound [including differential geometry viewpoint]; symmetry breaking [history].
@ Books: Goldstein 60; Von Mises & Friedrichs 71; Marchioro & Pulvirenti 94; Massey 98.
@ Books, astrophysics emphasis: Thompson 06.
@ General: issue JMP(07)#6 [mathematical aspects]; García-Colín et al PRP(08) [beyond the Navier-Stokes equation, Burnett hydrodynamics].
@ Relativistic: Geroch JMP(95) [dissipative]; Geroch et al JMP(01)gq [Lagrange formulation]; Sklarz & Horwitz FP(01) [continuous media]; Calzetta & Thibeault PRD(01) [interacting with scalar field]; Hawke et al PRD(05)gq [general relativistic hydrodynamics, excision methods]; Ehlers et al PRD(05)gq [pressure and gravity]; Andersson & Comer LRR(07)gq/06 [rev]; Ivanov a0905-in [geometrical modeling]; Rangamani CQG(09)-a0905-ln [fluid-gravity correspondence]; Molnár et al a0907 [dissipative, numerical methods]; > s.a. gravitating matter; time in gravity.

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