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    Fluid Mech. 18 10 0Copyrght 978 by Rvw' rghs

    RELATIVISTIC FLUD

    MEHANCS

    aubMathemaics Department, Universiy of Califoria Berkele California 7

    NOON

    1 Inducy Suvy

    6

    P-lavty mcacs ay b caact as a toy tat scbs tsa o a by mas o v ctos o at ost o vaablsT latt vaabls a o two s t vaabs tmg a o a tsoal ca spac a a ot o abg absolt t o soda uvel cloc. v unc te deede vble t ee w d: ee e e e vbe vi 123) eompos o t vloy vco o som cooat sysm atwo of them are thermoynamc arables, for example he pressure p an the densty

    of the u From a knowlege osuh arabes a oher theroynam arables

    uh a mau c nna ng an cc no S a balcla t at oft bg cos may b scb by tfcoal pc of as a fto op a .

    T t vaabls a as soto o v cosvatoqu. I e cde e ee e e ce a

    + (V) = t covsao o omtm

    a the coneaon oenergyEiE = yj+h

    w w av s t ollowg otatos t smao covo fi =

    = = ' ' s t coc o vscosy

    1ij AT

    d e e otvty

    0066489/78/0150301$0.0031

    An

    nu.

    Rev.

    FluidMech.

    1978.1

    0:301

    -332.

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    302 AUB

    T v covato qato gv abov a vaat a taatgo tat i gat by th -aat go of cla oto

    wth h abay a th h o of a othogoa a t taato

    t' = t+Twth abtay a th Gala taoato:

    wi h bi abiay oa a i aviy i ma ai tcl of wtoa atvty

    Ratvtc yac ao a thoy volvg v t ctothat cb t tat of th h of th katc vaab a twohoyac o owv h t vaabl whch a tl fo b o ot f to a ot a thoa ca ac aabolt t bt ta a ot (ca a vt) a fooacot ot a ac-t a ac wth a t tc wth agat ta w al tak to b (- ++ +). I al latvty ac-t ca Mow ac a t chaact by t act that t xt cooat yt tal o whc th tac btw th vt wtcooat (t,x,y,z) a (t+dt,x+dxy+dyz+dz) gv by

    ds2 c2dt2+dr+dy2+d2.

    a wt t ato a

    ds = dx dx",wh / 0, ,2 ,3 ,

    bbbO = l = x = y x3 = z a t ato ovto bg

    h go of taoato tat cay tal ooat yt to talo t taat Poca go cotg of taoato o th o

    w a a cotat a th cotat fo a ot atx ch that

    v covato law chaactg cal atvtc chac avaat th go

    a ga ooat y ag fo a a o by a afoato

    = y z) = (x) / = 0, , 2 3 ,w a v

    Annu.

    Rev.

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    w hav

    d = g"v d dewh

    oxU ax'v = u, a" av

    REAIVISC ID MECHANICS 33

    (.

    hs ga cooa sysm h hso symbos a

    rz tgU'(g.+g,l-g,) 0 wh

    ()owv h Rma cva so

    14)

    .. Mows sac s a a a ss s a abso sacm of s co

    Gaavsc mchacs s om ha o sca avy hah vaabs of h cosao qaos o a cv sacm h mtc enso o hs sactm s t as b o h

    resene o a raiaona e hose soure is eerine aious hsias sch as lcomac os o som cass by h sgavag whos moo s b s. o lcs h avaoal l o whos moo s n m o s sa o b a wh h moo o as hs ga lavsc chacs h a wo ys o oblms1) o m h moo o a s a gv a gavaoa a o m h gavaoa o h sc o a sgavag (a oh s) a o m smaosy h !oo o h (a h sa o h oh s ha may b s).

    I h a y o obms o mus m h sa o h a h

    gavaoa s by sovg h soca Es aos. hs ascb bow

    K Cmi Cit hs a h mmay olowg scos w sha scss som gomcalos o sbmaos o a sac-m Mch o o scsso w o o whh o o h sacm s a a o shal assm ha w aag wh a a cooa sysm wh cooa abs x(p

    =023)

    whch h m s gv by

    2.1)

    Annu.

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    30 AUB

    In a oonate yem e o o

    x=xl'(w) 22epeen a e wh w a paamete at labe an eent on e e. The angen

    eo to e gen by

    dxv=-.

    dw

    Te ae ote qanty

    (23

    eemne e nae of he e: If V VI' < 0 the e meke f VI =0

    n an f

    > 0 t paeke. Te pae pat of a n moon

    ae ebe by meke e. Lgt ay ae on nll e. Pale patae ao ae wo-ne of pale.o a v e aam may away b o o a

    L=_1.

    a ae w ofen epae by te ete an ale the pop me o

    The mee e epeenng the patle pah of a egon of he eemne a omalze eo el ae the foeoy eo whh afnton of een n paeme. In he oonae yem e aboe we hae

    an te oluo o a o" uds

    ebe he wolne of the "paleWe may wte e oon of aon .5) a

    . _03= x'; s 3weeg i 0

    2.

    2.

    (2.

    ae eqe to be e paamet eqaton of a hypefae he nal hypefae on w he pae e loae a ope me = Te fo aabe, s wh we al enoe a , fom a omong oonae ytm n paetme. Tey ae mla o agangan oonae of peeay yoynamb te na yefae nee no be the ypefae 0 een n e ate paetme Mnkowk pae an e oonae ae neal oneUne the anfomaon

    Annu.

    Rev.

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    AIVII I MI 30

    X* = t uion o t inti yu s = 0 bs

    X*4

    X*4X*i;0).

    T x* e gene vng nes. Ens 2) e en be nes

    ax uxx*

    ee n e eenn e X e e nsn f en ese vese e ne s seee

    n e gene vng ne sse e ve e nens f e f

    vey ve gven bv

    Hn

    whee g:v omonn o mti tno n t omovn oonytm.

    rn r l ln rA ve e wt omonn v n lon y wt ngen ve s s neg e ns ng y i sses e en

    (3.)

    w w v u mioon o not t ovin iviv wit t to mti g o tim tt i < i n y uion uv iy

    2

    o.Ve es iy t ution

    V = U1 33 to uo m-l not. t i onun o t nitiono# n t t t ovint ivtiv o mi no vi

    Annu.

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    6 TA

    Hence the tnent vecto u to cuve uderoe Fermi-Wke trnort etes Eqution (33 t w not undergo pel tnport e tfy Equton ule e cuve eec

    A e f fu vec

    th fy he eu34)

    wh

    t- 28 (35

    o fom n rhnorml er. We m cntuct uch tetd on cuve r b chn ut f he bve eun n even of he cuvewth mtr vu 5 = 5 an wth

    =u( nd thn tot he tetrd to te event wh meter vue Fem rt Such tt contn n thorm t a 2ohol to r Th fm "tl fe of eference f n erv whoew-lne me f efeence e elvc ezn hNew cep f "tg fme

    e e

    rje rbtr vecto c-e n t t f)

    We hve

    (3

    In ddon

    9

    m be eee he mec he heeme ce h hevect x) he even xhe cmone f he fu vec }f. fm 4 mx We h ee

    b Aa the eement f he nvee m ht

    n

    }rA =

    b v w }' u, }O - hu

    d the m e cred ou ove = ,,

    Annu.

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    REATIVI I MEI 07

    4 Sc ScmW ha that th lt t ld u d a thpaat al pa thh a hpfa L pat a lt f th

    euaindx = uxd

    Wh th lt a wtt a

    te euan

    = 0

    = 12

    a th paat at fLI w ha a dbd b th at

    i ),

    th th at btad (4 b bttt 42 naey

    x# x()s x(s)

    1

    (42

    (4

    dene a ienina uace n acetie If te cue gien by uatin2 i a ced cue te tdinina uace cad a tue

    Te ec edax

    }I=_ tat t th f paat th fa It a fEuain (4 at

    aA# au

    as a(44

    e eent beled by cnant ue ie n cue n e cnguencegen by uain 1 Tu een n e dine and n te dine + eeen neigng aice a i a diaceent ec gien by Te caa

    2 (4

    wh a th pal t f a thal ttad a b dda te patal coote o the ecto lat t h wldl

    Wh a that th d Wal tapt ay hwby dtat Eat (4. wth pt t ad Eat ( ad

    (4.4 that

    (6

    Annu.

    Rev.

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    08 TAB

    wh

    ad u; d h aa da h d

    W a dp u; b g h p pa h a lw

    wh

    2( =(u#v+ Uv#)hdh -%8h"

    u+u+auaui8h"

    (48

    (4.9)

    wh h a d h da da I a hd ur ad ha

    10

    Ea 6 a h a aa da a dgg a da Th a b alg alg

    h wdl abld b ad g Wa ppagad ax w#># ad8 db pl h a ha ad pa h ld ghbg pal ad a db h ala

    h

    11

    wh + h da h ad f' h aaag d h agla h al pa ch a b # mm a h c c W ha

    W#u# = .h cd

    W# 0

    h a ad cd ha

    u# pJ:#

    12

    wh ad fa caa c h a u# ppal h al h hpa

    fx = a

    Annu.

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    Wav Vlciy

    an

    XXX 0

    RAVII I MC

    51)

    dn a yr ra n a-ti i.. a trdnna a. atng wit at

    X = cntandm a a a tw-dma a in atim T may gadd a ty t t wd w t y a X vai.

    T ma t i gvn y 4 ad it may wn a

    (5.2

    where Ul is e tangent vector o te world-line of an observer eauated a apoin ofL,

    53

    and

    H

    (5.4)

    T v W dtmi ' ata dit agai wi t t tv u t wi

    (ww)j/2t

    and av

    Te vetor

    vk u

    wi a diamt k =

    (5.5)

    I c at nditn atid n i t wav vity ac a mard y v It i a nc t av ain tat

    uP [gV uuV)vJl/'

    hu

    6)

    )

    Annu.

    Rev.

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    TAUB

    Hece 2 a y i g 0 .e. e ra s saceie r ll reivaey, is elie is ee we ay te a t aln p wi nn 1 a e

    n 1 u)2] /

    a

    6 Symmetries

    Fr eac vale f te araeter te fr eqatis f e fr* = a

    8

    (5.9)

    (1

    ee a aig sace-e ta ss e ev te evet x*. Weese eais are sis e eais

    d* = *

    wee

    te trasfrats ee by qa (1 r a earaeter gr sa t begeerae b te vecr e

    A sace-te wt etric gv s ivariat er sc a gr .e. ais e

    gr ts a f e Klg eqats

    are ase were

    =g

    vC

    .2

    a as befre te seicl ees te cvariat ervatve wt rsec gTer els are ivariat er e gr i ter e ervatives wi resect

    vas. Tese ervtves re ee as

    T" n T''':

    s fr eale r a sala ci we ave

    f =

    .

    Annu.

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    RELATVC FU MCHN 3

    a ecr u e ae2u P- Pp

    a a caa e

    2wv= Pwv;+wpf+wPf'Ne a wv w e ay re

    !w wv; w"v w"P+w +wP";vece

    wv= w- wlv"

    e

    Te aer eas are e cs a esre a cally

    Wv= v- = v-.v

    r a vecr e .

    65

    6.6

    We c vcr s e rvecy e e rles e arc c rbs gr . e s ara er s g e s saary

    elu FormuaeWe clse e scss eacs a cllec o e ea ge as o e ovia vativ o t o-voity vco. We ee

    s e a cseece e e a a e reres e evCa alerag esrs a

    72

    a

    (7.3)

    ee e y e rg s ea s e geerae Krecer ea. is e a cseece Eas ( a

    7

    a ece

    !VWurW/ = 0 75

    (Jwt= )p1 ha'

    Annu.

    Rev.

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    3 TAB a a

    and a

    SECIALELAIISIC LID DAICS The Stress-Energy Tnsor

    (7.6

    77

    n eavc ee d and e aea cnna ae decbed b evec vec ed uPand b a ecndde ec en Tpv nn ae eeeg e e ed n e an na ead Ai 01 Aio = uPe ay dene

    -1i4 -14; TpAu

    144) = TpPu.

    i , )

    e x ndeenden ane 1;J) deene e e e en a eaedby an beve vng n a dne e angen vec uP e a e eec e ed) and ng A a aa axe e ee ane 1i4) =4 gve e e den eeen eaed b beve e

    caa 144 gve e e den eneg eaed by beve Te adjecvee ed eace e ae "a eaed b e beve vec up

    An eeneg en a be en a

    ee

    ee a bee

    u"up 1

    Hence

    n e ng e a e e decn e eeneg en a

    d gven b Ean ee ane decn baed n e evae and e vec e eneg en e n e A and X ange ean

    Annu.

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    RELVISTC FLUID MECHNCS 33

    A g 196 a Tvi c a

    a ac V h x a ii vc T Thi vc d ah h "a vc vc h d h ca a hacdc d h h d d h LadaLhz aLada & Lhz 5 ha h ad Ea ha v Ecka 0

    hd d ha a a cc Ea havTuV = -(wu:w).

    Hc h w 0 u a vc va - h ca hLadaLi ad ca aac cicd

    T g a d i a qai ( i c ai i h ciaaivic vc ig i a ,

    +e) Ih+ Ta)

    V c{)- 26Vh e, a ad V a a v h vc vc a c av ad a h cc vc h cc hacdcv

    ad

    a ga a gav

    h a

    di i cic ia g id; i ; ad T a a

    Ti dci a aivc d a id aviicc , dciig di a a cci ac dgigc ad dcid a di ci ag avcaai h Ba a A a hi h a di h ac h 9

    Na h ic h ad h za a ad a dad h cva a d h hav d ad

    a a h a hdaic h a ad avc ic h

    h dc ha ak ha h dcb hhav h d accc ig h cvai a ad havc a h a hdac

    d v a dg d c a cac daic vaa c a p ad cc a ci vaia

    a ga di c a 9 a a

    6)T a avic d caic qai a cd aviaii h c ha h vc d ad hvcii hc av a ha h vci ih

    Annu.

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    314 AB

    n eeav e e nena eneg a eec ga gven b

    s=-lP/,

    ee y a cnan ea e a e ecc ea e ga Reavcnec e e a a eec ga e a ncn e eeaeane eve en en n e abve n nge a cnanTe neae ( a ne ave

    y < 5/3.nge (1957) a n a a eavc eec ga e ave

    w+p = (+e+p/) = G(x)

    px xT kG(x)=K3(x)/Kz(x),

    e L(x) pxL(x)

    logL(x)= -xG(x)-og(K2;X),ee k a cnan eae e Bann cnan an So a cnan Ten ae cen a e vec g n n ace ne an eKn(x) ae Bee ncn gven b

    K(x) = t ex ( x c I c I dI age eeae e ave

    5(x) = +:x+

    L(x) =

    x a g eeae)G(x) = + ,x1 4 L(x) = e x +

    9

    Te ean ecng e n a eavc ae gven e vecnevan a

    (91

    Annu.

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    RLAIVSC D MCHANICS

    a

    TV = 9)qai (1 is kw as csvai-mass qai. T r qaiscai i qais (2 a cmbiais f qais a rc i eeav e he ea ceva ee a eWe Ea 9) by jec he a he dec he vecvec u# a a he dec eedca h vec

    a ceece Ea 9 ha(9.3

    a i iviy y w a ma a s scic y ia y rss a siy by qai

    dS de p de a e a 9) 9) (5) ad ) e Ea (9) a

    w # is y c vc

    w#

    # pu# -

    (9.4)

    (9.5)

    (.6

    he ea 95) eee he ecd a heac eavcd he kec he dec he d ead he de hedac eb a ha ae he d hch

    # = O a cas w s av

    a eec hch

    = = 0 Ea 9 ) 95) a 96) a

    .#u#

    (9.7)

    9)

    (

    Ta is s scic y is csv a w-is i. Tss sas aws csva ass a ms sssgy s w sa s imi -wi eav echac

    We a e he eee e he vc heacdc a

    #V = r "+ f

    Annu.

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    316 AB

    w : -ngy n f a pf ud = uV

    pn u namy

    P ehv and v pn ngy mmnum aad w a w

    I a nun f dnn gn ab a nan uan

    may b wn a

    a" hvffw

    and w py pn f du u and a w

    0 ionriy nd Thrmodynmic Equiibrium

    ud bdy and pam n w m a ad b anay f y u, mdynam aab and m n anaan und a n-dmnna d-pn f gup f anfman fpam n f w m b n paua m n v and npy un S a da ang mlk d angn b f gup mn. wn by fllwfm Sk m

    f sndv= f S(_g)1/Z d4X, 101D D ac fm f nd aw f mdynam and gp naana anay mp

    (02

    pf n ng b a dman n pam bundd by wpa k ypufa L l andL and a mk yp ufa B a ypufa akn b naan und gup and L

    l L

    f m mn

    f gup fu aumd a ud adabaaly lad n a n npy u ug B (w may b a nny)ndbm 96 a fu wn a f a gnala ud ang nn

    Annu.

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    ELISC LUD CHNICS 17 ccis f vscsi nd h cnducin, is i s f hmdnmieubum e he eun f se f he uid s bp nd e i f hefue f he ud he empeue is Kiing ve ed

    he Kilng vecr el whih hezes he acee a a aare ie

    hen he exs ss nd suh h

    = nd he e s ppnl he nm he spelke hpeufes

    can

    hs se he speime s sid e s n

    (1

    A heem due hnew (1) nd geneed Ce (12 impies ' w dd u wh ' ds ss ) h a aa ac acee ee e rce e avaaled is a ud, hee e w ndependen me ike Killing es pesen, ne fwhh s ppin h fuei f he ud

    Prfct Fui

    A r c he eien f eral cducii and he eens f

    vc ad va, e e c = = ' =,

    s sid e pefe uid F suh uid we hve

    PlU"U+pg,wh s h s ds,

    s h ssu, d

    wpIt = +sp/ =

    (1

    (112

    whee e e f gh is ken be un nd he es spetl t f f tw tmm bl e eun f se is gen b his funn dependene:

    s sip, A a c ee vaale e c a e vec d n he ui

    s ess hn h f f he uid hs inei-he desipin fm whih hemsp ibes gn bve m be deemined hen i mus sisf heiuiis 86)

    Annu.

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    18 AB

    e conservaon equaons sased b e vaabes descrbng e sae o eud ae e cosevao o ma

    (u' 0

    and e conservaon of enermomenum

    e lae aons ma be wren as

    TS. =

    and as

    ) =

    asJollows om e dscusson n Secon 9

    .3

    11.)

    11.)

    )

    Baooic ows in relaivisic dodnamics are descibed as olows: Te udmoon s suc a an equaon o sae o.e om

    =) 11.7)

    olds rouou e ud and durin e moon. Te equaons ovenn e Eu 4 u u Equ (6 nomalzao equaons

    118)

    seve o deemne e sae of e ud f one s ven e na sae Equaons11.) and 11) do no od. However, s a consequence of Equaon 11.) 11),and 11.8) a

    (s);

    wee e emodnamc varabe s s dened b e euaon

    ds dw

    Equaons 116) ae relaced b

    were +Q=og -

    11.)

    11.1)

    11.11)

    11.1)

    An exame of baoroic ow is ovided b isenroic ow in wic case sconsant. en Equaton 11) is dencal sased. ma be vered a s = sases Equaon 111) n s case and fue Equaon 11.) and 11.) becomedenca.

    We dene+V -p- 111)

    Annu.

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    RELATVISTC FUD MECHNCS

    in a quan (.) and () bain and

    w+pV= u

    n e baic cae. I a be eai eed a

    ae

    n e ae and

    (.)

    5

    (.)

    .

    in nd n. Fu quan (.) and () ae eeciely eaen (.) and ().W dn

    ee ean mpy tat

    Hence

    In ew an (6) we ae

    Qap -vuYIlaP+ T(pS,a-Up).

    Wen an ) ban

    e ay w

    (.8)

    (.9)

    .2)

    (.2)w = wen is ivn y (.) nd = wen i i in y (.4). I i a in v d, a is wcn

    1;vV=0

    w ae a a neen () r ()

    () =wt

    v

    (22)

    (.2)

    a nan ang wrd-n ud and need dung ew.

    Annu.

    Rev.

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    20 TA

    In h crdna ym n whch .. h angn vcr hcurv paramr xo, Euan 2 bcm

    =Vo=Uo cnanang h wrdln h ud. I may vary rm wrdn wrd-n. aub959 ha hwn ha n a wak gravana d wh a Nwnan pnal Vh ar uan may b wrn a

    + + V cnan 24

    n h apprman n whch pwr / hghr han h cnd ar glcd,whr

    90lgov)

    q gpv

    --- U U

    gooan

    goo=+2V.

    uan 14 h uua rm h Brnull uanh rul ad abv nab n gv h rlavc gnrazan h

    nn crcuan and Brnull' hrm.

    dn a ub n pac-m (c Scn 4 whr r d and varab w hav awrld-n h ud and r d s and varab r w hav a cd curv uch ha

    h

    h nga vn by

    Cs JxV"APdr .2 h ravc crculan I a cnunc S hrm ha hncy n ucn cndin

    Cs 0

    r ub dn by h vcr d by man arbrary nal urac anarbrary curv n h urac ha

    .26

    n viw uain ( h cndn uvaln h rurmn ha hw b uch ha

    I=0 and S O

    Annu.

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    RAIVIC I MCAIC 31

    h s f hs uaons is h samn ha h ow b ioaional (i.. 0and h scond is h samn ha h ow b isnopic

    Whn uaion (11 hods w ma wi

    8R

    x (11o som scl ncon R Tu 99 has uh shown h o ioonsnopic ow n a wak aaonal d wh a wonan pona V Euaons117 mpl ha

    w+p 8R V+zq p 8x 11.n h sam appomaon as was sd abov. aon 118 s h cassca

    fom of h Bnouli uaion fo a nonsad ioaional isnoic ow in aaiaiona ld wih ponia V

    follows from Equation 15, he detio of ad

    d

    C=

    - f2Q"UA"dr= -f"TS."A"d as folows fom uaions .1. Hnc a ncssa and sucn condion hadC/d 0 fo ubs dnd b h co ld u and abia iniia sfacs andaba cus n his sufac s ha

    (Sl(S ,i. ha h w b baoopic.

    2 Examp of pca Ravc w

    In his scion w shall discuss wo uid-dnamic pobms in spcal laii.Thus h uid moion aks plac in a passind spac-im naml Minkowsispac-im and aiaiona cs a inod. Th poblms w shal a a1 h on-dimnsiona isnopic moion of a pfc uid and . h saionaaismmic moion of a aiisica incompssib uid.

    W sha dscib h s pobm in inia coodinas in Minowski spacin which

    21

    All uid aabl a assmd o b funcions of Xl x and Xo = 1 and2 3 = O hn w my wi

    uaions (1. and (11.4 duc o

    pl- u2) 1/2], pu- u2) 1/ 0and

    (1

    1.

    Annu.

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    3 TAUB

    wee e suscs and denoe aa deeaon w esec o hesevaabes

    We dene he varabe quan

    G s' 2whee w+ l/ and he deenaon s aken wh he eno S consanand se

    =f; oows ha [c Tab (98] Eaons (.3 a be wen as

    D

    = 0

    whee u1/ = o ,u = o 1 + U1/2-u

    and

    (1

    (2.

    (.7

    (8

    Thu and are the specia elatvistc aalogues of the Rema fuctos,whic occur n he classcal heo o oaaon o onedensonal waes one alude.

    Proressve waves are dened as n classcal heo as hose or whch and [are consan o consderaon o such waves one concludes ha s he velocosound n uns wee he veloc o l s one ne ue nds a oaaeswh a veoc ha s an nceasn uncon o hus as n he cassca caseshock waves us occu. hese wl be dscussed n e nx secon

    e now un o e second exale in wc e ow wil be aken o bebaooc wh

    . (9o suc a ow e veoc o sound is equal o e veloc o l since a ven b Euaon (2 un. Fd o wc e equa f ae venb Euaon (9 ods ae caled exee uds o reavisca ompblones. The uan dened b Eaion 0 s

    (20

    and hence

    +/ W1/ 1

    Annu.

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    RTIVITIC UID MCHNIC 2

    nc quaon ( bcoms

    (WIuP). [wI-g)UJ 0. _)

    an ( bcoms

    Qvu = Vv- V)U 0whr

    122

    )

    )

    Snc w ar assumng ha h ow s saonary and asymmrc hr swo Kng vcors in inowsi spac say and I such ha h drivaivoh ud ariables wh rspc o hs vcos vanshs n sphca coodnas

    n inows spac h n mn coms2 gvx"V = -dc2+r2+r2(e2+sin2e),

    wh X X r e and 3 = n hs coordna sysm w may ak

    and a varabs ar rqurd o b ndpndn o and I foows from h rss of Scon ha

    Vo 2wi/2o 2H

    IV = V3 = WjU3 Kee and a cnant an h dn f h uid i

    uHv = u"K = O

    Sn in hia dnat

    (_g)1 sn 0, s a consqnc f Eqatn 2.2 ha

    u1 (r2sin8wI/2)-}0=Uh

    2.5

    6)

    (27)

    (21

    whr r 0) s h sram funcion and h subscrps dno para drnaon quaons ) mpy

    ui Hr+ u2Ho= ('oHr- Ho)2sin2tw1/2)-I =0,

    H

    H

    and smay

    K K')

    1219

    2.20

    Annu.

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    TAUB

    c

    u3 = (r2s

    8-IU3 = (w1/ZrZs

    81.

    Th ct

    gvuuV cs 2 1 2 )wH ( +'0.

    s

    at 2.1 rcs t th sgl at

    s ( 0) , , " +-- -. 8+KK = HH s 8 sm(

    21

    122

    .

    Sts t ths at r a H rstrct t lar cts havb gv by Pkrs 9 hs scss th scarlatvstc aal cks shrcal vrtx cks 199.

    13 Shok Wv

    Th scss sa t a Mkwsk sac tl th rvs sct a scss tal Tabs 19 ar shws thatshcks r r crssv ts jst as casscal thry. Wh thsccrs ats a 11 at b a at th shcks a t brac by statts that rat th varabs scrbg th w attrt a gy acrss th shcks. Ths ar th ratvstc akgt ats. Thy ar rv as llws ats 1. a 1. arvat t th statts that

    pu; ufa

    1.1

    132

    r arbtrary cts a vctr s } that hav cts rst rvatvsI vw Stks thr w ay wrt ths ats as

    .ppfun dv=fpu(- g)I/2d4x 1a

    1

    h tgras th rght-ha ss ths ats rr t a arbtrary rsa vl a ths th lt rr t a c hyrsrac wth t

    ra n clsg ths v. Ths ats ar agl v wh thtgras ar scts W ass thy h cas p u a T arscts acrss a hyrsrac Mkwsk sac whch s th hstry a twsa saclk src.

    Annu.

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    RTIVITIC ID CI 35

    B ncon an arrar ornL rrac n a n or-dmnonavm and n m a vom rn ro w ma w a

    r f[u#] n#dv 0

    J3,andf ;#vJ d 0

    wr n# e n noma oL and w av d noaon[

    wr F F ar ondar valu o on wo d oL

    (35)

    (36)

    Snc nra n qaon (15 and (1.) rr o arrar rion oLand inc f and A# ar arrar w m avuJn#

    O(37)

    (38)

    n T#V nin in a quain i ivn quain Equain37) and ( a cad aiviic anin-Hunio quain drivain ivn abv d in a na acim

    n viw quain (. w ma wi quain (. and (. a

    and

    /(-!) -(p+p_)n#

    wr

    wp# = u#

    wo-r m vcr c Y#n# = 0 and Y#Y# =

    (39)

    3.0)

    3

    o nomalid vc in uac L Fo uc vco olow rmquain ( a

    Hnc i

    0 (1.12)

    or

    " V!Y" (3.)

    or o. ca 1 0 rrn a li-ram diconini and 0 rna c wav In ormr ca n mar cro rac dcnn

    Annu.

    Rev.

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    36 TAB

    h hpufac mad up o amn o t ud = 0 nd Equation(33) o w ha ca t icontinuity a nity icontinuity n ti ca wmut av O. Suc a icontinuity cib t ituation tat oan wna gavang d o a ui i boun by a vacuum.

    Wn # 0 Equaon (33) wit two inpnnt vau of pn twoof t fou Euation (30) n tmining t maining two it i convnint todn

    = = (w) = -()

    wit bing t pcic ntapy in unit wit t vlocity o ligt qual to on anto call tat

    VV =

    t tn olow om uccivly multipying Equation (30) by V+ an V_ tat - p -_)(r+L = O (34)

    n mupng quation (30) n w nd

    1n2 + (35)Equation (3) (3) an (3) a t ativitic RaninHugoniot

    quation Fo ma ui vociti an fo i ma ty uc to to o t

    caica toy. Equation (3) tmin a cuv in t p plan tat i tanalogu of t Hugoniot cuv of caical toy. t gnaliation to magntoyoynamic a bn icu in tai by icnowic (67 7) un taumption tat ( s uc tat

    or r < 0 -> 0 and 0> O (36)op o

    Ia (60) a own tat t inquaiti may b iv om intic toy.h a h avc gnazaon o asumpon mad Hman W ni icuion of t Hugoniot cuv.

    It can b own tat, a a ult o t inquaiti 36) t vocity o toc font i tan t vlocity of igt an t inca in ntopy aco aoc i ti o in t pu ump Fut if t ubcipt not tmium aa of t oc an tat bin t oc tn

    p >

    >

    and a t tngt of t oc inca inca and S -S+ ncahon (73) a own tat t ut obtain wn t inquality oroS> 0 pac by oroS 0 an t lativitic Hugoniot cuv i connct.icnowic (75) ha pond ou ha t att conition o not hod navc magnohddnamc

    Annu.

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    FGVTTG F

    RELATIVITIC UID MECHANIC

    Genea-Reavt u DnamAs was poin out in t inouction t a wo yps o uiynamcapoblms n gnal avty: ) Tos in wic t gaviational l o t ucan b nglct bu he e ue som ot ma o ngy must bake accu; a 2 hse whch he gavaa el he u paysa e A mpa subcass f pbems f ype s he sua whee heony mae a gavaa es pese ae he u a s w gavaa

    o poblms of yp 1 o a atconucing viscous ui bas quaonsa (9.1) an (9.) w T is gvn by quaions (8.) an (8.) an covaantvativ s o a mc tnso gv min by nsn quations

    (1

    T mc tnso g min t gavtational l at by t ou wc c by t -ny tno f' n aon 4.) t ntn ataional constant in unts w t wtonan gavtatonal 1 an spcalatvty vocity o lig ; R i Ricc tno n in ms o cuvatu tcno givn in quaons (.4) by

    and R is saa cuvau givn by

    R = R"

    Tus quations (4.1) a a s o scono nonina paal ntiaqaions o g"

    obms o yp ( 1) in gna laivy i fom tos in spcial aivyin at unying spacim is cang om a inowsi on o on wosmtic nso saiss quaion (4.). Tus cocins in ntiaqations sas by vaiabls a n many cass muc mo compicat

    tan tos tat occ n spcal eavy.In pbems ype (2 we suce he gavaa e is jus tu sef Equas (.) ol wi

    (.)In cas u is a pc on T = T o cton 9 an quaons (1) an(1.4) o gions w t ow vaiabs a coninuous an av continuousvavs an quations (1.) an ol acoss isconnuis i yesce t conons a ol acoss socs o a bounas o ui Bcauso Banci niis ta cuvau tnso mus saisy quations (114 a

    a consqunc o quaions (1.). T lat quaons may b cons as a singa o t om ons s obsvaion as cVi 196 to a mtoo sovin poblms n plativiy yoynamcs n "sovng Euas

    Annu.

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    328 A

    (14.), (13), ad Tv = Tv we are required o deemie he e comoe of hemeric eor l he hree ideede comoe of he fouveociy vecor uP,ad wo caa hermodyamic fucio ay ad he caa fucio iuoed o be kow a a fucio of

    ad

    whe he aure of he uid

    ve h e eed ove he e Ea (14.) whe ehe hehhad de he ehad de ae ve We hall d ehd d h h aae gp ad he ala Tp ad b vea (4) v dae ad hee ed he deedevaae he gp ae

    a of mho above we ha dicu a ihy moe eera formof quaio 4) amey he equaio

    RR - g+Ag KT> (43)where A a oa he o-ca comooca coa dcu he aceime decibi he uivere a a whoe from he oi of view of Eiei' heoryof rava he a i made ha he e eee of aee f he

    R ds = dt ( l k/4) (dx dy dz ), (44)

    whee Z ad k - 10 or 1 hi fom for h ie eeme may bededuced he euieme ha ace-e coai hree-dimesioa hyerurface ha admi a ixrameer rou of moio. hee hyeruface rereehe a of oberver whoe word ie are ive by he curve of aameer cad ha ae ohooa o he hyerurface. he hyeurface are ch ha eveyoi i hem equivale o every oher oi ad evey diecio a a oi i hemi equivae o every ohe direcio. hee eomeica euieme erve odeermie he l ha eer io Euaio (14.4)

    If we ow comue from Euaio (14.3) we d haT' (w +p)uIUp

    l

    hee

    KW - 3( RK A-/R-(k+)/R .

    (14.5)

    he do deoe he deivaive wih eec o hu he g ive by (14.4) deemiea aceime a uivere coaii a efec uid wih eery deiy w euep, ad fourveociy ive by Equaio (145). he coordiae yem i which 144)hod i a co ovi oe

    n f Eqn f Mn n Cvng CneWe ow ur o he robem of exrei he uid variabe i em of hee oe by he ue of comovi coodiae whee

    Annu.

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    RELATVISTC FLUID MECHNIC 9

    uP= (_goo)liZOb

    (151

    and hence

    u g / We sha discuss the ase wherew = H is an appopatey gven functon of p and and

    (puPL" ( _ ;) I / Z [(-g) I/Zpu"= o.Hence

    g /Z g Z;Eqan 5 )bm

    23.

    Now n eneal

    i ,

    I n he comong coodnae ssem

    v [ go [( /J ]u;vu / IZ + goo " goo gOt .and as away wp TSv Hence Equaons 6 may be en as

    (5

    [(W + p)(

    gP) 12]

    =_(w+p( gOO) IZ) - (_ goO) lZ TS,w (15 g P ,

    Let 0 b deed b he eaton

    0 = g z TThen

    go / T Bo BoFhe le

    (wp gOi )V= - -I ) 12+OS,i. - 5 5

    Annu.

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    0 A

    n olows fom quaions ( a

    (w p 1/2),O =

    -goo . '

    Hn

    e

    ad

    Fijkjk i+ki = O

    Tuij Cij-Cji

    w

    H

    ijk = 2 3

    v=-w+p(

    gO\ 1 /2+OS'i)=Ci(x)1,;

    goo

    (5.

    whee ad S ae co o xl x x aoe ad a co of a o x'quaion 5. n eadO=(w+ _gOO 12) P ,J

    e

    w p _ go / o+kO h we have

    gOO =-- [I,o + k(xO)] .w pT lin lmn of saim ma n b in as

    ds = oodXOg0 dx dX+g dX dx

    (5.7Annu.

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    RTVTC U MCHC 331

    +{gij +W:2p) [21.i(Cj+eSJI.ilj} dX dxjf w ak th ansfoaion to h nw vaias

    XO = I(x") + S k dxo 2 3

    h coodinat syst ains a coovng on fo

    oxa u"- bg.ax" (-go) / 2

    W aso hav

    d _ 2 d-O) 2p2

    ( IS )d -i d -o-

    d-i d S - (W+ p 2 X - (W+p)2 Ci+ . x X +gj X x ,e

    - p [C(xi + .Jw + p I .

    '

    /2 PU = go = -.wpEaton ( 5 thn cos

    g/2W+ = f(i .

    5.

    9

    h nctonsf(x and Ci(X)ay dtnd o th nta condons o thpo a a to t votcty ot ow as was pond o n th papy MacC & a 7 n that pap xpct oas a gvn o tEnstn atons n ts o t antts j , an (x)

    UMMY

    n th pcdng dscsson w hav sn tha th v consvaton aws thatchaactz h havo of a d n p-ivstc hoy ay takn ov nospcia and gna atvistic d dynaics h fact tha th vocy o popagaon o sond st ss than hat o ght sticts th atons hat st stwn vaios hodynac vaas n patca th st spcc ntnangy canno an aay ncton o th pss and st patc dnstyhs stictons ay dvd fo a ativstc foation of ktc thoy

    h sty of pogssv wavs n spcaatvstc pctd ows shows tasoc wavs st occ o sc otons p n th d vaas acosssocs ay v o consvaton tos as n cassca toy

    aon cng s or gavang n gna av a v o o o pca o ng pncp o vanc Tat son ss act at t aon scng a oon n a gna coona

    Annu.

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    332 TAUB

    syse in inosi spae are oe a s oain en e id is in agriaiona ed

    e Einein eqions or a egriaing id a be regarded a rs inegrao e onseraion eqions deribing e oion is be sed odeerine e id ariae en e eri i non ro oer onsiderionWe o e e reedo of oie of e oordine e in egriingrobe o oe e onerion eqion or e id ribe i er o eeri one nd en by reding e ner o deenden rie o e roe

    e Einen ed eqion rein o be oed re noniner eqiono e e oeen V ei soion re of grea inere in asroyiaprobe rnging ro e eaior o rs o gie o e nierse oeO speia inere re e singariie o e oions o ese eins

    Lieraur Cited

    Crr, B 921n Blak Hole d C D Wi,B S DW pp. 57-24, sp. 55 NwYrk : Grdon & Breach

    Eat C 40 Phys. Rev. 58 : 24lrs, J. 7 In General Relaiity and

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    Rev.

    FluidMech.

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    Annual Review of Fluid MechanicsVolume

    CONTENTS

    SOME OTE ON TE STUDY O FLUID ECANIC IN AMBRIDGE,

    NGLAND, A. M Binnie

    ONTE ARLO SIMULATION O A FLOW, G A Bird 1

    HYDRODYNAMI PROBLEM O SI IN ETRICTED AER, E O. Tuck 33RAG EDUION BY POLYMER, Neil S Berman 47

    VIOU RANONI FLOW, Oleg S Rzhov 65

    DUST XPLOSIONS, Wayland C Grifth 9

    BJTI ETOD OR ATER PREDIION, E Leith 107

    IER EANDERNG, R A Callander 129

    OBY AE-ONGPERIOD ILATION O EAN AND TMO

    ERE, Robert E Dickinson 19

    FLOW O EMATI IQUID RYTAL, James T Jenkins 197

    E STRUCTURE O VORTEX REAKDOWN, Sidney Leibovich 221

    FLOW ROUG SREEN E M Las and J. L Livese 247

    URBULENE AND IXING IN SABLY STRATIIED ATER, rederick S.

    Sherman, Jor Imerer and Gilles M. orcos 67

    PROECT OR OMUTAIONAL FLUID EANI, G S Patterson Jr 289

    LAI LUD AN, A. H a 01

    URBUNRATD OI N PI FLOW, Gerhard Reeho 333

    IER E, George D Ashton 369

    UMERAL OD N ATRAE RACTON AND ADIATION,

    hiang Mei 393

    UMERICAL TOD IN OUNDARyAYER EORY Herbert B Keller 417

    AGNEOYDRODYNAMI O TE ART' YNAMO, F H Busse 4

    UOR ND

    UMULATI ND O ONTRIBUTING UTOR, VOLUME 10

    UMUAI ND O AR ITL, VOLUME 6-10

    463

    Annu.

    Rev.Flu

    idMech.

    1978.1

    0:301-332.

    Dow

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    views.org

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