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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.
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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)
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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.
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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
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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
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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
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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.
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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 =
.
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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'
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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
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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
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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
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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
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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)
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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)
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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.
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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.
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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
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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
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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.
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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
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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
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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
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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
Comology, ed. R Scs pp. 7w r Aadm. x 87 pp
Hks W M 8 Philo. Tran. R. SoLondon Sr A 92:
Irl W 6. Proc R oc. ondon 25 24
Landau L D Lfshz E M FluidMehn p. XV Rdng, Mss Addsn Wly + 56 pp.
ihnrwiz 55 Thore Reaede la Graiaion e de l"Elero-mnm, p 4 Prs Mn +28 pp
nrwi 67 Relaiii ydrodc d MddcNw r Bnjmn + 6 pp
inwiz . 7 In Relaiii luidDynaiCnn Ma Esv Jun
d. C. Caan, pp 4 RmCrmns 424 pp
hnrw A 7 R ad Si PariSe. 28 : 24
Lndblm L 76 Arophy. 8 878McCllm, M . H T, . H 72
Commun Mth Phys 25 78Mi G C. 56 Genera Relaiiy and
Cosmology. ap. V V. w YkWly x + 8 pp
krs C 76 Pro. Nal ad. Si.
US 3 68 ; Sng J. 6 Rliviy: Th SpclTheory msrdm Nrh Hllandxv+45 pp.
Syn J L. 7 The Relaiii Ga.msdm -Hllnd. + 8 pp
Tu, A H. 48 Phy. Re 74 284au . H 5 rh. Raion eh nal.
: 224n S 7 roh J 7 87
907
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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
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