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(1p PH If:Joo Pritt) ie) c)F fLULOS' H'W#' S bc-cJ71I1l-Jø
1 Ctt l:l f 'l~l
~/h\,øL' fV'~
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IVØíY no &J ..
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(
II
t'l"
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29T
HE
ST
AB
ILIT
Y O
F C
OU
ET
TE
FL
OW
Cha
p. V
II
fi of the two integrals included in 14 is positive definite for p, ? 0; but
the
seco
nd is
com
plex
. How
ever
, the
rea
l par
t of 1
4 is
pos
itive
def
inite
for
p, ?
0; i
n fa
ct,
Ii /dV vjZ
re(14) = rrP(r) dr -r dr.
'T
For,
exp
andi
ng th
e in
tegr
and
in (
185)
, we
have
~ I I
I rrp(r)I~~ -~r dr = I rp(r)(rl~~IZ + I~IZ) dr- I rP(r) d~~IZ dr;
'T 'T 'T
but I 1 I
I dlv
lZ I
(I )
d1vl
Z I
i dlv
lzrP(r)Tdr=(I-p,) r2-KTdr=(I-p,) r2Tdr. (187)
'T 'T 'T
The
refo
re, t
he r
ight
-han
d si
de o
f (1
86)
is, i
ndee
d, th
e re
al p
art o
f 14
,R
etur
ning
to e
quat
ion
(180
) an
d eq
uatin
g th
e re
al p
arts
of t
his
equa
tion,
we
obta
in
re(a
)(II-
.raz
I3)+
IZ-.
raZ
(a2f
-tre
(14)
) =
O. (
188)
Whe
n p,
? 7
)z, .
r -:
0 a
nd th
e co
effc
ient
of r
e(a)
in e
quat
ion
(188
) is
posi
tive
defi
nite
; and
so
also
are
the
rem
airu
ng te
rms
in th
e eq
uatio
n.T
here
fore
,re
(a)
-: 0
for
p, ?
7)z
, (18
9)an
d th
e fl
ow is
sta
ble;
this
res
ult i
s en
tirel
y to
be
expe
cted
on
phys
ical
grou
nds.
Nev
erth
eles
s, it
app
ears
to b
e th
e on
ly o
ne w
hich
can
be
esta
blis
hed
by g
ener
al a
naly
tical
arg
umen
ts. I
n pa
rtic
ular
, it d
oes
not
seem
that
one
can
ded
uce
the
gene
ral v
alid
ity o
f th
e pr
inci
ple
of th
eex
chan
ge o
f st
abilt
ies
for
this
pro
blem
. For
exa
mpl
e, b
y eq
uatin
g th
eim
agin
ary
part
s of
equ
atio
n (1
80),
we
obta
in (
cf. e
quat
ion
(176
))i
im(a
)(II+
.raz
I3)
= -
2TaZ
im I
~ d
;r*
dr, (
190)
'T
and
no g
ener
al c
oncl
usio
ns c
an b
e dr
awn
from
this
equ
atio
n; w
hen
p, -
: 0,
even
13
is n
ot p
ositi
ve d
efin
ite!
71. T
he s
olut
ion
for
the
case
of
a na
rrow
gap
whe
n th
e m
argi
nal
stat
e is
sta
tiona
ryH
the
gap
Rz-
Ri b
etw
een
the
two
cylin
ders
is s
mal
l com
pare
d to
thr
mea
n ra
dius
.HR
z+R
i), w
e ne
ed n
ot (
as in
§ 6
8 (b
)) d
istin
guis
h
'0 r
' s: c
' ¡, /
1 /lL
j/? I
I r-
€ k
l- A
- )
YI!#d/J':¡'~ ~C" P?0 17YtJ/)D/MP'Á-ø'iì¿
f'7/ ~?I y f ._(1
85)
(186
)
§ 71 THE STABILITY OF COUETTE FLOW 299
between D and D* in equations (161) and (162); and we ca alo replace
(A+
Blr
Z)
whi
ch o
ccur
s on
the
righ
t-ha
nd s
ide
of e
qua.
tion
(161
) by
( r-Ri J
01 1-(I-p,) Rz-Ri . (191)
In r
ewri
ting
equa
tions
(16
1) a
nd (
162)
in th
e fr
amew
o:ro
f~~p
roxi
-mations, it wil be convenient to measure radial ~.:f the
surf
ace
of th
e in
ner
cylin
der
in th
e un
it d
= R
z-R
i. 'l~
;l-""
"",;:
;;; .
~ =
(r-
Ri)
ld, k
= a
id, a
nd 0
' = ix
/v. (
192)
we
have
to c
onsi
der
the
equa
tions
20 dZ
(DZ-aZ-a)(DZ-aZ)u = --aZ(I-(I-p,)ij-
v 2AdZ
and
(DZ
-aZ
-a)v
= -
'1.
v
By
the
furt
her
tran
sfor
mat
ion
20 dZaz
'1 -+ I '1,
v
the
equa
tions
bec
ome
(DZ
-aZ
-a)(
DZ
-aZ
)u =
(l+
tX~
)v
(DZ
-aZ
-a)v
= ~
Taz
u,
T = _ 4AOi d4
vZ
and
whe
re, n
ow,
and
tX =
-(I
-p,)
.E
quat
ions
(19
6) a
nd (
197)
mus
t be
cons
ider
ed to
geth
er w
ith th
e bo
un:
conditions r d
'1 =
Du
= v
= 0
for.
. = 0
an
1.W
e ar
e pr
imar
ily in
tere
sted
in th
e so
lutio
ns o
f eq
uatio
ns (
196)
(197
) (s
ubje
ct to
the
boun
dary
con
ditio
ns (
200)
) fo
r va
riou
s va
lues
of.
.fo
r w
hich
the
real
par
t of
0' is
zer
o. T
he m
etho
d de
scri
bed
in C
hapt
er I
I.§
31 in
a d
iffe
rent
con
nexi
on is
app
licab
le to
this
pro
blem
. Thu
s, w
Öm
ust o
btai
n so
lutio
ns f
or tw
o ca
ses:
whe
n 0'
is z
ero
and
the
mar
gist
ate
is s
tatio
nary
and
whe
n 0'
is im
agin
ary
and
the
mar
gina
l sta
te is
osci
lato
ry. I
n th
e la
tter
case
for
eac
h va
lue
of a
, ia
mus
t be
dete
rmne
dby
the
cond
ition
that
T is
rea
L. t
In
eith
er c
ase,
we
mus
t fin
d th
e m
ini-
mum
of
T a
s a
func
tion
of a
; and
dep
endi
ng o
n w
hich
oft
he tw
o m
irum
a.is lower, we shall have the onset of
inst
abilt
y as
a s
tatio
nary
sec
onda
ryflow or as overstabilty. Careful experiments on the onset of
inst
abilt
y
t It is, of course, possible that under certain circumstances solutions with thi
pry
do n
ot e
xist
.
I'/JÆ
ÇS
r 19
(1)
TaE STABILITY OF COUETTE FLOW
Cha
p. V
II
and others have failed to reveal any suggestions of over-
stbilty. For this reason, the case a = 0 is the only one which has been
'Cns
ider
ed in
the
liter
atur
e. H
owev
er, a
s no
gen
eral
arg
umen
ts f
or th
e-i
iidity
of
the
prin
cipl
e of
the
exch
ange
of
stab
iltie
s ha
ve b
een
foun
dfo
r th
is p
robl
em, t
he c
ase
of o
vers
tabi
lty r
equi
res
inve
stig
atio
n. W
ereturn to this question in § 72.
Whe
n th
e m
argi
nal s
tate
is s
tatio
nary
, the
equ
atio
ns to
be
solv
ed a
re
(D2_
a2)2
u =
(1+
o:')v
and
(D2_a2)V = - Ta2u,
toge
ther
with
the
boun
dary
con
ditio
ns (
200)
.
(a)
The
sol
utio
n of
the
char
acte
ristic
val
ue p
robl
em fo
r th
e ca
se a
= 0
It c
an b
e re
adily
ver
ifie
d th
at th
e ch
arac
teri
stic
val
ue p
robl
em p
rese
n-te
d by
equ
atio
ns (
200)
-(20
2) is
not
sel
f-ad
join
t in
the
usua
l sen
se. F
orth
is r
easo
n, th
e m
etho
d to
be
desc
ribed
bel
ow w
as p
atte
rned
afte
r th
eon
es w
hich
hav
e be
en f
ound
suc
cess
ful i
n ca
ses
whe
re th
e pr
oble
ms
are
self
-adj
oint
. How
ever
, Rob
erts
has
rec
ently
fou
nd a
var
iatio
nal b
asis
for
the
met
hod;
this
is c
onsi
dere
d in
App
endi
x iv
.T
he m
etho
d of
sol
utio
n w
e sh
all a
dopt
is th
e fo
llow
ing.
. Sin
ce v
is r
equi
red
to v
anis
h at
, =
0 a
nd I,
we
expa
nd it
in a
sin
ese
ries
of
the
form
co
v =
L e
m s
in m
7T'.
(203
)m
=l
Hav
ing
chos
en v
in th
is m
anne
r, w
e ne
xt s
olve
the
equa
tion,
co
(D2_a2)2u = (1+0:') L Cmsinm7T',
m=
l
obta
ined
by
inse
rtin
g (2
03)
in (
201)
, and
arr
ange
that
the
solu
tion
satis
fies
.the
four
rem
aini
ng b
ound
ary
cond
ition
s on
u. W
ith u
det
erm
ined
in this fashion and v given by (203), equation (202) wil
lead
, as
we
shal
lpr
esen
tly s
ee, t
o a
secu
lar
equa
tion
for
T.
The solution of equation (204) is straightforward. The general
solu
tion
can
be w
ritte
n in
the
form
co C
(u=" m A(m)coshaY+B(m)sinhaY+A(m)YcoshaY+
¿ (m27T2+a2)2 1 ~ 1 ~ 2 ~ ~
m=
l
+B
~m),
sin
ha'+
(I+
o:')s
in m
7T' +
~0:
:i7T
2 c
osm
7T'),
m 7
T +
a
where the constants of integration Aim), A~m), Bim), and B~m) are to be
(201
)
(202
)
(204
)
§ 71 THE STABILITY OF COUETTE .FLOW 301
determined by the boundary conditions u = Du = 0 at , = 0 and 1.
The
se la
tter
cond
ition
s le
ad to
the
equa
tions
:
Aim
) =
4mm
x -,m
27T
2+a2
aBim)+A~m) = -m7T,
Aim
) co
sha+
Bim
) si
nha+
A~
m)
cosh
a+B
~m
) si
nha
= (-I )m+1 4m7T0:
m27
T2+
a2'
Aim
) a
sinh
a+
Bim
) a
cosh
a+
A~
m)(
cosh
a+
a si
nh a
)+
+B
~m
)(si
nha+
acos
ha)
= (
_I)m
+l(l
+o:
)m7T
. (20
6)
On
solv
ing
thes
e eq
uatio
ns, w
e fi
nd th
at
Aim
) =
40:m
7T
m27
T2+
a2'
Bim) = ~7T ta+ßm(sinh
a+a
cosh
a)-Y
m
sinh
a)
,
A~m) = _ m7T tsinh2a+ßma(sinha+acosha)-Ymasinha),
~
B~m) = :7T t(sinha
cosh
a-
a)+
ßm
a2
sinh
a-
Ym
(a
cosh
a-
sinh
a)), (2
07)
whe
re.6
= s
inh2
a-a2
,
ßm =
2 ~
o: ?
((-I
)m+
1+co
sha)
,m 7T +a.
and
Ym
= (
-1)1
+1(
1+0:
)+ 2
~o:
2as
inha
. (20
8)m
7T
+a
Now
sub
stitu
ting
for
v an
d u
from
equ
atio
ns (
203)
and
(20
5) in
equ
a-tio
n (2
02),
we
obta
inco L Cn(n27T2+a2)sin n7T'
n=l
co C
(= Ta2 " m A(m)
cosh
aY+
B(m
) si
nhaY
+A
(m)y
co
shaY
+¿ (m27T2+a2)2 1 ~ 1 ~ 2 ~ ~
m=
l
+B~m)'sinha'+(I+o:')sinm7T'+ ~CX:i7T 2cosm7T'). (209)
m 7
T +
a
Multiplyig equation (209) by sin n7T' and integrating over the ra of
.t. w
e ob
tain
a s
yste
m o
f lin
ear
hom
ogen
eous
equ
atio
ns f
or th
~"!
~
t ,t ~~ t 11
TH
E S
TA
BIL
ITY
OF
CO
UE
TT
E F
LOW
Cha
p. V
II
~.. = Om/(m21T2+a2)2; and the requirement that these constants are
not al zero leads to the secular equation
~(n2.;a2)((I+( _1)"Hl cosha)Aim)+(( _1)n+i sinha )
Bim
) +
+(-I)n+l(cosha- 2a sinaJA(m)+
n21T2+a2 2
+ ((-I)n+lSinha- 2a P+(-I)n+icoshaJJB&m))+
n21T
2+a2
+o:Xnm+l3nm~l(n21T2+a2)3:~TII = 0, (210)
( 0
if m
+n
is e
ven
and
m i=
n,
whe
re X
nm =
ì if
m =
n,
~nm
2(
2; 2
- 2(
21
2))
if m
+n
is o
dd. (
211)
n -m
m 1
T +
a 1T
n-m
On
usin
g th
e fir
st tw
o eq
uatio
ns o
f (20
6), e
quat
ion
(210
) si
mpl
ifies
toth
e fo
rm
II n1T f 4m1T0: (( l)m+n 1)
n21T
2 +
a2\m
21T
2 +
a2
2a (( _i)n+lf A (m) sinh
a+
B(m
) co
sha1
+B
(m))
) +
n21T2+a2 t 2 2 J 2
+o:Xnm+l3nm-l(n21T2+a2)3:~T" = 0; (212)
and on substituting for the constants A&m) and B&m; their explicit solutions
give
n in
(20
7), w
e fi
nd th
at e
quat
ion
(212
) si
mpl
ifie
s gr
eatly
and
we
are
left
with
II 4mn1T20: (( i)m+n I)
(n21
T2+
a2)(
m21
T2+
a2)
( 2 2 2;~~:i2 2)J(SinhaCOSha-a)(1+(l+0:)(-I)m+nJ+
n 1T +a sl a-a \
+
(sin
ha-a
co
sha)
(( -
i)n+
i+(I
+o:
)( _
i)m
+i)
_
t:~nh
~)(
sinh
a+a(
_i)m
+l)(
( _i
)m+
n_i))
+m
1T
+a
+l3nm+o:Xmn-l(n21T2+a2)3:~ITII = O. (213)
§ 71
TH
E S
TA
BIL
ITY
OF
CO
UE
TT
E F
LO
W30
3A
firs
t app
roxi
mat
ion
to th
e so
lutio
n of
equ
atio
n (2
13)
is o
btan
ed b
yse
tting
the
(I, I
)-el
emen
t of
the
mat
rix
equa
to z
e. W
e fi
d
l(1T2+a2)3 T~2 = ìo:+l-
2a1T
2(2+
0:)
.(1T2+a2)2(sinh2 a-a2) ((smh a cosh a-a)+(sa-ccoa)).
(2"1
4)O
n fu
rthe
r si
mpl
ific
atio
n, th
is g
ives
2 (1
T2+
a2)3
. ...
.. ..
'. .
T=
_ _
(215
)2+
0: a
2p-I
6a1T
2cos
h2la
j((1
T2+
a2)2
(sin
a+a)
J) .
We observe that, apart from the factor2j(2+0:), this expi:~..?
is id
entic
al w
ith w
hat w
as fo
und
in C
hapt
er II
(§
17, e
quat
i (m
Jl_
the Rayleigh number for the simple Bénard problem by the"~f1rlll
met
hod
in th
e fis
t app
roxi
mat
ion
for
the
case
of t
wo
rigid
~c
Con
sequ
ently
, in
this
app
roxi
mat
ion
(cf.
equa
tion
(199
)),
2 34
30 ..
, ,T" = - X 1715 = - and amin = 3'12.....
2+0:
I+p.
We
shal
l see
bel
ow (
§ (b
)) th
at f
or 0
.. p
. .. I
equ
atio
n (2
16)
givalues for the critical Taylor number which do not differ from tI
obtained in the higher approximations by more than one per cent. 1i
reason for this relatively high accuracy of the solution in the fi
appr
oxim
atio
n w
il be
mad
e ap
pare
nt in
§ (
d).
(b)
Num
eric
al r
esul
tsA
met
hod
of s
olvi
ng th
e in
fini
te o
rder
cha
ract
eris
tic e
quat
ion
whi
ch(2
13)
prov
ides
for
T w
ould
be
to s
et th
e de
term
inan
t for
med
by
the
fin rows and columns of the secular matrix equal to zero and let n tae
incr
easi
ngly
larg
er v
alue
s. I
n pr
actic
e, th
e us
eful
ness
of
this
met
hod
wi
depend largely on how rapidly the lowest positive root of the resultin
equa
tion
of o
rder
n te
nds
to it
s lim
it as
n -
+ 0
0. I
t app
ears
that
for
the
prob
lem
on
hand
, the
pro
cess
con
verg
es q
uite
rap
idly
.In
Tab
le X
XX
II th
e va
lues
of
T o
btai
ned
with
the
aid
of e
quat
ion
(213
) in
the
diff
eren
t app
roxi
mat
ions
are
list
ed f
or th
ose
valu
es o
f a
(for
the
assi
gned
p.'s
) at
whi
ch it
was
fou
nd (
by tr
ial a
nd e
rror
) th
at T
atta
ined
its
min
imum
val
ue. F
rom
an
exam
inat
ion
of th
is ta
ble,
itWould appear that for p. ? -1.0, the third approximation provides T
to well within one per cent of the true value. For - 3.0 ~ p. ~ -1,0,
the
calc
ulat
ions
wer
e ca
rrie
d ou
t to
as h
igh
appr
oxim
atio
ns a
s se
med
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(a)
(b)
Figu
re 7
.6.3
. l\1
otio
n in
a r
otat
ing
dish
of
wat
er 4
in. d
eep
due
to s
low
tran
slat
ion
of a
circular cylinder (E) of height i in. fronì right to left across the bottom of the dish,
view
edfr
om a
bove
. In
(a)
the
dye
has
been
rel
ease
d at
a (m
ovin
g) p
oint
A a
bove
the
top
of th
ecy
linde
r an
d di
rect
ly a
head
of
it, a
nd B
(a
divi
ding
poi
nt),
C a
nd D
are
sub
sequ
ent f
)osi
tions
of th
e dy
e; in
(b)
the
poin
t of
rele
ase
(A)
is \v
ithin
the
upw
ard
proj
ectio
n of
the
cylin
der
and the dye remains in the blob D. The flow evidently has a two-dimensional character.
(Fro
m T
aylo
r ,1
923.
)
OY
A/i?
/hie
S //
f ( If
/V -
:-...
./Tn
ö l)J
c. T
l 0 -
l To
FL
U (
0
rJ (
c- d
, k; l
l ~(L
f-,ç
l-/?
Ff( () r'
cÐ~l11? /~c¡ /l'kÇr
/ 9' 6' .7-
556
Flo
w o
f effe
ctiv
ely
invi
scid
flui
d w
ith v
ortic
ity (
7.6
is to say, if the basic rotation in the lateral plane is anti-clockwise, the
Cor
iolis
for
ce te
nds
to tu
rn th
e di
rect
ion
of m
otio
n of
an
elem
ent r
elat
ive
toth
e ro
tatin
g fr
ame
to th
e rig
ht. M
oreo
ver,
the
Cor
iolis
forc
e is
line
ar in
the
velo
city
, and
wil
tend
to c
hang
e th
e di
rect
ion
of th
e co
mpo
nent
of
u in
ala
tera
l pla
ne a
t the
sam
e ra
te fo
r al
l mag
nitu
des
and
dire
ctio
ns o
f tha
tco
mpo
nent
. Thu
s a
mat
eria
l eie
men
t who
se m
otio
n is
dom
inat
ed b
y th
eC
orio
lis f
orce
mov
es o
n a
path
who
se p
roje
ctio
n on
a la
tera
l pla
ne is
a c
ircl
e,th
e w
hole
cir
cle
bein
g co
mpl
eted
in a
tim
e of
ord
er .0
-1. T
he C
orio
lis f
orce
evid
ently
tend
s to
res
tore
an
elem
ent t
o its
initi
al p
ositi
on in
the
late
ral p
lane
.N
ote
that
ther
e is
no
spec
ial s
igni
fican
ce a
bout
the
posi
tion
of th
e ax
is o
fro
tatio
n, s
o fa
r as
the
Cor
iolis
for
ce is
con
cern
ed.
zC
ompo
nent
of
u in
late
ral
7"'ti~ W w"""" -
"", Coriolis force
Dir
ectio
n of
rotation of a
xes
"
Figu
re 7
.6. i
. To
show
the
dire
ctio
n of
the
Cor
iolis
for
ce w
hich
act
s in
a r
otat
ing
refe
renc
esy
stem
. The
(y,
z)-
plan
e is
nor
mal
to th
e ax
is o
f rot
atio
n.
Sinc
e th
e m
otio
ns o
f di
ffer
ent e
lem
ents
of
the
flui
d no
rmal
ly e
xerc
ise
ast
rong
mut
ual i
nflu
ence
thro
ugh
the
actio
n of
pre
ssur
e gr
adie
nts,
it is
desi
rabl
e to
con
side
r al
so th
e co
llect
ive
effe
ct o
f C
orio
lis f
orce
s on
dif
fere
ntel
emen
ts. S
uppo
se th
at r
elat
ive
to th
e ro
tatin
g ax
es th
ere
is g
ener
ated
am
otio
n w
hich
lead
s to
a n
on-z
ero
and
posi
tive
valu
e of
the
rate
of
expa
nsio
nin
the
late
ral p
lane
, tha
t is,
to a
pos
itive
val
ue o
f
ôv/ôy+ ôw/ôz
(in
term
s of
the
co-o
rdin
ate
syst
em o
f fi
gure
7.6
.1)
over
a c
erta
in r
egio
n of
the
flui
d. T
he a
rea
encl
osed
by
the
proj
ectió
n on
a la
tera
l pla
ne o
f a
clos
edm
ater
ial c
urve
in th
is r
egio
n w
il th
en b
e in
crea
sing
. The
eff
ect o
f th
e C
orio
lisfo
rce
acco
mpa
nyin
g th
is g
ener
al o
utw
ard
mov
emen
t of
the
proj
ecte
d cu
rve
is to
gen
erat
e a
tang
entia
l mot
ion
of th
e m
ater
ial c
urve
, whi
ch m
akes
ane
gativ
e co
ntri
butio
n to
the
circ
ulat
ion
roun
d it.
Thi
s ch
ange
in th
e ci
rcul
a-tio
n of
the
mot
ion
rela
tive
to r
otat
ing
axes
is o
f co
urse
sim
ply
the
chan
gere
quir
ed to
kee
p th
e ci
rcul
atio
n re
lativ
e to
an
abso
lute
fra
me
cons
tant
dur
ing
a m
ovem
ent l
eadi
ng to
incr
ease
of
the
proj
ecte
d ar
ea o
n a
late
ral p
lane
. Now
the
new
tang
entia
l mot
ion
of th
e m
ater
ial c
urve
itse
lf le
ads
to a
Cor
iolis
forc
ein
a d
irec
tion
norm
al to
the
curv
e; a
nd in
asm
uch
as th
e ne
w ta
ngen
tial
mot
ion
mak
es a
net
neg
ativ
e co
ntri
butio
n to
the
circ
ulat
ion,
the
asso
ciat
edC
orio
lis f
orce
is d
irec
ted
mos
tly in
war
ds a
nd s
o te
nds
to p
rodu
ce a
red
uctio
n
7.6) Flow systems rotating as a whole 557
in th
e ar
ea e
nclo
sed
by th
e pr
ojec
ted
curv
e. I
n ot
her
wor
ds, a
t pla
ces
in th
eflu
id w
here
ther
e is
a p
ositi
ve v
alue
of ô
vjôy
+ ô
wjô
z th
e ef
fect
of C
orio
lisfo
rces
is to
tend
to p
rodu
ce a
neg
ativ
e va
lue,
and
vic
e ve
rsa.
Thu
s th
e ne
tef
fect
of
Cor
iolis
for
ces
is to
opp
ose
disp
lace
men
ts o
f fl
uid
elem
ents
whi
chto
geth
er le
ad to
cha
nge
of th
e ar
ea e
nclo
sed
by th
e pr
ojec
tion
of a
mat
eria
lcu
rve
on a
late
ral p
lane
, tha
t is,
to a
non
-zer
o ex
pans
ion
in a
late
ral p
lane
.T
he e
xten
t to
whi
ch th
e re
stor
ing
effe
ct o
f Cor
iolis
forc
es r
estr
icts
the
disp
lace
men
t of
flui
d el
emen
ts e
vide
ntly
dep
ends
on
the
rela
tive
mag
nitu
des
of C
orio
lis f
orce
s an
d ot
her
forc
es a
ctin
g on
the
flui
d; a
nd in
the
pres
ent
cont
ext t
hese
oth
er f
orce
s ar
e in
ertia
for
ces.
If
U is
a r
epre
sent
ativ
e ve
loci
tym
agni
tude
(re
lativ
e to
rot
atin
g ax
es)
and
L is
a m
easu
re o
f th
e di
stan
ce o
ver
whi
ch u
var
ies
appr
ecia
bly,
the
ratio
of t
he m
agni
tude
s of
the
term
s u.
V'u
and
zSl x
u in
(7.
6.1)
is o
f or
der
UjL
.o.
The value of this ratio, known as the Rossby number in recognition of the
wor
k of
the
Swed
ish
met
eoro
logi
st, p
rovi
des
a co
nven
ient
mea
sure
of
the
impo
rtan
ce o
f Cor
iolis
forc
es. W
hen
Uj L
.o ~
x, C
orio
lis fo
rces
are
like
ly to
caus
e on
ly a
slig
ht m
odif
icat
ion
of th
e fl
ow p
atte
rn; b
ut w
hen
UjL
.o ~
x,
the
tend
ency
for
Cor
iolis
for
ces
to o
ppos
e an
y ex
pans
ion
in a
late
ral p
lane
is li
kely
to b
e do
min
ant.
And
in th
e in
term
edia
te c
ase
whe
n U
j L.o
is o
f or
der
unity
, an
inte
rest
ing
mix
ture
of
effe
cts
is to
be
expe
cted
, som
e hi
nt o
f w
hich
was
pro
vide
d by
the
disc
ussi
on o
f ste
ady
axis
ymm
etric
flow
with
sw
irl in
§ 7
.5.
Steady flow at small Rossby number ~ t !
The
dom
inan
ce o
f Cor
iolis
forc
es in
flow
at v
alue
s of
UjL
.o s
mal
l com
-pa
red
with
uni
ty h
as s
tran
ge c
onse
quen
ces
whe
n th
e fl
ow is
als
o st
eady
rela
tive
to r
otat
ing
axes
, as
was
fir
st p
oint
ed o
ut b
y J.
Pro
udm
an (
xQx6
). I
nst
eady
flo
w a
mat
eria
l ele
men
t of
flui
d m
oves
alo
ng th
e sa
me
stre
amlin
e,w
ithou
t rev
ersa
l of d
irect
ion,
at a
ll tim
es. B
ut th
e st
rong
Cor
iolis
forc
es .
oppo
se a
ny d
ispl
acem
ent o
f fl
uid
elem
ents
lead
ing
to a
non
-zer
o ex
pans
ion
in a
late
ral p
lane
. It f
ollo
ws
that
, in
the
limit
U L
.o ~
0 th
e fo
rmg-
lhe
stre
amlin
es m
ust b
e co
nsis
ent
wit
a ze
ro r
ate
0 ex
pans
ion
in a
lae
can
esta
blis
h th
is r
esul
t for
mal
ly b
y no
ting
that
whe
n ôu
/ôt =
0 a
ndth
e te
rm u
. V'u
is n
eglig
ible
by
com
paris
on w
ith th
e C
orio
lis fo
rce,
the
equa
tion
of m
otio
n (7
.6.1
) be
com
es1
zSl x
U =
--
V'p
, (7.
6.2)
p. x (ÔP ôp ÔP)
that is, p ôx' ôy' ôz = (0,2.oW, -2.oV)
with
the
co-o
rdin
ates
sho
wn
in fi
gure
7.6
. x. E
limin
atio
n of
P th
en g
ives
ôv ôw
ôy + ôz = 0 (7.6.3)
in s
tead
y fl
ow r
elat
ive
to r
otat
ing
axes
, and
, as
a co
nseq
uenc
e of
the
mas
s-
i. E t: j) ~ it ~
l-. y
. , ~
.._ 1
'l -
'2 l
iAJ
~)'r
Flow of effectively inviscid fluid with vorticity (7.6
\ lôu/ôx = o. I. (7.6.4)
The
cur
ious
pro
pert
y of
thes
e ap
prox
imat
e eq
uatio
ns w
hich
hol
d w
hen
U/L
D. ~
1 is
that
the
mot
ion
in th
e la
tera
l or
(y, z
)-pl
ane
is n
ot c
oupl
ed w
ithth
e m
otio
n pa
ralle
l to
the
axis
of r
otat
ion.
Fur
ther
mor
e, n
óne
of th
e flo
wpr
oper
ties
depe
nds
on x
. Pro
udm
an's
theo
rem
is s
omet
imes
sta
ted
as b
eing
that
'slo
w' s
tead
y m
otio
ns r
elat
ive
to r
otat
ing
axes
mus
t be
two-
dim
ensi
onaL
.Si
nce
in th
is b
ook
we
have
reg
arde
d th
e te
rm tw
o-di
men
sion
al m
otio
n as
impl
ying
that
the
velo
city
vec
tor
ever
yher
e lie
s in
a c
erta
in p
lane
, it w
ould
be m
ore
appr
opri
ate
here
to s
ay th
at s
tead
y m
otio
ns a
t sm
all R
ossb
y nu
mbe
rm
ust b
e a
supe
rpos
ition
of
a tw
o-di
men
sion
al m
otio
n in
the
late
ral p
lane
and
an a
xial
mot
ion
whi
ch is
inde
pend
ent o
f x.
The
val
ue o
f the
vel
ocity
com
pone
nt u
par
alle
l to
the
axis
of r
otat
ion
isev
iden
tly d
eter
min
ed b
y th
e bo
unda
ry c
ondi
tions
. It w
il of
ten
happ
en th
atev
ery
line
in th
e fl
uid
para
llel t
o th
e ax
is m
eets
a s
tatio
nary
bou
ndar
y; in
such
cas
es th
e ab
ove
rela
tions
req
uire
u =
0 e
very
whe
re, a
nd o
nly
the
two-
dim
ensi
onal
mot
ion
rem
ains
. The
pho
togr
aphs
on
plat
e 23
(m
ade
by G
. 1.
Tay
lor
man
y ye
ars
befo
re th
e su
bjec
t of
rota
ting
flui
ds h
ad a
ttrac
ted
muc
hno
tice)
of t
he fl
ow in
an
open
flat
dis
h of
wat
er w
hich
is r
otat
ing
show
that
Cor
iolis
for
ces
do in
deed
mak
e th
e m
otio
n tw
o-di
men
sion
al in
thes
e ci
rcum
-st
ance
s. I
n fi
gure
7.6
.2 (
plat
e 23
) a
drop
of
colo
ured
liqu
id h
as b
een
draw
nou
t int
o a
thin
she
et b
y a
'slo
w' m
otio
n im
part
ed to
the
rota
ting
flui
d, a
ndth
e tw
o ph
otog
raph
s, ta
ken
by a
cam
era
plac
ed o
n th
e ax
s of
rot
atio
n of
the
dish
, sho
w th
at th
e sh
eet i
s ev
eryw
here
par
alle
l to
the
axis
and
that
the
com
pone
nt o
f vel
ocity
in a
late
ral p
lane
is in
depe
nden
t of x
. The
flow
rev
eale
dby the streak of dye released from oint A in figure .6. late 2 is more
start mg. e motion re ative to rotatmg axes is ue ere to a portion of a
circ
uIa
cylin
der
E b
eing
dra
wn
slow
ly a
cros
s th
e bo
ttom
of
the
dish
. The
dept
h of
wat
er is
4 in
. and
the
cylin
der
is 1
in. h
igh,
and
in a
non
-rot
atin
gfl
uid
the
wat
er w
ould
pas
s ov
er th
e to
p of
the
mov
ing
cylin
der
as w
ell a
sro
und
the
side
s. H
owev
er, t
he d
ye e
mer
ging
fro
m a
poi
nt il
in. a
bove
the
top
of th
e cy
linde
r an
d di
rect
ly a
head
of
it (f
igur
e 7.
6.3a
) di
vide
s at
poi
nt B
,as
if it
had
met
an
upw
ard
exte
nsio
n of
the
cylin
der,
and
pas
ses
roun
d th
isim
agin
ary
cylin
der
in tw
o sh
eets
, t th
e sh
eet o
n on
e si
de (
D)
even
sho
win
gse
para
tion
and
the
form
atio
n of
edd
ies.
In fi
gure
7.6
.3b
the
dye
is b
eing
're
leas
ed f
rom
a p
oint
just
insi
de th
e cy
lindr
ical
reg
ion
vert
ical
ly a
bove
the
body
, and
col
lect
s in
a b
lob
whi
ch m
oves
with
the
cylin
der.
It s
eem
s th
at th
efl
ow o
utsi
de th
e up
war
d pr
ojec
tion
of th
e cy
linde
r is
app
roxi
mat
ely
the
sam
eas
if th
e cy
linde
r ex
tend
ed f
rom
the
botto
m to
the
top
of th
e la
yer
of w
ater
,an
d th
at v
ertic
ally
abo
ve th
e cy
linde
r th
ere
is a
cyl
indr
ical
col
umn
of w
ater
whi
ch m
oves
with
it. T
hus
the
mot
ion
is tw
o-di
men
sion
al in
the
way
that
isco
nsis
tent
with
tran
slat
ion
of th
e cy
linde
r, e
ven
thou
gh th
e he
ight
of
that
cylin
der
is o
nly
one-
quar
ter
of th
e de
pth
of w
ater
.
t Ano
ther
str
ikin
g ph
otog
raph
of
this
phe
nom
enon
is r
epro
duce
d in
Gre
ensp
an's
boo
k.
558
conservation equation,
7.6) Flow systems rotating as a whole 559
Whe
n th
e fl
uid
is n
ot e
nclo
sed
by s
tatio
nary
bou
ndar
ies
inte
rsec
ted
bylin
es p
aral
lel t
o th
e ax
is o
f ro
tatio
n, th
e va
lue
of th
e ax
ial v
eloc
ity c
ompo
nent
in th
e fl
uid
wil
usua
lly b
e de
term
ined
by
cond
ition
s at
an
inne
r bo
unda
ry.
An
inte
rest
ing
and
fund
amen
tal c
ase
is fl
ow d
ue to
tran
slat
ion
of a
rig
id b
ody,
with
vel
ocity
U p
aral
lel t
o th
e ax
is o
f ro
tatio
n, th
roug
h fl
uid
whi
ch is
un-
boun
ded
in th
at d
irec
tion.
It s
eem
s th
at h
ere
the
abov
e re
quir
emen
ts f
orfl
ow w
ith U
/LD
. -+
0 c
an b
e sa
tisfi
ed o
nly
if a
ll th
e fl
uid
in th
e cy
linde
r ci
r-cu
msc
ribi
ng th
e bo
dy m
oves
par
alle
l to
the
axis
with
the
body
, the
com
-po
nent
of v
eloc
ity in
a la
tera
l pla
ne b
eing
zer
o ev
eryh
ere.
Exp
erim
ents
do
in f
act s
ugge
st th
at a
col
umn
of f
luid
is p
ushe
d ah
ead
of a
bod
y m
ovin
gpa
ralle
l to
the
axs,
alth
ough
the
flow
beh
ind
the
body
see
ms
not t
o be
who
llyin
acc
ord
with
the
abov
e si
mpl
e th
eory
. Fur
ther
ref
eren
ce to
thes
e ex
peri
-m
ents
wil
be m
ade
late
r in
this
sec
tion.
In th
e ca
se o
f bo
dies
mov
ing
eith
er p
aral
lel t
o th
e ax
is o
f ro
tatio
n or
nor
mal
to it
, the
abo
ve th
eory
for
flow
at s
mal
l Ros
sby
num
ber
lead
s to
the
conc
lusi
onth
at a
so-
calle
d 'T
aylo
r co
lum
n' o
f fl
uid
para
llel t
o th
e ax
is a
ccom
pani
es th
ebo
dy. A
t the
edg
e of
the
colu
mn
ther
e ar
e sh
ear
laye
rs w
here
the
vort
icity
isla
rge.
It
is to
be
expe
cted
that
the
appr
oxim
ate
linea
r eq
uatio
n (7
.6.2
)is
not a
pplic
able
eve
ryhe
re in
thes
e la
yers
, alth
ough
the
cons
eque
nces
for
the
who
le f
low
fie
ld a
re n
ot w
ell u
nder
stoo
d.
Prop
agat
ion
of w
aves
in a
rot
atin
g fl
uid
We
have
see
n th
at a
ny d
ispl
acem
ent o
f th
e el
emen
ts o
f a
flui
d in
rig
id-
body
rot
atio
n w
hich
lead
s to
a n
on-z
ero
expa
nsio
n in
a la
tera
l pla
ne is
acco
mpa
nied
by
Cor
iolis
for
ces
whi
ch te
nd to
elim
inat
e th
is e
xpan
sion
.Si
nce
ther
e is
no
diss
ipat
ion
of e
nerg
y in
an
invi
scid
flu
id, i
t fol
low
s th
at a
disp
lace
men
t of
this
kin
d w
hich
is g
iven
to th
e fl
uid
initi
ally
may
set
up
anos
cila
tion.
Thi
s ra
ises
the
poss
ibili
ty th
at a
trai
n of
wav
es c
an p
ropa
gate
thro
ugh
a ro
tatin
g fl
uid,
with
dif
fere
nt p
hase
s of
the
wav
e be
ing
asso
ciat
edw
ith p
ositi
ve a
nd n
egat
ive
valu
es o
f th
e ex
pans
ion
in a
late
ral p
lane
. We
can
exam
ine
this
pos
sibi
lity
by s
eeki
ng s
olut
ions
of
the
equa
tion
gove
rnin
gde
part
ures
fro
m a
sta
te o
f ri
gid-
body
rot
atio
n w
hich
are
per
iodi
c in
tim
ean
d in
cer
tain
spa
tial c
o-or
dina
tes.
We
shal
l con
side
r fi
rst t
he p
hysi
cally
sim
ple
case
of
an a
xisy
mm
etri
c w
ave
mot
ion
with
pro
paga
tion
in th
e di
rect
ion
of th
e ax
is o
f ro
tatio
n. R
elat
ive
toro
tatin
g ax
es, t
he w
ave
mot
ion
is s
uper
impo
sed
on s
tatio
nary
flu
id a
nd s
o,fo
r a
sim
ple
harm
onic
wav
e, a
ll fl
ow q
uant
ities
var
y si
nuso
idal
ly in
tim
e w
ithan
gula
r fr
eque
ncy
ß s
ay (
perio
d 27
T/ß
) an
d si
nuso
idal
ly w
ith r
espe
ct to
the
axial co-ordinate x with wave-number ex say (wavelength 27Tjex). The
gove
rnin
g eq
uatio
n fo
r fl
ow r
elat
ive
to r
otat
ing
axes
is (
7.6.
1), i
n ax
isym
-.' inetric form, and, following
the usual pattern of
inve
stig
atio
n of
wav
e m
otio
ns,
''we
mig
ht p
roce
ed b
y ne
glec
ting
term
s in
this
equ
atio
n ø
f deg
ree
high
er th
an'th
e fi
rst i
n qu
antit
ies
repr
esen
ting
the
depa
rtur
e fr
om th
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