Analytic expression for Taylor-Couette stability boundary

7 Pages·0.2 MB·English

by  A. Esser

Analytic expression for Taylor–Couette stability boundary
Alexander Esser and Siegfried Grossmann
Fachbereich Physik der Philipps–Universit¨at, Renthof 6, D-35032, Marburg, Germany
(February 1, 2008)
The radius ratio η =r1/r2 characterizesthe geometryof
the system. For counterrotation a nodal surface exists
We analyze the mechanism that determines the boundary
where u =0,
ϕ
of stability in Taylor–Couette flow. By simple physical argu-
ment we derive an analytic expression to approximate the
stability line for all radius ratios and all speed ratios, for R1 ηR2
6 co- and counterrotating cylinders. The expression includes rn =r2sR1 −η−1R2, R2 ≤0. (2)
9 viscosity and so generalizes Rayleigh’s criterion. We achieve −
9
agreement with linear stability theory and with experiments
1
inthewholeparameterspace. Explicitformulaearegivenfor
2. Derivation of the stability criterion
n limiting cases.
a
J PACS 47.15.Fe, 47.20.Gv Let us start with Rayleigh’s stability argument for in-

1 viscidflow[17]: Considertwoneighbouringfluid ringsat
1 r and r+δr of the laminar flow. Stability is guaranteed

iftheir hypotheticalexchangeby virtualmotionthrough
3 1. Introduction
the surrounding flow costs kinetic energy. Neglecting
v
momentum loss by viscosity, the angular momenta per
6
3 The Taylor–Couettesystemhas beenstudied formore mass L(r)=ruϕ(r) of the rings are conserved, and with
1 than 100 years, beginning with the first experiments by their kinetic energy per mass being u2/2 we arrive at
ϕ
9 Couette[1],Mallock[2],andthepioneeringworkbyTay- Rayleigh’s famous stability criterion L2(r)′ > 0 when
0 lor [3]; for a recent review see Tagg [4]. considering the energy balance
5
We here focus on the question when and why the la-
9 minar flow becomes unstable. This can and has been 1 L(r+δr) 2 1 L(r+δr) 2
t/ evaluated with linear stability theory [3,5–9], and was −δEkin = 2 r − 2 r+δr
a
m measured for a variety of radius ratios [3,10–13]. The 1(cid:16) L(r) 2 (cid:17) 1 L((cid:16)r) 2 L2((cid:17)r)′
2
- aim of this note is to elucidate the physical mechanism + 2 r+δr − 2 r = r3 (δr) >0.
d and derive a simple approximate analytical formula for (cid:16) (cid:17) (cid:16) (cid:17)
n the stability boundary in the whole parameter space. δr has been choosen infinitesimal and the prime denotes
o Earlierattempts of this kind have been made by Don- the r-derivative. Stability is predicted for corotation if
c
nelly andFultz [10],Coles [14],Joseph[7],andEckhardt R1 <R2/η,andinthecaseofcounterrotationforR1 =0
:
v and Yao [15]. However, these authors did not consider only.
i all relevant aspects. Those include the smooth crossover Nowwetakeintoaccountviscosity. Duringthemotion
X
from a solid boundary (for corotation) to a free one (for to exchange the two fluid rings as described above they
r
a sufficiently strong counterrotation) and the localization looseangularmomentumandthus kinetic energyby dis-
of the initial perturbation in the center of the relevant sipation. The frictional energy loss per mass is given by
gap. We elaborate that in Sects. 2–4. the viscous force per mass times the shift δr,
Weconsideranincompressiblefluidwithkinematicvis-
ν
cosity ν flowing between two concentric cylinders with δEvis ∼ ℓ2 δur δr.
inner radius r1 and outer radius r2, and angular veloci-
(cid:0) (cid:1)
ties ω1 > 0 and ω2, positive for corotation and negative ℓdenotesthetypicaldiameterofthering(inradialaswell
for counterrotation. This fluid flow is characterized by
asinaxialdirection). δu =δr/δtistheradialvelocityof
r
the Reynolds numbers R = ω r d/ν (i = 1,2), where
i i i the shift which takes place within time δt. The ‘ ’ sym-
d=r2 r1 is the gap size between the cylinders. bol means ‘up to a factor oforder unity’. For ins∼tability,
−
Thewell-known[5,16]laminarflowispurelyazimuthal
δEkin hastoexceedδEvis. Itslowerlimitisrelatedtothe
and has the velocity profile
upperlimitforδt,whichissetbythediffusivetimescale,
on which the rings loose angular momentum, δt ℓ2/ν.
B ∼
uϕ(r)=Ar+ with (1) We now obtain from δEkin δEvis
r ∼
A=−η21ω1−η2ω2, B = ω11−ηω22 r12. − L2r(3r)′ (δr)2 ∼ νℓ42 (δr)2 (3)
− −
1
2
as a relation which characterizes the stability boundary, 1 (1+η)
or, after inserting the profil (1), i.e., L=Ar2+B, R1,c(η)= α2 2η (1 η)(3+η), (7)
−
1+2 η (ηR1−R2) 2 rrn22 −1 ∼ dℓ 4. (4) wsohrebreedailnl cαo.nstWanetfifaxctαorbspymmeaanttchbiyng‘∼R’1,hca(ηv)e btoeelninaeabr-
h i h i (cid:16) (cid:17) stability theory in the narrow gap limit, because in this
Here rn2 = −B/A, which for negative R2 coincides with limit R1,c is known very accurately, e.g. [3,5,16]. The
the squared position of the nodal surface. combination 4(1 η)R2 /(1 + η) = T is the critical
In contrast to the inviscid case the timescale for the − 1,c c
Taylor number for R2 = 0 whose value for η 1 is
motionandthesizeoftheringsenterhere;theexpression 2/α4 = T 3416 [5]. Thus we have α = 0.1556→corre-
c
on the rhs. of (3) has also been used in [14]. As we will ≈
sponding to ℓ = d/6.427, a reasonable value, which was
see soon, the radial position of the rings is important as
also found by the same reasoning in [14].
well.
Since R1,c diverges for η 1 and η 0, one better
ℓ is some small but finite fraction of the part of the → →
rescales R1 and R2 to discuss the stability curve R1(R2)
gap which is susceptible to instability. For positive R2 for general radius ratio η, cf also [6]. We employ 1 =
this is the whole gapbetweenthe cylinders,therefore we R
R1/R1,c and 2 = R2/(ηR1,c), which will happen to be
writeℓ=αdwithαaconstantconsiderablysmallerthan R
usefulalsointhelimitwhentheReynoldsnumbersgoto
1 [14]. We shall determine it by comparison with linear
infinity.
stability analysis.
After eliminating α =ℓ/d from (4) with (7) we arrive
In the counterrotating case L2(r) decreases in the
at the final formula for determining the stability bound-
range from the inner cylinder up to the nodal surface
ary:
only. We therefore compare ℓ with dn = rn r1, which
−
is an old idea, cf also [10]. We further take into account 2 r2 r2 (1+η)2 d
that the flow patterns appearing at the instability line R1−R2 nr−2 p (1 η)(3+η) =∆(a dn)−4. (8)
extendbeyondthenodalsurface[3]. Thismaybeunder- (cid:16) (cid:17) p −
stoodasachangeinthetypeofboundaryfromsolid(for
We now calculate a as explained above and get
R2 0) to sort of free (for R2 < 0). The simplest way
of d≥escribing this crossoveris −1
(1+η)3
a(η)=(1 η) η . (9)
ℓ=αd∆(adn) with ∆(x)= x , if x<1 . (5) − "s2(1+3η) − #
d 1 , if x 1
(cid:26) ≥
a increases weakly with η from 1.4 to 1.6. We always
Thus, ℓ=αadn for R2 ≤R2∗, where R2∗ is det∗ermined by tookthisa(η),butdifferencesto≈takinga=constwithin
the condition ad /d = 1. If a > 1 we get R < 0. This
n 2 this range could hardly be noticed in our following fig-
delayinthe decreaseofℓ whenR2 becomes negativecan ures. Since ad can also be understood as the radial
n
also be seen in Fig. 12 of Ref. [3] for the axial spacing of
size ofa Taylorvortexforcounterrotatingcylinders,this
vortices. We expect this crossover to be even a smooth
rangeof a can be confirmedby inspecting severalfigures
process. Therefore we demand that the stability curve
in Refs. [3–5,8,9].
R1(R2) is at least continuously differentiable. This will
fix the parametersa andR2∗. Colesuses the sameansatz ∗ η2+4η+1 3+η
ℓ = αadn but for the whole range R2 ≤ 0 [14], which R2(η)=− 2(1+η)2 1+3η (10)
makes the stability boundary discontinuous at R2 =0. r
Now we address the remaining question at which po- is also weakly dependent on η, ranging between 0.75
−
sition rp the virtual motion of the rings has to be con- and 0.90. The continuousdifferentiabilityofthe sta-
≈−
sidered. At firstsight, one mightthink to choose it as to bility curve 1( 2) at adn/d = 1 can not trivially be
R R
maximize the virtual energy gain. This leads to rp =r1. satisfied. Butitispossiblebecause∆enterstwicein(8),
But more reasonable seems a position where the fluid is namely on the rhs. via ℓ, cf (4) and (5), and on the lhs.
able to move in radial direction if instability indeed sets viar ,cf(6). Theargumentof∆is the sameinbothin-
p
in. For Taylor vortices radial fluid motion is strongest stances. Thecurvatureoftheboundarylinehasopposite
near the gap center. This suggests choosing sign to the left and to the right of ∗ for all η.
2
R
Eq. (8) together with (2), (5)–(7), and (9) constitute
d d
n
rp =r1+ ∆(a ). (6) the main result of this work. It is an implicit equation
2 d
determiningthe fullstability curve 1( 2)forallradius
R R
Experimental data [18] indicate that this might be rea- ratiosη. Itiseasilysolvable;analyticallyinspecialcases,
sonablealsofor flowpatterns otherthan Taylorvortices. and numerically in general.
Let us first evaluate the critical Reynolds number for Althoughweintroducedseveralparameters,onlyαhas
resting outer cylinder defined as R1,c =R1(R2 =0). For been adjusted to linear stability theory. The othershave
R2 = 0 we have rn = r2, ℓ = αd and rp = (r1 +r2)/2, been fixed by simple physical arguments and are not ar-
yielding from (4) bitrary fit parameters.
2
non-differentiable corner-type minimum at R2 = 0 and
3. Comparison with data and stability theory to considerably higher curves for R1(R2) at counterrota-
tion, as was also found in [15]. Further on, taking a>1
∗
We now discuss the contents of (8), i.e., how the sta- and R2 =0 as in [14] gives a minimum at R2 =0 which
is not even continuous.
bility boundary globally looks like.
Fig. 1 shows R1,c(η), Eq. (7), compared with R1,c(η) It is not surprising that our minimum’s position
from the Navier–Stokes linear stability theory and some
1 η 3+η
selected experimental data. The good overall agreement 2,min(η)= − ,
for η approaching 1 and even for small η comes because R −2(1+η)r1+3η
Eq. (7) has the correct asymptotic behavior, cf [3,5] for
its depth andwidth do not matchthe measurements too
η 1, [6] for η 0, and [9]. The 1/η law may be
→ → wellsincetheminimumislocatedinthe crossoverregion
considered as a consequence of our choice for rp, since R2∗ R2 0. We have marked R2∗ with an arrow in
with rp =r1 [15],for example, it is R1,c =constfor η → the ≤figures≤. Choosing some other, smoother crossover
0. Alsotheprefactorcomesoutremarkablywell. Wefind
function ∆ in (5) allows to modify the location and the
ηR1,c =11.9inthelimitη →0,whereasηR1,c ≈11from width of the minimum.
linearstabilitytheory[6],cfalsotheinsetofFig.1. Some
Interestingly, for η 1 the minimum is exactly at
previousauthorsgavephenomenologicalscalingformulae →
R2 =0, cf Fig. 4, independent of the value of a (as long
which, however, do not show the correct exponents of
as a > 1) and of the precise form of ∆, and nicely com-
the asymptotic laws for η 0 [19,14,12] or η 1 [12].
→ → patible with experiment [3,12] and theory [3,5,6,8,9].
In particular, Coles finds R1,c ∝ 1/(η ln1/η) for η → For R2 far left of the minimum our curves calculated
0 [14] which does not agree with linear stability theory
from (8) are clearly systematically too high. Before dis-
p
(see the inset of Fig. 1 which clearly does not show a
cussing this, we complete our analysis by looking at in-
logarithmic singularity). The origin of this logarithmic
teresting special cases, in which explicit formulae follow.
singularityisthathetookanintegralmeanof L2(r)′/r3
−
over the range r1 r rn for R2 < 0, and r1 r
≤ ≤ ≤ ≤
r2 for R2 0, respectively, instead of choosing a well- 4. Asymptotic formulae
≥
localized position r as we do.
p
In Figs. 2–4 the corresponding comparison is offered
As long as the vortex flow fills the whole gap, i.e., for
for the stability boundary (8), in the original variables
∗
R1(R2),forη =0.20,0.50,and0.964,coveringthewhole R2 ≥ R2, we have ∆(adn/d) = 1; then Eq. (8) can be
range for which experimental data seem to be available. solved for R1:
Intermediate values of η as in Refs. [3,11–13] give essen-
2
tially the same satisfying picture. 1 η 1+η
1 = − 2+ 1+ 2 2 , (11)
For R2 0 we find excellent quantitative agreement, R 3+η R s 3+η R
≥ (cid:18) (cid:19)
and for R2 <0 there still is satisfactory agreement. The for 2 ∗2.
remainingdiscrepanciesmaybeattributedtoseveralrea- R ≥R
−2
sons. First, the experimental data are affected by the fi- This implies a leading correction relative to the
2
∝ R
nite lengthofthe cylinderswhichbecomes importantfor Rayleigh line 1 = 2 for large 2.
sufficiently negative R2 [20,12]. This can be clearly seen For 2 < R∗2, theRquantity R
R R
in Fig. 3 as a deviation of the Donnelly–Fultz data from
linear stability theory, the latter being calculated with rn 2 1 η2 1
ε= 1= − R (12)
aspect ratio infinity. Snyder’s data in Figs. 2 and 4 are r1 − η2 1 2
R −R
supposed to be free of finite length effects [12]. Still, the (cid:0) (cid:1)
is a suitable parameter for discussing the strongly coun-
difference of his data to our curve in Fig. 4 near R2 =0
terrotatingcaseaswellasthenarrowandwidegaplimits.
is significantly larger than in other cases if we take the
η value reported in [12], η = 0.959. But the deviation For ε 0, corresponding to rn r1, we find from (8)
→ →
is much below the claimed 1% error in the experimen-
tal value for η [12]; we indicated the correspondingerror η2 2 = R1 5/3 η2 1, 1 (1 η2)3/2C5/2, (13)
− R C − R R ≫ −
in R1,c by a bar. Since the deviation could be caused by (cid:16) (cid:17) 5 1/5
thestrongη dependence 1/√1 η,cfFig.1,wematch 2 (3+η)
∼ − C(η)= .
the R1,c values of the experimental data and of the lin- a4(2 a)(1+η)7
ear stability theory (and thus our Eq. (7)) by choosing (cid:18) − (cid:19)
η = 0.964. Now we find the same satisfactory overall The leading term for large 2 reads
|R |
picture as for the other η values.
Our curves correctly show a minimum at some nega- 1 =C(η) η2 2 3/5, 2 . (14)
R | R | R →−∞
tive R2. This can be attributed to the crossover in the
The exponent 3/5 was already found by Donnelly and
boundary condition from solid to free. a = 1 leads to a
Fultz [10] by a dimensional argument. That, of course,
3
did not yet give the complete η dependent prefactor Treating p as a free fit parameter we find the re-
η6/5C(η) and also not the non-trivial corrections con- maining discrepancies between our stability boundary
tained in (13). These turn out to be non-neglegible and that of experiment or linear stability calculations
in the range of 2 accessible to experiment. Since nearly removed for p=0.47,0.46,0.43,corresponding to
ε η2 2 −2/5 foRr sufficiently large η2 2 , the conver- η = 0.964,0.5,0.2, respectively. This is exemplified in
∝ | R | | R |
gencetothe leadingbehavior(13)and(14)isratherbad Fig.2bythedottedline;fortheotherη valuestheagree-
in terms of 2 for fixed η and becomes worse with de- ment is similarly good. Positionshifts of this magnitude
R
creasing η. are confirmed by several figures in Refs. [3,5,8,9] which
Coles [14] has chosen the value of a by matching his show that the center of each Taylor vortex is situated
analyticformulatolinearstabilitytheoryforη 1inthe nearer towards the inner cylinder. Analytically we find
→
limit R2 . This choice leads to an η independent roughly a 1/p and C approximately p if p is near
→ −∞ ∼ ∼
value of a 1.3. This means he adjusted his formula to 1/2 and for η not to small. We could not figure out a
≈
two values from linear stability theory whereas we only simple physical condition for fixing p(η). For example,
use one. second order continuous differentiability of the stability
∗
On the other hand, for η 1 Eq. (13) becomes exact boundary at turns out not to be realizable.
2
∗ → ∗ R
for all 2 2. In this limit we have 2 = 3/4 and Our argument does not take into consideration non-
R ≤ R R −
a=8/5 from (10) and (9). Furthermore, the position of axisymmetric perturbations of the laminar flow. As we
the minimum approaches 2,min = 0, and the stability knowfromexperiment[18]andlinearstabilitytheory[8],
R ∗
boundary becomes symmetric for 2 2 , cf Fig. 4. the relevant perturbations are indeed non-axisymmetric
|R |≤|R |
Therefore,inFig.4theasymptoticformula(13)would beyond a finite negative R2. They lead to a somewhat
be indistinguishable from the full curve, whereas in lower stability boundary than that determined with ax-
Figs. 2 and 3 it would be a rather poor approximation. isymmetric disturbances. The difference for the values
Ofparticularinterestisthewidegaplimitη 0,since of R2 considered here is pretty small. It turns out that
→
it has not yet been examined very much. It is accompa- ourdeviations fromthe correctstability boundaryareof
nied by large ε for any finite 2 interval, cf (12). For the same order of magnitude. Furthermore these devi-
ε the relevantexpansionRparameter in (8) is ε−1/2. ations start already near R2,min. We therefore think it
→∞
We keep the two leading terms and consider η small in tobeacceptablethatthepossibilityofnon-axisymmetric
them: disturbances is neglected.
Ofcourse,alsoothercorrectionshavetobeconsidered.
1 3+2√2 2
1 = √3+ η 1 2, (15) For example, energy dissipation is curvature dependent.
R 2 1+√2 s − √3R Therefore it seems natural to extend the rhs. of (3) to
for 1/η2 2 ∗2. (ν2/ℓ4)(δr)2(1+bℓ/rp) with b a constant. ℓ/rp vanishes
−R ≪R ≤R for η 1 or for 2 and is rather small even
Here ∗2 = √3/2 and a = √2 for η 1, cf (10) and for η → 0; for exRamp→le −in∞the case 2 = 0 we have
(9). BRecause−of the badconvergencein≪terms of ε−1/2 to ℓ/rp =→α 0.16. Thus (13) remainRs valid, but with
≈
this limiting behavior the line 1( 2) according to (15) a different a in C. Note, without this curvature cor-
is clearly visible only for η < 0R.01R. Of course, for fixed rection the value for a(η), Eq. (9), is close to a = 8/5
η >0andsufficientlylargely∼negative 2equation(13)is whichminimizes C forfixedη but variablea. Ingeneral,
R
valid again because then ε 0 instead of . As can this means the curvature correction shifts the stability
be seenfrom(14)and(15)t→he mostreason→abl∞evariables boundary upwards if 2 is well below ∗2 counteracting
R R
for strongly counterrotating cylinders, i.e., 2 , the influence of the rp shift.
are 1 andη2 2. NotethatthemeaningoftRheir→rat−io∞is
η2 R2/ 1 =ωR2/ω1. This frequency ratio as a parameter
R R 6. Summary
has already been used in [3,5].
Similar arguments as are presented here for Taylor–
5. Improvements Couette flow have been invoked to elucidate the phys-
ical mechanism of the Rayleigh–B´enard instability [21].
Nowwediscusspossiblecorrectionstoourmainresult. But in the Taylor–Couette system it carries much fur-
Itturnsoutthattheprecisevalueofrp hasasurprisingly ther since this has two additional parameters, η and R2,
well observable influence on the stability curve. whose influence on the shape of the stability boundary
To see this, we generalize (6) to we could successfully explain. At Rayleigh–B´enard in-
stability a corresponds to the ratio of the α’s for the
d
rp =r1+pd∆(a n) (16) two distinct cases of free and solid boundary conditions.
d From[5]wefindforthisratio 1.13. Weunderstandthis
≈
with the new parameter p describing the position of the valuecloserto1thanwefoundhereforthecounterrotat-
disturbance; p = 1/2 gives the previous case, cf (6), to ingTaylor–Couettesystemfromthestrongerinfluenceof
locate it in the center of the relevant gap. thesurfacetensioninanopenRayleigh–B´enardsystemin
4
contrastto the completely missing surfacetensionatthe [7] D. D. Joseph, Stability of Fluid Motions I, 1st ed.
nodal surface r in Taylor–Couette flow. Therefore this (Springer-Verlag, Berlin Heidelberg, 1976).
n
nodal surface is much softer, allowing for a larger exten- [8] P. G. Drazin and W. H. Reid, Hydrodynamic Stability,
sion of the Taylor vortices beyond r . One could check 1st ed.(Cambridge UniversityPress, Cambridge, 1981).
n
this by putting another fluid layer on top of the origi- [9] T. Gebhardt and S. Grossmann, “The Taylor–Couette
nal Rayleigh–B´enardfluid; if this diminishes the surface eigenvalue problem with independently rotating cylin-
tension, a larger a should result. ders”, Z. Phys.B 90, 475–490 (1993).
Toconclude: Whileaccurateresultsfortheonsetofthe [10] R.J.DonnellyandD.Fultz,“Experimentsonthestabil-
ity of viscous flow between rotating cylinders. II. Visual
Taylor–Couetteinstabilityderivedbylinearstabilitythe-
observations”,Proc.Roy.Soc.(London)A258,101–123
ory are available, in this paper we have drawn a picture
(1960).
that gives comprehensive physical insight into the insta-
[11] D. Coles, “Transition in circular Couette flow”, J. Fluid
bilitymechanismandyieldssimpleanalyticalexpressions
Mech. 21, 385–425 (1965).
for the stability boundary in the full parameter space.
[12] H. A. Snyder, “Stability of rotating Couette flow. II.
These agree astonishingly well with experimental data
Comparisonwithnumericalresults”,Phys.Fl.11,1599–
and with the rigoroustheory,especially in regardof how
1605 (1968).
straightforwardthe argumentis. The positionwhere the
[13] C. D. Andereck, S. S. Liu, and H. L. Swinney, “Flow
initial perturbationoccurs and the change of the type of
regimes inacircularCouettesystemwith independently
boundary when the nodal surface moves inwards turned rotatingcylinders”,J.FluidMech.164,155–183 (1986).
out to be important for the wide gap behavior and for [14] D. Coles, “A note on Taylor instability in circular Cou-
the existence of a minimum of the stability boundary ette flow”, J. Appl.Mech. 34, 529–534 (1967).
for negative R2. The values found for the parameters α [15] B. Eckhardt and D. Yao, “Local stability analysis along
and a are reasonable and agree with experiment as well. Lagrangian paths”, Chaos, Solitons and Fractals (1995),
Assumingaradiusratiodependentsmallshiftoftheper- in press.
turbation’s initial positiontowardsthe inner cylinder we [16] L. D. Landau and E. M. Lifschitz, Fluid Mechanics,
were able to understand the remaining quantitative dis- vol.VIofCourse of Theoretical Physics,2nded.(Perga-
crepancies. mon Press, Oxford, 1987).
We believe that these simple ideas should be appli- [17] Lord Rayleigh, “On the dynamics of revolving fluids”,
cable to instabilities in more complicated rotating fluid Proc. Roy.Soc. (London) A 93, 148–154 (1916).
flows, where the standard linear stability analysis is too [18] H.A.Snyder,“WaveformsinrotatingCouetteflow”,Int.
difficult. J. Non-Lin.Mech. 5, 659–685 (1970).
Acknowledgments: We thank Thomas Gebhardt [19] L. Prandtl, Abriss der Str¨omungslehre, 1st ed. (Vieweg,
Braunschweig, 1931), Essentials of Fluid Dynamics
andMartinHolthausfortheirinterestandhelp. S.G.ac-
(Stechert-Hafner, NewYork,1952).
knowledgesstimulatingdiscussionsaboutsimilaranalysis
[20] H. A. Snyder, “Stability of rotating Couette flow. I.
of thermal convection with Stefan Thomae, quite some
Asymmetric waveforms”, Phys.Fl. 11, 728–734 (1968).
time ago, during a sabbatical stay at the Max-Planck-
[21] C. Normand, Y. Pomeau, and M. G. Velarde, “Con-
Institut fu¨r Mathematik in Bonn.
vective instability: A physicist’s approach”, Rev. Mod.
Phys. 49, 581–624 (1977).
[22] T. Gebhardt,private communication.
FIG. 1. Critical Reynolds number R1,c as a function of
η from Eq. (7) (solid line), from linear stability theory [9]
[1] M. M. Couette, “Surunnouvelappareil pour l’´etudedu (dashed line), and from various experiments [3,10–13] (sym-
frottement des fluids”, Comptes Rendus 107, 388–390 bols). For the error bar see Fig. 4. Inset: The same for R1,c
(1888). rescaled toR1,cη√1 η.
−
[2] A. Mallock, “Determination of the viscosity of water”,
Proc. Roy.Soc. (London) A 45, 126–132 (1888).
[3] G. I. Taylor, “Stability of a viscous liquid contained be- FIG.2. ThestabilitylineR1(R2)forη=0.2fromEq.(8)
tweentworotatingcylinders”,Phil.Trans.Roy.Soc.Lon- (solid line), and from experiment [12] (symbols). The dotted
don A 223, 289–343 (1923). line is our result with the perturbation’s position parameter
[4] R.Tagg,“TheCouette–Taylorproblem”,Nonlin.Science set to p = 0.43, see Eq. (16). The arrow marks R2∗. Inset:
Today 4, 1–25 (1994). The same on a magnified scale.
[5] S.Chandrasekhar,HydrodynamicandHydromagneticIn-
stability, 1st ed. (Clarendon Press, Oxford, 1961).
[6] J.Walowit,S.Tsao,andR.C.DiPrima,“Stabilityofflow FIG.3. ThestabilitylineR1(R2)forη=0.5fromEq.(8)
betweenarbitrarilyspacedconcentriccylindricalsurfaces (solid line), from linear stability theory [9,22] (dashed line),
includingtheeffect of aradial temperaturegradient”, J. andfromexperiment[10](symbols). Thedashedlinemaybe
Appl.Mech. 31, 585–593 (1964). reproduced with p=0.46. The arrow marks R2∗. Inset: The
same on a magnified scale.
5
FIG. 4. The stability line R1(R2) for η = 0.964 from
Eq.(8)(solid line),andfromexperiment[12](symbols). The
vertical bar denotes the error in R1,c due to η = 0.959 1%
±
[12]. p = 0.47 fits the experimental data. The arrow marks
R2∗. Inset: The same on a magnified scale.
6
400 600
45
200
R R1,cη 1-η R R1
1,c 30 1
100
15
0
200 300 -100 -50 0 R 50
00.0 0.5 η 1.0 2
p=0.43
~1/η ~1/ 1-η η=0.2 Snyder
Rayleigh
0 0
0.0 0.5 η 1.0 -600 -300 0 R
2
Figure 1 Figure 2
800 600
η=0.5 η=0.964
R R
1 1
Donnelly & Fultz
Gebhardt & Snyder
Grossmann
400 300
200 300
R R
1 1
100 230
Rayleigh
0 160
-200 -100 0 R 100 Rayleigh -200 0 R 200
2 2
0 0
-4000 -2000 0 R -1000 -500 0 R 500
2 2
Figure 3 Figure 4