H.K. Moffatt- Local and Global Perspectives in Fluid Dynamics

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LOCAL AND GLOBAL PERSPECTIVES

IN FLUID DYNAMICS

H . Keith Moffatt

Isaac N ew ton Insti tute fo r Mathematical SciencesUniversity of Cambridge, United Kingdom

[email protected]

1. INTRODUCTION

It is a great honor to present this Closing Lecture at what has been an

outstanding Congress, ICTAM 2000. These Congresses have a proud tra-

dition, going back to Delft 1924, when Prandt l, von KBrm&n,G . I. Taylor

and Burgers initiated the series and laid the ground rules. Since then,every four years, with a brief interlude during World War 11, the world

mechanics community has gathered together to share its knowledge and

to grapple with new emerging problems. This is truly a global enterprise,

and we can all take pride and pleasure in our involvement in it . The

first ICTAM that I was privileged to attend was the tenth in the series,

held at Stresa in 1960. And here we are now at the 20th ICTAM and

actually the 10th that I have been personally involved in.

The global participation at these Congresses is achieved in each caseonly through an effective local organization. Each Congress, whether at

Stresa or Stanford, Lyngby or Haifa, Moscow or Toronto, has had its

own distinctive local flavor that remains in the heart of each participant

long after the details of individual lectures have faded from the memory.

Chicago will be no different in this respect, and I would like to record

my personal thanks to Hassan Aref and his great team on the Local

Organizing Committee here, who have succeeded in making this 20th

and millennia1 Congress so uniquely memorable.Locality and globality: the two complementary perspectives whose

interplay is essential to the successful running of a World Congress. It

is a different kind of locality and globality that I propose to discuss in

this lecture; but certain parallels may emerge, and I shall return to the

Congress scenario in my closing remarks.

52 1

H. Aref and J.W. Philips (eds. ) ,Mechanics fo r a Ne w Millennium, 521-540.

0 2001 Kluwer Academic Publishers. Printed in the Netherlands.

www.moffatt.tc



522 UJ-CLOSING L E C T U R E :H. K E I T HMOFFATT

In fluid dynamics, the concept of locality emerges most clearly through

what we describe as local similarity solutions of the governing Navier-

Stokes equations, and I shall give some examples of these. Although

such solutions are generally accurate only within a small sub-region of

a fluid domain, nevertheless an understanding of them can be of crucialimportance for two reasons: first, they provide a check on the accuracy of

numerical solutions computed throughout the fluid domain; and second,

they can act as organizing centers for the whole flow field; and knowledge

of their structure can provide important clues concerning global features

of the flows.

But what are these global features? They are those features that

are essentially topological in character, structural properties that are

‘robust’, i.e. insensitive to small perturbations. Again, an understand-ing of such topological features provides a guide for computation and

a possible check on computational accuracy. By way of simple exam-

ple, consider an incompressible two-dimensional flow on the surface of a

sphere. At any instant there are a number of stagnation points, which

are of either elliptic or hyperbolic type. Topological considerations tell

us that if these stagnation points are ‘non-degenerate’ and if N is the

number of elliptic points and M the number of hyperbolic points then

N-A4 = 2 . A numerical search for stagnation points for a given complexflow can be conducted, and their classification as elliptic or hyperbolic

can be achieved, at least in principle. If this process leads to values of N

and M with M < N - 2 , then we know that we must have missed some

hyperbolic points. As a first priority, we should always seek to ensure

that computed flow properties are compatible with such fundamental

topological constraints.

2. LOCAL SIMILARITY SOLUTIONSI want to focus first on a fascinating problem in which I have been

intermittently involved since 1963-this is the problem of flow near a

sharp corner. Our Chairman has made a vital distinction (Barenblatt

1979) between two types of similarity solution, which both crop up in this

context: similarity solutions of the first kind for which dimensional argu-

ments play a key role; and of the second kind for which a key parameter

has to be determined through solution of an eigenvalue problem. The

two prototype situations are sketched in Figs. l (a ) and l (b) . The first

shows G . I. Taylor’s (1960) ‘paint-scraper’ configuration in which one

plane boundary is scraped relative to the other with constant velocity

U . Sufficiently near the intersection, the flow is a Stokes flow and the

streamfunction $ (T , 0) is determined on dimensional grounds in the form



Local and global perspect ives in fluid d y n a m i c s 523

$ = U r f ( O ) ,where

1a2- k e ) sin 8 - 8 sin2a CO S ea2 - sin2a

, k = T ( 2 a - s in2a) . (1)( 8 )=

This is evidently, in Barenblatt’s terminology, a similarity solution of

the first kind.

1

ruV

+O. 1

U (> 0 )

Figure l ( a ) Solid curves are the streamlines for the Taylor similarity solution (1);

dashed curves show the first-order inertial correction obtained by Hancock, Moffatt,

and Lewis (1981).

Figure l ( b )

$J N r’f(0) with X complex. [Courtesy C. P. Hills]

Corner eddies described by a similarity solution of the second kind,



524 LJ-CLOSINGL E C T U R E :H . KEITHMO FFA T T

In Fig. l( b ) by contrast, both boundaries are at rest and fixed at

angle a , and a two-dimensional flow is driven by some agency (e.g. a

rotating cylinder) far from the corner. In the Stokes approximation,

the streamfunction sufficiently near the corner satisfies the biharmonic

equation

with no-slip boundary conditions

v*+= 0: (2)

1+!1=+0=0 on O=O:a. ( 3 )

Here similarity solutions of the form II,= ~ ' f ( 0 ) ay be found in which

the parameter X must be determined from the eigenvalue problem result-

ing from ( a ) , ( 3 ) . This is evidently a similarity solution of the secondkind. The most interesting feature of the solution is that, for all acute

angles a (and actually over the wider range cy < 147"); all relevant eigen-

values X i are complex, and the corresponding flows are 'eddying' flows

as T + 0 (Moffatt 1964a;b). The fundamental structure is precisely as

indicated in Fig. l(b).

Figure 2

plane boundary (Taneda 1979).

Corner eddies in a Stokes flow over a cylinder at a small distance from a

Although this was all well understood in 1964, it was not until 1979

that the first photographs of such corner eddies were published (Taneda

1979). Taneda's beautiful photographs, many of which are reproduced

in Van Dyke's Album of Fluid Motion (1982), show a great variety of

situations in which corner eddies can arise. For example, shear flow over

a cylinder resting on a plane boundary shows the eddy sequence in the



Local and global perspec t ives in f l u i d d y n a m i c s 525

two cusp-shaped regions. If the cylinder is separated from the plane by

a very small amount) then a finite number of these eddies survive as

attached ’wall-eddies’ on both the plane and the cylinder (Fig. 2 ) . If the

cylinder is rotated slowly, then these eddies still survive, and may in some

circumstances move into the interior of the flow (Jeffrey and Sherwood1980). For all such situations, it is evident that an understanding of the

basic prototype problem of Fig. l(b) is the prerequisite for an analysis,

whether analytical or numerical, of the more complicated configurations.

Fzgure 3similarity structure (Hills and Moffatt 2000) .A three-dimensional configuration for which the Stokes flow admits a double

We have recently found an example of a steady three-dimensional

Stokes flow for which both types of similarity solution play a part (Hills

and Moffatt 2000). The configuration is shown in Fig. 3: two fixed planes

c p = +,f~ re ‘honed’ by rotation of the plane boundary 8 = 7r/2 with

constant angular velocity R (using spherical polar coordinates ( r ,8,v)).The fluid domain is 0 < 8 < ./a, 1 ~ 1 p. In the ‘Stokes’ region near

T- = 0 , on dimensional grounds, the velocity field has the form

U k ?8’ p ) = flT-W?p)? (4)

where U ( 8 ’ p ) is dimensionless, a similarity structure of the first kind. I

use here the word ’structure’ rather than ‘solution’,because the function

u ( 8 : c p ) is as yet unknown. A solution can be sought in the form U =

X ( z T ( r , @ , p ) )here, consistent with (4)’ (r,8?cp) flrL?(e,cp).It

turns out tha t, near 8 = 0 , the solution has the form

W ’ )= exf(P) ( 5 )

where X is determined by precisely the same eigenvalue problem as in

the two-dimensional case described previously. The flow again exhibits

corner eddies (when 2p < 1 4 7 O ) , which now scale with radius r (as well



as with arigle B according to ( 5 ) ) . We thus see a similarity solution of

the second kind ( 5 ) imbedded wi th in one of the first kind ( 4 ) .

F i g w e 4rotating a horizontal plate under a fixed vertical plate (Hills and lloffatt 2000)

Closed streamlines (revealed by injection of dye) in the flow generated by

,4n exp erim enta l realization of th e si tua t ion w ith 2p = T is shown

in Fig. 4. The streamlines are closed. being the intersections of 'Taylor

surfaces' r f ( Q )= const and spheres r2+ z 2 = const . A ful l analyt ical

solution of th e Stokes problem has been found for this case.

3. FREE SURFACE CUSPS

The corner flows considered above are non-analytic (or 'singular ' ) in

m y rieighborhood of the corner as a result of the imposed geometrical

s ingular i ty on the boundary . I now wish to briefly consider a different

type of singular behavior t h a t occurs at a free surface an d t h at is intr insic

to th e prope rt ies of the f luid. T his is the phenomenon of 'cusping ' th at

occurs w hen flow is induced to converge tow ards a line on a free surface

( Jo se p h e t al. 1991. Jeong a nd M offatt 199 2). T h e pro toty pe configu-ration is shown in Fig. 5: a flow driven by some stirring mechanism

in the interior of the fluid of characteristic scale L say, is supposed to

converge sym metrically an d th e surface typically dips downw ards form-

ing a cus p as ind ica ted . An exa mp le in which th e flow is driven by

two co unte r-rota t ing cyl inders im mersed in sy rup is shown in Fig . 6. i n



Local a n d global perspec t i ves in f l u i d d y n a m i c s 527

which incidentai iy corner eddies can also be observed in the corners of

the containing vessel .

FIguie 5 here.

the floxv is generated b) a vortex dipole of strength C I placed at depth L below the

undisturbed free-surface level (Jeong and SIoffatt 1992)

Prototype configuration for the formation of a free surface cusp

F i g w e 6 Free-surface cusp formed i n go!den syrup by the counte r-ro tation of

immersed cylinders (the left cylinder rotating clockwise, the right anti-clockw

The Reynolds number is small. Note the viscous eddies in the bottom corners of

ta nk . [Ph oto courtesy of 1'.Kimura]

tM'O

ise).

' the



528 UJ-CLOSING L E C T U R E :H . K E I T HMOFFATT

Of course, one would expect surface tension y to ‘round off’ the appar-

ent cusp singularity. The small radius of curvature R at the cusp may

be estimated by a purely local argument (first noted by E. J. Hinch) as

follows: the surface tension provides a virtual ‘line force’ 2y located at

the center of curvature, which we take at T = 0 (Fig. 7) . This drives aStokeslet flow with upward velocity on the plane of symmetry

where c is a constant of order unity. There is also a local downward

velocity U say, associated with the imposed stirring. These must equili-

brate at the stagnation point on the free surface at T = R, i.e.

U =Y n ($)2TP

or equivalently, - = c exp (-,>TPU .

L

Thus the radius of curvature decreases exponentially with local capillary

number (@IT).

Circle ofcurvature

radius R

Figure 7 Representation of the flow near the free-surface stagnation point as a super-

position of a two-dimensional Stokeslet and a locally uniform stream induced by

remote forcir,g.

The details have been worked out for the idealized Stokes flow in

which the stirring is induced by a vortex dipole of strength (U at depth L

below the (undisturbed) free surface (Jeong and Moffatt 1992). In this

case, the cusp forms at depth 2L/3 (i.e. at height L/3 above the vortex

dipole), and the exact solution yields U = 16a/L2 and c = 256/3. Hence

(6) becomes

R 256- =- xp( - 32~C ) ,L 3

(7 )



Local a nd g lobal per spec t i ves in f l u i d d y n a m i c s 529

where C = p a / L 2 y , the capillary number based on the ‘given’ flow

parameters Q and L. This number represents a ratio of downward viscous

drag to restoring surface tension. If C = 1 , we might anticipate that R / L

should be of order unity, and so in a mathematical sense it is; however,

numerically, (7) then gives

which must surely be the smallest 0 ( 1 ) number ever encountered in a

fluid dynamical context! In reality therefore a cusp does form, and there

is an associated stress singularity that is presumably resolved only if

non-Newtonian and/or non-continuum effects are taken into account.

In a region R (< r << L , the streamfunction of the ‘cusp’ flow within

the fluid is given in polar coordinates ( r , ) by

The second term shows a self-similar structure, the same as that associ-

ated with a symmetric flow near the edge of a flat plate 0 = & T .

Although the cusping phenomenon has been analyzed only within theStokes approximation, I believe that it occurs quite generally in high

Reynolds number flow also, and that many observed phenomena can be

explained only through an understanding of the cusping mechanism. I

show one example in Fig. 8 , which shows a spherical cap bubble rising

in glycerine in the bubble tube at DAMTP, Cambridge. This bubble is

in the ‘wavy-skirted regime’ characterized by the very thin film of air

entrained into the glycerine from the sharp edge of the bubble. The

flow is converging near this sharp edge, and it is therefore presumablycusped. The skirt forms because of the large air pressure generated in

this cusp region.

Similarly, when water runs smoothly into a deep bath (the ‘Japanese

bath problem’), it may be observed that, above a critical flow rate,

bubbles of air are entrained and the pouring suddenly becomes noisy as

a result of bubble oscillation. I believe that this entrainment of bubbles

must occur through the circular cusp that forms where the incoming jet

of water impinges on the free surface (Fig. 9) .

4. GLOBAL INVARIANTS

Let me pass now to the opposite situation, namely the unsteady

fully three-dimensional flow of a strictly inviscid, incompressible fluid (of

course an idealization). This is the classical fluid dynamics of Helmholtz



530 UJ-CLOSIAGL E C T U R EH K E I T H f O F T A T T

Figuie 8 The rise of a T $ ~ \ T . s k i r te d bubb le t h r ough t he c om m on r oom of D A L I T PC a m br i dge T he bubb l e t ube o f d i a m e t e r 17 c m c on ta i n s g l \ ce r i ne T he po r t r a i t 1s

of X S E dd i ng t on

Figuie 9 Th e Japanese ba t h prob lem a l amina r s t ream of w a t e r flows i n to a deep

ba th , above a c r it ica l flow ra t e Q c , bubbles a re en t ra ined th rough the c i rcu la r cusp

formed at the free surface

and Kelvin, for which we know t ha t t he c i rcu la t ion round eve ry ma te -

r ia l ( i .e . Lag rang ian) c i rcui t is conserved. A closely related result ( J . - J .



Local and g lobal perspect ives in fluid dynamics 531

M oreau 1961, M offatt 1969) is th a t t h e helicity

Fl= 1 , u . w d V

for any m ater ial volume V . on whose surface w . n = 0. is also conserve d:here. w = c u r l u is th e vorticity field. T hi s conserv ation of helicity

ad m its topological in terpreta t ion : i t i s. in a sense that can be ref ined

(X rno l’d 19 74 ). th e degree of l inkage of co nsti tu ent vortex tub es in th e

flow. a linkage that. as recognized by Kelvin, is indeed conserved.

An ana logous resu lt ho lds in the mag ne tohydrodyn amics (M HD ) of

perfectly co nd uc tin g fluids, in which lines of m agne tic field B (or ‘B-

l ines’) are f rozen in th e f lu id . T h e analogous conserved qu ant i ty is the

magnetic helici ty

X.11= l7 . B d V

where B = c u r l A . an d i t ad mi t s s imi la r topolog ical in te rp re ta t ion . I t

is in this MHD context that the concept of helici ty has proved to have

profound s ignif icance. in two quite complementary s i tuations:

4.1. The turbulent dynamo

I be l ieve tha t the mos t impor tan t fundamenta l b reak th rough o f the

last half-century in our understanding of turbulent processes relates to

the dynamo problem: under what c i rcumstances wi l l a magnet ic f ie ld

B ( z . ) systematically increase in intensity un der th e dis tort in g an d dif-

fusive action of a field of s ta t i o n a ry h om og en eo us t u rb u le n c e u ( ~ , t ) ?

T h e ques tion was posed first in th is form by Batchelor (195 0) . T h e sim-

ple answer is tha t a suff icient condit ion for such dy na m o acti on is tha t

the turbule nce should . lack reflectional sy m m etry ’ : th e s imples t mea-

s u r e of such lack of reflectional sy m m et ry is th e m ean helicity of th e

tu rbu lence ( U w ) . an d if this mean helici ty is non-zero. the n dy na m o

action will in general occur (Steenbeck, Krause, and Radler 1966. Mof-

fa t t 1970) . Th is result h as huge impl icat ions for curren t un ders tan ding

of the process by which magnetic f ields are generated in planets , s tars

an d galaxies: i t is in more senses th a n one, a t ru ly g lobal phen omen on .

4.2. R,elaxation to equilibrium states

The relaxation process is the converse of the dynamo process and is

bes t i l lus t r a ted by s imple example . S uppose th a t a t some in i t i a l ins tan t

t = 0 , a magnetic field Bo(z )having some nontr ivial knotted or l inked

s t ruc tu re i s imbedded in a perfectly con du ctin g. bu t viscous. incom-

pressible fluid a t res t . T h e pr ot oty pe consis t ing of two l inked, un kn ot-



532 UJ-CLOSING L E C T U R E :H. EITHNOFFATT

ted flux tubes, is shown in Fig. 10. The current corresponding to Bo is

j o = curlBo, the Lorentz force distribution Fo = j , x Bo is in general

rotational, and the fluid must flow in response to this force. Thus mag-

netic energy is converted to kinetic energy as in an electric motor, and

this is dissipated by viscosity. However, the topology of the field B ( z , )is conserved, since the lines of force are frozen in the fluid and deform

continuously with the flow. In particular , the magnetic helicity, which

in this special case is XM = 2QlQ.2 where Q1 and cP2 are the fluxes in

the two tubes, is conserved.

Fegure 10 Pro toty pe conf igurat ion for the problem of magne t ic re laxat io n: two l inked

f lux tubes c ontrac t in a perfectly cond uct ing f lu id , wi th f luxes an d volumes conserved,

unt i l th e l inkage prevents fu r ther c ontrac t ion (Moffat t 1985) .

We are thus faced with the problem (Arnol’d 1974, Moffatt 1985) of

minimizing magnetic energy subject to conservation of field topology;

more precisely, of finding minimum energy states that are topologically

accessible from the initial state. In these states, the Lorentz force is

irrotational, since the fluid must come to rest, i.e.

j x B = V p , j = c u r l B ,

for some scalar field p (evidently the pressure field). This process of

magnetic relaxation is brought about in physical terms through relax-

ation of the Maxwell stress in each flux tube, as illustrated in Fig. 10.

The tubes contract in axial length, and since volume is conserved, their

cross sections increase in area. The topological constraint arrests the

relaxation process when the tubes ultimately make contact: a singular-

ity (here a surface discontinuity) inevitably appears as t -+cc (it may

occur at finite time, but this seems unlikely).

Equation (8) has precisely the same structure as the Euler equation

for steady flow of an inviscid incompressible fluid:

u x w = V h , w = c u r l u , ( 9 )



Local and global perspect ives in f l u i d d y n a m i c s 533

where h = p / p + i u 2 a n d p is the f luid densi ty (assumed co ns tan t) . To

every so lution of (8) , the re corresponds a solution of (9) via th e analogy

B + u , j + w , p + h o - h ,

where ho is an arb i t rary con s tant . Th us , magnet ic relaxation provides anindirect means of proving the existence of steady Euler flows (solutions

of (9)) of arb itra rily complex streamline topology; consideration of their

s tabi l i ty is quite another matter (Moffat t 1986; see also Vladimirov,

M offatt , and I lin 1999).

5. EXISTENCE OF SMOOTH SOLUTIONS

OF THE NAVIER-STOKES EQUATIONS

FOR ALL T > 0

T h e problem of existence an d smoothn ess for th e Navier-Stokes (N S)

equa tions was addressed by Leray (1934) an d has att ra ct ed intense effort

in the mathematical community s ince then. Despite these effor ts , the

problem remains open; it has recently achieved millennia1 status as one

of th e seven problems identified by th e Clay In sti tu te for which a prize

of one million d ollars is offered. Cha rles Fefferman has given d etails for

the NS problem on the websi te of the Clay Inst i tute.I do n’t propose t o solve thi s problem toda y! I do wish however to

make some observat ions about i t , because i t seems to me that both

local and global considerations are likely to play a part in its solution.

Leray himself was unable to prove the existence of smooth solutions

of the N S equations in 3 dimensions for all t > 0, nd recognized the

al ter nat iv e possibi li ty th a t a s ingular i ty may develop a t f inite time, t*

say. If such a singularity develops at a point that we may take to be

z =0, then i t seems plausible that the approach to this s ingular i ty (ast + *) hou ld be a t least locally self-similar. Leray n oted a remarkable

similar ity t ransfo rm ation of th e NS equations, or equivalently of the

vorticity equation

dW

at- v x ( U x w ) + v v 2 w

Th is t r ans format ion is

where I? is a constant with the dimensions of circulation (velocity x

length ) . Th e vor tici ty the n t ransforms as

w = c u r l u = -Cl((X) . (11)1

t* -0 = cur1,U



534 UJ-CLOSISG L E C T U R E :H . KEITH IOFFATT

and equation (10) transforms to

0= v x (U + ;x) f-2 + E V f - 2 , (12)

whereE= u/r, dimensionless parameter that we may assume to besmall. The (x,) problem of (10) is thus converted to a 'steady' problem

( 1 2 ) in the scaled space variable X . The term dw/d2 in (10) transforms

to the term i V x ( X x f-2) in (12). which represents outward transport of

the vorticity field f-2 by the steady spherically symmetric 'compressible

velocity field' i X . The question now is whether there exists a distribu-

tion of vorticity such that the corresponding velocity U in conjunction

with diffusion can compensate this outward transport.

For reasons developed in a recent paper (Moffatt 2000a), it is relevantto seek solutions of (12) in an inner region (in the sense of matched

asymptotic expansions) for which O ( X )matches with w ( x , ) (the outer

solution) in an overlap region X + m, x + 0. At leading order, this

seems to require that

1 11

. R ( X ) - L a s X + x ,IN2

( X )-1x1

with corresponding outer behavior

(This, incidentally, is suggestive of Stokeslet behavior, which might be

thought to require a source of momentum in the inner region. and thus

to be incompatible with unforced evolution!)

It has been shown by NeEas, R6iiEka and SverAk (1996) that no

smooth nontrivial solution of (12) exists (with E > 0) for which IUI E

L 3 ( X 3 ) . he behavior (13) is jus t at the boundary of this function space,

and is not excluded by the theorem.

A vorticity singularity as represented by (11) can occur only through

stretching of vortex lines at a rate that also becomes infinite as t + t * .Such a process can conceivably occur through the nonlinear interaction

of vortex tubes each one of which provides the strain field acting on the

other. Some mechanism must be present to cause all length scales in the

inner region t o decrease to zero (like (t*- ) ' I 2 ) as t + t * . Scenariosinvolving the collision of non-parallel vortex pairs seem plausible candi-

dates (Pelz 1997, Moffatt 2000a), the collision zone being then the inner

region in which the Leray transformation is relevant. An alternative

scenario is sketched in Fig. 11, which shows a vortex pair knotted in

the internal zone and arranged in the external zone in such a way as to



Locul u n d global perspec t i ves in fluid dy numic s 535

.pul l th e knot t ig ht ‘ . Th is is cer ta in ly suggest ive of how a finite-time

s ingula r i ty may ar ise if th e ini tia l condit,ions are sufficiently ingeniously

contr ived.

Figuie 11 Conceptual sketch of a vortex pair configuration fo r which propagation of

the pair in the external region leads to tightening of any knot that may be contained

within t he inner region (here represented simply by a ’black box‘).

Topology evidently plays a pa r t in th i s . Note tha t th e con t r ibu t ion to

helicity from any region V ( t ) :1x1 < R ( w i th R co ns ta nt) of the inner

zone is given by

?i/ c . . u d 3 x = r l R U , 2 d 3 X ,

an d th a t th is is ind epe nd ent of t im e. Th is suggests conservation of

topology of vortex tubes in the inner region d e s p i t e the znfluence of

viscoszty. I t i s pe rha ps here th at one may f ind a clue for t h e relevance of

global ( i .e . topological) considerations for t h e NS s ingularity p roblem.

6. A TOY MODEL OF A FINITE-TIMESINGULARITY

I t may be app ropr ia te to conc lude th i s lec tu re wi th a real finite-times ingu la r i ty ! Th is is exh ibite d by t h e toy ’Euler’s disk’ , a heavy disk that

ro lls on i t s edge (F ig . 1 2 ) ’ and exh ib i t s a paradoxical increase of rolling

speed as i ts energy decreases (Bendik 2000) . T he energy E is equal to

:Mgact. (M = m as s , a = r ad ius ) when th e ang le Q between th e disk and

th e table is sma l l ( i . e . du r ing th e f inal ‘ shuddering’ phase of m ot ion) .



536 U J -C L O S I N G L E C T U R E :H. K E I T HMOFFATT

The rolling (precessional) speed R is related to Q during this phase by

the equation

(14)R 2 a = 49,

a

a consequence of Euler’s angular momentum equation, coupled with the

rolling condition. As a tends to zero, tnen for so long as (14) holds (the

‘adiabatic approximation’), R increases like ciP1l2, apparently without

limit. The interesting feature of this toy is that the phenomenon can

indeed be observed at finite time (of the order of 102s) after the disk is

set in motion; I demonstrate this here with the aid of a microphone to

amplify the increasing frequency R and the very abrupt stop. Reflection

of a laser from the surfa,ce of the disk onto a screen can be used to

visualize and measure the decrease of a as a function of time.

\ I

\ I

\ I

\ I

7/

F z g w e 12 Euler’s disk: a heavy c i rcular d isk rol ls on i t s r im wi th precessional angu lar

velocity fl. T h e sy s t e m e x h i b it s a f in ite- time s ingu lar i ty induced by weak diss ipat iveprocesses (Moffat t 2000b).

The energy equation is - - @ ( E ) ,E

dt

where @ is the rate of dissipation of energy, whatever the dominant

dissipative mechanism may be. If @(E)( E X wkere X < 1, (15) implies

that E goes to zero at finite time, t* say, and hence that, within the

adiabatic approximation, R + cx) as t + t*. If X < 0, then obviously

d E / d t + cc also as t + t*, and the settling process may reasonably

be described as ‘dramatic’. The behavior is familiar to anyone who has

spun a coin upon a table. (It is to be contrasted with, for example,



Local and g loba l perspect ives in f l u i d d y n a m i c s 537

linear damping of a simple pendulum, for which X = 1, and the decay is

exponential.)

The viscosity of the air in the thin layer between the disk and the table

in the final shuddering phase provides one obvious dissipative mecha-

nism for which @ ( E )may be calculated (Moffatt 2000b). A lubricationapproximation gives X = -2; an improved analysis allowing for Stokes

layers on the disk and table (L . Bildsten, private communication) gives

X = -5/4. Either way, the finite-time singularity is dramatic.

There is also a contribution to dissipation from rolling friction, which

is apparently much more difficult to quantify. On the simple assumption

that for this mechanism @ c( R , (14) with E 0: (Y gives X = -3 , and again

the behavior is dramatic. The difficulty here is to obtain the coefficient

of proportionality between @ and R in terms of the physical propertiesof the disk and the table-a challenging problem that perhaps someone

in this large audience knows how to solve.

It is likely that different dissipative mechanisms dominate at differ-

ent stages of the settling process, and that the effect of air viscosity is

dominant only in the very final stage before the sudden stop. The pre-

dicted singularity is resolved during the final few microseconds, when the

downward vertical acceleration of the center-of-mass of the disk becomes

comparable with the acceleration of gravity (Moffatt 2000b).

7. CONCLUSION

The smoothness, or alternatively the finite-time singularity, of the

Navier-Stokes equations offers a challenge that will continue to make

great demands on both analytical ingenuity and computational power.

If, as computer simulations continue to indicate (Kerr 1997), a finite-time

singularity does occur and if this is generic behavior, then of course we

shall have to understand by what mechanism these putative singularities

are resolved. Since the pressure gradient must also become unbounded as

a singularity is approached, the incompressibility assumption, on which

most analyses of this phenomenon are based, becomes no longer tenable.

The infinite stress at a singularity can be relieved by cavitation in liq-

uids, and by acoustic radiation in gases. The Japanese bath provides a

congenial environment for the contemplation of such problems!

As I mentioned in my introductory remarks, this Congress exhibits a

most fruitful interplay between local and global characteristics. I would

maintain now that it also has some features of a finite-time singular-

ity: there was certainly a decreasing length-scale as we all converged on

Chicago; and the Organizers were acutely aware of a decreasing time-

scale in the last few weeks of hectic preparations. We should be relieved



tha t the Narr io t t Hote l has ne i ther imploded nor exploded under the

extrenic pressure of activity th a t i t has experienced this week. Now

we are at the stage where the singularity must be resolved: we shall

returri to our homes aro und th e world like acoust ic pulses radia t ing from

this source. and th e local ly acquired imp act of th is C ongress will surelyinform and inspire future research in theoretical and applied mechanics

on a global scale . O n th at note , an d l ike a finite-t ime singularity, I mu s t

bring this ta lk t o a sudden end .

Acknowledgments

I t han k S arah Ki rkup an d Jon a than Chin fo r he lp in process ing th e t ex t a nd

f iguies , and SIu stap ha Xm rani for help in prepar in g the pow erpoint version of t h e

lec tu re

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H.K. Moffatt- Local and Global Perspectives in Fluid Dynamics

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