H.K. Moffatt- Behaviour of a viscous film on the outer surface of a rotating cylinder

Journal de Mkcanique, Vol. 16, No 5, 1977.

Behaviour of a viscous film on the outer surface of a rotating cylinder*

by

H. K. MOFFATT**

ABSTRACT. - The problem of the dynamics of a thin film of viscous fluid on the outer surface of a horizontal rotating roller is considered by the techniques of lubrication theory. The condition for the existence of a steady solution is first obtained, and the maximum load that can be supported for given rotation rate is found on the basis of the steady solution. Kinematic wave theory shows that the steady state is not approached in general from arbitrary initial conditions, but that discontinuities in depth appear, like backward breaking waves relative to the direction of rotation. Weak diffusion can eliminate this tendency and can restore an approach to the steady state.

A qualitative experiment is described showing that the steady state can be realised, but that it is in general unstable as the rotation rate is increased. The instability evolves into a sequence of rings of liquid around the cylinder, each ring carrying one or more “lobes” exhibiting depth discontinuities of the kind suggested by kinematic wave theory. The observed behaviour of the rings is described and discussed.

RESUME. - On considere le problhme de la dynamique d’un film mince d’un fluide visqueux sur la surface exterieure d’un cylindre horizontal en rotation, par les methodes de la theorie de lubrification. D’abord, on obtient la condition d’existence d’une solution stationnaire, ensuite on determine la charge. niaximale que I’on peut supporter, pour

.’

un taux de rotation donne, a partir de cette solution stationnaire. La theorie des onpes cinematiques montre que, a partir de conditions initiales arbitraires, on n’arrive pas

” I .;F‘ . *

en general a 1’6tat stationnaire, mais que, au contraire, apparaissent des discontinuites de profondeur, semblables a des ondes deferlant vers l’arriere relativement a la direction de rotation. UneJaible diffusion peut eliminer cette tendance et peut rktablir 1’6volution vers l’etat stationnaire.

Une experience qualitative est presentee, qui demontre que 1’8tat statioanaire p u t ttre rhlise, mais qu’en general il devient instable quand le taux de rotation est augmentb. Cette instabilite engendre une sequence d’anneaux de liquide autour du cylindre; chacun de ces anneaux emporte au moins un G lobe v qui presente une discontinuite de profondeur comme suggerk par la thkorie preckdente. Le comportenient de ces anneaux est decrit et discute.

5)

* This work was presented at the British Theoretical Mechanics Colloquium, St Andrews, in April 1974. It is now submitted to the Journal de M&znique in retros- pective gratitude for the year 1975-1976 that I spent as Professeur Associe at the Labora- toire de MCcanique Theorique, Universite Paris VI.

** School of Mathematics, University ofnBristol, BriStol, BS8 1 TW, England,

*. ..

1 *s I “ JOURNAL DE MECANIQUE

652 H. K. MOFFATT

1. Introduction

It is a matter of common experience that if a knife is dipped in honey and then held horizontally, the honey will drain off; but that the honey may be retained on the knife by simply rotating it about its length. The question arises: what is the maximum load of honey that can be supported per unit length of knife for a given rotation rate?

The geometry of a knife does not of course lend itself to simple mathematical analysis, and it is a natural idealisation to consider instead the problem of the behaviour of a viscous film of liquid coated on the outer surface of a circular cylinder which rotates about its horizontal axis with constant angular velocity R. If R = 0, then of course the liquid still drains off the cylinder; but if R # 0, it is at least plausible that the effect of gravity may on average be nullified by the viscous shear forces generated in the layer by the rotation; and in this case a steady solution of the Navier- Stokes equations may be possible.

My intercst in this problem was stimulated by a lecture delivered by V. V. Pukhnachev [l] in October 1973 at the Autumn Course on Mathe- matical and Numerical Methods in Fluid Dynamics at the International Centre for Theoretical Physics, Trieste. Pukhnachev was concerned with the general question of the existence of steady solutions of the Navier- Stokes equations when part of the fluid domain is bounded by a rigid surface, and part by a free surface; by a skilful use of contraction mapping techniques in the appropriate function space, he was able to show that the above problem does indeed have an exact steady solution whenever the kinematic viscosity v of the fluid is sufficiently large (all other parameters being regarded as fixed). Of course when v--t CO, the fluid effectively becomes rigid, and rotates with the cylinder, and provided the free surface is a circular cylinder coaxial with the rigid rotating cylinder, the flow and the configuration are steady. When v is large but finite, Pukhnachev’s steady solution is near to this state of rigid body rotation, and indeed the contraction mapping procedure starts with the rigid body rotation as a first approximation to the solution.

Pukhnachev’s analysis was valid for a liquid layer of arbitrary thickness (which might even be as large as or larger than the cylinder radius). To actually find the analytical form of the exact steady solution (as opposed to simply proving its existence) appears to be quite beyond present techniques. The problem is greatly simplified however if it is assumed that

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VISCOUS FILM ON A CYLINDER 653

the film thickness h (0) (0 5 0 5 2 n) is everywhere small compared with the cylinder radius a. Under this condition it is reasonable to make the usual approximation of lubrication theory, and progress is then relatively straightforward. We follow this procedure in Section 2, and derive the condition for the existence of a steady solution to the problem: this can be expressed either in terms of a maximum load for given rotation rate, or a minimum rotation rate for given load.

In Section 3 we consider the question of whether the steady state (when it exists) is approached, starting from arbitrary initial conditions. The stability theory of Yih [2], which neglects gravity and assumes rigid body rotation of the film, is not relevant when the rate of shear within the layer is large; but the kinematic wave theory of Lighthill and Whitham [3, 41 is applicable, and this describes the evolution of perturbations and the development of discontinuities in depth (in the &direction). The behaviour is analogous to the breaking of a wave - except that the wave breaks backwards relative to the direction of rotation of the cylinder !

In Section 4, a simple experiment is described which succeeds in providing at least a qualitative demonstration of (i) the existence of a steady state for a suitable range of parameters, and (ii) the evolution of disturbances in the manner described above. The experiment goes much further however than the theory can follow at present. The disturbances that always occur as the cylinder rotation rate is increased invariably evolve into rings of remarkably regular spacing along the cylinder axis (see, for example, Fig. 6). The rings are never perfectly axisymmetric; typically, a ring has one or two cclobes” which rotate at somewhat less than the cylinder angular velocity; moreover these lobes tend to oscillate about the plane of the parent ring with a frequency governed by the rotation rate of the cylinder. The lobes can be eliminated from a ring by artificial inter- ference, e. g. by the insertion of a micrometer needle, which leads to a further remarkable steady state configuration [Fig. 8 (a)].

The experiment is purely qualitalive, the aim at this stage being merely to provide photographic evidence of a remarkable phenomenon. Quanti- tative experiments designed to determine precisely what controls the ring size and spacing are in progress.

The work of this paper was motivated primarily by mere mathematical and physical curiosity. One does not need to look far however for possible areas of application: (i) Yih’s [2] analysis was motivated by the need to understand the fluid flow on the rollers of a paper-making machine.

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654 H. K. MOFFATT

(ii) Paint-coated rollers are a popular alternative to brushes as spreaders in the paint industry: if the roller rotates too slowly, the paint falls off, while if too rapidly it flies off. (iii) The same phenomenon is apparent on the rotating spit of a barbecue, with boiling oil as the working fluid. (iv) Rotating shafts frequently require a layer of lubricating fluid in engi- neering machinery. (v) Film coating techniques using rollers are of practical importance in the photographic industry. Finally, it is tempting to conceive of possible applications in such fields as (vi) molten glass technology or even (vii) chocolate technology. Here we are content to describe the phenomenon, confident that applications will follow in due course.

The problem that we shall describe has certain clear points of contact with the complementary problem that arises when the viscous film is on the inside surface of a hollow circular cylinder rotating about its horizontal axis. The steady solution obtained in Section 2 in fact applies also to this situation, with minor modifications in interpretation. When viscous forces dominate, the interior problem is rather similar to the exterior problem; but when the rotation rate increases, centrifugal forces are stabilising for the interior problem, but destabilising for the exterior problem ; an experimental investigation of the interior problem would be of great interest. The inviscid limit for the interior problem was studied by Phillips [5] (') and further aspects of the stability problem for both interior and exterior flows were investigated, again in the inviscid limit by Pedley [ 6 ] . Certain aspects of the viscous limit for the interior problem have been studied by Ruschak and Scriven [l] who (like Pukhnachev) treated the flow as a small perturbation from rigid body rotation, an approximation that is not valid for the range of conditions treated in the present paper.

2. Existence of a steady state solution

We consider first the two-dimensional configuration sketched in Figure 1 (a) (the exterior problem) and Figure 1 (b) (the interior problem). In either case the cylinder of radius a rotates with angular velocity SZ, so that its circumferential velocity is U = Q a, and the free surface is given by

r = a+_hjO, t ) ,

( l ) Photographs showing the instability to which this flow is liable have been published by Karweit and Corrsin [8].

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the upper and lower signs referring to the exterior and interior problems respectively. The mean film thickness is

i f 2 n

ho = 2 J h(0, t )d0 , 27F 0

and this quantity must clearly be constant in time, the fluid being assumed incompressible. The load per unit length of cylinder is

(3) W = 2npaho,

where p is the fluid density. We assume that ho -4 a, and, at least until the solution of the problem indicates a breakdown, we use the thin film 0

Free surface frza-h@,t)

(4 (b) Fig. 1. - A film of viscous fluid coats (a) the outer or (b) the inner surface of a circular

cylinder whose axis is horizontal; the cylinder rotates about its axis with angular velocity i2.

(lubrication) approximation in which V2 is approximated by 8’/8r2 in the viscous layer, and inertia effects are neglected. A necessary condition for the validity of this approximation is

v a

but, of course, even if (4) is satisfied, the approximation may break down at any points where h (0, t ) or ah/a0 become locally large. We shall also neglect surface tension effects; the criterion for this is most easily obtained in retrospect [see (27) below]. The normal stress is then continuous across the free surface. There is a negligible contribution to this normal stress from viscous forces [see (28) below] and the pressure, which in the lubrication approximation is impressed through the viscous layer

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656 H. K. MOFFATT

may therefore be taken as constant. The equation of motion for the circumferential component of velocity U ( r , 8) in the layer is then simply

where 8 is measured from the horizontal (Fig. 1) and g cos 8 is then the tangential component of gravitational acceleration. The relevant boundary conditions are

u = U on r = a ,

au - = O on r = a + h ( 0 , t ) ar

and the solution is gcose

U = U + -(y -2yh), 2 v

(7)

where exterior for the { } problem. interior

+(r -a ) (8)

The flux Q (0, t ) at the section 8 is then given by

(9) Q(8, t ) = [ :udy = Uh- --h g 3 cos€). 3 v

This type of cubic relation between Q and h is well known in a number of contexts, e. g. the film coating context (Deryagin and Levi [9]). Note that at 8 = 0, Q is maximum (as a function of h) when

and then :

(11) 1 2 3 3

Qmar=Uhl- -Uhl = - U h , .

The velocity on the free surface y = h is given from (7) by

gcos0 3Q 1 2v 2h 2

u,(e, t ) = U- -.A = ~ - -U.

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When h = hl at 0 = 0, g uv 1

2v g 2. U =u---.-=-u.

In this state of maximum flux the fluid velocity therefore decreases (parabo- lically) from U on the cylinder to U/2 on the free surface (at 0 = 0); there is therefore substantial shear in the layer, and the motion can certainly not be approximated by a rigid body rotation; in this respect, the conditions are very different from those considered by Ruschak and Scriven [7]. *

I

Fig. 2. - The equation f (v ) = -2 a cos f3 has a continuous solution if and only if a 5 1.

Suppose now that the flow is steady; then clearly Q is constant and h (0) is given in terms of Q from the solution of (9). Likewise the dimen- sionless surface velocity D (0) = U , (€))/U is given [eliminating h from (12)] by the cubic equation

f@) =(U-1)(2~+1)2 = - 2 ~ ~ 0 ~ 8 ,

Equation (14) yields the condition for the existence of a steady solution in its simplest form: the function f ( u ) is sketched in Figure 2, and it is evident that as 0 varies between 0 and 271. we can have a single-valued continuous solution v (0) of (14) satisfying 2 o (0) + 1 > 0 (i. e. h (0) > 0) if and only if

a s i , i .e. Q S ~ ( - - ) 2 u3v =Q,,,.

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658 H. K. MOFFATT

If a > 1, then there is no steady solution to the problem of the form proposed.

This does not quite answer the question posed at the outset concerning the maximum load that can be supported for a given rotation rate. To do this we have to translate the condition (16) into a condition involving the load W rather than the flux Q. Under the critical condition of maximum flux a = 1, equation (9) may be written:

(17) 4 - ~ 0 ~ 8 q ~ - q + i = 0, 27

where q = U h/Q. This defines q as a function of 8 (the root of the cubic for which q (0) = 3/2, q (7c/2) = 1 being the relevant one). This function is plotted in Figure 3 (’). Numerical evaluation of the area under the curve gives

so that

2~ ho = ? = jr h d8 = 4.428 (Y)”’ = 1.057 x 2n-. Q P a U (19)

Note that although the film thickness varies widely between extremes of 1.419 ho at 8 = 0 and 0.846 ho at 0 = n, the load supported is only

(2) Note that when 3 2

a = 1, q = ~ -0.866181 +0(O2).

or equivalently

on the free surface near 8 = 0. This suggests the presence of a sharp corner on the free surface at 8 = 0 when the load is maximal; however, since

the singularity at this point is weak, the angle of the corner being very near to 180”. When a < 1, the free surface is certainly smooth at 8 = 0.

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1.5

1

Fig. 3. - The function 17 (8) as calculated from equation (17). For II: 5 8 5 2 II: q (0) = q (2 x-8) . Note the finite slope of the curve at 8 = 0; although this occurs only under the critical condition a = 1 , it does not imply a genuine singularity in the shape of the free surface at 8 = 0, see footnote on page 658.

marginally greater than it would be if the film thickness were uniform (in which case Q = Uho).

If now the flux is not a maximum, i.e. a -= 1, equation (9) becomes

11 = ~ + x c o s ~ ~ ~ ,

where

The solution of this equation which equals 1 when 0 = n/2 may be obtained as a power series

The first few coefficients are

a. = a, = 1, u2 = 3, a3 = 12, a4 = 55,

a6 = 1428, a, = 7887, . . . ,

The series converges most slowly when 0 = 0, and its radius of conver- gence is then 1c = 4/27. The load is then given by

1 1.3

= e p a ( 3 ~ ) 1 ’ 2 ( l + 0 . 0 3 3 a 2 + 0 . 0 1 0 a 4 + 0 . 0 0 5 a 6 + . . .).

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660 H. K. MOFFATT

For given values of W/p a and v U/g satisfying

;( pu)”’ 5 4.428,

equation (23) determines a unique value of a and hence of

Q = - U g ) 2 auv ,

3

If the condition (24) is not satisfied, then there is no steady solution to the problem. For the external problem, some of the load must then simply fall off the cylinder; one may surmise that if, for given W, U is decreased below the critical value U, for which equality pertains in (24) then W decreases (by load shedding) to the value W, given by

WC=4.428pa -‘ , r: >”’ (or, more realistically, to a value just a little less than W,).

For the interior problem, there is of course no question of load-shedding. In this case, if (24) is not satisfied, then presumably there is an accumu- lation of liquid in the lower portion of the cylinder, and lubrication theory must cease to be valid in some neighbourhood of 0 = 3 42 .

The solution obtained in this section is of considerable interest in that it provides an example of a fluid dynamic configuration for which a steady solution exists only in a certain region of parameter space [given by (24)]. Analogous problems have been considered in purely static contexts : e. g. a pendant drop of density p and volume V can exist in static equili- brium only if V (p g/T)3/2 is not too large, where T is surface tension (Lohnstein [lo]). But it is believed that the flow treated in this section is the first fully dynamic example of such a phenomenon, amenable to analysis.

Before concluding this Section, a comment is necessary concerning (i) the neglect of surface tension T, and (ii) the neglect of the viscous contribution to the normal stress at the free surface. The variation of curva- ture of the free surface as given by (21) may clearly lead to a tangential pressure gradient

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VISCOUS FILM ON A CYLINDER 66 1

Since U 5 4/27 for a steady solution, this is certainly negligible compared with the gravitational contribution to ( 5 ) provided

Secondly the viscous contribution to the normal stress at the free surface is of order p u/a and this gives rise to a tangential pressure gradient

(28)

4 Hence, in the thin film approximation, h, < a, this also has negligible effect on equation (5).

3. Evolution of unsteady perturbations

The existence of a steady solution does not of course guarantee that the solution will be realised in practice ; the behaviour may depend critically on the initial conditions. To examine the evolution of an unsteady film, equation (9) must be supplemented by the equation of mass-conservation in the form

The theory of kinematic waves (Lighthill and Whitham [3]) applies precisely to situations such as this, for which the flux is determined uniquely by the local "density" h. Regarding h as a function of Q and 0 [given by (9)], we have

ah -=(-) ah aQ = - - - ) 1 aQ at aQ eat a ae

and since d Q/d t = aQ/dt+ (aQla0) (d Oldt), when Q is constant we find:

a-=-a-- d0 (aQ,a&- 1 dt a4aQ (ahlaQ>e

(3 1)

Q from (30). Hence if we follow a given value of Q round the cylinder, the time taken to go once round is

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662 H. K. MOFFATT

where Qo = (U3 v/g)”’. The series converges if the series (23) converges, i. e. if a < 1, or equivalently Q/Qo < 2/3.

Now T (Q) is clearly an increasing function of Q. Hence high Q-values take longer to get round the cylinder than low Q-values and so the gradient of Q must steepen in any region where aQ/d0 > 0. The low Q-values tend to overtake the high Q-values, so that discontinuities of depth, in which h increases suddenly in the direction of rotation, must tend to develop (’). Lubrication theory is of course no longer valid in the neighbourhood of such discontinuities.

the cylinder surface (as well as of 8 and t ) provided the scale of variation in the z-direction remains large compared with the mean thickness h,. This remark is relevant to the interpretation of the experiment described in section 5.

6

The above theory is still valid if h is a function of axial distance z along i

and so, from (15) and (23) (which still obviously applies under unsteady conditions),

(33) T(Q) =%(I+ -( 5 Q ->*+ E( “r+ -( 7735 -yz+. Q . .), 6 Qo 24 Qo 972 Qo

4. Inclusion of weak diffusion effects in 0-direction

The theory of Sections 2 and 3 neglects gradients of velocity in the &direction, on the grounds that these are certainly small compared with gradients across the layer when the layer is sufficiently thin. The conclusion of section 3 however that discontinuities of depth in general tend to develop suggests that a second-order lubrication theory taking some account of gradients in the 0-direction may be required for a full description. Equations (9) and (30) give the non-linear evolution equation

&-

(34)

and although this equation admits steady solutions [when (24) is satisfied]. section 3 indicates that such steady solutions will inevitably be unstable,

(3) Thc, situation is similar to that arising in the traffic flow problem studied by Lighthill and Whitham [4].

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Weak diffusion in the @-direction may have a stabilising effect. for example the modified equation

Consider

(35) a ah ah g a a2 h U at ae 3vu ae ae2 - - + - = __ -(h3cose)+E-,

in which the diffusion term E a2 h/a02 (with E < 1) has been included in an attempt to represent qualitatively the various effects neglected in lubrication theory (at lowest order). It is not suggested that equation (35)”

Fig. 4. - Computed solution of equation (35) with hg g/v U = 0 . 3 3 , E = 0.03, and h (9,O) = h o .

can be obtained by systematic expansion procedures, but merely that solutions of (35) will resemble qualitatively the exact behaviour of h (8, t ) for a thin film. The small parameter E must then be regarded as a function of ho/a, tending to zero as h,/a + 0. Figure 4 (a, b) shows the behaviour of the computed solution of (35) subject to the initial condition h (e,O) = ho (i.e. uniform film thickness initially) and for values of the parameters 4

~- h’g - 0 . 3 3 and E = 0.03 . vu (36)

Figure 4 (a) shows h (8, t) /ho after 2 revolutions and after 12 revolutions of the cylinder (U t /a = 4 n, 24 TC respectively) and Figure 4 (b) shows the variation of thickness as a function of time at the point 8 = 0 where that variation is most marked. It is clear that the solution is settling down to a steady state, and that weak lateral diffusion is sufficient to overcome the tendency for discontinuities to develop described in Section 3 above.

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664 H. K. MOFFATT

5. Experimental observations

Figure 5 shows the apparatus used for a simple qualitative experiment. A perspex roller, 4.08 cm in diameter and 30 cm long was mounted on a steel shaft which could be rotated about its axis by a variable speed electric motor. The shaft and the motor were mounted on flanges in such a way that the roller could be lowered into the trough of viscous fluid and raised clear of the fluid while still rotating. The liquid used was golden syrup (pure sucrose) diluted with a small quantity of water till its viscosity was approximately 80g.cm-l . s-’ at 15°C. A layer of syrup was first coated on the roller by rotating it while partially immersed in the trough; the roller was then raised from the trough while still rotating, and it was then possible to observe the behaviour of the film for a range of rotation speeds.

Figure 5 (a-d) shows a sequence of 4 photographs taken at rotation rates ranging from 21.1 rpm to 48.8 rpm. In (a) and (b) the film thickness is very nearly independent of the axial coordinate z, except for slight bulging towards the ends of the roller (non-uniformity of depth in these regions is of course an inevitable consequence of the finite length of the roller). In (c) at 38.2 rpm these bulges are much more marked, while in (d) at 48.8 rpm they have developed into characteristic rings of syrup around the roller and are accompanied by three new similar rings in the central region. It should be observed that, whereas in (U), (b) and (c) there is no sign of any tendency for depth discontinuities to develop, these signs are apparent in the rings of ( d ) which are by no means axisymmetric but which generally exhibit a “lobe” which rotates at an angular velocity slightly less than that of the cylinder. Figure 7 (a-d) shows the passage of a typical well-developed lobe round the cylinder, viewed from along the cylinder axis; the cylinder here rotates in a clockwise sense.

The mean film thickness in Figure 5 (a) was approximately 2.5 mm. At 21.1 rpm, the flow apparently remained steady (as far as simple visual observations were concerned), and without any tendency to develop instabilities, for as long as the rotation was continued (several hours); it was quite clear that the steady state observed was that described by the steady solution given in Section 2. It must be inferred that lateral diffusion analogous to that considered in Section 4 was sufficient at this low angular velocity to suppress the tendency of Section 3 for the development of depth discontinuities. By contrast in Figure 5 ( d ) at the greater rotation

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666 H. K. MOFFATT

rate 48.8 rpm, at least one (and sometimes two or even three) depth disconti- nuity (i. e. lobe) develops on each syrup ring.

Figure 6 ( a d ) shows the development of syrup rings when the rotation speed is maintained at a relatively high value (77.4 rpm). The photographs show the viscous film 1, 3, 5 and 30 min. after the commencement of rotation (the film at the initial instant being approximately uniform). The instability first appears as a number of bumps on the surface more or less randomly distributed. It seems likely that this appearance of bumps is due to the destabilising action of the centrifugal force; an analogous instability due to gravity arises if a plate containing a thin layer of syrup is suddenly turned upside-down. The bumps on the rotating cylinder rotate slightly less rapidly than the cylinder itself (a manifestation of the fact that high Q-values take longer to get round the cylinder, Section 3) and after 3 minutes [Jig. 6 (b)] each bump has evolved into a ring around the cylinder, the bump itself surviving as a lobe on the ring. At t = 5 min. the rings are more developed and some regularity of spacing along the cylinder axis is apparent. At t = 30 min. (after more than 2,000 revolutions of the cylinder) the rings have attained an asympto- tic state that survives during subsequent hours of rotation. Of course if the rotation is suddenly stopped, the syrup immediately drains from each ring into the trough.

The slow development of the rings as described above is again presumably controlled by centrifugal forces. These make a contribution p u2/r to the radial pressure gradient, and since p is constant on the free surface, the pressure is consequently lower under a ring than elsewhere (at the same value of r ) . Consequently each ring tends to suck fluid in from either side until the surviving film between the rings is so thin that the growth of the rings effectively ceases. The film thickness between the rings in Figure 6 (d) , measured with a micrometer screw gauge, was 0.043 mm.

Figure 8 shows enlarged photographs of rings in various configurations. The micrometer needle inserted in the large ring of Figure 8 (a) had the effect of eliminating the lobe on the ring, and of creating an absolutely steady flow. The right-hand ring of Figure 8 (b) exhibits a large lobe which is c6top-heavy” at the top of its circular path and which is consequently trying to fall to the left of the central plane of the ring. This deformation corrects itself as the lobe moves down under the cylinder and reappears as it rises again to the top; there is generally a rather erratic tendency for such lobes to oscillate between left and right deformations in succes-

@,

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(a) t = 1 min.

(b) t = 3 min.

(c) t = 5 min.

f

(d ) t = 30 min.

Fig. 6. - Development of instabilities when R = 77.4 rpm. The instabilities start as small humps randomly distributed on the surface; these gradually evolve into twelve well-developed instability rings, each ring exhibiting the characteristic lobe.


658 H. K. MOFFf


(4 Fig. 8. - (U ) Stabilisation of a syrup ring by insertion of needle; (b) an example of a

lateral instability of a lobe on a ring due to “top-heaviness” at the top of its trajectory; (c) the two left hand rings of (b) have coalesced to produce a giant ring that cannot be stabilised by the needle.


670 H. K . MOFFATT

(a) 11.4rpm

(b) 58.2 rpin

(c) 48 rpm

Finally, Figure 9 shows a sequence in which, starting with a set of eleven fully developed rings on the roller at 77.4 rpm, the angular velocity was decreased in steps, the flow being allowed to settle for approximately 5 min. at each step before the photograph at the reduced angular velocity was taken. Note that the lobes disappear in reducing speed from 48 rpm to 36.4 rpm. At this latter speed the eleven rings are still apparent on the roller. With further reduction of speed however, there was a coalescence of rings and only 5 weak bulges survived the subsequent reduction of speed to 8 rpm.

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- Decay and coalescence of instability rings during reduction of rotation rate.

6. Conclusions

The observations described in the preceding section raise a number of theoretical problems of great difficulty. First there is the question of the observed realisation of the steady state when the rotation rate is sufficiently low [but not so low that the condition (2.24) is violated]; lubrication theory indicates that although this steady state exists, it will not in general be approached starting from arbitrary initial conditions, but discontinuities in film depth will tend to develop. Weak diffusion (Section4) is sufficient to defeat this tendency and to replace it by an approach to

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672 H. K. MOFFATT

the steady state ; but an improved second-order lubrication theory is really required to elucidate the nature of the real 0-diffusion effects.

Secondly, there is the question of the observed instability of the film as the rotation rate is increased. As mentioned earlier, Yih's [2] theory, which neglects gravity and shear within the layer, is not applicable in the present context in which gravity and viscous shear provide the primary force balance. Centrifugal acceleration, though weak compared with gravity at the rotation speeds considered, can have a slow cumulative effect over a large number of revolutions, and it seems certain that centrifugal forces are in some way responsible for the instability. Surface tension effects must also be significant in determining the scale of the initial bumps [Fig. 6 (a)] and the final shape of the instability rings [e. g. Fig. 6 ( d ) ] . But the detailed stability problem, involving as it does the strong shear and che 8-dependence of the basic flow must necessarily be extremely difficult.

Thirdly, there is the question of the dynamics of the finite amplitude motions within the fully developed instability rings. The lobes on these rings can be understood at least qualitatively in terms of the kinematic wave theory of Section 3; but questions regarding the thickness of the rings and their detailed shape and their response to external stimulus are as yet wide open.

L "("

Acknowledgements

I am greatly indebted to Professor V. V. Pukhnachev who persuaded me that the problem had a solution, to Sir James Lighthill who drew my attention to the applicability of kinematic wave theory and to the method of solution adopted in Section 3, to Dr. Robert Harding for computing solution curves of equation (4.2), and to many colleagues who have shown interest in the experiment and made constructive comments concerning its interpretation. I am also grateful to Mr David Cheesley for his expert technical assistance in constructing the apparatus and to Mr Chris Chalk for help with the photography.

- ~

REFERENCES

[l] PUCKNACHEV V. V., Problems with a Free Boundary for the Navier-Stokes Equations (Proc. Autumn Course on Mathematical and Numerical Methods in Fluid Dynamics, 1973, Int. Centre for Theor. Phys., Trieste).

[2] YIH C.-S., Instability of a Rotating Liquid Film with a Free Surface (Proc. Roy. SOC., Vol. A258, 1960, pp. 63-89).

VOLUME 16 - 1977 - NO 5


[3] LIGHTHILL M. J. and WHITHAM G. B., On Kinematic Waves. I. Flood Movement in Long Rivers (Proc. Roy. Soc., Vol. A229, 1955, pp. 281-316).

[4] LIGHTHILL M. J. and WHITHAM G. B., On Kinematic Waves. II. A Theory of Trafic Flow on Long Crowded Roads (Proc. Roy. Soc., Vol. A 229, 1955, pp. 317-345).

[5] PHILLIPS 0. M., Centrifugal Waves (J. Fluid Mech., Vol. 7, 1960, pp. 340-352). [6] PEDLEY T. J. The Stability of Rotating Flows with a Cylindrical Free Surface (J.

Fluid Mech., Vol. 30, 1967, pp. 127-147). [7] RUSCHAK K. J. and SCRIVEN L. E., Rimming Flow of Liquid in a Rotating Horizontal

Cylinder (J. Fluid Mech., Vol. 76, 1976, pp. 113-125). [8] KARWEIT M. J. and CORRSIN S., Observations of Cellular Patterns in a Partly Filled,

Horizontal, Rotating Cylinder (Phys. Fluids, Vol. 18, 1975, pp. 111-112). [9] DERYAGIN B. V. and LEVI S. M., Film Coating Theory: the Physical Chemistry of

Coating Thin Layers on a Moving Support, Translated by W. R. EICHLER and E. A. SUTHERNS (Focal Library), London (Hungary pr., 1964).

[10] LOHNSTEIN T., Zur Theorie des Abtropfens mit besonderer Riicksicht auf die Bes- timmung der Kapillaritatskonstanten durch Tropversuche (Ann. d. Phys., Vol. 20,

\t- 1906, pp. 237-268).

(Manuscrit reGu le 25 mars 1977.)

JOURNAL DE M~CANIQUE

H.K. Moffatt- Behaviour of a viscous film on the outer surface of a rotating cylinder

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