Jae-Tack Jeong and H.K. Moffatt- Free-surface cusps associated with flow at low Reynolds number

8/3/2019 Jae-Tack Jeong and H.K. Moffatt- Free-surface cusps associated with flow at low Reynolds number

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J . Fluid .Mech. (1992), vol. 241, p p . 1-22Printed in Great Br ita in

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Free-surface cusps associated with flow at lowReynolds number

By JAE-TACK JEONGT A N D H. . MOFFATTDepartment of Applied Mathematics and Theoretical Physics, University of Cambridge,Silver Street, Cambridge, CB3 9 E W , CK

(Received 7 August 1991)When two cylinders are counter-rotated at low Reynolds number about parallelhorizontal axes below the free surface of a viscous fluid, the rotation being such asto induce convergence of the flow on the free surface, then above a certain criticalangular velocity R ,, the free surface dips downwards and a cusp forms. This paperprovides an analysis of the flow in the neighbourhood of the cusp, via an idealizedproblem which is solved completely : the cylinders are represented by a vortex dipoleand the solution is obtained by complex variable techniques. Surface tension effectsare included, but gravity is neglected. The solution is analytic for finite capillarynumber GR, but the radius of curvature on the line of symmetry on the free surfaceis proportional to exp ( - 327ccB) and is extremely small for GR 2 . 25 , implying (in areal fluid) the formation of a cusp. The equation of the free surface is cubic in (x, )with coefficients depending on V , and with a cusp singularity when W = CO.

The influence of gravity is considered through a stability analysis of the freesurface subjected to converging uniform strain. and a necessary condition for thedevelopment of a finite-amplitude disturbance of the free surface is obtained.. An experiment was carried out using the counter-rotating cylinders as describedabove, over a range of capillary numbers from zero to 60 ; the resulting photographsof a cross-section of the free surface are shown in figure 1'3.For R < R ,, a roundedcrest forms in the neighbourhood of the central line of symmetry ; for R > R,, thedownward-pointing cusp forms, and its structure shows good agreement with theforegoing theory.

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1. IntroductionNewtonian fluids at low Reynolds numbers have been recently published in thisJournal (Joseph e t al. 1991, hereafter referred to as JN RR). These photographs. provide compelling evidence for the formation of two-dimensional cusps on the freesurface in regions of convergence of the flow to what would otherwise be a stagnationline. We have repeated the experiment of JNRR using a Newtonian fluid and a pairof counter-rotating cylinders, in the symmetric configuration of figure 1 (forexperimental details, see $ 6 ) .For very slow rotation rates, there is a stagnation lineon the free surface, and in some circumstances a small rounded crest can form in theneighbourhood of this stagnation line (figure l a ) . (This phenomenon was noted in anearly investigation by Griggs (1939) in a paper concerned with the process ofmountain formation by convection in the underlying lithosphere. Griggs carried out

Present address : Department of Mechanical Engineering, Kum-Oh Sational Institute ofTechnology, 188 Shinpyung Dong, Kumi, Kyung Buk, Republic of Korea, 730-701.

b Some striking photographs of free-surface flows of both Newtonian and non-

www.moffatt.tc


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2 J.-T.Jeong and H . K . Mojfatt

F~QURE. Experimental configuration showing the observed form of the free surface (a )when therotation rate R is very small and ( b )when R is larger. The range of SZ in the experiment was 0-7 5-l.The fluid used was polybutene, and the range of Reynolds numbers Re = a r a / v was zero t o 0.25 ithe range of capillary numbers geX,see$ 6 ) was zero to 61.1. and of Froude numbers F r = R(rc/g)gzero to 0.34. he ratio WeXp/Fr= , u ( r ,g ) i / y had the value 180.a similar experiment to tha t described here. but with the cylinders rotatkd manuallyand with a layer of a mixture of heavy oil and sand on the fluid surface to simulatethe Ea rths crust.) When the rotation rate SZ is increased however. the surface dipsdownwards. and simple visual observation indicates the presence of a very sharpcusp on the free surface (figure l b ) . If powder is sprinkled on the free surface, thispowder is immediately swept through the cusp into the interior of the fluid. Thus,observation suggests that fluid particles on the free surface are similarly advectedthrough the cusp into the interior.

There are, however, certain fundamental difficulties in accepting this conclusion,

,


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Free-surface cusps 3convincing though the observational evidence may appear to be. First, a cusp is asingularity of curvature. and one would expect this singularity to be resolved bysurface tension forces in its neighbourhood. J S R R proposed a solution in which thevelocity is locally a small perturbation of a uniform stream I'parallel to the (double)tangent a t the cusp. the perturbation stream function being of the form $ = Inf O ) ,where h is a parameter dependent on the capillary number %? = pC/y (where p isviscosity, and y surface tension). We comment in more detail on this solution in thefollowing sections; for the moment it is sufficient to recall JS R R ' s conclusion thatwhen V = CO , then h = i. and the cusp has the local form y - 1x1;. The correctboundary conditions are then satisfied everywhere except at the singularity x = 0,y = 0 itself, which is in a sense where most of the interest of the problem resides!J S R R refer to an earlier discussion of cusp-type singularities by Richardson (1968)who proposed a local solution of the form

(in plane polar coordinates) which is singular at r = 0. and is associated with a pointforce of magnitude 2y exerted by the free surface on the fluid directed along thetangent at the cusp and out of the fluid. This solution suffers from the seriousdifficulties that the associated velocity is O(1nr) near r = 0, and the associated rateof dissipation of energy is infinite. Thus, although JX'RR regard Richardson'ssolution as being valid in some extremely small neighbourhood of the cusp, thismerely replaces one imperfection by another. and does nothing to resolve the realnature of the flow near the * cusp in a viscous fluid with non-zero surface tension.

There is a second major difficulty associated with the presence of the air outsidethe viscous liquid, which is subject to the no-slip condition a t the free surface. If thecusp is genuine and fluid particles on the free surface do move into the interior of thefluid. then air must be entrained into the interior also. There is however no evidencein the experiments for the entrainment of air bubbles (although small bubbles can ofcourse be deliberately injected to provide indicators of particle paths and rate ofstrain). This is a paradox reminiscent of that encountered in the famous movingcontact-line problem (see, for example, Dussan V . & Davis 1974) in which strictapplication of the no-slip condition is in flat contradiction with both observation andcommon sense.The object of the present paper is to provide a complete analytical solution of amodel problem which does indeed reveal the full nature of the flow and the extentto which a description in terms of a cusp is legitimate. The model problem is anidealization of figure 1, in which the rotating cylinders are represented by a vortexdipole at fixed depth d ( = 1)below the undisturbed position of the free surface (figure2 a ) . The outer fluid boundaries are supposed moved to infinity, and we adopt thenatura l outer boundary condition U -+ 0 as 1x1+ CO. The resulting problem is solvedusing complex analysis and conformal mapping techniques. The solution doesconfirm the formation of a cusp in the limit %'+ O. For finite % however. the solutionremains regular with a stagnation point on the free surface on the plane of symmetry.The radius of curvature R of the free surface at this stagnation point has theextraordinary behaviour Rld - exQ{- 2 ~ V ) (1.2)and is therefore extremely small when %? (defined by (2 .24) below) is of order unityor greater. It is this behaviour that presumably leads to the extremely sharp cusp-like structures observed in experiments.


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4 J .- T . Jeong and H . K . Moffatt

:-plane(a )

Vortex dipole

IFIQURE. ( a ) Idealized problem showing deformation of the free surface r by a vortex dipoleplaced at z = r+ y = - . ( b )The image.configuration in the


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Free-surface cusps 5of curvature, taken positive if the centre of curvature is on the air side of theinterface.It is well-known that $ can then be expressed in the form

$ = Im (f 4 W z ) ) , (2 .2 )where the overbar represents the complex conjugate, and f ( z ) , g(z) are analyticfunctions at all points z in the fluid domain 9 xcept at the singularity at z = - iwhere

iuf (4- zsi ( z+-- ) . The velocity components are then given by

U- w = (z) + ~ g ( z ) gO, (2 .4 )p-ipw = 4pg(z). ( 2 . 5 )

and the pressure ( p )and vorticity ( U ) fields are given by

It is easy to verify that, with these relations, the Stokes equation V p = p V 2 u issatisfied in the fluid. The condition ZL, +-0 as IzI + CO are satisfied providedf-cz, g - a as I z / + - c o , (2 .6 )

where c is an arbitrary constant. We shall find that the choice c = -iy/4p isappropriate. The symmetry conditions $ = 0, w = 0 on x = 0 clearly imply that

As shown by Richardson (1968) ,the boundary conditions on r take the formImf(iy) = 0, Reg(iy)= 0. ( 2 . 7 )

(2 .9 )where s is the arclength on fmeasured from the point of symmetry B (figure 2 a ) , andu0(z) s the (real) tangential. velocity a t an arbitrary point z of the free surface..Equation (2 . 9 ) is equivalent t o (2 . 1 ) . Manipulation of (2 .8 ) and (2 .9 ) yields theequations (for z E r )

(2.10)and f 2 )+zg(z) = 0. (2 .11)Equations (2 . 3 ) ,(2 . 6 )and (2 .8)-(2 .11)constitute the essential boundary conditionsthat f(z) and g(z) must satisfy.Now let z = w(5) be the conformal mapping that maps the fluid domain 9 o theunit disc 9 :51< 1 , and which places the (imaged) vortex dipole at 5 = 0, so thatw(0)= -i (figure 2b) . The points A, B, C, D of figure 2 ( a ) map to the points A, B,C, D of figure 2 ( b ) .Let (2.12)) (w ( 5 ) )= f (4 ,

Q(5 )= g(w(5))= g(z ) ,U ( 5 )= UO(W(5) )= uo ( z ) ,

(2.13)(2.14)


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6 J . - T .Jeong and H . K . Moffattso that (2.15)Moreover, if 1 ( = n - 8 ) represents arclength from B on the unit circle, then by theconformal mapping property ds/dZ = lw(


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Free-surfacecusps 7(2 .26) , he boundary condition (2 .22)can also be satisfied, as will be seen later.) Weshall find that , for suitable choice of a (dependent on the capillary number %), we cansatisfy all the conditions of the problem, so that (2 .26) does indeed provide therequired solution.Note first that, from (2 .26) , w(i) = 2ai, so that the point 5 = i is the image ofz = 2ai, the point B in figure 2(a) .Since this must lie above the vortex dipole atz = -i , we must have a > -a . Secondly, the function

2 (a+1)w ( 0 = a - ( f + i ) l (2 .27)must be non-zero in 161 < 1. The zeros of w (0 are at = - i f ( 2 ( a + l ) / a ) i , nd it iseasily shown that both roots lie outside the unit circle if either a < -1 or a > -4.Together with the condition a > -a , we see therefore that the relevant range of a is

a > -4, ( 2 . 2 8 )and we note that as a-+-4, a singularity appears on the boundary 151 = 1 and henceon r also.

-,

Substituting ( 2 . 2 6 ) in (2 . 23 ) (and using E = 1/5)now givesHence, as 5 - 0 , F ( 5 )h- -aG(O)/C, so that comparing with (2 .22)we find

ai - iG(O) = -- =aw ( 0 ) a ( 3 a + a ) .

(2 .29)

( 2 . 3 0 )This provides a first relation between G(0) and a.A second relation is needed to effecta complete solution. Note that G(0) is pure imaginary.Returning now to ( 2 . 2 1 ) , he real part of the right-hand side is now known (apartfrom the real constant a ) . Hence by a well-known corollary of Cauchys integraltheorem (Muskhelishvili 1953) we have that, for 151 < 1,

where b is a real constant. The symmetry condition Reg(iy)= 0 in fact implies thatb must be zero. Evaluating the second term of the integral in (2 .31)and rearranging,we thus obtain

(2. .32)Consider now the behaviour of this expression as 5 tends to a point on the unit circle.- With 5, = eieo and 5 = eie, we have from ( 2 . 3 2 )

yi fi 1 cos8,+cos8-- - ~ C O S ~ - + - - + -0 ) Y 1 - d@,. ( 2 . 3 3 )Cw(5) w(0) 4pIw (


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8 J.-T.Jeong and H . K. MoffattIII

a0.1 0.2 0.3

-3 -FIGURE. Graph of the function H ( a )defined by (2 .3 9 ) .For each value of the capillary number V(positive or negative) the real parameter a is uniquely determined by H ( a )= 47M (equation (2 .3 8 ) ) .Now, from ( 2 . 2 7 ) ,we may obtain

sin8 +( a + l )2 ,(w'(eie)I2= a2+2a(a+ 1)1+sin8 1+sin8 ( 2 . 3 5 )Hence uo(z)s given explicitly in terms of the parameter a, which is still unknown.This parameter may now however be determined from the condition that uo(z)-t 0as z + f CO (i.e. as O+-$ in (2 .34 ) ) .Near 8 = - in , we have

(2 .36 )Hence the term in square brackets in (2 .34 )must certainly vanish as 8+ - t z , i.e.

dboG(0)=- w'(eieO)I (sin8,+ 1) (2 .37)(Note that , since (2 .36 )has a simple pole and the term in square brackets in (2 .34 )has a double zero, uo(z) oes indeed tend to zero at infinity.)Substituting (2 .26 ) , 2 .30 )and (2 .35 ) nto ( 2 . 3 7 ) ,we obtain an equation of the formY\ , ( 2 . 3 8 )

from which a may be determined. The function H ( a ) s given (see Appendix A ) by- ( 3 a2 ) 2 ,K(m) for - g


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Free-surface cusps(4

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FIGURE. Streamlines a nd corresponding free-surface shape for various values of a. All streamlinespass throug h the singular i ty a t ( 0 ,- ) , nd the streamlines near th is point are c ircles touching t hey-axis. ( a )a = -0.2, '3 = 0.072 ( b )a = 0 ,W = 0 (po tent ial f low wi th undisturbed free surface) ; ( c )a = 0 .2 , V = -0.13; in this case (z < 0) the flow is diverging on r and the ,f ree surface is pushedupwards.The function H ( a )as given by (2.39)has been computed and is shown in figure 3. Theasymptotic behaviour of this function is easily calculated :

H ( a )= -27ra(1+$a)+O(a3) for a 0.With a known, we can now complete the solution of the problem. First , from (2 .32)and (2.371,we derive


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10 J.-T.Jeong and H . K . MoffattI

FIGURE ( a )As for figure 4. but with a = - 0 . 3 . W = 0 1 : ( b ) same figure expanded around thestagnation point on r.(4

FIGL-RE. (a )As for figure 4 . but with a = - V2 = CO : ( b ) same figure expanded around thecusp on rThis integral can be expressed in terms of complete elliptic integrals of the first andthird kinds (see Appendix A). Now from (2.2). 2 .1 2 ) . ( 2 .1 3 )and (2.29). he streamfunction @ is given in terms of G(C)by

(2 .47)from which the streamlines @ = const. may be plotted. Figures 4-6 show thestreamlines for a range of values of a (with corresponding values of 9? from figure 3 ) .Note in particular the tendency to form a cusp as a +-&(%'-+ m).

3. The free surfaceThe free surface r is given by z = w(C) with 5 = eiB: or?from (2.26);

The real and imaginary parts give the equation of r in parametric form :cos9x = acos8+(a+l) - 1+sin 8 '

( 3 . 1 )

( 3 . 2 a )y = a ( l+s in8) . ( 3 . 2 b )


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Free-.vrrlfctce c m p 8 11

xzy = ( 2 a - y ) ( y + a + l ) ' . (3.3)Elimination of 0 gives the equation of r as the cubic curveThis curve is sketched in figure 7 for a range of values of rc : only the range n 2 Qis relevant to the problem studied here. When cc = -Q. the curve has a cusp withvertical tangent at x = 0. y = -$.For a > -Q: the curve is locally parabolic near the point ( 0 . 2 ~ ) :

(y- n ) .3a+ 1)*- ax2 x (3.4)* The radius of curvature at ( 0 . 2 n ) is

R = (3a+ l ) z / (4 ~ ) . (3.5)As a+-$, the asymptotic form (2.44) together with (2.38) now gives ( 3 a f l )-yexp{- 6xV); hence, restoring the dimensional length r l . R is given asymptoticallyby Rld - y e x p - 2xV}. (3.6)as stated in the Introduction. This asymptotic behaviour is reasonably accurate for$5 2 .1 ( H ( a )2 1.2 in figure 3) , and indicates extremely small values ofR/d when Vis of order unity. Even for % = 0.1. we have Rld x .67 x 10-3. while for % = 0.25.Rld x 10-g, and for W = 1, Rld z 1.87x 10-42!Of course the continuum approxi-mation fails on such small lengthscales: from a continuum point of view. it seemsfair to state th at when d is of order 1 m or less. a cusp does indeed form when %? 2 .25(givingRld 5 10-g).Putting n = - i+s , we find from (3.3) hat for E 4 1 and y+Q % E . the free-surface

( 3 . 7 )shape isIf we further restrict y to the range for which e < y + Q < 1 (so that y % - Q ) . then (3.7)gives

X J x p-J* = y+$. (3.8)in agreement with the self similar form x - C f i obtained by JNRR. but with theadded bonus that the coefficient T , which is undetermined in JXRR's local analysis.is here determined as F = (g); x .2%. This value of G is particular to the idealizedvortex dipole problem ; in more general experimental configurations. F may beexpected to depend on the dimensionless parameters defining the geometry (e.g. r , /din the configuration of figure l u ) . J S R R used a coordinate system in which thecusp opened along the negative x-axis. and their result, equivalent to (3.8) wasy - X ~ . heir experiments with STP, silicone oil and castor oil showed qualitativeagreement with this theory. but the value of F was ill-determined. The capillarynumber used by J N R R is (using ( 3 . 1 2 ) )approximately 16 times the %? that we usein the present paper ; hence the value %j= 0.25 corresponds to [%?],J,RR x 4. n roughagreement with their critical value for cusp formation.

In spite of the dependence of c o n the geometry. the curves ( 3 . 7 )and (3.8) whensuperposed on the photograph obtained in our experiment (figure 8) at a capillarynumber ( % ) e x p = uQr,/y x 0 exhibit a remarkable convergence to the observedform of the cusp. Agreement is not to be expected far from the cusp where the precisegeometry of the experiment must certainly have some influence. The imperfection ofthe fit between theory and experiment at the very tip of the c~ispmay be due toimperfect scaling of the photograph.

x2y x ( y + $ ) 3 .

,

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12 J . - T . Jeong and H . K . JloffattI

-Y-:FIGURE. The curve ( 3 . 3 ) or a = -1. -j. -0.1.0. +0.2. Only the curves corresponding to a 2 - jare physically relevant here. The curve is cusped for a = -f. For a 2 4, the curve provides thefree-surface shape a t the capillary num ber % given by figure 3 .

FIGURE. The a symp tot ic curves (3.i) nd ( 3 . 8 ) uperposed o n the observed cusp in an ex perimen twith Vex*= 5 2 . 4 . The photograph is enlarged t o give the best fit with the curve 2 y = - ( Y + $ ) ~

The curve ( 3 . 3 )has a universal form when a = -g+e. 0 < E 4 . This is obtainedby the substitutionsRetaining only leading-order contributions (of order e 3 ) . his yields the curve

x = E", Y = -2 a = 7. ( 3 . 9 )f' = ?T(T + 3)2, (3 .10 )

which exhibits the parabolic behaviour t 2- y7 for 7 4 . and the cuspidal behaviourt2 k3or 7 % 1 (figure9 ) .The curve (3 .10 )has inflexion points at 7 = 1 . 6 = + 2 .\/6marking the transition between these regimes.

Consider now the tangential velocity uo(z) n r.which is given from (2 .34 )and(2 .37 )by

'

1 1-uo(z)= - C(e'0) = -- 1+sin 8)cos8 Iw'(e'')I I(sin8 ;a)U a 4n w (3.11)where I is still as defined by (2 .46 ) (where the principal value must now be used).With 8 related to x by ( 3 . 2 ~ ) .3 .11 )determines uo /a mplicitly as a function of x.This function is shown in figure 10 for various values of a in the range


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Free-surface usps5 t

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' regimeFIGURE. The universal curve 5"= $(? +312 epresenting transition fromparabolic to cuspidal regimes.

FIGURE0. The tangential velocity uo(z) n r normalized with respect to the strength U of thevorte x dipole, for a = -i, -0.3, -0.2, -0.1,0,0.2. Note the singular struc ture as a+-&, and thelimiting value uo(0)/u= f 6.I -4 < a < 0.2(co 2 W 2 0.13). he curve corresponding to a = 0 coincides withthat obtained by a linearized analysis (Appendix B, equation (B 16)).The free-surface velocity is directed towards or away from the stagnation point on the freesurface according as a < or > 0. A singular stru ctu re is evident a s E = a+++- , ithrap id deceleration in th e region (21= O ( E ~ )here th e cuspidal shape of the free surfacegives way to th e parabolic shap e. Note th e limiting behaviour.

lim lim (uo/a) & 16,/+o E+ O

(3.12)indicating th at the (dimensional) fluid velocity through th e cusp into the fluidinterior has magnitude 16a / d 2 ,As noted above, for any capillary number greatert ha n a bou t 0.25, the parameter E is extremely small, and this limiting descriptionmust certainly be appropriate.We may note tha t , when % 2 .25, the 'parab olic region' is on a subm icron scale,so th a t th e question of whether fluid particles on the free surface do or do not move

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14 J . - T . Jeong and H . K . Moffattinto the interior of the fluid has a rather philosophical character. In situations suchas this where the continuum approximation breaks down, the concept of a fluidparticle is ill-defined, and the true behaviour in the neighbourhood of the cusp mustsurely require consideration of intermolecular forces and related non-continuumeffects. The same comment applies to the behaviour of the air trapped in the cuspregion. These considerations are unlikely, however, to affect the 'outer ' cuspdynamics, and the shape of the cusp as given by (3.8).4. Asymptotic form of stream function as a --f -+

We have already seen in (2.44) hat the function H ( a )has a logarithmic singularityas a+-& (i.e.V+ CO). The same singularity appears in the function I((c-c-l)/2i;a)appearing in the solution for G(5) (equation (2.45)) .I

(see Appendix A). Hence from (2.45):and so from (2.47): G(5)- ia(5+3i) (c+i )>with 5 related to z , when a = -4, by

(4.1) '

(4.3)

Near the cusp point 5 = i ( z = -gi), (4.4) becomesi6z+gi x -(c-i)2+ ... (4.5)

and, with z +$i = ir eis (so that r is distance from the cusp and 8 is measured from thedirection of the cusp tangent), (4.3) gives the asymptotic form of $ near the cusp as$ - 16ax-886arisin3($B) ( - n < 6 < n ) , (4.6)

agreeing with the local similarity solution proposed by J N R R (a t least when V = C O ) .This stream function represents the uniform stream -16a into the fluid, with asuperposed perturbation velocity of order r i near r = 0. Note however that theassociated rate of strain is O(r-4). This is resolved on the scale r = O(s) = O(e-lsnv).Hence a maximum strain rate of order eEZvs implied. The corresponding stress in thefluid is extremely large for quite modest values of V. owever, the O ( r - l ) singularityin the rate of viscous dissipation is integrable.

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5. Influence of gravityIt is obvious that , in an experiment like that of figure 1,gravity exerts a stabilizinginfluence on the free surface. tending to keep it horizontal against the influence ofviscous stresses. The effect of gravity was neglected in the model problem of $2. We

may however estimate its effect by a local stability analysis similar to that used byLister (1989) in a rather similar axisymmetric situation.


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Free-surface usps 15We consider the stability of the free surface y = 0 under the action of theconverging uniform strain

Consider a disturbance of the free surface of the formU = (-sx, y). (5.1)

( 5 . 2 )It is obvious th a t k( t ) = koe S t , (5.3)due t o th e compression of the w ave-form (figure 11a). f grav ity a nd surface tensionwere absen t, then th e disturbance am plitude d ( t )would also increase as ea t, purelykinematic effect. Both gravity and surface tension have a countervailing influence.The stream function $(x,y, t ) , satisfying V4$ = 0 a n d $+O as y+- CO, m ayreadily be shown to have the form

(5.4)an d th e (linearized) kinematic boun dary condition DF/Dt = 0 on y = ~ ( x ,) whereF = y-q(x , t) gives

y = q(x, ) = f ( t )

.a $ = (A +By) Rye,

_ -f - f- ikA.dt ( 5 . 5 )The tangential stress is zero on y = q ; hence

B = -2is7j. ( 5 . 6 )The linearized normal stress condition crnn= y a z q / a x zon y = 7 yields

2ik2pA = (yk2 p g ) f . ( 5 . 7 )Hence, from ( 5 . 5 )and (5 .7) ,

where (5.9)Th e form of the curve C = C(K) s shown in figure 11(6) for various values of S . Notet h a t E,, = S- , so t h a t C s positive for a range of wavenumbers if

$ > 1. (5.10)As a disturbance of initially large wavelength (small k} is compressed, it is thenexponentially amplified as its w avenum ber k( t ) passes through this unstable range.We may regard (5.10) as a necessary cond i t i on for the development of a finite-amplitude disturbance of the interface in a neighbourhood of x = 0.If th e condition (5.10)is satisfied, and if a depression of the free surface forms atx = 0, then the local strain rate tends to increase. In fact, as discussed in $4,maximum strain rate of order ad-3 esxrg ay be expected in the parabolic region. Thissuggests th a t when th e disturbance grows into th e nonlinear regime, the local strainra te s m ay in fac t increase in such a w ay as to keep in ste p with the increasing effectof surface tension near th e stag natio n point. C ertainly, gra vity becomes insignificantin this neighbourhood, and the results of the previous sections may be expected tobe asymptotically valid near x = 0 even when gravity is present.Note t h a t, as shown in figure 13 and described in the following section, it is alsopossible for a crest to a pp ear a t x = 0 when the strain ra te s is small. The linearizedanalysis of Appendix B indicates that this occurs when the horizontal separation of

,.


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16 J.-T. Jeong and H. . MoflattI y

y = $ ( t ) cos k ( t ) x- x

FIGURE1. (a)Compression of free-surface wave form under the action of uniform strain ; ( b ) hedispersion relation (5.9)showing an unstable range of wavenumbers when the condition (5.10) s. satisfied.the vortices representing th e rotating cylinders is greater t ha n 2 d . The shape of thecrest is (presum ably) influenced by b ot h grav ity a nd surface tension.6. Experimental details

The ex perim ent depicted in figure 1 was carried ou t in a Persp ex box of horizontaldimensions 250 x 250 mm. The radius of each cylinder is rc = 23 mm and theseparation of centres is 2c = 100mm . The d ep th of fluid d is of course variable; weused d = 40 mm for th e experime nt described here. Th e speed of rota tion of the twocylinders can be independently varied, b u t we focus here on th e symm etric situationwith angular velocities fa, nd f2 in the range zero to approximately 7 s-l. TheFroude number F r = f2(rc/g)i was then in the range [0,0.34].The fluid used w as a reasonably Newtonian polybutene of density p = 883 kg/m3,viscosity ,U = 12.9 kg m-l s- and surface tension y = 0.034N m-l. The range ofReynolds numbers Re = ps;ZrE/p, was [0,0.25],o that the low-Reynolds-numberapproximation Re 4 is applicable. The range of capillary number WeXp pQr,/ywas [0,61.1].he flow is accura tely two-dimensional exce pt in layers on th e fron t an drear walls of the box where the no-slip condition is satisfied. The free surface wasilluminated by a weak diffuse source of light from behind the box and this enabledphotographs of the free surface shape with the camera positioned as in figure 12.

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F r e e - s u r f a c e c u s p s 17

DiLight amerasource lens

FICCRE 2. Configuration of l igh t source and camera in the exper im ent ; no te th a t the to ta linte rna l reflection of l ight from th e free surface in th e rear-wal l layer mean s th a t no l igh t reachesthe cam era lens from th e region of ai r imm ediately above the free surface.Total internal reflection of light from the free surface in the rear boundary layermeans that no light gets to the camera from the air just above the free surface, andthis region therefore appears black in the photographs. The ratio We,,/Fr = p ( r , g ) i / yhad the value 180.The sequence of photographs (a-h) in figure 13 shows the effect of increasing theangular velocity slowly from zero. For small values of R, there is a crest at x = 0, andas R increases this crest is progressively more concentrated near x = 0. As Rapproaches a critical value R,, the crest disappears and for Q > Q,, a downward-pointing cusp appears on the free surface. The Froude number at which thistransition took place was F r , = 0.136. and the capillary number was %?,N 24. Withfurther increase of R , the cusp becomes more pronounced, but there is no furtherqualitative change in the flow structure for the range of R covered by thisexperiment. Further experiments using a range of different fluids. and varying thecylinder radius r , , are needed t o determine the locus in the (Fr, ?)-planewhere thetransition from crest to cusp occurs.As indicated previously (figure 8), the form of the cusp is well represented by theequation 2 = c 2 Pwith c = (i)i although since this value of c was determined for theidealized dipole configuration, the agreement is to some extent fortuitous).

The similarity solution for cusp flow given by J N R R gave a cusp of the formx - cYAwhere h is a function of %? tending to as %+ 0 . Our exact solution of thevortex dipole problem gives h = t even for finite %? (see the universal behaviour offigure 9) ; this means that the flow in the immediate neighbourhood of the cusp (i.e.the parabolic region) has an important effect on the cuspidal region where the free-surface behaviour x - f i s valid. There are some similarities here with the problemof viscous fingering (Saffman 1986),for which the gross properties of the fingers aresensitively dependent on conditions very near the tip.

.

@

8 This work was motivated by observations of a flow exhibiting what looked like acusp during a visit of one of us (H.K.M) o the Ecole Normale Superieure in Lyonat the invitation of Stefan Fauve in June 1990.The work described in $85 and 6 ofthis paper was presented a t the British Theoretical Mechanics Colloquium in Oxfordin April 1991.and we gratefully acknowledge the helpful comments of Philip Drazin,41ex Craik and others following this presentation ; also the equally helpful commentsof John Lister and other colleagues at DAMTP who have taken an interest in theexperiment. We thank David Cheesley for helping in the design of the experimentand for constructing the apparatus; and Jason Newling for help with thephotography. J.-T.J. has been financed by a Fellowship of the Korea Science andEngineering Foundation 1990.


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18 rJ .-1

- -v)D9IIc!-

s -n9Q


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Free-surface usps 19Appendix A. Calculation of some integrals required in 92obtain H ( a )= 4 x p a / y whereSubstituting w(0)= 3 a + 2 from ( 2 . 2 7 ) .G ( 0 ) rom ( 2 . 3 0 ) ,and ( 2 . 35 ) nto ( 2 . 3 7 ) ,we

(A 1)Evaluating the integral (see, for example, Gradshteyn & Ryzhik 1980),(A 1) can bereduced to ( 2 . 3 9 ) .Now, consider ( 2 . 4 6 ) ,where the integral can be expressed in termsof complete elliptic integrals as follows

H ( a )= -@(3a+2) dB01a(3a+2 )sin28,+2a ( 2a+ 1)sin 6, + a2 + ( a+1)211n 2 dt

( t - 5 ) (1- z ) i [ a ( 3 a+2 ) ~ +a ( 2a+ 1) +aZ+ ( a+ 1 )2 1 4

where

=I for - i< a < 0,for a 2 0, ( A 2 )

2(5- t , ) ( a 5 + 2 a +and m , m are defined in ( 2 . 4 0 ) . In (A a ) , K(m)and n ( n ,m) are complete ellipticintegrals of the first and third kinds (Gradshteyn & Ryzhik 1980: Byrd & Friedman1971)

n = (5+1) (1-t,) (1-$) ( a+ l ) 3 a+ 1)

(I Appendix B. Linearized analysis for a small perturbation of the freesurfaceConsider the flow due to vortices of strengths f placed at (Tc, 1) respect-

ively, in otherwise quiescent fluid. Suppose that the free surface is y = ~ ( x ) ,hereT (f O) = 0 and 7 and 1are both assumed small. We include here the effects of bothgravity and surface tension, represented by the non-dimensional numbers0 = pg/2,LLK, % = 2,LLK/y. (B 1)

q? = 0: czy-0 on y = O , (B2 )The linearized form of the boundary conditions at the free surface is


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20 J. -T . Jeong and H . K . MojJuttand ryy Y Y ( X ) on Y = ~ ( 4 , (B 3)

( ~ y v ) y - a= - P)y-7+2Pu(av/w,-o = p g r- P(a2+ /az Y),-O. (B 4)here(potential) flow satisfying (B 2) isWith image vortices + K a t ( + c , 1 ) respectively, the stream function for the

{ ( x - c ) ~ + ( y - 1 ) ~ } { ( x + c ) ~ + ( y +)2}{ ( 2 + c ) 2 + ( y - 1 ) 2 } { ( x - c ) 2 + ( y + 1 ) 2 }$(x:y ) = gKln

an d the conditions (B 3), (B 4) may then be combined to give the equa tion for ~ ( x )

whereth e tangential velocity on th e free surface.The solution is r ( 4= W g ( 4+9( - 4 1 . (B 8)wherean d ,8 = (%G)i.It will be sufficient to consider two limiting cases.( i ) ,8 + 1In th is l imit ,

g(z) x - 2 t a n - ( x + ~ ) + 2 t a n - ~ ( x - c ) (B 10)so tha t ~ ( x )-2%{tan- (x+c)- tan- ( 2 - c ) } . (B 1 1 )Figure 14 shows the free-surface shape for c = 1 .gives (with % = ,ua/y)I n th e special limit of a vortex dipole ( c - t 0 , K - + CO with ~ C K a = const . ) , (B 1 1)- %r ( 4-which agrees with the limiting form of (3.3)when I% \ 1 , namely

2a -4%y----x2+1 z2+1since U - 2% when ]%?I is small. Figure 14 also shows the limiting form (B 12).(ii) ,8+ CO ,-In this l imit , (B 6) gives immediately 7 = G - l ~ - l d u o / d ~ ,.e .

(B 14)8~(2 c + 1 ) (c + 1- 2)+~cx -c { ( x 2 + ~ 2 + 1 ) 2 - 4 c 2 ~ 2 } 2A crest is present a t x = 0 only for c > 1 . Figure 15 shows this form of th e free surfacefor c = 2 an d 0.5 an d for th e limiting dipole case (c -+0, ~ C K a ) for which

8 1-32r - - - q& i j ?with G = pg /pa .


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Free-surface cusps 21

, FIGURE4. Linearized free-surface shape when ,Ll< 1, V = 0.05, fo r c = 1 and for the dipolelimit c +0.m I y c - 2a

c k o - 1 T osFIQURE5. Linearized free-surface shape when /3 % 1, G = 20, for c = 2 , 0 . 5 and for the dipolelimit c + O .

Finally, note that in the dipole limit (B 7 ) becomes

which as expected agrees with the limiting form of ( 3 . 1 1 )as a+O.Note added in proof. Following a seminar in DAMTP on the topic of this paper,Dr John Hinch provided the following simple and elegant explanation for theexponential dependence ( 3 . 6 )of radius of curvature R on capillary number W :The surface tension exe rts a force 2 y per unit length of cusp on the fluid in the cusp

region. This cre ates a Stoke s flow with s trea m function ( 1 . 1 ) and associated upwardvelocity on z = 0r

for some dimensionless co nsta nt c , . A t a distance $dfrom th e vortex dipole (i.e. in thepeighbourhood of the location where th e cusp forms) there is a downward flow whichwould be 8 . 6 4 a / d 2 if the free surface were flat, but is in fact increased to c z a / d 2( c 2> 8 . 6 4 )due to the dow nward stream lining of th e free-surface shape. These twoflows must balance at r = R where the free surface is stationary; this balance ofvertical velocity gives

r = c1d ex p {- c x p a / d 2 y }= c1d exp (- c, x W ) .The exact solution shows that in fact c , = an d cz = 16. Note that the Stokesletsingularity is at a distance R above the free surface (outside the fluid). Thecombination of Stokeslet and uniform stream gives the re ar stagnation point flowin the parabolic region.

..


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22 J.-T.Jeong and H. . MoffattR E F E R E K C E S

B YR D.P. F. 8: FRIEDMAK,. D . 1971 Handbook of Elliptic integrals for Engineers and Scientists,2nd Edn. Springer.DUSSAK.. . B. 8: DAVIS.S. H. 1974 On the motion of a fluid-fluid interface along a solidsurface. J . Fluid Mech. 65, 71-95.GRADSHTEYN.I.S. 8: RYZHIR, . M. 1980 Table of Integrals, Series and Products (Corrected andEnlarged Edn) . Academic.GRIGGS: . 1939 A theory of mountain building. A m . J . Sci . 237. 611-650.JOSEPH.. D.: KELS ON. . , R E N A R D Y . . B R , E NAR DY,. 1991 Two-dimensional cuspedLISTER, . R . 1989 Selective withdrawal from a viscous two-layer system. J . Fluid Mech. 198,MUSKHELISHVILI,?;. I. 1953 Some Ba sic Problems of the Mathematical The ory of Elastici ty, 3rd Edn.RICHARDSON,. 1968 Two-dimensional bubbles in slow viscous flows. J . Fluid M ech. 33, 47-93.SAFFMAP;:. G . 1986 Viscous fingering in Hele-Shaw cells. J . Fluid Mech. 173, 73-94.

interfaces. J . Fluid Mtch. 223; 383-409.231-254. I-

gP. Koordhoff.

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Jae-Tack Jeong and H.K. Moffatt- Free-surface cusps associated with flow at low Reynolds number