A class of exact solutions of Hele-Shaw flows without surface tension in a
rotating cell is reported. We show that the interplay between injection and
rotation modifies drastically the scenario of formation of finite-time cusp
singularities. For a subclass of solutions, we show that, for any given initial
condition, there exists a critical rotation rate above which cusp formation is
prevented. We also find an exact sufficient condition to avoid cusps
simultaneously for all initial conditions. This condition admits a simple
interpretation related to the linear stability problem.Comment: 4 pages, 2 figure

A. Rocco

D. A. Kessler

D. Bensimon

F. X. Magdaleno

J. Casademunt

Ll. Carrillo

M. Mineev-Weinstein

M. Siegel

P. G. Saffman

P. Pelcé

S. D. Howison

S. K. Sarkar

S. P. Dawson

S. Tanveer

V. M. Entov

W. Dai

English

arXiv

A class of exact solutions of Hele-Shaw flows without surface tension in a rotating cell is reported. We show that the interplay between injection and rotation modifies the scenario of formation of finite-time cusp singularities. For a subclass of solutions, we show that, for any given initial condition, there exists a critical rotation rate above which cusp formation is suppressed. We also find an exact sufficient condition to avoid cusps simultaneously for all initial conditions within the above subclass

Magdaleno Escar, Francesc Xavier

Rocco, Andrea

Casademunt i Viader, Jaume

Diposit Digital de la Universitat de Barcelona

RAPID COMMUNICATIONSPHYSICAL REVIEW E NOVEMBER 2000VOLUME 62, NUMBER 5Interface dynamics in Hele-Shaw flows with centrifugal forces:Preventing cusp singularities with rotationF. X. Magdaleno, A. Rocco, and J. CasademuntDepartament d’Estructura i Constituents de la Mate`ria, Facultat de Fı´sica, Universitat de Barcelona, Avenida Diagonal 647,E-08028 Barcelona, Spain~Received 31 January 2000!A class of exact solutions of Hele-Shaw flows without surface tension in a rotating cell is reported. We showthat the interplay between injection and rotation modifies the scenario of formation of finite-time cusp singu-larities. For a subclass of solutions, we show that, for any given initial condition, there exists a critical rotationrate above which cusp formation is suppressed. We also find an exact sufficient condition to avoid cuspssimultaneously for all initial conditions within the above subclass.PACS number~s!: 47.20.Hw, 47.20.Ma, 47.15.Hg, 68.10.2mThe dynamics of the interface between viscous fluids con-fined in a Hele-Shaw cell @1–3# has received attention forseveral decades from physicists, mathematicians, and engi-neers. In particular, it has played a central role in the contextof interfacial pattern formation @4,5#. As a free boundaryproblem it has the particular interest that explicit time-dependent solutions can often be found in the case with nosurface tension @6–9#. As a consequence, the issue of the roleof surface tension as a singular perturbation in the interfacedynamics has received increasing attention @10–14# for itspotential relevance to a broad class of problems. However, towhat extent the physics of the real problem ~with finite sur-face tension! is captured, even at a qualitative level, by theknown solutions is still poorly understood.As an initial-value problem, the zero surface tension caseis known to be ill-posed @11#. An important aspect related tothis fact is that some smooth initial conditions develop finite-time singularities in the form of cusps of the interface @6,7#.After this blow-up of the solution, the time evolution is nolonger defined. Generation of finite-time singularities is initself interesting in connection with other singular perturba-tion problems in fluid dynamics, such as in the case of theEuler equations. In the present problem, surface tension actsas the natural regulator curing this singular behavior, butunfortunately the problem with surface tension is much moredifficult and usually defies the analytical treatment. Moti-vated by this fact, and inspired by recent experiments onrotating Hele-Shaw cells @15,16#, we address here the pertur-bation of the original free boundary problem by the presenceof a centrifugal field. This new ingredient enriches the prob-lem in a nontrivial way but, as first discussed in @17#, it stilladmits explicit integration, so it may provide new analyticalinsights. Here we will focus on the possibility that rotation,although not fully regularizing the problem, may prevent theemergence of singular behavior in the form of cusps at finitetime, thus enlarging the class of exact solutions without sur-face tension which are potentially relevant to the physicallyrealizable situations.We study an interface between a fluid with viscosity mand density r and one with zero viscosity and zero density ina Hele-Shaw cell with gap b. The cell can be put in rotationwith angular velocity V and fluid can be injected or ejectedfrom the cell through an orifice at the center of rotation, withPRE 621063-651X/2000/62~5!/5887~4!/$15.00areal rate Q. The cases Q.0 and Q,0 correspond respec-tively to injecting or ejecting fluid. As in the traditional Hele-Shaw problem, the flow in the viscous fluid is potential, v5f , but now with a velocity potential given by @15#f52b212m S p2 12 rV2r2D . ~1!Incompressibility then yields Laplace equation „2f50 forthe field f ~but not for the pressure!. The two boundaryconditions at the interface which complete the definition ofthe moving boundary problem are the usual ones, namely,the pressure on the viscous side of the interface p52sk ,where s and k are respectively surface tension and curva-ture, and the continuity condition for the normal velocityvn5nf . The crucial difference from the usual case is inthe boundary condition satisfied by the Laplacian field on theinterface due to the last term in Eq. ~1!.This problem is well suited to conformal mapping tech-niques @2#. The basic idea is to find an evolution equation foran analytical function z5 f (v ,t), which maps a referenceregion in the complex plane v , in our case the unit diskuvu<1, into the physical region occupied by the fluid in thephysical plane z5x1iy , with the physical interface beingthe image of the region boundary, uvu51. We consider twotypes of situations, one in which the viscous fluid is insidethe region enclosed by the interface, and one in which it isoutside. It can be shown @15# that the evolution equation forthe mapping f (v ,t) in the rotating case can be written in acompact form asIm$] t f *]f f %5Q2p q112 V*]fHf@ u f u2#1d0]fHf@k# ,~2!where V*5b2V2r/12m , d05b2s/12m , and where we havespecified the mapping function at the unit circle v5eif. Thecurvature is given by k52Im$]f2 f /(]f f u]f f u)% and the Hil-bert transform Hf is defined byHf@g#512pPE02pg~u!cotanF12 ~f2u!Gdu . ~3!R5887 ©2000 The American Physical SocietyRAPID COMMUNICATIONSR5888 PRE 62F. X. MAGDALENO, A. ROCCO, AND J. CASADEMUNTIn Eq. ~2!, q511 and q521 correspond to the cases withthe viscous fluid respectively inside or outside the regionenclosed by the interface.In this context a class of solutions is defined by a func-tional form of the mapping which is preserved by the timeevolution, allowing for a finite number of time-dependentparameters. Some explicit time-dependent solutions forf (v ,t) with d050 but V*Þ0 have been previously reportedin Ref. @17#. Here we report additional explicit solutions forthe rotating case, which all fit into the general rational formf ~v ,t !5vqa0~ t !1( j51N a j~ t !vj11( j51N b j~ t !v j, ~4!although not any mapping of this form is necessarily a solu-tion. Remarkably, we have found that an important class ofsolutions of the usual case (V*50), which is free of finite-time singularities, namely, the superposition of logarithmicterms @9#, is no longer a solution for V*Þ0, while mappingswith poles turn out to include solutions for both V*50 andV*Þ0. A more detailed study will be presented elsewhere@18#. The polynomial case (b j50 for all j’s!, which in thenonrotating case V*50 is known to always yield finite-timesingularities in the form of cusps @6,7#, for V*Þ0 is also asolution @17#. However, we will study in the rest of thisRapid Communication that the scenario of cusp formation inthese solutions is modified in a nontrivial way by the pres-ence of rotation.We focus on the role of rotation in preventing cusp for-mation in the subclass of polynomial mappings of the formf ~v ,t !5a0~ t !vq1an~ t !vn1q. ~5!For an!a0 this describes an n-fold sinusoidal perturbation ofamplitude an superimposed on a circular interface of radiusa0 . It is convenient to introduce the dimensionless parameter«5(n1q)an /a0 . The range of physically acceptable valuesof an and a0 is given by the condition 0,«,1 for all n. Wealso introduce a scaled mode amplitude d5a0an which turnsout to be useful to characterize the interface instability. Tosee this, let us first compute the standard linear growth rate.Inserting Eq. ~5! into Eq. ~2! and linearizing in an , we get,a˙ nan5qnV*2~qn11 !Q2pa02 2d0a03 n~n221 !. ~6!The term 2Q/2pa02, independent of both n and q , has apurely kinematic origin, associated with the global expansion~or contraction! of the system. This quantity would be thegrowth rate of an interface mode which followed the ~undis-torted! flow field with radial velocity v5Q/2pr @which inturn would imply an(t)a0(t)5const#. Accordingly, the mar-ginal modes for d ~which in the rotating case may occur forall n) will be those for which the flow field is undistorted bythe interface perturbation, although such perturbation maygrow or decay in the original variables an . In this way,growth or decay of d will correspond unambiguously to thestability of the flow configuration with the radial velocityfield. In this sense it may be justified to qualify the interfaceinstability as described by d as ‘‘intrinsic,’’ as opposed to the‘‘morphological’’ one as described by the amplitude an . Inthis way the intrinsic growth rate takes the simpler formd˙d5qn~V*2Q*!, ~7!where we have defined Q*5Q/2pa02 and have dropped thesurface tension term, since hereinafter we will focus on tothe zero surface tension case. We introduce the relevant di-mensionless control parameter of our problem, expressingthe ratio of centrifugal to viscous forces, asP5V*2pR2Q 5prb2R2V26mQ , ~8!where R is a characteristic radius of the interface.Equation ~7! clearly exhibits the competing effects of ro-tation and injection, although their roles are not quite sym-metric. In fact, notice that Q*, which may have both signs,contains a dependence on a0 . In practice this means that Q*depends effectively on time. An immediate consequence isthat the growth of linear modes is not really exponential @15#and may even be nonmonotonic. The asymmetry betweeninjection and rotation shows up also in the fact that the signof Q determines which of the two effects dominates asymp-totically in time. In fact, for positive injection rate the typicalradius of the inner fluid is growing while typical interfacevelocities are decreasing, so centrifugal forces will dominateat long times. On the contrary, for negative injection rate,typical velocities increase while typical radii decrease, soinjection will asymptotically dominate over rotation.In view of Eq. ~7!, the most interesting configurations willbe those in which Q.0, so that injection and rotation havecounteracting effects. In the case q511 ~viscous fluid in-side!, which was experimentally studied in Ref. @15#, rotationis always destabilizing. A positive injection rate in this casetends to stabilize the circular interface. However, for fixed Q,Q* will decrease with time, so eventually the interface willreach a radius after which all modes are linearly unstable. Itis thus expected that, in this case, the formation of cusps canonly be delayed but not avoided @18#.The most interesting case from the point of view of pre-venting cusp formation is q521 and Q.0, the usual con-figuration in viscous fingering experiments. In this case, asmall rotation rate will only slightly affect the linear insta-bility, but could eventually stabilize the growth at long times,so it is conceivable to have a nontrivial evolution startingfrom an unstable interface but not developing finite-time sin-gularities.As an example, we now study the fully nonlinear dynam-ics of polynomial mappings. Inserting Eq. ~5! into Eq. ~2!with d050 we obtain two ordinary differential equations de-scribing the evolution of a0(t) and an(t). These can be in-tegrated analytically and yielda02~ t !1q~n1q!an2~ t !5Qpt1k0 , ~9!a0n1q~ t !anq~ t !5knenV*t, ~10!RAPID COMMUNICATIONSPRE 62 R5889INTERFACE DYNAMICS IN HELE-SHAW FLOWS WITH . . .where k0 and kn are constants to be determined by initialconditions, and where n>2 for q511 and n>3 for q521.Physically acceptable solutions require that the points inthe v-plane where ]v f (v ,t)50 ~noninvertible! should lieoutside the unit disk. The occurrence of a cusp is associatedwith such a point crossing the unit circle uvu51 at a finitetime tc ; that is,U qa0~ tc!~n1q!an~ tc!U51. ~11!If we take the initial value a0(0) as the characteristic lengthR, which coincides with the radius of the perturbed circle ifwe are in the linear regime, and define the dimensionlesstime t5V*t , condition ~11! readsanS 2R2tcP 1k0D5ebntc, ~12!wherean5~n1q!n/n12qn12q kn22/~n12q!, bn52nn12q . ~13!Our aim is now at finding conditions such that an initiallysmooth interface remains smooth for an infinite time. Thuswe have to impose that Eq. ~11! should not have any solutionfor tc.0. The transition between the regions with and with-out cusps will be defined by the conditions that both Eq. ~12!and its time derivative are satisfied, such that the curves oneach side of Eq. ~12! have a common tangent. These twoconditions allow us to eliminate tc , and yieldx log x2x52ank0 , ~14!where x52an /Pbn , and with R5a0(0) in Eq. ~8!. We nowsearch for solutions of Eq. ~14!. For q511 it can be proventhat this equation has no solutions, and therefore all initialconditions must eventually develop a cusp at finite time, asFIG. 1. Critical lines Pc for different values of n. The regionfree of cusp singularities for a given n is the one above the corre-ponding curve.expected from the linear analysis. On the other hand, for q521 the quantity on the rhs of Eq. ~14! takes the simpleformank05n21n22 S 12 «2n21 D «2/~n22 !, ~15!and a nontrivial critical line Pc(«;n) can be found for eachn>3. This implies that, in the configuration with the viscousfluid outside, for any initial condition @within the class ofpolynomial mappings of the form Eq. ~5!# there is always acertain rotation rate above which there is no cusp formation.The numerical determination of these curves is shown inFig. 1.The leading behavior for initial conditions in the linearregime, «!1, can be found by expanding the lhs of Eq. ~14!around x5e and is given by Pc’@(n21)/ne#«2/(n22). No-ticeFIG. 2. Evolution of the interface in the case q521 ~viscousfluid outside!, with n53, a0(0)51.0, e(0)50.5, for ~a! V*50.025 ~cusp formation! and ~b! V*50.045 ~cusps prevented byrotation!.RAPID COMMUNICATIONSR5890 PRE 62F. X. MAGDALENO, A. ROCCO, AND J. CASADEMUNTthat there are qualitative differences for small values of n.For n53 the curve starts horizontal at the linear level, im-plying that a very small rotation rate is sufficient to preventcusp formation. For n54 the threshold curve starts with afinite slope and for n.4 it has an infinite slope at «50. Amore detailed description and analysis of this diagram willbe presented elsewhere @18#. An example of rotation prevent-ing cusp formation is shown in Fig. 2.In Fig. 1 we also see that for any given « , the critical Pcincreases monotonically with n. If we take the limit n→‘ atfixed « we get ank0→1. From Eq. ~14! this implies x51and consequently we obtain an absolute upper bound Pcmax51 for all values of n and « . This implies that, for all initialconditions @within the class Eq. ~5!# there is a critical rotationrateVc5S 6mQprb2R2D1/2~16!above which cusps are always eliminated. Although Eq. ~16!has been derived for the class Eq. ~5!, with the identificationa0(0)5R , one might expect that the existence of a certainVc and the scaling with physical parameters given by Eq.~16! could be more general. Notice that P51 corresponds tothe intrinsic marginal stability of the circular shape, Q*5V*. Therefore, the sufficient condition, valid for all initialconditions of the form Eq. ~5!, for not developing cusp sin-gularities is that a circular interface with radius given bya0(0) be intrinsically stable, in the sense of Eq. ~7!. Whetherdeeper consequences can be drawn in a broader context fromthis inner connection between the linear problem and thepossibility of cusp formation remains an open question.We acknowledge financial support by the Direccio´n Gen-eral de Ensen˜anza Superior ~Spain!, Project No. PB96-1001-C02-02 and the European Commission Project No. ERBFMRX-CT96-0085.@1# P. G. Saffman and G. I. Taylor, Proc. R. Soc. London, Ser. A245, 312 ~1958!.@2# D. Bensimon, L. P. Kadanoff, S. Liang, B. I. Shraiman, and C.Tang, Rev. Mod. Phys. 58, 977 ~1986!.@3# S. Tanveer, in Asymptotics Beyond All Orders, edited by H.Segur et al. ~Plenum Press, New York, 1991!.@4# P. Pelce´, in Perspectives in Physics ~Academic Press, NewYork, 1988!.@5# D. A. Kessler, J. Koplik, and H. Levine, Adv. Phys. 35, 255~1988!.@6# S. K. Sarkar, Phys. Rev. A 31, 3468 ~1985!.@7# S. D. Howison, J. R. Ockendon, and A. A. Lacey, Q. J. Mech.Appl. Math. 38, 343 ~1985!; S. D. Howison, J. Fluid Mech.167, 439 ~1986!.@8# M. Mineev-Weinstein and S. P. Dawson, Phys. Rev. E 50, R24~1994!; S. P. Dawson and M. Mineev-Weinstein, Physica D73, 373 ~1994!.@9# S. P. Dawson and M. Mineev-Weinstein, Phys. Rev. E 57,3063 ~1998!.@10# W. Dai, L. P. Kadanoff, and S. Zhou, Phys. Rev. A 43, 6672~1991!.@11# S. Tanveer, Philos. Trans. R. Soc. London, Ser. A 343, 155~1993!.@12# M. Siegel and S. Tanveer, Phys. Rev. Lett. 76, 419 ~1996!; M.Siegel, S. Tanveer, and W. S. Dai, J. Fluid Mech. 323, 201~1996!.@13# F. X. Magdaleno and J. Casademunt, Phys. Rev. E 57, R3707~1998!; F. X. Magdaleno, E. Paune´, and J. Casademunt ~un-published!.@14# M. J. Feigenbaum, I. Procaccia, and B. Davidovich, e-printchao-dyn/9908007.@15# Ll. Carrillo, F. X. Magdaleno, J. Casademunt, and J. Ortı´n,Phys. Rev. E 54, 6260 ~1996!.@16# Ll. Carrillo, J. Soriano, and J. Ortı´n, Phys. Fluids 11, 778~1999!; Ll. Carrillo, J. Soriano, and J. Ortı´n ~unpublished!.@17# V. M. Entov, P. I. Etingof, and D. Ya. Kleinbock, Eur. J. Appl.Math. 6, 399 ~1995!.@18# A. Rocco, F. X. Magdaleno, and J. Casademunt ~unpublished!.

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Interface dynamics in Hele-Shaw flows with centrifugal forces. Preventing cusp singularities with rotation

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Interface dynamics in Hele-Shaw flows with centrifugal forces: Preventing cusp singularities with rotation

Interface dynamics in Hele-Shaw flows with centrifugal forces.
  Preventing cusp singularities with rotation

Interface dynamics in Hele-Shaw flows with centrifugal forces. Preventing cusp singularities with rotation

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