Physica D 237 (2008) 652–664www.elsevier.com/locate/physd

Geometric approach to stationary shapes in rotating Hele–Shaw flows

Eduardo S.G. Leandroa, Rafael M. Oliveirab, Jose A. Mirandab,∗

a Departamento de Matematica, Universidade Federal de Pernambuco, Recife, Pernambuco 50740-540, Brazilb Departamento de Fısica, LFTC, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil

Received 6 July 2007; accepted 16 October 2007Available online 22 October 2007

Communicated by H. Levine

Abstract

We consider the geometry of a special family of curves associated with the motion of a two-fluid interface in a rotating Hele–Shaw cell.This family of stationary exact solutions with surface tension consists of interface shapes which balance exactly the competing capillary andcentrifugal forces. The result is the formation of patterned structures presenting fingers that eventually assume a teardrop-like shape, and tend tobe detached from the main body of the rotating fluid (occurrence of a pinch-off event). By using the vortex–sheet formalism, we approach theproblem analytically, and show that the curvature of these particular curves can be very simply expressed as a function of the radial distance to therotation axis. Motivated by this fact, and through a simple geometric interpretation, we show that the exact solutions for the rotating Hele–Shawproblem satisfy a first-order ordinary differential equation which is readily solved, so that the shape solutions are determined up to quadratures.The ability to probe a number of key morphological features of such solutions analytically is demonstrated, and a gallery of plots is provided tohighlight these findings. The possibility of accessing a criterion to predict the occurrence of pinch-off is also discussed. Finally, we prove that thegeneral solutions are dense, and use this fact to guarantee the very existence of the neat, highly symmetric physical patterns.c© 2007 Elsevier B.V. All rights reserved.

Keywords: Pattern formation; Interfacial dynamics; Hele–Shaw flow; Geometry

1. Introduction

When a less viscous fluid pushes a more viscous one in thenarrow gap of a Hele–Shaw cell, the interface between the fluidsdevelops the Saffman–Taylor instability [1], leading to theformation of complex fingerlike patterns. This hydrodynamicinstability has been extensively studied over the last fiftyyears [2], and is now a paradigmatic problem for the formationand evolution of dynamic structures in the area of nonlinearphenomenology.

The continuing interest in the Saffman–Taylor problem isjustified in part by the fact that it provides a relatively simpleexample of morphological instability, which involves nontrivialbut sometimes solvable pattern selection problems [3]. Anothercontributing factor for the lasting enthusiasm toward theproblem is the existence of explicit time-dependent solutions

∗ Corresponding author. Tel.: +55 81 21267610; fax:+55 81 32710359.E-mail address: jme@df.ufpe.br (J.A. Miranda).

0167-2789/$ - see front matter c© 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2007.10.005

which can often be found in the zero surface tension limit [4–6].The understanding and elucidation of the mechanisms ofselection and the singular character of the surface tension havebeen a subject of lively discussion in the literature [7–10] andare related to the more general subject of pattern selection innonequilibrium phenomena [11–17].

Although the inclusion of a nonzero surface tension into theSaffman–Taylor problem is desirable and of much interest, it isconsiderably more difficult than the zero surface tension situa-tion, and usually defies analytical treatment. In contrast to thezero surface tension case, only a couple of examples of exactsolutions are known when a finite surface tension is considered:one of these is due to Kadanoff [18], which applies to a configu-ration somewhat different from the traditional Saffman–Taylorproblem; a second noteworthy example, first reported by Nye,Lean and Wright [19], and further investigated more recently byAlvarez-Lacalle, Ortın and Casademunt [20], is truly interestingand of practical relevance, presenting an appealing connectionwith the so-called Euler’s elastica problem [21].

E.S.G. Leandro et al. / Physica D 237 (2008) 652–664 653

The elastica problem has a distinguished history [21,22],dating back to the Bernoullis, who suggested to Euler todetermine the shape of planar bending rods. Euler obtained thedifferential equations describing the curvature of a thin elasticand inextensible rod bent in a plane, and the problem ended upbecoming a milestone in the development of the mathematicaltheory of elasticity [23,24]. In addition to its historicalimportance to the theory of elasticity, the elastica problem is ofcurrent interest, being related to phenomenological descriptionsof stiff polymers and filamentary biomolecules like DNA[25,26], and with suggestive applications to nanotechnology[27,28]. Amazingly enough, it is also related to fluid dynamics,notably to the Saffman–Taylor problem [19,20].

The family of exact solutions with nonzero surface tensionof the usual Saffman–Taylor problem studied in Refs. [19,20]consists of interface shapes which balance exactly thecompeting capillary and viscous forces, resulting in solutionswhich coincided, in the case of rectangular (channel) geometry,with the classical elastica solutions of ideal bending rods. Thesituation in the rectangular Hele–Shaw cell setup has beenstudied both theoretically and experimentally in Refs. [19,20],where patterned structures of teardrop-like shapes arise. Ina rather interesting paper, Alvarez-Lacalle et al. [20] haveidentified these elastica solutions as the unstable subcriticalbranch of the Saffman–Taylor instability which ends upat a topological singularity (interfacial pinch-off). Thisphenomenon occurs when the individual teardrop-like fingeringstructures are about to be detached from the main body of thefluid.

In Ref. [20] it is also briefly mentioned that a similar familyof stationary shape solutions could also be obtained in rotatingHele–Shaw flows [29,30] when centrifugal and capillary forcesare exactly matched at the interface. The rotating Hele–Shawproblem is a variation of the traditional viscosity-drivenSaffman–Taylor instability, in which the cell rotates and thecentrifugal force induces interface destabilization driven by thedensity difference between the fluids. Although the numericalsolution for the beautiful petal-shaped patterns occurring in thisrotating case are depicted in Ref. [20], not much is reported onthe very determination of these structures, or on the possibilityof probing key morphological properties of this particular set ofexact solutions analytically. Therefore, a thorough investigationof the elastica-like solutions in rotating Hele–Shaw cells stillneeds to be addressed. This is precisely the main purpose ofthis work.

The rest of the paper is organized as follows. Section 2introduces the physical system and utilizes the vortex–sheetformalism to access the stationary shape solutions of therotating Hele–Shaw problem. We show that the curvature ofthe resulting curves has a very simple dependence on thepolar radius. Motivated by this fact, in Section 3 we approachthe problem geometrically, and show that it admits explicitintegration, allowing new analytical insights. In particular, wefind that the general set of curves describing the exact solutionsare opportunely expressed in terms of ψ , a parameter with avery simple geometric interpretation, being the angle betweenthe radial and tangent directions to the curve. A representative

Fig. 1. Sketch of a rotating Hele–Shaw cell.

collection of possible solutions (both non-self-intersecting andself-intersecting) is presented and thoroughly discussed indetail in Section 4. A systematic criterion for predicting theoccurrence of pinch-off is presented. Moreover, the unstablenature of the patterns discussed in Refs. [19,20] is reinterpretedgeometrically in the sense that a generic choice of parametervalues and initial conditions leads to dense curves. The latteris used to discuss the very existence of the predicted physicalpatterns. Section 5 summarizes our main results. Appendix Ahighlights important aspects related to the geometry of planarcurves, focusing on the calculation of their curvature andinflection points. Finally, Appendix B describes the procedurewe used to access the exact solutions numerically.

2. Statement of the problem and governing equations

In the rotating version of the Saffman–Taylor problem,a Hele–Shaw cell of gap spacing h turns around an axisperpendicular to the plane of the flow with constant angularvelocity ω (Fig. 1). Inside the cell an initially circular droplet(radius R) of the more dense fluid 2 is surrounded by an outerfluid 1. The densities and viscosities of the fluids are denoted byρ j and η j , respectively ( j = 1, 2), and between the fluids thereexists a surface tension σ . We define our Cartesian coordinatesystem (x, y, z) in such a way that its origin is located at thecenter of the effectively two-dimensional droplet (located in thex–y plane), whereas the cell rotates about the z axis.

Our analytical study is based on the vortex–sheetrepresentation for Hele–Shaw flow [31], which explores thejump in the tangential component of the fluid velocity asone crosses the fluid–fluid interface. This approach offersa particularly useful method to probe the morphology ofthe pattern-forming structures which arise under the actionof centrifugal forcing [20,32,33]. The basic hydrodynamicequation of the system expresses the two-dimensional velocityfield given by Darcy’s law [29,30]:

v j = −h2

12η j

∇p − ρ jω

2r r, (1)

where p is the hydrodynamic pressure, r is the distance tothe rotation axis, and r denotes a unit vector pointing radiallyoutward. The problem is then specified by two boundaryconditions: (i) the pressure jump at the interface p2 − p1 = σκ ,where κ denotes the interface curvature; and (ii) the kinematic

654 E.S.G. Leandro et al. / Physica D 237 (2008) 652–664

boundary condition n · v1 = n · v2, which refers to thecontinuity of the normal velocity across the interface (n isthe unit normal vector pointing from fluid 2 to fluid 1). Thetangential component of the velocity, however, is discontinuousand gives rise to the vortex–sheet strength Γ = (v1 − v2) · s,where s = ∂sr is the unit counterclockwise tangent vector alongthe interface, and s denotes the interface arclength.

By writing down Darcy’s expression (Eq. (1)) for both fluids,then by subtracting the resulting expressions, we solve for thevortex–sheet strength to obtain a dimensionless expression forthe vorticity:

Γ = 2AV · s + ∇

[1Ωκ − r2

]· s. (2)

In deriving Eq. (2) we have also used the pressure jumpand the kinematic boundary conditions described earlier. InEq. (2) lengths and velocities are rescaled by R and U =

[b2 R(ρ2 − ρ1)ω2]/[12(η1 + η2)], respectively. The parameter

Ω = [ω2 R3(ρ2 − ρ1)]/2σ represents the dimensionlessrotational Bond number, and A = (η2 − η1)/(η2 + η1) is theviscosity contrast. The self-consistent equation for V(s, t) isgiven by the Birkhoff integral formula [31,34]:

V(s, t) =1

2πP∫

ds′z × [r(s, t)− r(s′, t)]

|r(s, t)− r(s′, t)|2Γ (s′, t), (3)

where P means a principal-value integral and z is the unitvector along the direction perpendicular to the cell. The velocityintroduced in Eq. (3) is an average velocity of the interfacedefined as V = (v1+v2)/2, where v1 and v2 are the two limitingvalues (from both sides of the interface) of the solenoidal partof the velocity at a given point. To obtain the evolution ofthe interface, Eq. (2) has to be solved with V given by Eq.(3), yielding a complicated integro-differential equation for thevorticity. Once Γ is known, Eq. (3) is used again to obtain V,and then its normal component is used to update the position ofthe interface.

Here, instead of trying to solve the complicated integro-differential equation for the vorticity, we focus on the exact(static) solutions of Eq. (2). In a stationary state, v1 = v2 = 0by definition. So, by taking V = 0 in Eq. (2), and consideringthe condition of zero vorticity (Γ = 0) we find that thecurvature of the fluid–fluid interface must satisfy a third-ordernonlinear ordinary differential equation of the form [20,32,33]

∇

[κ − Ωr2

]· s = 0. (4)

The solutions of this equation balance the centrifugal forceagainst the surface tension, and reveal very importantmorphological features of the emerging patterns in thestationary case.

Clearly, Eq. (4) represents the directional derivative of thefunction between the squared brackets, which we represent byg, along the unit tangent vector of the curve γ describing theinterface. Notice that Eq. (4) keeps its form if we replace theunit tangent velocity vector by any tangent velocity vector. So,if we restrict g to γ parameterized by a generic parameter u,

then Eq. (4) can be simply written as dg(γ (u))/du = 0, whichcan be immediately integrated to obtain

κ = κ(r) = Ω(r2− r2

I ), (5)

valid along γ , where rI is a radial distance at which thecurvature vanishes. The points of inflection of γ are locatedon the circle of radius rI (see Appendix A). Notice that in Eq.(5) the curvature is very simply related to the radial distancer . This simple expression motivated us to follow an approachwhich is sort of the “inverse” of the one usually employedto describe the shape of planar curves: in general, a curve isgiven and one is asked to calculate its curvature. Here theexpression for the curvature is known and written in terms ofsome parameter (in Eq. (5) κ is a simple function of r ), and wewant to search the curves that satisfy such an expression. As wewill show in the following sections, this single fact opens upthe possibility of systematically probing the stationary shapesof the rotating Hele–Shaw problem analytically, and throughgeometric means.

3. Geometric approach to the stationary shape solutions

We wish to study the family of planar differential curveswhose curvature has the form

κ = κ(r), (6)

where r stands for the polar radius. Let us consider aparameterized curve γ : x(u) = (x1(u), x2(u)), u being anarbitrary parameter. The curvature of γ is given by [35]

κ(u) =(x′(u), x′′(u), z)

‖x′(u)‖3 , (7)

where the prime indicates differentiation with respect to u,(·, ·, ·) represents the triple product of its arguments, and‖x′(u)‖ is the norm of x′(u). A brief review of the definitionof curvature as well as a deduction of Eq. (7) can be found inAppendix A.

Due to Eq. (6), it seems natural firstly to express thecurvature in terms of polar coordinates (r, ϕ). If we write

x(u) = r(u)(cosϕ(u), sinϕ(u)),

and use Eq. (7), a straightforward calculation gives

κ(u) =r2ϕ′3

+ r ′(2r ′ϕ′+ rϕ′′)− rr ′′ϕ′

(r ′2 + r2ϕ′2)3/2. (8)

A fundamental geometric object throughout our discussionis the angleψ between the radius vector r and the tangent vectors to γ . From Fig. 2, it is clear that ψ = θ − ϕ; hence

tanψ = tan(θ − ϕ) =(x(u), x′(u),k)(x(u) · x′(u))

, −π

2< ψ <

π

2,

since we have that tanϕ = x2(u)/x1(u), and tan θ =

x ′

2(u)/x ′

1(u). In terms of polar coordinates, it is easy to checkthat

tanψ =rϕ′

r ′. (9)

E.S.G. Leandro et al. / Physica D 237 (2008) 652–664 655

Fig. 2. Cartesian coordinate axes and the angle ψ formed by the radial segmentfrom O to P ∈ γ and the tangent s to the planar curve γ at P . The polar angleϕ and the auxiliary angle θ are also shown.

Notice that in order to study the situations where ψ assumes thevalues −

π2 ,

π2 , it suffices to replace Eq. (9) with

cotψ =r ′

rϕ′. (10)

As we mentioned previously in Section 2, the main goal ofthis paper is to study a family of planar curves whose curvaturesare given as a function of the polar radius. In order to achieveour goal, we begin by choosing u = r as the parameter of γ .From Eq. (8), it is immediate that ϕ = ϕ(r) must satisfy thesecond-order differential equation

rϕ′′+ ϕ′(2 + r2ϕ′2) = κ(r)(1 + r2ϕ′2)3/2, r > 0. (11)

Motivated by Eq. (9), let us make the change of variablesrϕ′

= tanψ . This change of variables proves to be veryuseful since it permits a convenient reduction of the order ofthe original differential equation [Eq. (11)] from two to one,leading to

ψ ′+

tanψr

= κ(r) secψ. (12)

After manipulating a little with Eq. (12) we find, not withoutsome surprise, that it can be written in an even simpler formas (r sinψ)′ = rκ(r). This means that the differential equation(12) can be readily integrated to give

ψ = arcsin[

1r

(∫ r

r0

tκ(t)dt + r0 sinψ0

)]. (13)

Inverting the change of variables mentioned above, there resultsa simple first-order differential equation whose solution is

ϕ(r) = ϕ0 +

∫ r

r0

1t

tanψ(t)dt. (14)

The latter expression is quite general, in the sense that theplanar curves whose curvature is given by Eq. (6) and which

are parameterizable by the polar radius r are determined upto quadratures. Notice that the rotating Hele–Shaw situationdiscussed in Section 2 defines an important special case of thesetype of curves for which κ(r) = Ω(r2

− r2I ) (Eq. (5)), so that

Eq. (14) can be a valuable theoretical tool for their study. It isalso worth noting that the general set of curves described byEq. (14) are opportunely expressed in terms of ψ , a parameterwith a very simple geometric interpretation, being the anglebetween the radial and tangent directions to the curve. Eqs. (13)and (14) are central results of this work, not only intrinsicallypeculiar due to its relatively simple geometric interpretation,but also for yielding analytical access and description ofthe exact stationary solutions which arise in the rotatingHele–Shaw problem.

We are thus in the setting previously studied and, if weconsider curves parameterized by r , by substituting Eq. (5) intoEq. (13) we deduce that

sinψ =14r

[4r0 sinψ0 − 2Ωr2

I (r2− r2

0 )+ Ω(r4− r4

0 )],

r > 0. (15)

Indeed, we can write Eq. (15) in the form

sinψ =P(r)

r, with P(r) = Ar4

+ Br2+ C, (16)

where A = Ω/4, B = −Ωr2I /2, and C = r0(sinψ0 −

Br0 − Ar30 ). As will become evident in Section 4, since sinψ

is a limited function, it is precisely the interplay between thepolynomial of degree four P(r) and the linear function r shownin Eq. (16) that will allow us to describe the stationary exactsolutions under study in an analytical fashion.

Finally, notice that if we make the substitution t = r2 andput P(t) = P(

√t), then Eq. (14) turns into

ϕ = ϕ0 +12

∫ r2

r20

P(t)

t√

t − P(t)2dt. (17)

Integrals such as the one above are known to be of elliptictype [36], so that a general expression for the curve describingthe interface can be found using elementary functions, andelliptic integrals of the first, second and third kinds.

4. Discussion: Collection of possible solutions

The analysis of our main results are divided in three separateparts. In Section 4.1 we examine a set of exact solutionswhich are directly connected to static shapes arising in therotating Hele–Shaw problem. These are closed curves that donot cross themselves. In a limiting case, successive interfacialundulations touch one another, and a pinch-off phenomenonoccurs. Beyond this limiting case, it is still possible to findbeautiful closed curves, but these cross themselves at leastonce, so they do not correspond to physical solutions for thephysical problem under consideration. These self-intersectingsolutions are presented in Section 4.2. In addition to its inherentaesthetic appeal and mathematical relevance, this later setof self-intersecting solutions will help us to find a way

656 E.S.G. Leandro et al. / Physica D 237 (2008) 652–664

Fig. 3. Strategy and notation. Left panel: typical pattern and the various circles of radii ri with i = 1, 2, 3, 4, 5; right panel: polynomial P(r) and the lines y = ±r ,plotted as a function of r (top), and variation of ϕ′(r) with radius distance r (bottom).

to determine and predict the physically relevant pinch-offevents in a systematic fashion. Moreover, the study of thesesolutions further illustrates the usefulness and generality ofour geometric approach. Section 4.3 discusses the possibilityof dense solutions in which curves are not closed, and fill upan annulus. Finally, the very existence of the closed, periodic,physical solutions of the problem is investigated.

By using the approach presented in Section 3, we canexplore the richness behind a family of curves having curvatureproportional to the radius (as prescribed by Eq. (5)) by varyingthe magnitude of a single control parameter. We obtain avariety of morphologies as Ω is changed, while the remainingparameters (rI , r0, and ψ0) are kept fixed. We show thata number of key features of the exact solutions apparentlyonly accessed by (“brute force”) numerical calculations (seeRef. [20] and Appendix B), can be predicted and clearlyrevealed by properly analyzing the relation between P(r) and rgiven by Eq. (16), and also by investigating how the polar angleϕ varies with r . As a by-product, a systematic and geometricallybased criterion for determining the particular value of Ω atwhich pinch-off takes place is presented. As commented above,our approach also guarantees the existence of closed, non-self-intersecting, periodic solutions.

Let us consider the polynomial P(r) = Ar4+ Br2

+ Cdefined in Section 3. In order for Eq. (15) to be consistent, thereare two possibilities:

(i) If C 6= 0, then r ≥ |P(r)|, i.e., we must determine theintervals on the positive r -axis on which the graph of P(r)is in the sector bounded by the lines y = r and y = −r ;

(ii) If C = 0, then we must determine the intervals on thepositive r -axis on which |Ar3

+ Br | ≤ 1.

Since A is positive, the intervals in either case are clearlycompact. Moreover, it is a simple exercise to check that,

depending on the initial conditions, there are either one or twointervals if C 6= 0 and precisely one interval if C = 0. Theend points of the former intervals are all positive, while thelatter has r = 0 as an endpoint. Since r = 0 lies outside thedomain where the differential equation (11) is defined, the caseC = 0 leads to singular solutions and we will not discuss it inthis article.

Before presenting and discussing the possible stationaryshape solutions, as depicted in Figs. 4 and 6, we introducethe basic notation and definitions that will be used toextract valuable information from the various plots presentedthroughout this work. This is done in Fig. 3. In the left panel wepresent the interface shape of interest (obtained by numericallysolving Eq. (4) — see Appendix B). In the right panel (on thetop) we plot the polynomial P(r) (gray curve) and the linesy = ±r, 0. Finally, the panel on the bottom depicts how thederivative of the polar angle ϕ′ varies with the radial distance r .As it will become evident in Figs. 4 and 6, the various possibleshapes can be obtained by varying Ω (while rI , r0, and ψ0remain fixed), which results in a modification of the relativeposition of P(r) with respect to the lines y = ±r, 0.

As illustrated in Fig. 3, the general behavior of theshapes can be described by a “generating arc” containing fiveparticularly important points: ri , with i = 1, 2, 3, 4, 5. Points r1and r5 are those obtained by solving P(r) = r (points at whichthe polynomial intersects the line y = +r ). So, r1 and r5 definethe minimum and maximum radii of the circles delimiting theinterval in which the curve is drawn (see the light gray solidcircles in Fig. 3). On the other hand, points r2 and r4 are thoseobtained by solving P(r) = 0. In the left panel of Fig. 3 thesepoints are determined by the intersection of the dark gray solidcircles and the interfacial curve itself. Finally, the remainingpoint r3 is given by the minimum of P(r), and because P ′(r) =

rκ , that coincides with those points on the curve at which the

E.S.G. Leandro et al. / Physica D 237 (2008) 652–664 657

Fig. 4. Possibilities for the region r ≥ |P(r)| for five increasingly larger values of Ω : (a) 0.731, (b) 2.360, (c) 5.924, (d) 6.790, and (e) 7.427.

curvature is zero, or r3 = rI (see Eq. (5)). The latter can beidentified in Fig. 3 as the intersection points between the dashedcircle and the curve representing the exact solution. Note thatr3 is an special point, establishing the location at which thecurvature of the curve changes its sign along the generating arc

specified within the interval r1 ≤ r ≤ r5 (see Appendix A).The location of all these points (ri , with i = 1, 2, 3, 4, 5) is alsoindicated in the graph ϕ′ versus r at the bottom of the right panelin Fig. 3. The usefulness of this graph resides in the fact that itimmediately gives information about the sign of ϕ′, something

658 E.S.G. Leandro et al. / Physica D 237 (2008) 652–664

that will be quite handy in identifying the occurrence of pinch-off.

The key issue here is that a number of very importantaspects of the exact solutions can be extracted and revealedsimply by analyzing the interplay between P(r) and r , andalso the behavior of ϕ′ (Fig. 3). This is indeed quite convenientsince this approach does not require the explicit solutionof complicated elliptic-like integrals (Eq. (17)) or nontrivialdifferential equations (Eq. (4)).

4.1. Closed solutions without self-crossings

We begin our study by focusing on stationary solutionswhich are physically realizable under the rotating Hele–Shawcell setup. Fig. 4 presents a representative collection of closed,non-self-intersecting solutions when rI = 1, r0 = 1.405,and ψ0 = π/2, for five increasingly larger values of Ω . Wepoint out that these values for rI , r0 and ψ0 will be utilizedas our reference parameters throughout the rest of this work.Following the notation established in Fig. 3, in the left panel ofFig. 4 we present the interface shape in black, and the delimitingcircles of radii r1 and r5 in light gray. The middle panel plotsthe polynomial P(r) in gray and the black lines y = ±r, 0, andthe panel on the right depicts how the derivative of the polarangle ϕ′ varies with r .

Fig. 4(a) illustrates the simplest pattern possible: a circle,obtained when Ω = 0.731. This is easily obtained from Eq.(5) by setting r = r0 and κ = 1/r0. This situation occurswhen P(r) = r or when P(r) touches y = +r at a singlepoint, so that r1 = r5 = r0. In this case it is easy to seefrom Eq. (16) that ψ = π/2, so that from Eq. (10) we obtainr ′

= 0. It is also evident that ϕ′(r) diverges. This divergenceis due to the singularity of the parameterization by r at pointswhere ψ = ±π/2. If instead of r we use ϕ as a parameter,we eliminate the singularity, and the circle is described simplyby r ′

= 0, ϕ′= 1. Fig. 4(b) illustrates a situation for a larger

value of Ω , for which P(r) crosses the line y = +r at twodistinct points, but it still does not cross the horizontal liney = 0, so that 0 < P(r) < r . In this case, r = r(ϕ) andthe resulting patterns are regularly deformed n-fold symmetricstructures (here n = 3) bounded by an annulus determined bythe radii r1 and r5. Notice that ϕ′ is always positive, meaningthat the polar angle ϕ increases as r is increased. It is also clearthat at the points r1 and r5 ϕ

′→ ∞.

As we increase the value of Ω , the minimum of P(r)approaches the line y = 0, and eventually touches it whenP(r) = 0. This is the situation shown in Fig. 4(c). Now thefingers formed present an inflection point (r = r3 = rI ) atwhich the radial unit vector to the curve coincides with the unittangent vector to it, so that ψ = 0. Consequently, ϕ′ now can beboth positive and zero. Under such circumstances, the resultingfingers look more straight on their sides than those depicted inFig. 4(b). From this point on, more distorted equilibrium shapesare expected, and the emerging interface will not be a single-valued function of ϕ.

A fourth possibility is depicted in Fig. 4(d) when Ω isincreased further: now P(r) crosses the line y = 0 at two points

(r2 and r4). One noteworthy aspect of this situation is the factthat now ϕ′ can also assume negative values. In fact, ϕ′ < 0for r2 < r < r4. As a result ϕ(r) is an increasing function ofr within the intervals r1 < r < r2 and r4 < r < r5, but adecreasing function of r in the range r2 < r < r4. Therefore,the emerging patterns develop meanders, so that the fingers tendto be wider near their tips, and narrower close to their bottom.

From our discussion of Fig. 4(d), if one keeps increasing thevalue of Ω it would be expected that the neck located at thebottom of a finger would tend to become thinner and thinner.Eventually, at a given critical value of Ω = Ω∗, the oppositesides of the neck would touch one another. As represented inFig. 4(e) for Ω = 7.427, the phenomenon of droplet pinch-offis about to take place. Since the limiting value Ω∗ defines atopological singularity, it would be quite desirable to determineit for any given set of values for other relevant parameters in theproblem (rI , r0, ψ0). It is worth mentioning that in Refs. [19,20] no systematic way to determine Ω∗ is reported, other thanby successive (trial and error) approximations. In contrast toRefs. [19,20] we show how to obtain Ω∗ based on simplegeometric arguments.

Generically speaking, the overall appearance of the graphsP(r) and ϕ′ in Fig. 4(d) and (e) are fairly similar to eachother. So, one may wonder what could differentiate the exactstationary shape with pinch-off from the one without it. Withrespect to this important point, a valuable piece of informationcan be extracted by inspecting the graphs for ϕ′ in Fig. 4(d) and(e). Notice that, for two given radii ri and r j , the area betweenϕ′(r) and the line y = 0 is very simply given by∫ r j

ri

ϕ′(t)dt = ϕ(r j )− ϕ(ri ), (18)

where ϕ(r j ) − ϕ(ri ) = 1ϕi j quantifies the variation in polarangle as r varies from ri to r j . However, at pinch-off we havethat ϕ(r2) = ϕ(r5), so by using Eq. (14) we can establish a veryuseful condition for determining Ω∗, given by

1ϕ25 =

∫ r5

r2

P(r)

r√

r2 − P(r)2dr = 0, (19)

where r2 is the smallest positive root of P(r; rI ,Ω , r0, ψ0) =

0, and r5 is the largest positive root of P(r; rI ,Ω , r0, ψ0) = r .An alternative interpretation for Eq. (19) can be obtained byobserving the graphs ϕ′(r) versus r in Fig. 4(e), and noticingthat at pinch-off the area delimited by ϕ′(r) and y = 0 withinr2 < r < r4 exactly matches the equivalent area for the intervalr4 < r < r5. In summary, for a given set of parametersrI , r0, ψ0, we can use Eq. (19) to accurately determine thecritical value for the parameter Ω = Ω∗ which leads to apinch-off situation. This is clearly illustrated in Fig. 5 whichplots 1ϕ25 as a function of Ω : pinch-off is characterized byΩ = Ω∗

= 7.481 which corresponds to the situation 1ϕ25 = 0(Eq. (19)).

At this point we would like to contrast the way we determinethe value of Ω = Ω∗ which leads to pinch-off in the rotatingcase, and how the authors of Ref. [20] addressed a similarissue for the rectangular flow situation. There is a subtle,

E.S.G. Leandro et al. / Physica D 237 (2008) 652–664 659

Fig. 5. Variation of 1ϕ25 with Ω . The zero of 1ϕ25 determines the value ofΩ = Ω∗

= 7.481 at which pinch-off occurs.

but important difference between our points of view regardingpinch-off determination. In Ref. [20] pinch-off is analyzedby plotting a typical distance H as a function of a relevantdimensionless parameter (B∗) for their problem (Fig. 4 inRef. [20]), where H is the distance between the maximumand minimum heights of the rectangular flow interface. Fromtheir analysis pinch-off would occur when the typical distanceH hits its highest value (Hmax ) as their parameter B∗ isvaried. As recognized by the authors of Ref. [20] themselves,this description is somewhat incomplete, since in principle Hwould keep increasing for increasingly larger values of B∗.In the rectangular case, if H > Hmax the successive fingersintersect and cross themselves, corresponding to nonphysicalsolutions for the problem of interest. Although the latter istrue, ultimately there would be no a priori way to determinethe value of B∗ at which H = Hmax simply by plotting Hagainst B∗ and searching for a “maximum”. Note that we donot have this kind of problem in our current rotating Hele–Shawsituation. We define the occurrence of pinch-off precisely when1ϕ25 = 0, so that when 1ϕ25 is plotted against Ω (as inFig. 5), Ω∗ is readily determined. In Fig. 5 any Ω > Ω∗ wouldbe associated with a negative 1ϕ25 leading to self-intersectingsolutions (which will be discussed in Section 4.2). Here the zerofor the polar angle displacement 1ϕ25 is well defined, while inRef. [20] their Hmax is not a global maximum of H . Of course,a typical distance similar to the one presented in Ref. [20] canalso be defined in our rotating case, being H rot

= r5 − r1.However, as was the case for its rectangular geometry analogH , H rot is not bounded and a global maximum is not welldefined, since H rot keeps increasing as Ω is increased (seeFig. 4(a)–(e) and Fig. 6(a)–(e)). In this work, we present analternative view for this pinch-off determination matter, whereinstead of examining a difference between typical distances inthe patterns, we focused on a proper angular displacement asgiven by Eq. (19).

4.2. Self-intersecting solutions

In this section we consider the patterns which present self-intersection. As previously observed, such patterns fulfill thecondition1ϕ25 = ϕ(r5)−ϕ(r2) < 0. Fig. 6 shows five of thesepatterns corresponding to five increasing values of Ω > Ω∗.

In Fig. 6(a) we see a typical self-intersecting solution. Theshape of the generating arc is similar to the one shown inFig. 4(e). However, it has a more accentuated S-shape due tothe larger variations of ϕ caused by an increase in the differencebetween area A24 below the interval r2 ≤ r ≤ r4 and abovethe graph of ϕ′(r), and the area A45 above r4 ≤ r ≤ r5 andbelow the graph of ϕ′(r). If Ω is kept increasing we observethat the successive loops fall exactly on top of one another, sothat a single 8-shaped curve results, as depicted in Fig. 6(b).At this point, the total area A15 in the graph of ϕ′(r) is zero,corresponding to 1ϕ15 = 0. For the cases where 1ϕ15 > 0(Fig. 4(a)–(e) and Fig. 6(a)), if one sits on the curve and followsits arclength, one will spin around the annulus (r1 ≤ r ≤ r5) inthe counter-clockwise direction. This direction is reversed forsituations in which 1ϕ15 < 0 (Fig. 6(c)–(e)).

After reaching the peculiar 8-shaped situation, the succes-sive loops begin to separate from each other, and the distancebetween them tends to increase as Ω is ramped up, as illustratedin Fig. 6(c). Fig. 6(c) shows a situation where A24 increaseseven further while A45 decreases slightly. The generating arc ofthe corresponding pattern shows ϕ with a larger decrease as rvaries from r2 to r4.

Fig. 6(d) shows an interesting solution presenting only twoloops, which is special since it corresponds to a parameterΩ such that the point of minimum on the graph of P(r) isvery close to the line y = −r . Such proximity causes ϕ′(r3),the minimum value of ϕ′, to approach −∞, since ϕ′(r) =

P(r)/[r√

r2 − P(r)2], so that the generating arc of the solutiontends to spin around the origin at very high speed with respectto variations of r (in the clockwise direction if r is increased).

Finally, we see in Fig. 6(e) a solution which is contained inthe narrow annulus bounded by two circles of radii r3 and r5,where r3 is the largest root of P(r) = −r . A similar situationalready occurred in Fig. 6(d). The situation described here iscurious because there is another annulus which could containthe solution, namely the one determined by the circles of radiusr1 and the smallest root of P(r) = −r . The dashed curves inFig. 6(d) and Fig. 6(e) imply the possibility of a second patternlocated on a different annulus, corresponding to the same valuesof the parameters Ω and rI , but different initial conditions r0,ψ0. Since the solution is a continuous curve and the two annuliare disjoint, the choice of initial conditions determines whichof the annuli will contain the solution. This is illustrated inFig. 6(e), where the inner annulus solution is obtained by settingr0 = r1 = 0.472, whereas the outer annulus pattern is obtainedby taking the usual value r0 = r5 = 1.405. In order to producethe two patterns in Fig. 6(e) from the same polynomial P(r), wenotice that P(r) is uniquely determined by Ω , rI and either rootof the equation P(r) = r , as long as we set ψ0 = π/2. Here weuse the fact that a polynomial of degree four is determined byfive constants, for instance its coefficients. Since our P(r) does

660 E.S.G. Leandro et al. / Physica D 237 (2008) 652–664

Fig. 6. Self-intersecting solutions occurring for Ω > Ω∗: (a) 7.836, (b) 8.745, (c) 9.596, (d) 10.029, and (e) 10.802.

not have a cubic term, only four constants are needed for itscomplete determination. From the observation that P(r0) = r0,and fixing Ω , rI and ψ0 = π/2, we must pick r0 = r1 toproduce the inner pattern, and r0 = r5 to produce the outerone. Finally, we observe an interesting feature in Fig. 6(e):

the absence of the point of inflection in the generating arc,and hence its changing from an S-shaped to a rather C-shapedcurve. Actually, any solution for which P(r) crosses the liney = −r will no longer have an inflection point, characterizinga C-shaped generating arc.

E.S.G. Leandro et al. / Physica D 237 (2008) 652–664 661

4.3. One last important issue: Dense solutions

At first glance, self-intersecting solutions such as thosediscussed in Section 4.2 might be seen as being of “lessimportance” in practical terms, since they are not directlyrelated to the physical patterns studied in Section 4.1. In thissection we show that this is not entirely true. Although the self-intersecting solutions are not physical, it is exactly through theanalysis of solutions like those (and also the so-called densesolutions studied in this section) that the very existence ofthe physical, closed, and non-self-intersecting solutions can beassured. The investigation of these issues is the main purposeof the more mathematically rigorous discussion we present inthis portion of our work.

We get started by recalling a standard result fromanalysis [38].

Lemma 4.1. Let Rθ : S1→ S1 be a rotation of the unit circle

by an angle θ which is incommensurable with 2π . Then for allx ∈ S1 the orbit O(x) = Rn

θ (x)/n = 0, 1, 2, . . . is dense inS1.

An important application of the lemma above is the theoremstated below, which contains an important analytical result ofthis work.

Theorem 4.1. The solutions of Eq. (11) are bounded and areeither closed or dense in one of the (possibly degenerate)annuli determined by the non-negative roots of the polynomialequations P(r) = ±r . For a generic choice of initial conditionsand parameter values, the corresponding solution fills up anannulus.

Proof. The relation (10) implies that ψ = ±π/2 if and onlyif r ′

= 0. So the points where P(r) = ±r correspond topoints where r ′ vanishes. In order to analyze the solutions to ourproblem in a neighborhood of such points, let us consider thedifferential equation obtained from Eq. (8) when the parameteru is ϕ. This equation can be put in the form

r ′′=κ(ϕ)(r ′2

+ r2)3/2 − r2+ 2r ′2

r,

κ(ϕ) = κ(r(ϕ)), r > 0, (20)

which is suitable for applying the existence and uniquenesstheorem of solutions of ordinary differential equations [37].

Let C 6= 0 (see remark (i) in Section 4). As we commentedabove, the solutions of Eq. (11) are defined for r belongingto a bounded interval. Thus the solutions are bounded by twocircles S1 and S5 whose radii are designed by r1 and r5, with0 < r1 ≤ r5 (see Fig. 3). The solution γ must satisfy Eq. (20)near the points of tangency to S1 and S5, and at these points wehave ψ = ±π/2 so, as we observed, r ′

= 0. Now, from theuniqueness of the solution of Eq. (20), it follows from the factthat κ is a function of r that the reflection of γ about the raysr = r1 and r = r5 must coincide with γ . This key observation isanalogous to the approach used to describe bounded trajectoriesin planar central force fields [38].

Let us follow a segment of γ from r1 to r5. Let ϕ1 andϕ5 be the initial and final polar angles. From the discussion

above, it is clear that the difference 1ϕ15 = ϕ5 − ϕ1 must becommensurable with 2π in order for the solution γ to be closed.However, if such a condition is not verified, then Lemma 4.1implies that γ is dense in the annulus bounded by S1 and S5.

Proposition 4.1. The function 1ϕ15 = 1ϕ15(r0, ψ0,Ω , rI ) isanalytic.

Proof. Recall that

1ϕ15 =

∫ r5

r1

P(r)

r√

r2 − P2(r)dr. (21)

Let us write Q(r) = r2− P2(r). Observe that r1 and r5 are

simple roots of Q(r), so we have

Q(r) = (r − r1)S1(r) = (r5 − r)S5(r), Si (ri ) 6= 0, i = 1, 5.

Notice that the integral in Eq. (21) is improper.Since both P(r) and Q(r) depend analytically on r0, ψ0,Ω

and rI , the implicit function theorem for solutions of analyticequations [39] says that both r1 and r5 are analytic functions ofthe initial conditions and parameters. Let us write

1ϕ15 =

(limδ→0+

∫ c

r1+δ

+

∫ c′

c+ limε→0+

∫ r5−ε

c′

)P(r)

r√

r2 − P2(r)dr,

(22)

where c and c′ are chosen within the intervals of convergenceof the power series representing the function

F(r) =P(r)

r√

Si (r)

near r = r1 and r = r5, respectively. We examine the firstintegral in Eq. (22). For δ > 0 sufficiently small, we have∫ c

r1+δ

F(r)

r√

r − r1dr = F(r1)[(c − r1)

1/2− δ1/2

]

+ O((c − r1)3/2)+ O(δ3/2).

Thus the limit when δ tends to zero in Eq. (22) exists and isan analytic function of r1, and hence of r0, ψ0,Ω and rI . Asimilar argument applies to the third integral in Eq. (22). Theproposition is proven.

The rather strong commensurability condition mentioned inthe proof of Theorem 4.1 represents a serious constraint on theinitial conditions and parameter values from which a closed γcan be obtained. From Proposition 4.1, the condition above issatisfied only for a set of initial conditions and parameter valuesof zero Lebesgue measure (denumerable union of manifolds ofdimension less than 4). In general, i.e., with probability onein the measure sense, γ will not close; instead it will fill updensely the annulus determined by the circles S1 and S5 as itturns around the origin. This fact is illustrated in Fig. 7, showinga curve which is becoming dense as the arclength is increased.

As a consequence of the analytic dependence on the initialconditions and the parameter values, there exists, arbitrarilynear a dense solution γ1 of our problem, a closed solution γ2whose shape mimics that of γ1 up to the first complete turn, i.e.,

662 E.S.G. Leandro et al. / Physica D 237 (2008) 652–664

Fig. 7. Occurrence of dense solutions as the arclength is increased. The initialpattern (top left) is an almost closed curve, quite similar to the closed periodicsolution shown in Fig. 4(d), but with a slightly larger value of Ω (here Ω =

6.800).

the smallest multiple of 1ϕ15 which is greater than or equal to2π . Indeed, if the variation 1ϕ15 along the generating arc ofγ1 is an irrational number in a sufficiently small neighborhoodof 2π/k, for some k = ±1,±2, . . ., then we expect, basedon the strong numerical evidence provided by Fig. 8, that wecan choose γ2 which closes precisely after the first turn aroundthe origin. For instance, since Fig. 8 shows that 1ϕ15 changessign for fixed r0, ψ0, rI and varying Ω , it certainly attains allvalues 2π/k for k large enough. Note that for the physicalpatterns (Ω < Ω∗) k = 2n, where n is the number of fingeringstructures (or “fingers”) formed. The fact that (1ϕ15)

−1(2π/k)is a nonempty set for suitably chosen values of k justifies theexistence of the beautiful patterns seen in Sections 4.1, 4.2, andin Refs. [19,20].

5. Concluding remarks

A geometric formulation of the pattern-forming stationarystructures in rotating Hele–Shaw cells has allowed us to accesskey features of these exact solutions analytically. We havedemonstrated that a complete description of a general emergingcurve can be properly performed with the identification of aspecial portion of it, defined as a generating arc. From basicgeometric properties of such an arc we have been able todescribe the exact solutions of the problem as a one-parameter(Ω ) family of curves having curvature proportional to thesquare of the radial distance. As a result, a vast gallery ofpossible curves has been identified and discussed, includingthose directly related to the physical problem at hand (i.e.,closed, periodic, non-self-intersecting solutions), as well as theones presenting self-intersections, and those which are dense.

Fig. 8. Variation of 1ϕ15 with Ω . The horizontal lines in gray correspond toseveral values of k. From top to bottom we have k = 6, 7, 8, . . ., where thepatterns which close precisely after the first turn around the origin are labeled byk even. In particular, the value k = 22 (black dashed horizontal line) indicatesthe situation at which pinch-off occurs, corresponding to a pattern with 11fingers such as the one depicted in Fig. 4(e). The vertical dotted line indicatesthe location of Ω∗. When k → ∞ we obtain the 8-shaped pattern shown inFig. 5(b) for which 1ϕ15 = 0. Beyond this point k assumes negative values,and the curves are like those illustrated in Fig. 6(c)–(d).

Among these the dense solutions are the most likely and generalcase. From our analysis a particularly important physicalfeature can be systematically determined, namely the criticalvalue of the relevant parameter Ω = Ω∗ for which interfacepinch-off takes place. All this is approached analytically, usingfundamental geometric concepts, and avoiding the solution ofcumbersome differential equations and nontrivial integrals.

On a more mathematically rigorous note, we point outthat the practical existence of the physical, highly symmetricfingered patterns is not necessarily obvious, and should notbe taken for granted. After all, earlier studies [19,20] haveidentified this type of solution as unstable. In this work, fromthe study of the dense solutions, we provide a geometricallybased interpretation for this unstable behavior, and concludefrom more formal grounds (both analytically and numerically)that the existence of such physical patterns is indeed ensured.

In a forthcoming work, we plan to address the importantissues raised and examined in this paper for other systemsin which exact stationary configurations may arise. Oneinteresting possibility refers to those Hele–Shaw systemsinvolving the confinement of magnetic fluids [42] subjected toan external magnetic field. In these systems a balance betweenthe competing capillary and magnetic forces can be achievedby tuning the strength of the applied magnetic field. Of course,the inclusion of nonlocal magnetic body forces would probablycomplicate the structure of the integro-differential equationsinvolved. However, the nontriviality of the task makes theproblem even more challenging and attractive. This sort ofstudy would certainly be of interest, particularly because theinvestigation of exact solutions involving magnetic fluids inHele–Shaw cells has been largely neglected in the literature.

E.S.G. Leandro et al. / Physica D 237 (2008) 652–664 663

Acknowledgments

J.A.M. thanks CNPq (Brazilian Research Council) forfinancial support of this research through the CNPq/FAPESQPronex program, and CNPq program “Instituto do Mileniode Fluidos Complexos” contract No. 420082/2005-0. R.M.O.acknowledges financial support from FACEPE (Fundacao deAmparo a Ciencia e Tecnologia do Estado de Pernambuco)through the contract BPD No. 0007-1.05/07.

Appendix A. Definition of curvature and the inflectionpoint

Let x = (x1(s), x2(s)) be the parameterization of planardifferentiable curve γ by arc length. The vectors

s = (x ′

1(s), x ′

2(s)), and n = (−x ′

2(s), x ′

1(s)) (A.1)

form an orthonormal frame s, n along γ with the sameorientation as the canonical basis e1, e2 of R2. Let M(s) ∈

SO(2) be the rotation matrix which takes e1, e2 to s, n.From the fact that the inverse of M(s) is its transpose, it isimmediate to verify that M ′(s)M(s)−1 is a skew-symmetric2 × 2 matrix, which we write as

C(s) =

(0 κ(s)

−κ(s) 0

).

C(s) is known as the Cartan matrix of γ [35]. The off-diagonalentries of C(s), with an appropriate sign convention, defineuniquely the curvature of γ . An interesting interpretation of thecurvature arises from this formula: if θ(s) is the angle of therotation M(s), then κ(s) is simply the angular velocity θ ′(s) ofthe motion of a rigid plate attached to the frame s, n.

Using that x ′

1(s)2+ x ′

2(s)2

= 1, we obtain

κ(s) = x′

1(s)x′′

2(s)− x′

2(s)x′′

1(s) = (x′, x′′,k). (A.2)

If we consider γ parameterized by a different parameter u,Eq. (7) can be deduced from the chain rule and the relations′(u) = ‖x′(u)‖.

We introduce coordinates (x(s), y(s)) along the orthogonalaxis defined by the vectors s(s) and n(s) at the point γ (s). Fixa point P on γ and set s = 0 at P . Using the Serret–Frenetformulas for planar curves [40], we deduce

x ′(0) = 1, x ′′(0) = 0, x ′′′(0) = −κ2, (A.3)

y′(0) = 0, y′′(0) = κ, y′′′(0) = κ ′. (A.4)

Consider the Taylor expansion of y as a function of x nearx = 0:

y(x) = y′(0)x +y′′(0)

2x2

+y′′′(0)

6x3

+ O(x4). (A.5)

After several applications of the chain rule, Eq. (A.5) reads

y(x) =κ(0)

2x2

+κ ′(0)

6x3

+ O(x4). (A.6)

If γ is parameterized by r instead of s, then κ ′=

cosψ (dκ/dr). So, in the special case discussed in this work,

we have

y(x) =Ω(r2(0)− r2

I )

2x2

+2Ωr(0) cosψ(0)

6x3

+ O(x4).

(A.7)

Therefore, we draw the following conclusions: in a neighbor-hood of P , the curve γ approaches a parabola with positive con-cavity if r(P) > rI or negative concavity if r(P) < rI . More-over, if r(P) = rI , then γ approaches a cubic if cosψ(P) 6= 0,and the concavity changes sign as we move along γ . In the lattercase, P is a point of inflection of γ .

Appendix B. Arclength parameterization and the numeri-cal solution for the stationary shapes

To plot the exact solutions numerically, we take the arclengths as the parameter u, so that the curvature (Eq. (8)) can berewritten as

κ(s) = (1 + r ′2)ϕ′+ r(r ′ϕ′′

− r ′′ϕ′). (B.1)

Inserting Eq. (B.1) into Eq. (4), we obtain a coupled third-orderordinary differential equation for the stationary curves:

(r ′r ′′− rr ′′′)ϕ′

+ (1 + 2r ′2)ϕ′′+ rr ′ϕ′′′

= 2Ωrr ′. (B.2)

Due to the coupling between radial and polar angle variablesin Eq. (B.2), we must consider a second differential equationin order to solve the problem. The own choice of the arclengths as the relevant parameter immediately gives us this auxiliaryequation, being precisely the normalized velocity vector

r ′2+ r2ϕ′2

= 1. (B.3)

With Eqs. (B.2) and (B.3) we have a system formed by athird-order differential equation and a first-order one. To solvethis system numerically we use the artifice of differentiatingEq. (B.3) twice with respect to s, transforming it into a systemof two third-order equations.

The key feature of our numerical strategy is to keep dealingwith the original system formed by Eqs. (B.2) and (B.3).This can be accomplished by noticing that, out of the sixinitial conditions, there are two pairs that are closely related:(i) ϕ′(s = 0) relates to r ′(s = 0) through Eq. (B.3); and(ii) ϕ′′(s = 0) is conveniently connected to r ′′(s = 0) bydifferentiating Eq. (B.3) with respect to the arclength. It isexactly such restrictions on the initial conditions that make thesolutions of the transformed system match the desired ones,i.e., the solutions of the original system. The implementation ofour numerical approach to solve these differential equations hasbeen performed with the help of the software Mathematica [41].

Another important aspect in the success of our studyis the connection between the initial conditions of thedifferential equations arising in the numerical procedure[r(0), r ′(0), r ′′(0), ϕ(0), ϕ′(0), ϕ′′(0)], and the parameters(r0, rI , ψ0,Ω ) of the polynomial of degree four P(r), obtainedanalytically from our geometric approach in Section 3. FromEqs. (9) and (B.3) we get

664 E.S.G. Leandro et al. / Physica D 237 (2008) 652–664

r ′(0) = cosψ0, (B.4)

ϕ′(0) =sinψ0

r0. (B.5)

In addition, from Eqs. (5) and (B.1), together with the firstderivative of Eq. (B.3), we obtain that

ϕ′′(0) =r ′(0) [Ω(r2

0 − r2I )− 2r2

0ϕ′(0)3 − 2r ′(0)2ϕ′(0)]

r0,(B.6)

r ′′(0) = −r0ϕ

′(0) [r ′(0)ϕ′(0)+ r0ϕ′′(0)]

r ′(0). (B.7)

Thus, putting all these ingredients together, we can produce allthe beautiful curves presented in the left panels of Fig. 4(a)–(e),and Fig. 6(a)–(e), and also in Fig. 7.

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