Capillary and geometrically driven fingering instability in nonflat Hele ...

PHYSICAL REVIEW E 95, 033104 (2017)

Capillary and geometrically driven fingering instability in nonflat Hele-Shaw cells Rodolfo Brandão and José A. Miranda* Departamento de F´ısica, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901 Brazil (Received 30 December 2016; published 8 March 2017) The usual viscous fingering instability arises when a fluid displaces another of higher viscosity in a flat Hele-Shaw cell, under sufficiently large capillary number conditions. In this traditional framing, the reverse flow case (more viscous fluid displacing a less viscous one) and the viscosity-matched situation (fluids of equal viscosities) are stable. We revisit this classical fluid dynamic problem, now considering flow in a nonflat Hele-Shaw cell. For a specific nonflat environment, we show that both the reverse and the viscosity-matched flows can become unstable, even at low capillary number. This peculiar fluid fingering instability is driven by the combined action of capillary effects and geometric properties of the nonflat Hele-Shaw cell. Our theoretical results indicate that the Hele-Shaw cell geometry significantly impacts the linear stability and nonlinear pattern-forming dynamics of the system. This suggests that the geometry of the medium plays an important role in favoring the occurrence of fingering patterns in nonflat, confined fluid flows. DOI: 10.1103/PhysRevE.95.033104 I. INTRODUCTION

It is known that a morphological interfacial instability can occur during the displacement of two fluids, when a fluid of low viscosity displaces a more viscous fluid. In the context of a Hele-Shaw cell flow, i.e., in a flow confined in the narrow gap separating two parallel glass plates, this may lead to the development of complex fingerlike patterns of the less-viscous fluid growing into the more viscous one. This phenomenon defines the celebrated viscous fingering (or Saffman-Taylor) instability [1–4] which is driven by the viscosity difference between the two contiguous fluids. Interestingly, the reverse flow situation, wherein a more-viscous fluid moves a lessviscous fluid in a Hele-Shaw cell, is stable. Likewise, for a viscosity-matched flow situation, when the fluids have equal viscosities, the fluid-fluid interface is also stable. These basic viscosity-driven interfacial responses related to the Saffman-Taylor instability can be characterized by two dimensionless quantities. The first one is the viscosity contrast parameter (dimensionless viscosity difference between the fluids) [3] η2 − η1 , (1) A= η2 + η1 where η2 (η1 ) is the viscosity of the displaced (displacing) fluid, and −1 A 1. In this framework, the fluid-fluid interface can become unstable (stable) if A > 0 (A 0). Note that the viscosity contrast is negative for the reverse flow situation and zero for the viscosity-matched flow case. Ultimately, the stability behavior of the two-fluid interface results from the interplay between viscous and capillary forces. The competition between these two physical effects can be described by another relevant dimensionless parameter of the problem, namely the capillary number [2], 12(η1 + η2 )U R02 , (2) σ b2 where U is a characteristic velocity of the interface, R0 is a characteristic length, σ is the surface tension between the Ca =

*

[email protected]

2470-0045/2017/95(3)/033104(10)

fluids, and b is the small gap thickness separating the cell’s plates. From the dynamic interaction between capillary and viscous forces one has the formation of interface shapes which can vary from a steady single-fingered pattern [1–4] to branched fronts in which fingerlike structures split at their tips (the so-called finger-tip-splitting phenomenon) [5–13]. Generally speaking, one can say that the interface instability is favored for positive viscosity contrast A and for higher values of the capillary number Ca. The viscous fingering instability has received much attention over half a century not only due to its practical importance but also because it represents an archetype for a wide range of fields, including research in oil recovery processes [14,15], fluid mixing [16,17], metallurgy [18], and even biology [19–21]. Depending on the circumstances, one might want to either suppress or stimulate the formation of fingering instabilities. For example, in petroleum extraction one usually wishes to restrain the fingering instability since it significantly reduces oil recovery efficiency. This type of important practical difficulty has motivated a number of recent studies that propose different strategies intended to inhibit the emergence of viscous fingering. One particularly successful controlling technique has been posed in Refs. [9,22–25], where proper control of the shape of the emergent fingered patterns has been achieved through the employment of time-dependent injection schemes. It has also been shown that viscous fingers can be stabilized if one of the flat Hele-Shaw cell plates is tilted [26,27] or if one of the rigid plates is replaced by an elastic membrane [28,29]. On the other hand, for distinct physical circumstances or industrial purposes, the emergence of viscous fingering can be something beneficial. For instance, it has been recently verified that intense fingering destabilization in Hele-Shaw cells does favor fluid mixing in confined systems, such as in microfluidic devices [16,17]. As these confined fluid arrangements typically have a small Reynolds number (a measure of the ratio of inertial to viscous forces), inertial effects are negligible, and turbulence does not set in. Therefore, achieving enhanced fluid mixing via fingering instabilities is something really handy. Other studies have also investigated the possibility of triggering the fingering instability under conventionally

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©2017 American Physical Society

˜ AND JOSE´ A. MIRANDA RODOLFO BRANDAO


stable conditions: (i) a reverse Saffman-Taylor instability has been reported in the literature if surfactants [30–33] or small particles [34] are present on the the walls of the Hele-Shaw cell; (ii) a viscosity-matched fingering instability has been revealed by experiments [35,36], theory [37], and numerical simulations [38] if chemical reactions occur at the fluid-fluid interface. In this work, we propose a different way to produce fingering instabilities that do not rely on adding external agents (e.g., surfactants or particles) on the cell plates or by requiring the occurrence of complex chemical reactions at the fluid-fluid interface. Additionally, in contrast to what is done in Refs. [9,22–25], we keep the injection rate constant in time. We demonstrate that a fluid fingering instability can be properly generated by simply modifying the shape of the Hele-Shaw cell, so such a confined, effectively two-dimensional (2D) flow environment acquires specific geometric features. It is shown that this unique fluid fingering instability can arise even under conventionally stable reverse (A < 0) and viscosity-matched (A = 0) settings, for low-capillary-number conditions. Although the overwhelming majority of studies in HeleShaw flow research deal with effectively 2D flow between flat glass plates (see, for instance, Refs. [1–13], and references therein), a growing number of both theoretical [39–48] and experimental [49–51] works have been devoted to the investigation of pattern formation phenomena between parallel, narrowly spaced, nonflat, rigid surfaces. In these studies the dynamics of pattern forming structures in nonflat backgrounds have been investigated in a variety of configurations, including spherical, cylindrical, conical, toroidal, and saddle-shaped Hele-Shaw cells. In spite of the validity and importance of these investigations, they focused on understanding how traditionally unstable viscous fingering flow (A > 0 and large Ca) reacted to changes in the geometric properties of the nonflat Hele-Shaw cell. Conversely, in this work we concentrate on taking advantage of the peculiar geometric features of a nonflat Hele-Shaw cell to activate a fluid fingering instability, even under customarily stable circumstances (A 0 and low Ca). The rest of this paper is outlined as follows. In Sec. II we present the fluid fingering problem in an arbitrarily shaped Hele-Shaw cell and utilize a perturbative weakly nonlinear method to derive a mode-coupling differential equation that describes the evolution of the interfacial perturbation amplitudes. This is done for both linear and early nonlinear dynamical regimes. In Sec. III, we examine how the geometry of a particular family of nonflat Hele-Shaw cells (that possesses a “warped-cone” shape) influences the linear stability of the fluid-fluid interface, as well as how it impacts intrinsically nonlinear finger-tip-splitting events. Finally, Sec. IV briefly summarizes our findings and presents our chief conclusions.

II. LINEAR AND WEAKLY NONLINEAR INTERFACIAL EVOLUTION: GOVERNING EQUATIONS A. Arbitrarily shaped Hele-Shaw cell

In this section, our main goal is to describe analytically the evolution of the interface separating two fluids while they flow on a nonflat Hele-Shaw cell, represented by an arbitrarily shaped 2D surface. To accomplish this task, we employ a

perturbative, second-order, mode-coupling theory that has been introduced in Ref. [10]. This theoretical approach enables one to derive a differential equation describing the linear and weakly nonlinear evolution of the interfacial perturbation amplitudes. Consider two immiscible, incompressible viscous fluids flowing on a curved surface. The geometric properties of this system can be described by using geodesic polar coordinates [52], a generalization of the usual polar coordinates for curved spaces. The parameters are defined as follows: r is the geodesic distance between the origin (an arbitrary point on the surface) and some other point on the surface, and ϕ is the polar angle between a geodesic and a given reference geodesic (0 ϕ < 2π ). The metric on such a surface takes a simple form [53–55], ds 2 = dr 2 + ρ(r,ϕ)2 dϕ 2 .

(3)

Despite its mathematical simplicity, metrics of this form can be used to describe any regular surface at least locally. To facilitate our analysis, we focus only on surfaces that exhibit polar symmetry, so ρ(r,ϕ) = ρ(r). This allow us to describe the flow dynamics on an arbitrary surface of revolution, such as the ones previously investigated in the nonflat Hele-Shaw cell literature [39–51], as in many other instances, including developable surfaces, and generalized helicoids [54]. On this curved environment, a fluid of viscosity η1 is injected at the origin of the coordinate system into a fluid of viscosity η2 (see Fig. 1). The rate of injection is given by Q, which gives us the area increase per unit time. The hydrodynamics of the system is governed by the gap averaged Darcy’s law [41,56], vj = −

b2 ∇pj , 12ηj

(4)

where vj = vj (r,ϕ) and pj = pj (r,ϕ) are, respectively, the velocity and pressure in fluids j = 1 and 2. The Hele-Shaw cell gap thickness b [defined in Eq. (2)] is smaller than any other

FIG. 1. Schematic illustration of the fluid flow on an arbitrarily shaped nonflat Hele-Shaw cell. A fluid of viscosity η1 is pumped into a cell with injection rate Q, and pushes a fluid of viscosity η2 , creating interfacial deformations ζ at the fluid-fluid interface (solid curve). A polar coordinate system (r,ϕ) is defined on the nonflat cell, where R = R(t) is the time-dependent radius of the unperturbed interface (dashed curve), and ζ = ζ (ϕ,t).

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CAPILLARY AND GEOMETRICALLY DRIVEN FINGERING . . .

length in the problem, and therefore the system is effectively 2D. The gradient operator in Eq. (4), associated with the metric (3), is [53] 1 ∂ ∂ ˆ rˆ + ϕ, ∇= ∂r ρ(r) ∂ϕ

(5)

and the unit vectors rˆ and ϕˆ point in the direction of increase of r and ϕ, respectively. Due to a fingering instability, the initially circular fluidfluid interface deforms, and its perturbed shape is written as r(ϕ,t) = R = R + ζ (ϕ,t), where R = R(t) denotes the time-dependent unperturbed radius, and ζ (ϕ,t) represents the net interface perturbation with Fourier amplitudes ζn (t) and integer azimuthal wave numbers n. We note that by using Eq. (3), it is possible to relate the unperturbed radius R, and the injection rate Q at a given time t as R Qt = 2π ρ(r)dr, (6) R0

where R0 represents the initial unperturbed radius. At this point, it is convenient to rewrite Eq. (4) in terms of velocity potentials defined by vj = −∇φj . After doing that, subtract the resulting expression for one fluid from the same equation for the other fluid and divide by the sum of the two fluids’ viscosities. This yields an equation of motion for the system, valid at the two-fluid interface, φ1 + φ2 φ1 − φ2 b2 (p1 − p2 )|r=R A , − =− 2 2 12(η1 + η2 ) r=R r=R (7) where the viscosity contrast A has been introduced in Eq. (1). We proceed by defining Fourier expansions for the velocity potentials φj . Far from the interface, the velocity field should approach the unperturbed steady flow with a circular interface of radius R. Thus, for r → 0 and r → ∞ the velocity potentials φj approach φj0 , the velocity potentials for purely radial flow, satisfying Laplace’s equation in metric (3) [54,57] Q

(r) + Dj , (8) 2π where Dj are independent of ρ and ϕ, and (r) is given by r d . (9)

(r) = R ρ() φj0 = −

The general velocity potentials obeying all these requirements are φj = φj0 + φj n (t) exp[(−1)j −1 |n| (r)] exp(inϕ). (10) n=0

To determine the coefficients φj n , we need to specify two boundary conditions [2–4,41]: the first one, commonly known as the Young-Laplace pressure boundary condition, includes the contributions coming from surface tension [presented in Eq. (2)], and expresses the pressure jump across the fluid-fluid interface, (p1 − p2 )|r=R = σ κ|r=R ,

(11)

where κ designates the interfacial curvature on the nonflat surface. In terms of the metric given by Eq. (3), κ can be


written as [54]

∂r 2 ∂2r − ρ(r) ∂ϕ ρ (r)ρ 2 (r) + 2ρ (r) ∂ϕ 2 , κ= ∂r 2 3/2 ρ 2 (r) + ∂ϕ

(12)

where the primes denote derivatives with respect to r. The second boundary condition, often referred to as the kinematic boundary condition [2–4], connects the velocity of the fluids with the motion of the interface itself ∂φj 1 ∂r ∂φj ∂R = − . (13) ∂t ρ 2 (r) ∂ϕ ∂ϕ ∂r r=R

r=R

This equation manifests the fact that the normal components of the fluids velocities are continuous across the interface: n · ∇φ1 = n · ∇φ2 , with n = ∇[r − R(ϕ,t)]/|∇[r − R(ϕ,t)]| representing the unit normal vector at r = R. To complete our derivation we use the kinematic boundary condition (13) to express the Fourier coefficients of φj in terms of ζn . Substituting these relations and the pressure jump condition (11) into Eq. (7), always keeping terms up to second order in ζ , and Fourier transforming, we find the equation of motion for the perturbation amplitudes ζn (for n = 0)

F (n,m)ζm ζn−m + G(n,m)ζm ζn−m , ζn = λ(n)ζn + m=0

(14) where, from this point onward, the primes are used to indicate derivatives with respect to the unperturbed radius R. The linear growth rate is given by λ(n) =

1 [A|n| − ρ (R)] ρ(R) 1 − |n|[n2 − ρ 2 (R) + ρ (R)ρ(R)], Caρ 2 (R)

(15)

where the capillary number Ca has been defined in Eq. (2), with U = Q/(2π R0 ). Notice that we utilize the primes notation just to simplify our analysis. By doing this, we do not need to invert Eq. (6). However, if one is interested in the time evolution of ζn , Eq. (6) also provides a simple way to rewrite the expressions in terms of time derivatives. In addition, the second-order mode-coupling terms are ρ (R) 1 F (n,m) = 2 |n| − sgn(nm) ρ (R) 2 |n| 1 ρ (R) 3 ρ (R)ρ (R) ρ 3 (R) − + 3 − Ca 2 ρ (R) 2 ρ 2 (R) ρ (R) ρ (R) m 1 ρ (R) − 3 (3m + n) − (16) ρ (R) 2 2 ρ (R) and G(n,m) =

1 {A|n|[1 − sgn(nm)] − ρ (R)}. ρ(R)

(17)

The sgn function equals ±1 according to the sign of its argument. Notice that in Eqs. (14)–(17) distances and velocities are rescaled by R0 , and U , respectively. Equations (14)–(17) generalize previous works that derived equivalent expressions describing viscous flows for specific

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FIG. 2. Illustrative surfaces associated with different values of the parameter β: (a) regular cone (0 < β < 1), (b) flat disk (β = 1), and (c) warped cone (β > 1).

Hele-Shaw cell geometries, defined in flat [10], spherical [41], and hyperbolic [47] spaces having constant Gaussian curvatures. In fact, Eqs. (14)–(17) are also applicable to HeleShaw flows occurring under variable Gaussian curvature circumstances, such as in the cases of toroidal [48] and Gaussian-shaped profile [51] Hele-Shaw cell configurations. It should be noted that Eqs. (14)–(17) are fairly general, being valid to flows in arbitrarily shaped Hele-Shaw cells defined in nonflat spaces described by the metric (3), under the condition that ρ(r,ϕ) is only a function of r. With a particular metric of this type at hand, and by using Eqs. (14)–(17), one can extract useful analytical information about key fluid dynamic features of the system at both linear and weakly nonlinear regimes. Specifically, at the linear level (i.e., at first-order in ζ ), one can assess the stability properties of the two-fluid interface, as well as predict the typical number of emerging fingering structures. On the other hand, at the weakly nonlinear level (at second order in ζ ), one can capture important aspects related to intrinsically nonlinear, complex phenomena connected to the morphology of the interface, such as the appearance of finger-tip splitting.

B. Warped-cone Hele-Shaw cell

In this section, we explore the generality of Eqs. (14)–(17) but now focus on a particularly interesting metric related to a generalized cone. This metric is quite simple but possesses conspicuous geometric features that lead to unusual fluid dynamic behaviors. A generalized cone can be parameterized by a closed curve on the unit sphere. The conical surface can be obtained joining the points of this curve to the center of the sphere using lines. Thus, making use of ordinary spherical coordinates, we can describe the metric of this surface as [58] ds 2 = dr 2 + r 2 [sin2 (θ (ν)) + θν (ν)2 ]dν 2 ,

parameter β is defined as 2π 1 sin2 [θ (υ)] + θυ (υ)2 dυ. β= 2π 0

This definition ensures that 0 ϕ < 2π . In this way, the metric given by Eq. (18) can be rewritten as ds 2 = dr 2 + β 2 r 2 dϕ 2 ,

(20)

where the function ρ(r) = βr. Roughly speaking, we can say that the generalized cone is a singular 2D surface which is flat everywhere except at its apex, where all the Gaussian curvature is concentrated [44]. This singular point defines what is known as a “point defect” or a “topological” or “conical” defect [58–60]. If the parameter 0 < β < 1, then the metric given in Eq. (20) defines a regular cone [Fig. 2(a)]. For β = 1 the cone turns into a flat disk [Fig. 2(b)]. However, if β > 1, one obtains a skirtlike, warped-cone surface [Fig. 2(c)]. Previous works have already investigated various weakly nonlinear aspects of the Saffman-Taylor instability for cases in which 0 < β < 1 (conical Hele-Shaw cell case) [44], as well as for β = 1 (flat Hele-Shaw cell situation) [10]. Nevertheless, a similar study for the situation β > 1 is still lacking, so we exploit it in this work. As we will see in Sec. III, the warped-cone case β > 1 exhibits a surprisingly subtle fluid dynamic behavior in which a fluid fingering instability may arise, even under ordinarily stable conditions (i.e., A 0 and low Ca, as discussed in Sec. I). Now we use the metric (20) to express Eqs. (15)–(17) for the warped-cone case β > 1, 1 1 [A|n| − β] − |n|[n2 − β 2 ], βR Caβ 2 R 2 1 1 − sgn(nm) |n| F (n,m) = βR 2 2 1 m 2 − (3m + n) , |n| β − Ca β 2 R3 2

λ(n) =

(18)

where r is the radius of the sphere, which represents the distance to the apex of the cone, and ν is the azimuthal angle. In Eq. (18) θ (ν) is the spherical polar angle, written as a function of ν, defining a parametric equation of a curve on the unit sphere. Moreover, θν (ν) represents the derivative of θ with respect to ν. In this context, it is convenient to perform the change of ν variables ϕ = (1/β) 0 {sin2 [θ (υ)] + θυ (υ)2 }1/2 dυ, where the

(19)

(21)

(22)

and G(n,m) =

1 {A|n|[1 − sgn(nm)] − β}. βR

(23)

While dealing with Eqs. (18)–(20), it should be clear that there are infinitely many possible functions θ (ν) for a given value of β. In this work, we use θ (ν) = C sin (N ν), where

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FIG. 3. Linear stability diagram in the parameter space CaR − β, for (a) A = 0.95, (b) A = 0, and (c) A = −0.95. Weakly nonlinear patterns are also shown in (b) and (c). The weakly nonlinear shapes corresponding to points I–IX in (a) are presented in Fig. 4.

C is a constant and N is an integer. This choice for θ (ν) has the advantage that by using Eq. (19), one finds that by increasing N , β is also increased. This means that in a legitimately 2D problem there is no bound for β. However, in view that the Hele-Shaw cell has a small but finite thickness b, the parameter β cannot be arbitrarily large. To comply with this spatial requirement, in the present work we set β 6.5. We have obtained such a range of allowed values for β by performing simulations to verify the parallel plate geometry of the cell’s plates, utilizing typical experimental values for the gap thickness b [2–13]. This specific choice of β values guarantees that there are no self-intersections or any kind of discontinuity on the Hele-Shaw cell geometry. Keeping this in mind, in the next section we examine how the nontrivial geometry of the warped-cone Hele-Shaw cell affects the stability of the two-fluid interface and the early nonlinear pattern formation dynamics of the system.

III. TRIGGERING FLUID FINGERING INSTABILITY VIA CAPILLARY EFFECTS COUPLED WITH THE GEOMETRY OF THE HELE-SHAW CELL A. Linear stability behavior

We begin our discussion by presenting a linear stability diagram for the system. This is accomplished by considering the stability curves for which the linear growth rate, given by Eq. (21), vanishes. Figure 3 plots the stability diagram in the parameter space CaR − β, for three values of the viscosity contrast: (a) A = 0.95, (b) A = 0, and (c) A = −0.95. In this figure, the stable regions [for which λ(n) < 0] are colored in white. Notice that we have to deal with a multidimensional parameter space, containing the parameters A, CaR, and the parameter β. The quantity CaR is a particularly convenient parameter, making the diagram valid for any given value of the unperturbed radius R of the evolving interface (or, equivalently, any value of time). In order to make the stability diagram even more illustrative, Fig. 3 also depicts representative weakly nonlinear patterns that arise in the various regions of the parameter space CaR − β. This is done by considering the action of two characteristic cosine Fourier modes, namely, a fundamental mode n given by the closest

integer to the mode of largest amplitude for that R, 1 2 log R β + CaβA , nζmax = 3 R−1

(24)

and its harmonic mode 2n. Equation (24) is obtained by integrating the purely linear part of Eq. (14) [i.e., ζn = λ(n)ζn ], and then by using Eq. (21), to find the value of n that maximizes the interfacial amplitude ζn . Under such conditions, the perturbed shape of the evolving two-fluid interface is given by R = R + ζ (θ,t),

(25)

with ζ (θ,t) = ζ0 + an (t) cos(nθ ) + a2n (t) cos(2nθ ),

(26)

where an = ζn + ζ−n and a2n = ζ2n + ζ−2n are real valued, with ζ0 (t) = −[1/(4R)][|an (t)|2 + |a2n (t)|2 ]. The patterns are obtained by solving the mode-coupling differential equation (14), with the functions λ(n), F (n,m), and G(n,m) given by Eqs. (21)–(23). We point out that the theoretical results presented throughout this work are obtained by utilizing parameter values that are consistent with those used in typical experimental realizations of radial flows in flat Hele-Shaw cells [2,3,5,7,9]. Moreover, without loss of generality we may choose the phase of the fundamental mode so an > 0. In all plots displayed in this work, we take the initial conditions an (R0 ) = 5 × 10−4 , and a2n (R0 ) = 0 with R0 = 1, so mode 2n is initially absent. These initial perturbation amplitudes are selected in such a way to avoid artificial growth of the first harmonic mode 2n (responsible for setting the finger shape as wide or narrow), imposed solely by the initial conditions. The reasons for taking these two specific cosine Fourier modes will become clear in the discussion of Sec. III B. The stability diagram illustrated in Fig. 3 reveals that by varying the values of CaR and β, for a given A one can identity interesting dynamical behaviors and still unexplored morphological responses. We initiate by inspecting what happens under large, positive viscosity contrast circumstances, namely for A = 0.95 [Fig. 3(a)]. First, it is worth noting that for the flat cell limit (i.e., for β = 1), we observe the standard phenomena associated with the conventional Saffman-Taylor

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FIG. 4. Representative weakly nonlinear patterns that arise at points I–IX of the linear stability diagram depicted in Fig. 3(a) for a positive value of the viscosity contrast A = 0.95: traditional Saffman-Taylor instability, tip-splitting patterns (I, IV, and VII), expected (stable) circular shapes (II, V, and VIII), and unexpected unstable structures that come to light due to the geometric properties of the nonflat, warped-cone Hele-Shaw cell (II, VI, and IX).

instability: The system is unstable for higher values of CaR, becomes stable when CaR is decreased, and reaches a certain critical value. Completely analogous dynamical responses are observed for flows in regular conical Hele-Shaw cells (0 < β < 1). However, a different picture is revealed for the warped-cone Hele-Shaw cell case (β > 1). First, let us consider a fixed value of β > 1 (i.e., β = 2.5) and three decreasing values of CaR related to points I, II, and III in Fig. 3(a). By the way, the interfacial patterns that arise at points I–IX in Fig. 3(a) are represented in Fig. 4 for R0 R Rf , where the initial radius R0 = 1, and the final radius Rf = 10. Each one of the patterns shown in Figs. 3 and 4 are composed by 21 interfaces plotted in sequence, separated by an interval r = 0.31 in Figs. 3(b) and 3(c), and by r = 0.5 in Fig. 4. For point I in Fig. 3(a) (A > 0, and a large value of CaR), we have the formation of a traditional, viscosity-driven pattern, where a number of fingers emerge and split at their tips. This is a classical example of the usual finger-tip-splitting phenomenon commonly detected in experiments and numerical simulations in the radial Saffman-Taylor problem [5–13]. Moreover, as one could have anticipated, by decreasing the magnitude of CaR, the interface tends to become more and more stable, so at point II the resulting interface is perfectly circular. Nevertheless, something rather unusual happens if one keeps decreasing the value CaR. Surprisingly, the interface becomes unstable again, even at considerably low values of CaR. An example of such a fact is given by point III in Fig. 3(a) and by its equivalent pattern shown in Fig. 4, where

an evidently noncircular interfacial shape is unveiled. As a matter of fact, similar kinds of dynamical and morphological behaviors can be also observed for other values of β > 0 and CaR [see points IV–VI for β = 4.5, VII–IX for β = 6.5 in Fig. 3(a) and in Fig. 4]. Therefore, even though the behaviors of the situations associated with points I, II, IV, V, VII, and VIII of the stability diagram are expected for A > 0, the untypical, noncircular shapes related to points III, VI, and IX that arise at significantly low values of CaR emerge due to the geometric properties of the warped-cone cell. We advance our analysis by inspecting Fig. 3(b) that considers the conventionally stable viscosity-matched situation (A = 0), for which the fluids present the same viscosity. By examining Fig. 3(b) one can verify that there is indeed a vast stable region when A = 0. Of course, the unstable region in Fig. 3(b) has shrunk a bit in comparison with the corresponding region depicted in Fig. 3(a) for A > 0. One can also check that when A = 0, 0 < β < 1 (conical cell case), and β = 1 (flat cell case), the system is stable regardless of the value of CaR. Nonetheless, as β assumes values larger than 1 (warped-cone cell situation), one finds that the system can become unstable for lower values of CaR. This is exemplified by points X (β = 4.5) and XI (β = 6.5), and in the corresponding weakly nonlinear shapes indicated in Fig. 3(b) for 1 R Rf , where Rf = 6.2. It should be emphasized that the pattern-forming structures X and XI illustrated in Fig. 3(b) are not induced by the viscosity difference between the fluids (after all, A = 0). In this sense, it has nothing to do with the ordinary (viscosity-driven) Saffman-Taylor instability. In fact, patterns X and XI are driven by both capillary and geometric effects. Equivalent types of unanticipated conclusions about the stability of the fluid-fluid interface can be reached by observing Fig. 3(c) for the negative viscosity contrast case A = −0.95. Once again, one can observe that for both conical and flat Hele-Shaw cells (0 < β 1), this strongly negative viscosity contrast situation is highly stable, considering that the displacing fluid is much more viscous than the displaced one. This happens independently of the value of CaR. In spite of that, as in the case of Fig. 3(b), in Fig. 3(c) one can still find a considerably large unstable region. This happens for β > 1, and intermediate or low values of the parameter CaR, where one can encounter noncircular patterned structures [see, for instance, points XII (β = 4.5) and XIII (β = 6.5) in Fig. 3(c)] for 1 R Rf , where Rf = 6.2. We conclude this section by offering a physical explanation of the instability mechanism that is driven by the peculiar geometric properties of the nonflat Hele-Shaw cell. By inspecting the linear growth rate expression given by Eq. (21) one observes that the parameter β modifies the effect of the capillary term (n2 − β 2 ): It is clear that by tuning β one can decrease the stabilizing role of such a term, even to the point of making it destabilizing [when (n2 − β 2 ) < 0]. This interesting behavior can be understood by examining the difference between the lengths (perimeters) of deformed and unperturbed (i.e., circular) fluid-fluid interfaces (for the same enclosed areas) that lie on the warped-cone surface given by the metric (20) π 2 (n − β 2 )|ζn |2 , (27) L = Ldeformed − Lcircular = βR n=0

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2π where L = ds = 0 (dr/dϕ)2 + β 2 r 2 dϕ. Therefore, one concludes that the destabilizing mechanism is due to the fact that the warped-cone metric (for which β > 1) makes it possible that a deformed interface has a smaller length than the circular one. In this capillary-driven scenario, interface destabilization would be favored for such a specific Hele-Shaw cell geometry. B. Influence of capillarity and geometry on the finger-tip-splitting mechanism

For the canonical Saffman-Taylor problem under radial fluid injection in flat Hele-Shaw cells [2–13], the most emblematic morphological feature of the resulting patterns is the emergence of fingers that tend to bifurcate at their tips, generating the famous finger-tip-splitting phenomenon. As a consequence, the evolving fingers tend to multiply and proliferate through repeated subdivisions, ultimately forming complex branched structures. An advantageous aspect of our second-order perturbative approach presented in Sec. II is the fact that, through the coupling of just a few Fourier modes, one is able to extract key analytical information about the morphology of the two-fluid interface (e.g., finger-tip-splitting formation) at the onset of nonlinearities [10]. Here, we take advantage of this fact and use Eq. (14), plus Eqs. (21)–(23), to examine how geometric effects influence the development of finger-tip splitting in warped-cone Hele-Shaw cells. It turns out that finger-tip-narrowing, finger-tip-broadening, and finger-tip-splitting phenomena can be described by considering the interplay of just two particular Fourier modes: the fundamental mode n and its harmonic 2n [10]. In other words, to mimic the essential physics associated to these mechanisms, one just needs to analyze the influence of n on the growth of mode 2n. By utilizing Eq. (14), and Eqs. (21)–(23), the equations of motion for the cosine and sine modes of the harmonic can be written as a 2n = λ(2n)a2n + 12 T (2n,n)an2 ,

(28)

b 2n = λ(2n)b2n ,

(29)

T (2n,n) = [F (2n,n) + λ(n)G(2n,n)],

(30)

where

is the finger-tip function. From Eq. (29) we can see that the growth of the sine mode b2n is uninfluenced by an and does not present second-order couplings, so we focus on the growth of the cosine mode. A noteworthy point about the function T (2n,n) is that it controls the finger-shape behavior. The sign of T (2n,n) dictates whether finger-tip splitting or finger-tip narrowing is favored by the dynamics. From Eq. (28) we see that if T (2n,n) > 0, the result is a driving term of order an2 forcing growth of a2n > 0, the sign that is required to cause outward-pointing fingers to become sharp, inducing finger-tip narrowing. In contrast, if T (2n,n) < 0, then growth of a2n < 0 would be favored, leading to outward-pointing finger-tip widening, and eventually to finger-tip splitting. It should be

FIG. 5. Snapshot of a linear (dashed curve) and a weakly nonlinear-WNL (solid curve) fluid-fluid interface for Rf = 6.2, β = 6.5, A = 0.95, Ca = 0.212, and n = 3. It is apparent that the finger-tip-splitting phenomenon is only captured if one considers the solution of the whole WNL, second-order differential equation (28).

noted that, in the flat Hele-Shaw cell case [2–13], the finger-tip function is indeed negative, so finger-tip splitting is naturally favored by the dynamics. Before we continue our discussion about the effect of β > 1 on finger-tip bifurcation, it is worthwhile to stress that the finger-tip-splitting phenomena detected in Fig. 3 and in Fig. 4 are intrinsically nonlinear and could not be either predicted or captured by a purely linear perturbative description of the flat or nonflat Hele-Shaw cell problems. To illustrate this important point, in Fig. 5 we plot the linear interface shape (dashed curve), and the corresponding weakly nonlinear (WNL) interface shape (solid curve) in a warped-cone cell, for β = 6.5, A = 0.95, Ca = 0.212, Rf = 6.2, and n = 3. Actually, the illustrative interfaces shown in Fig. 5 are plotted by using physical parameters and initial conditions already utilized to produce the threefold bifurcating patterns depicted in Figs. 3 and 4. The only fundamental difference between the dashed and solid patterned structures shown in Fig. 5 is that, while the dashed curved is obtained by solving the purely linear part of Eq. (28) [i.e., a 2n = λ(2n)a2n ], the solid contour results from the solution of the entire, second-order mode-coupling Eq. (28). By examining Fig. 5 it is evident that one does not observe any tendency towards finger-tip-splitting formation in the linear description of the problem (dashed curve). Linearly, the shape of the pattern is dominated by the growth of fundamental mode n, while the harmonic mode 2n just cannot keep up with its growth. On the other hand, the weakly nonlinear tip-splitting morphology revealed by the solid curve in Fig. 5 emerges due to the coupling and enhanced nonlinear growth of mode 2n. This observation reinforces the necessity and relevance of considering the nonlinear interaction among participating modes [through the second-order mode-coupling Eq. (14)] in order to get finger-branched morphologies via such a perturbative scheme. These findings justify the fact that

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FIG. 6. Variation of (a) the finger-tip function T (2n,n), and of (b) the amplitude ration a2n (R)/a2n (R0 ), as the capillary number Ca is varied. Three values of the viscosity contrast are considered: A = 0.95 (solid black curves), A = 0 (dashed black curves), and A = −0.95 (solid gray curves). Note that here β = 4.5 and R0 = 1.

in Figs. 3 and 4 we have considered only two participating cosine Fourier modes (n and 2n) and also explain why we have chosen to illustrate the physically significant, bifurcating weakly nonlinear shapes. Now we turn out attention to investigating more closely how the geometric structure of the warped-cone cell (measured by the parameter β) impacts the occurrence of finger-tip-splitting episodes. This is done with the help of Fig. 6, which plots how the finger function T (2n,n) varies with the capillary number Ca for three values of the viscosity contrast A [Fig. 6(a)]. On the other hand, Fig. 6(b) illustrates how the amplitude ratio for the harmonic mode a2n (R)/a2n (R0 ) varies with Ca for the same values of A taken in Fig. 6(a). In Fig. 6, we consider a parameter β = 4.5. We begin by analyzing the conventionally unstable situation linked to a large, positive viscosity contrast A = 0.95 (solid black curves). As discussed in Sec. I, in the flat cell case (β = 1), if A > 0, then the interface is only unstable, and finger-tip splitting detectable, for larger values of the capillary number. Therefore, it is of interest to see what one can get for β = 4.5, as in the case of Fig. 6. By inspecting Fig. 6(a), it is apparent that, regardless the value of Ca, the finger-tip function T (2n,n) only assumes negative values. This indicates that finger-tip splitting would be favored under such conditions. However, by examining Fig. 6(b), one also concludes that, despite the fact that T (2n,n) is always negative, the amplitude

ratio tends to zero for a considerably broad range of values of the capillary number. So, in the end, finger-tip splitting is only actually seen for larger values of a2n (R)/a2n (R0 ), something that happens for values of Ca located outside the range of capillary numbers for which a2n (R)/a2n (R0 ) → 0. The important point here is that, as opposed to what occurs for β = 1, for β = 4.5 one can also detect finger-tip-splitting formation for lower values of Ca. This is possible due to the action of the geometric effects present in the warped-cone Hele-Shaw cell. We close this section by investigating the traditionally stable Saffman-Taylor situations depicted in Fig. 6, where the viscosity contrast is zero (A = 0, dashed black curves) or large and negative (A = −0.95, solid gray curves). It is well known that under such conditions, and for any finite value of Ca, the two-fluid interface is stable, and therefore there is no materialization of finger-tip splitting if the Hele-Shaw cell is flat (β = 1). But the scenario can considerably differ if the flow takes plane in a warped-cone Hele-Shaw cell. From Fig. 6, it is also true that finger-tip splitting should be absent for β = 4.5 if Ca > 1. In this case, the finger-tip function is positive, but the amplitude ratio a2n (R)/a2n (R0 ) tends to zero, indicating that neither finger-tip narrowing nor finger-tip splitting take place. Conversely, by observing Fig. 6, one can also find that, even for the cases A = 0 and A = −0.95, the finger-tip function T (2n,n) can be negative, and the amplitude ratio a2n (R)/a2n (R0 ) sizable, if Ca < 1. Therefore, one deduces that a capillary- and geometry-induced finger-tip-splitting event can indeed emerge for Ca < 1, when β = 4.5. The weakly nonlinear findings presented in this section are quite reassuring and reinforce the linear conclusions reached from the analysis of linear stability diagram shown previously in Fig. 3. IV. CONCLUSIONS

The classical viscous fingering (or Saffman-Taylor) instability in flat Hele-Shaw cells arises when the viscosity contrast A between the fluids is positive (A > 0), whenever the capillary number Ca is sufficiently large. Recent theoretical and experimental studies have proposed several strategies highlighting the possibility of triggering a fluid fingering instability under conventionally stable circumstances in which A 0 and Ca is small. Normally, these destabilizing schemes involve the addition of surfactants at the cell plates, the incorporation of solid particles into the flow, or the occurrence of complex chemical reactions at the two-fluid interface. In this work, we proposed an alternative destabilizing method which allows one to induce fluid fingering formation through the combined action of capillary and geometric effects. These effects are associated with the peculiar properties of a particular family of nonflat Hele-Shaw cells. By employing an analytical, second-order mode-coupling approach, we have been able to derive an equation of motion describing the evolution of the fluid-fluid interfacial deformations on arbitrarily shaped, nonflat Hele-Shaw cells. In this framing, our results have demonstrated that for a specific nonflat Hele-Shaw cell configuration, i.e., for a warped-cone Hele-Shaw cell, the linear stability and the weakly nonlinear dynamics of the system are governed not only by A, and Ca but also by

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a parameter β. It turns out that this controlling parameter is linked to the distinguishing geometric properties of the warped-cone Hele-Shaw cell arrangement. In conclusion, we have shown that in such a nonflat confined flow environment, a fluid fingering instability can be activated and leads to the development of eye-catching, nonlinear finger-tip-splitting events, even under traditionally stable Saffman-Taylor flow conditions. A natural extension of this theoretical work would be the investigation of our current capillary- and geometry-mediated destabilization scheme under dynamical scenarios that go beyond the linear and weakly nonlinear regimes of pattern evolution. This is clearly an important issue that could be assessed by fully nonlinear numerical simulations of the

problem in the warped-cone and possibly other Hele-Shaw cell geometries. On the experimental side, after successfully facing the challenge of building nonflat, cylindrical [49], spherical [50], and Gaussian-shaped profile [51] Hele-Shaw cells, we hope that experimentalists will feel instigated to perform laboratory experiments in a warped-cone Hele-Shaw cell and test our theoretical predictions.

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ACKNOWLEDGMENTS

We thank CNPq (Brazilian Research Council) for financial support under Grant No. 304821/2015-2. We are grateful to Fernando Moraes and Eduardo Leandro for useful discussions.

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