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PHYSICAL REVIEW LETTERS
PRL 115, 174501 (2015)
Controlling Viscous Fingering Using Time-Dependent Strategies Zhong Zheng, Hyoungsoo Kim, and Howard A. Stone* Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, USA (Received 19 July 2014; revised manuscript received 14 August 2015; published 20 October 2015) Control and stabilization of viscous fingering of immiscible fluids impacts a wide variety of pressuredriven multiphase flows. We report theoretical and experimental results on a time-dependent control strategy by manipulating the gap thickness bðtÞ in a lifting Hele-Shaw cell in the power-law form bðtÞ ¼ b1 t1=7 . Experimental results show good quantitative agreement with the predictions of linear stability analysis. By choosing the value of a single time-independent control parameter, we can either totally suppress the viscous fingering instability or maintain a series of nonsplitting viscous fingers during the fluid displacement process. In addition to the gap thickness of a Hele-Shaw cell, time-dependent control strategies can, in principle, also be placed on the injection rate, viscosity of the displaced fluid, and interfacial tension between the two fluids. DOI: 10.1103/PhysRevLett.115.174501
PACS numbers: 47.15.gp, 47.20.Gv, 47.54.-r, 47.55.N-
When a less-viscous fluid displaces a more-viscous fluid, the viscous fingering phenomenon often occurs due to a hydrodynamic instability at the fluid-fluid interface [1–4]. This unstable displacement occurs widely in industrial and geological processes, such as enhanced oil recovery [5] and CO2 geological sequestration [6]. The instability also serves as an example for pattern formation processes in many contexts, such as diffusion-limited aggregation [7], crystal growth [8], and biological growth [9]. The classical viscous fingering experiments in a circular Hele-Shaw cell, see Fig. 1(a), show that the injected fluid (e.g., air) initially maintains a circular shape and then develops into radial fingers as the radius of the interface increases. Each finger continues to grow and eventually splits into more fingers at later times. Linear stability analysis explains how surface tension fails to suppress the growth of the small perturbations at the fluid-fluid interface as the radius of the interface increases [10,11], and numerical solutions of the governing equations allow the nonlinear evolution of the fingering shapes to be studied [12–19]. Recently, it has been shown that by modifying the geometry of the classic Hele-Shaw cell, viscous fingers can be stabilized. For example, if one of the plates is tilted to provide a permeability gradient in the flow direction, the growth of the perturbations can be suppressed as a consequence of capillary stresses becoming significant [20,21]. In addition, if one plate is replaced by an elastic membrane, viscous fingering can also be stabilized [22–24]. In this Letter, we report the stabilization that can occur when the gap thickness between the two plates is a function of time, as occurs in a lifting Hele-Shaw cell [25–28], see Figs. 1(b)–1(c). We note that a soap film problem with an instability that is analyzed as a lifting Hele-Shaw configuration was studied recently by Goldstein et al. [29]. In previous work, air fingers propagated from outside the cell to the center. In our work, we inject air at the center while, 0031-9007=15=115(17)=174501(5)
simultaneously, the gap thickness increases. In particular, we show using linear stability analysis that when the gap thickness of a Hele-Shaw cell obeys the power-law form bðtÞ ¼ b1 t1=7 , the growth rate is time independent. In this case, either the fingering instability is suppressed or nonsplitting fingers with a constant number of fingers are maintained. We use linear stability analysis [10,11] in this Letter; the viscosity of the invading fluid μi is negligible compared (a)
(b)
(c)
FIG. 1 (color online). (a) Classic tip-splitting viscous fingering in a radial Hele-Shaw cell with constant gap thickness. (b) Injection without viscous fingering in a lifting Hele-Shaw cell with gap thickness obeying bðtÞ ¼ b1 t1=7 . In (a) and (b), the shapes from several different times are overlaid. (c) Sketch of the nonsplitting viscous fingering in a lifting Hele-Shaw cell with ¯ bðtÞ ¼ b1 t1=7 . RðtÞ is the average location of the air-liquid ¯ interface Rðθ; tÞ, and Aðθ; tÞ ≡ Rðθ; tÞ − RðtÞ represents the amplitude of the perturbations.
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© 2015 American Physical Society
PRL 115, 174501 (2015)
with the viscosity of the displaced fluid μ. For a small perturbation of wave number n from a circular shape ¯ of radius RðtÞ, the growth rate of disturbances σðn; tÞ is given by ðn − 1ÞQ nðn þ 1Þπγb3 1 − : ð1Þ σðn; tÞ ¼ ¯ ¯ 2 6μRðtÞQ 2πbRðtÞ
(a)
¯ 6μQRðtÞ J≡ : πγb3
Air Stationary top plate
Silicone oil Movable bottom plate
b(t) [mm]
(b) 2.4
ð3Þ
¯ which holds for any RðtÞ. We suggest the control of the viscous fingering is through the value of J, which contains information about the injection rate, gap thickness, viscosity of the displaced fluid, and interfacial tension between the two fluids. Note that J ¼ 5.75 corresponds to nm ¼ 1.5, and we take the nearest integer of nm for a physical system. Hence, we predict that if J < 5.75, all of the perturbations will be suppressed, and, hence, the displacement is stable. The control parameter J, as defined in (2), is usually time dependent. However, for constant flow rate Q, if the gap thickness obeys bðtÞ ¼ b1 t1=7 , J becomes time independent: J ¼ 6μQ3=2 =ðπ 3=2 γb7=2 1 Þ. From (3), the wave number nm , and, hence, the most unstable perturbation mode, is also time independent. Thus, according to linear stability theory, the number of fingers nm selected is a constant in a radial viscous fingering experiment. We conducted laboratory experiments in a lifting HeleShaw cell, which has a stationary top plate and a movable bottom plate [Fig. 2(a)], with the gap always immersed in a fluid bath. The bottom plate is mounted to a motorized linear stage with an accuracy of 1 μm. We control the position of the bottom plate such that the gap thickness follows bðtÞ ¼ b1 t1=7 with b1 ≈ 0.823 mm=s1=7 . The measured gap thickness is shown in Fig. 2(b), together with the designed value. The gap thickness follows the desired time dependence except during an initial adjustment period (≈20 s), when it is smaller than the designed value. This adjustment period is short compared with the total experimental time (≈550 s). We use 1000-cSt silicone oil (Sigma-Aldrich, USA) as the displaced fluid. The fluids wet the plates. To observe the
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Experiment Theory : b(t) = b1t1/7
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ð2Þ
The connection between the value of J and nm is
Motorized linear stage Top plate holder
¯ The radius RðtÞ can be calculated from the equation of R ¯ 2 ¼ t QðsÞds. The wave conservation of mass, i.e., πbRðtÞ 0 number corresponding to maximum growth rate, denoted as nm , can be determined through ½∂σðn; tÞ=∂njn¼nm ¼ 0, and nm (rounded to the nearest integer) represents the number of viscous fingers selected in the physical system. We now define a dimensionless control parameter J, as suggested from (1),
J ¼ 3n2m − 1;
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PHYSICAL REVIEW LETTERS
FIG. 2 (color online). (a) The lifting Hele-Shaw cell, filled with silicone oil, consists of a stationary top plate and a movable bottom plate with a circular wall around the edge to hold the displaced fluid. (b) Time evolution of the gap thickness of the parallel plates in a typical experiment. The values of the gap thickness from measurements are plotted with the designed control profile, which follows bðtÞ ¼ b1 t1=7 , with b1 ≈ 0.823 mm=s1=7 .
oil phase in the experiments, we used an oil-soluble red dye (Natural Sourcing, LLC, Oxford, UK). The physical properties of the dyed silicone oil are measured at room temperature (T ≈ 298 K) dynamic viscosity μ ¼ 0.95 Pa s, density ρ ¼ 965 kg=m3, and surface tension γ ¼ 20.5 mN=m. Air is injected at the center of the top plate. Once the bottom plate moves, the oil immediately fills the gap. Air propagates outward from the center of the cell, displaces the oil, and creates an immiscible fluid-fluid interface. The time evolution of the fluid-fluid interface is recorded from above (Fig. 3). First, as a comparison experiment, we maintained a constant gap thickness (e.g., b ¼ 0.2 mm). When air was injected at a constant rate (e.g., Q ¼ 40 mL= min), the classic tip-splitting viscous fingering was observed [Fig. 3(a)]. Next, we employed the timedependent strategy where the gap thickness follows bðtÞ ¼ b1 t1=7 . Compared with the classic tip-splitting fingering, different fingering patterns were observed, as shown in Figs. 3(b)–3(d) for different values of J. In particular, four different fingering patterns are identified in sequential time periods of a typical experiment, as follows: (i) Splitting fingers: The interface initially grows and destabilizes to form finger shapes; each finger then extends and splits into more fingers, and the whole interface eventually develops a well-branched fingering shape. This initial period lasts for 5–10 s. (ii) Shrinking fingers: The well-branched fingering shape starts to shrink, and the
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10 cm
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t=0s
t=0s
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t = 44 s
t = 24 s
t = 20 s
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t = 220 s
t = 110 s
t = 13 s
t = 330 s
t = 330 s
t = 220 s
t = 16 s
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t = 330 s
Thus, J was always greater than the designed value, and, hence, the classic tip-splitting fingering appeared. We used Fast Fourier Transforms (MATLAB subroutine FFT) to compute the time evolution of the Fourier coefficients cðn; tÞ for the nth Fourier components of the ¯ amplitude of the perturbation Aðθ; tÞ ≡ Rðθ; tÞ − RðtÞ, Rπ i.e., cðn; tÞ ¼ ð1=2πÞ × −π Aðθ; tÞe−inθ dθ. To demonstrate the dynamics of the selection process, we show an experiment of complete stabilization in Figs. 4(a)–4(b) and an experiment with five nonsplitting fingers in Figs. 4(c)–4(d). A broad range of Fourier components exist following the tip-splitting and shrinking periods, including the Fourier component that is later selected in the nonsplitting fingering period. Different values of J were chosen in the experiments, and different finger numbers nm in the nonsplitting finger period were observed. We have documented nonsplitting viscous fingering with 5 to 10 fingers in our experiments, in addition to the case of stable displacement (Fig. 5). The theoretical prediction of nm versus J [Eq. (3)] is also plotted. Very good agreement is obtained between the experimental observations and the theoretical predictions for the number of nonsplitting fingers. We note that we did not observe the nonsplitting fingers for nm ¼ 2; 3; 4 in our experiments; this is probably because of the finite size of the Hele-Shaw cell. Thus, we have verified the control effect when the gap thickness follows bðtÞ ¼ b1 t1=7 in a lifting Hele-Shaw cell. By choosing the value of J, we are able to either
FIG. 3 (color online). Viscous fingering in a radial Hele-Shaw cell [30]. (a) Constant gap thickness: classic tip-splitting viscous fingering. (b)–(d) Time-dependent gap thickness obeying bðtÞ ¼ b1 t1=7 : four different fingering patterns were observed in sequential periods of a typical experiment, i.e., splitting fingers, shrinking fingers, reorganizing fingers, and nonsplitting fingers. (b) J ≈ 5.7, (c) J ≈ 87, and (d) J ≈ 134.
rough interface approaches a circular shape at the end of this period. These dynamics follow the splitting finger period and last for 20–40 s. (iii) Reorganizing fingers: The interface starts to grow again and the fingers reorganize. The finger number approaches a constant at the end of this period. (iv) Nonsplitting fingers: The fingers continue to extend with a constant finger number until the end of the experiments. This time period constitutes the majority of the experimental time and lasts for several hundred seconds. There are various factors that influence the tip-splitting and shrinking fingers in the early times of the experiments. We note from (1) that for constant Q and b ∝ t1=7 , σ ∝ t−1 ; this highlights that any unstable contributions are worst at early times. Most importantly, the gap thickness was always smaller than the designed value in the first 20 s or so, i.e., bðtÞ ¼ b1 t1=7 − δðtÞ where δðtÞ > 0 [Fig. 2(b)].
(a) c (n, t) [mm]
(b)
t=6 s t = 10 s t = 28 s
(c) c (n, t) [mm]
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PHYSICAL REVIEW LETTERS
PRL 115, 174501 (2015)
5 cm
5 cm
t = 10 s t = 16 s t = 40 s
n
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FIG. 4 (color online). Time evolution of the Fourier coefficients cðn; tÞ for the nth Fourier components of of the R π the amplitude perturbation Aðθ; tÞ: cðn; tÞ ¼ ð1=2πÞ −π Aðθ; tÞe−inθ dθ. To demonstrate the dynamics, in (a),(b) we chose an experiment of complete stabilization [Fig. 3(b)]; in (c),(d) we chose an experiment with five nonsplitting fingers [Fig. 3(c)]. We note that n ¼ 1 corresponds to the translational perturbations.
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FIG. 5 (color online). The number of the nonsplitting viscous fingers nm as a function of the control parameter J. The triangles with error bars represent the experimental observations in the late times, while the discrete red lines represent the predictions of the linear perturbation theory. Five typical experimental pictures of the nonsplitting viscous fingers are also shown.
stabilize the displacement process or maintain a series of nonsplitting fingers, as predicted by linear stability analysis. Here we have assumed that the changes with time are “slow” such that the assumptions for the Stokes equation and the linear stability analysis still hold. In other words, b2 =ν ≈ 10−3 s for b ≈ 1 mm; this is much shorter than the experimental scales. In principle, time-dependent control strategies can also be placed on the injection rate [19,31–34], viscosity of the displaced fluid, and interfacial tension between the two fluids. Specifically, power-law strategies can be used to make J time independent, i.e., the injection rate QðtÞ ¼ Q1 tαQ , the gap thickness bðtÞ ¼ b1 tαb , the viscosity of the displaced fluid μðtÞ ¼ μ1 tαμ , and the interfacial tension γðtÞ ¼ γ 1 tαγ . Substituting the above expressions into (1), we obtain an expression for the growth rate σðn; tÞ, σðn; tÞ ¼
ðαQ þ 1Þðn − 1Þ½1 − nðn þ 1Þ=J ; 2t
J¼
6μ1 Q3=2 1
Z π 3=2 ðαQ þ 1Þ1=2 γ 1 b7=2 1 t
;
3 7 1 Z ≡ − αQ þ αb − αμ þ αγ − : 2 2 2
ð4Þ ð5Þ ð6Þ
Therefore, when Z ¼ 0, J becomes time independent. From (3), the selected number of fingers nm is predicted to be a constant in a radial viscous fingering experiment. In particular, if the gap thickness between the two parallel plates b, the viscosity of the displaced fluid μ, and the interfacial tension between the two fluids γ are all
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held constant, we can manipulate the injection rate in the form of QðtÞ ¼ Q1 t−1=3 such that J ¼ 541=2 μQ3=2 1 = ðπ 3=2 γb7=2 Þ is time independent; this is related to previous studies [19,32–34]. We note that the control strategy of [19] appears fundamentally different from the current study: Li and colleagues manipulated the form of the initial perturbations in the numerical simulation, and observed that the selected finger number was not present in the initial data, and this was “completely outside the prediction of linear theory.” In the current Letter, however, the final finger number nm, predicted as the most unstable perturbation mode from linear theory, does exist at earlier times. When a single control parameter J is determined, the final finger number nm is predictable. We thank the Princeton Carbon Mitigation Initiative for support of this research. The authors thank I. Jacobi, J. Nunes, and O. Shardt for helpful discussions. H. K. and Z. Z. contributed equally to this Letter.
*
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