# Fluid Morphologies Governed by the Competition of Viscous Dissipation and Phase Separation in a Radial Hele-Shaw Flow

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## Abstract

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## 1. Introduction

_{2}sequestration [3], frontal polymerization [4], and secondary and tertiary oil recovery [5]. It is generally accepted that when the displacing fluid is more viscous, no interfacial instability occurs, while a finger-like pattern forms at the interface in the reverse situation (Figure 1). The latter case, where a less viscous fluid displaces a more viscous one in porous media, is called Saffman–Taylor instability [6] or viscous fingering (VF) [7]. Saffman–Taylor instability, based on Darcy’s law [8,9], can be explained by Figure 2 and Equation (1):

_{2}SO

_{4}, and water, allowing quantitative and qualitative control over the thermodynamic stability of the fluid system as well as the hydrodynamic interfacial stability. We changed the progression of the phase separation by changing the concentrations of PEG and Na

_{2}SO

_{4}and investigated the effect on the fluid patterns to make a phase diagram of patterns. In addition, we measured the dynamic interfacial tension between the displacing and the displaced fluids to clarify the mechanism of pattern formation and to evaluate the patterns quantitatively.

## 2. Experiment

#### 2.1. Solutions

_{2}SO

_{4}solution. Figure 3 shows the phase diagram of the ATPS [36]. The partially miscible zone is the green Region II, where the system can be separated into two phases because of thermodynamic instability. Region II is called the spinodal region, where concentration fluctuations spontaneously grow due to thermodynamic instability as the system reduces the total free energy, but the interfacial free energy increases because the interfaces of domains gradually grow. After the domain growth, the domains coalesce with each other, and phase separation is eventually completed. The progression of phase separation becomes faster with the increase in the concentration of Na

_{2}SO

_{4}, according to the previous paper [10,31], as this leads the system far from equilibrium. Region I is a one-phase region where the system is fully miscible because of the thermodynamically stable region. Here, we consider that the progression of phase separation represents the extent of phase separation. The details are described in [10,31]. The solutions used here are shown in Table 1; seven more viscous solutions and 11 less viscous solutions were used for the fully and partially miscible systems and a combination of Phases L and H was used for the immiscible system. Since Phases L and H are the solution after the phase separation, they are thermodynamically stable. The composition of Phases L and H is 36.5 wt % PEG solution with 3.2 wt % Na

_{2}SO

_{4}and 1.4 wt % PEG solution with 16.0 wt % Na

_{2}SO

_{4}[36]. To visualize the displacement process, the more viscous solutions were dyed blue by 0.1 wt % indigo carmine, which cannot dissolve into Na

_{2}SO

_{4}solution because of the salting-out effect.

_{2}SO

_{4}solution, respectively, the concentration at the interface was considered to be 20 wt % PEG–5 wt % Na

_{2}SO

_{4}–75 wt % water. This assumption is very simple, but we think this is appropriate. In previous studies [10,31], the same assumption was taken into account, and it can be explained by the assumption that the result of the fluid displacement, calculation of free energy, and the result of the interfacial tension measurement all have good agreement.

#### 2.2. Displacement Experiment

_{2}SO

_{4}solution and then injected the more viscous PEG solution into the cell using a syringe pump. The PEG solutions were dyed blue to visualize the experimental process. The displacement experiments were recorded by a video camera from the bottom. The injection flow rate of the more viscous liquid was fixed as 3.35 mL/h. All experiments were performed at room temperature (25 $\pm $ 1 °C) and atmospheric pressure (approx. 1 atm).

#### 2.3. Physical Property Measurements

^{3}) is the density difference between the two fluids. Here, $\omega $ was fixed at 6000 rpm (628 rad/s).

## 3. Results and Discussion

#### 3.1. Physical Properties

_{2}SO

_{4}for easy observation. The viscosities of PEG solutions depend greatly on the concentration of PEG, while the viscosities of Na

_{2}SO

_{4}are very low compared to those of PEG solutions. Figure 5c shows the densities, which have a linear relationship with the concentration of the solution. In the horizontal Hele-Shaw cells, however, the density effect (gravitational effect) can be ignored when the gap between the cells is small enough.

#### 3.2. Fluid Displacements

_{2}SO

_{4}, System II is at 20 wt % PEG and 20 wt % Na

_{2}SO

_{4}, and System III is at 40 wt % PEG and 20 wt % Na

_{2}SO

_{4}. System I lies in Region I, whereas Systems II and III lie in Region II in Figure 3. Figure 6a shows the time evolution of the pattern formation of System I. A circular pattern expands as time proceeds because System I is absent from the thermodynamic instability and hydrodynamic interfacial instability. Similarly, the immiscible system, which is thermodynamically and hydrodynamically stable, approaches a perfectly circular pattern because of the high interfacial tension. The displacement patterns in the immiscible system have already been reported in [10].

_{2}SO

_{4}molecules diffuse from the interface into the displacing fluid to reduce the PEG concentration inside the diffusion region. At t = 50 s, many indigo blue domains newly form in the outermost part of the displacing fluid, and the light blue region is left inside the outer indigo domains to produce a light blue circular ring. The formation of the outer indigo blue domains and the light blue circular ring indicates the spontaneous formation of PEG-rich and Na

_{2}SO

_{4}-rich phases, respectively. The interface becomes distorted, and the annular-like pattern expands with distortion as time proceeds. System I has a similar viscous ratio to System II because the viscous ratio depends on the PEG concentration, as shown in Figure 5b. Thus, thermodynamic instability, i.e., phase separation, is estimated to contribute to the spontaneous formation of the outer indigo domains and to the distorted interfacial pattern.

_{2}SO

_{4}concentration, but they are different in terms of their viscous ratio. A high viscous ratio restricts the diffusion region to the periphery of the displacing fluid. Thus, phase separation occurs at the outmost interface of the displacing fluid, inducing the interfacial distortion. This fingering-like pattern is already reported by Suzuki et al. [10]. The distorted interfacial pattern is created by the Korteweg force exerted in a direction toward the higher region of PEG concentration [10]. Thus, the pattern of System II is a phase separation-dominated displacement in comparison with System III, where the process of viscous dissipation weakens the phase separation effect.

_{2}SO

_{4}concentration decreases, i.e., the composition goes to the upper left in Figure 7. In contrast, the finger-like pattern forms as PEG concentration decreases and Na

_{2}SO

_{4}concentration increases, i.e., the composition moves to the lower right in Figure 7. For PEG concentrations higher than 30 wt %, the light blue circular ring disappears, and the outer indigo blue interface becomes distorted. As the Na

_{2}SO

_{4}concentration increases, the interface becomes sharp because the interfacial tension increases with the increase in Na

_{2}SO

_{4}concentration [31].

_{2}SO

_{4}increases [10,31]. As shown in Figure 5b, the viscosity decreases as the concentration of PEG solution decreases, which means that the solutions with less viscosity easily move, and phase separation is thought to easily occur.

_{2}SO

_{4}concentrations are related to the pattern formation, which means that the morphologies are affected by the complexity of the hydrodynamic effect, such as the viscosity and thermodynamic effect such as phase separation. On the other hand, the patterns are circular (●) when the Na

_{2}SO

_{4}concentration is less than 5 wt % in Figure 8, regardless of the PEG and Na

_{2}SO

_{4}concentrations, because the displacement patterns are determined only by the hydrodynamic effect of viscosity.

_{2}SO

_{4}solution and PEG solutions with several concentrations. IFT, at all concentration ranges, increases with time. The steady value of the IFT increases with the decrease in PEG concentration. The IFT of the fully miscible systems decreases with time because the width of the interface becomes wider due to molecular diffusion [10]. In contrast, the IFT of partially miscible systems increases with time because the interface becomes sharp due to phase separation. Figure 9b depicts the rate constant, k, defined as $\gamma =\left({\gamma}_{0}-{\gamma}_{\infty}\right){e}^{-kt}+{\gamma}_{\infty}$, where ${\gamma}_{0}$ and ${\gamma}_{\infty}$ are initial and steady values of IFT in Figure 9a, respectively. The relaxation process of IFT corresponds to that of phase separation. Therefore, the rate constant, $k$, is thought to represent the progress of phase separation. The rate is higher with the decrease in PEG concentration because the formation rate of the interface decreases with the increase in the viscosity of the displacing fluid. Therefore, the progress of phase separation is important for forming patterns, for example, the annular-like pattern (Figure 6b) for high rates and the finger-like patterns (Figure 6c) for low rates. Moreover, the progression is affected by the mobility-like viscosity contrast, and the viscosity is affected by the concentrations of the components (here, PEG). Thus, we compared the patterns using a dimensionless number considering those effects, such as modified capillary number, ${\mathrm{Ca}}^{\prime}$, including the viscosity, interfacial tension, and flow rate.

^{3}/s) is the flow rate, $b$ (m) is the gap between the cells, and $\gamma $ (N/m) is an interfacial tension between displacing and displaced liquids [37,38,39,40,41]. It is noted that ${\mathrm{Ca}}^{\prime}$ at fully miscible systems cannot be defined because IFT in the fully miscible systems is almost zero. A dimensionless number, ${\mathrm{B}}_{\mathrm{f}}$, represents the relative effect of the body force driven by thermodynamic instability versus the pressure gradient related to Darcy’s law, which was introduced in [10]:

_{2}SO

_{4}-rich regions generated by phase separation and created by deformation, divided by the area of a circle with maximum radius of 42 mm as shown in Figure 10a. Here, we measured the patterns of the immiscible system, the conditions of which are described in [10], to better understand the mechanism of the pattern formation of all cases. The ${\rho}_{\mathrm{di}}$ of the immiscible system is almost zero because the patterns are perfectly circular, as mentioned in the Introduction.

## 4. Conclusions

_{2}SO

_{4}solution, show constant circular patterns. For the partially miscible systems, phase separation occurs at the interface between displacing and displaced solutions, creating a separated region with domain growth for the annular patterns and creating a deformed interface for the finger pattern. Moreover, the deformation index of the patterns can be scaled with ${\mathrm{B}}_{\mathrm{f}}$, which involves viscous dissipation, molecular diffusion, and phase separation. Therefore, the patterns in the partially miscible systems are proved to be formed by the competition of viscous dissipation and phase separation. ${\mathrm{B}}_{\mathrm{f}}$ is a crucial factor to describe the complex morphologies induced by the fluid displacement in partially miscible systems. The investigated morphologies and ${\mathrm{B}}_{\mathrm{f}}$ will directly contribute to predicting and/or controlling CO

_{2}-enhanced oil recovery, where more viscous water displaces less viscous CO

_{2}under the ground, the conditions of which are partially miscible.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Fluid displacements for (

**a**) a hydrodynamically stable condition of the interface, (

**b**) a hydrodynamically unstable condition of the interface. The blue solution displaces the surrounding white solution. A more viscous solution is displacing a less viscous one in (

**a**) while a less viscous solution is displacing a more viscous one in (

**b**). The patterns like fingering in (

**b**) are called “Saffman–Taylor instability” or “viscous fingering” (VF).

**Figure 2.**The explanation for Saffman–Taylor instability. The solution flows from left to right in the figures. (

**a**) The setup situation and the results under (

**b**) a hydrodynamically stable condition (${\mu}_{1}>{\mu}_{2}$) and (

**c**) a hydrodynamically unstable condition (${\mu}_{1}<{\mu}_{2}$), i.e., Saffman–Taylor instability. $p$ represents pressure at the position $z$.

**Figure 5.**(

**a**) Viscosity of PEG solutions against shear rate, (

**b**) viscosity of PEG and Na

_{2}SO

_{4}solutions against concentration, (inset) viscosity of Na

_{2}SO

_{4}solutions against concentrations, (

**c**) density of PEG and Na

_{2}SO

_{4}solutions against concentrations.

**Figure 6.**Time evolution of the pattern formation at (

**a**) System I, where 20 wt % PEG solution displaces 0 wt % Na

_{2}SO

_{4}solution; (

**b**) System II, where 20 wt % PEG solution displaces 20 wt % Na

_{2}SO

_{4}solution; and (

**c**) System III, where 35 wt % PEG solution displaces 20 wt % Na

_{2}SO

_{4}solution.

**Figure 7.**The results of the hydrodynamically stable displacement. The time shown in the right bottom corner is the time when the longest radius reached 42 mm.

**Figure 9.**(

**a**) Time evolution of the interfacial tension between PEG solutions and 20 wt % Na

_{2}SO

_{4}solution; (

**b**) the phase separation rate constant, k, of (

**a**).

**Figure 10.**(

**a**) The definition of ${\rho}_{\mathrm{di}}$. (

**b**) The relationship between the modified capillary number, ${\mathrm{Ca}}^{\prime}$, and the ${\rho}_{\mathrm{di}}$ for the patterns. (

**c**) The pattern evaluation with ${\mathrm{B}}_{\mathrm{f}}$. The fitted curves are for a better visualization of the effects of ${\mathrm{Ca}}^{\prime}$ and ${\mathrm{B}}_{\mathrm{f}}$.

Displacing More Viscous Liquid | Displaced Less Viscous Liquid |
---|---|

0 wt % Na_{2}SO_{4} solution | |

2 wt % Na_{2}SO_{4} solution | |

10 wt % PEG solution | 4 wt % Na_{2}SO_{4} solution |

15 wt % PEG solution | 6 wt % Na_{2}SO_{4} solution |

20 wt % PEG solution | 8 wt % Na_{2}SO_{4} solution |

25 wt % PEG solution | 10 wt % Na_{2}SO_{4} solution |

30 wt % PEG solution | 12 wt % Na_{2}SO_{4} solution |

35 wt % PEG solution | 14 wt % Na_{2}SO_{4} solution |

40 wt % PEG solution | 16 wt % Na_{2}SO_{4} solution |

18 wt % Na_{2}SO_{4} solution | |

20 wt % Na_{2}SO_{4} solution | |

Phase L | Phase H |

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**MDPI and ACS Style**

Suzuki, R.X.; Takeda, R.; Nagatsu, Y.; Mishra, M.; Ban, T. Fluid Morphologies Governed by the Competition of Viscous Dissipation and Phase Separation in a Radial Hele-Shaw Flow. *Coatings* **2020**, *10*, 960.
https://doi.org/10.3390/coatings10100960

**AMA Style**

Suzuki RX, Takeda R, Nagatsu Y, Mishra M, Ban T. Fluid Morphologies Governed by the Competition of Viscous Dissipation and Phase Separation in a Radial Hele-Shaw Flow. *Coatings*. 2020; 10(10):960.
https://doi.org/10.3390/coatings10100960

**Chicago/Turabian Style**

Suzuki, Ryuta X., Risa Takeda, Yuichiro Nagatsu, Manoranjan Mishra, and Takahiko Ban. 2020. "Fluid Morphologies Governed by the Competition of Viscous Dissipation and Phase Separation in a Radial Hele-Shaw Flow" *Coatings* 10, no. 10: 960.
https://doi.org/10.3390/coatings10100960