# Analyzing Impacts of Interfacial Instabilities on the Sweeping Power of Newtonian Fluids to Immiscibly Displace Power-Law Materials

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Numerical Approach

#### 2.2. Enforcing Boundary Conditions

#### 2.3. Modeling of Non-Newtonian Fluids

- Shear-thinning or pseudoplastic behavior;
- Viscoplastic behavior;
- Shear-thickening or dilatant behavior.

## 3. Numerical Modeling

#### 3.1. Code Validation

#### 3.2. Lattice Layout Study

#### 3.3. Numerical Procedure

- (1)
- Initialization of distribution functions and relaxation time;
- (2)
- Calculation of the macroscopic velocity and density;
- (3)
- Calculation of equilibrium distribution functions;
- (4)
- Calculation of the velocity gradient;
- (5)
- Calculation of viscosity using the structural equation of non-Newtonian fluids;
- (6)
- Calculation of the new relaxation time for non-Newtonian fluids;
- (7)
- Calculation of new distribution functions in the collision step;
- (8)
- Streaming distribution functions toward their adjacent points (propagation step);
- (9)
- Calculation of a new relaxation time, velocity field, and distribution functions for the next time step;
- (10)
- Repeating this process from step 2 to 9 for the next time step.

## 4. Results

- The location of max and min shear rates;
- The location of max viscosity;
- Max and min viscosity ratio at different cross-sectional areas.

#### 4.1. Impacts of Power-Law Index

#### 4.1.1. Displacement Shape Analyses

#### 4.1.2. Evaluation of Displacement Efficiency

#### 4.2. Impacts of Compatibility Factor

#### 4.2.1. Displacement Shape Analyses

#### 4.2.2. Evaluation of Displacement Efficiency

_{x}in Figure 29. In other words, because of enhanced interfacial instabilities and more severe mixing of the fluids, the mass fraction of the secondary fluid changes more strongly in every cross-section of the channel. It is worth noting that C

_{x}should fluctuate minimally (in the sections where the fingering structure exists) for a stable penetration with the least perturbations at the fluids’ interface.

## 5. Discussion

**Displacement total efficiency:**The viscosity ratio can regulate the displacement efficiency in two main ways; firstly, by altering the viscous forces against the invading fluid movement, and secondly, by adjustment of interfacial instabilities. Because of the higher viscosity of pseudoplastic fluids (compared to dilatant materials), it is more difficult for the injected Newtonian fluids to displace them. On the contrary, dilatant materials’ lower viscosity can magnify interfacial instabilities and violate the finger structure’s integrity, leading to higher efficiency. Therefore, the total displacement efficiency is not an appropriate criterion to determine the successful removal of the static wall layer. Consequently, we decided to investigate the displacement efficiency in each longitudinal and cross-sectional area of the channel over time.

**Static wall layers:**Removing static wall layers is an important issue when dealing with displacement flows and fingering problems. For dilatant materials, the viscous force mitigation against the injected fluid promotes the invading fluid’s tendency to go upward. Therefore, in comparison with other cases, it is expected there will be a thicker wall layer at the bottom of the channel and a thinner one adjacent to the top wall. However, as time elapses, the extension of the finger structure (i.e., interaction surface) magnifies interfacial instabilities. As a result, the static wall layer thickness does not behave according to the expectation for specific periods. In contrast, pseudoplastic fluids possess greater viscosities, and the maximum viscosity takes place around the centerline of the channel. Therefore, the invading fluid tends to move toward the channel walls leading to compression of the static wall layers. This compaction effect causes the static wall layers to initially possess smaller values in displacing pseudoplastic materials.

**Finger structure shape:**Buoyancy, viscous, and inertial forces control the invasion angle of the finger structure. While the buoyancy force pushes the injected fluid towards the channel walls, the viscous force moderates the finger structure’s upward and downward movement. Accordingly, small fluctuations of the fingertip in displacing pseudoplastic fluids represent the stabilizing effect of the forces. However, due to the low viscosity of dilatant fluids and viscosity ratio, the finger structure looks asymmetric along the longitudinal axis.

**Displacing pattern:**By tracing the tails of the finger structure, the sweeping power adjacent to the channel walls can be investigated. In pseudoplastic fluids, the higher viscosity of displaced fluid prevents the injected fluid from moving upward. On the other hand, the maximum viscosity of the displaced fluid occurs at the channel centerline, directing the invading fluid toward the channel walls. The two factors counterbalance the effects of each other. As a result, the location of the finger tail at the top of the channel does not change. The higher viscosity of pseudoplastic leads to compaction of the finger structure. Accordingly, the injected fluid tends to completely fill the cross-sectional areas at the beginning of the channel and move ahead. However, the injected fluid tends to reach the end of the channel rather than filling the cross-sectional areas when a Newtonian fluid is injected to displace a dilatant fluid.

**Compatibility factor vs. power-law index:**There is a linear relationship between viscosity and the compatibility factor. As expected, increasing the compatibility factor of the displaced fluid results in a reduction in displacement efficiency. However, compared to the nonlinear impact of the power-law index, compatibility factor alteration has a negligible effect on the location of the fingertip and top tail.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**A schematic illustration of the geometrical configuration (not to scale) and initial flow structure.

**Figure 3.**Viscosity changes versus shear rates in power-law fluids for different n ($\lambda $ = 0.4).

**Figure 4.**A comparison between the results of the present study and Redapangu et al. [19].

**Figure 6.**The effects of changes in the lattice layout on the mean thickness of the static layer left on the lower wall of the channel at different times.

**Figure 12.**A schematic demonstration of the displacement shape. ${X}_{t}$ is the location of the invading fluid front at the top of the channel, and ${X}_{b}$ is a similar location at the bottom of the channel. ${X}_{L}$ is called the attacking front, which is the front of the finger-like structure in the longitudinal direction. ${f}_{t}$ and ${f}_{b}$ represent the length of the finger structure (${f}_{t}={X}_{L}-{X}_{t},{f}_{b}={X}_{L}-{X}_{b}$). ${H}_{L}$ is the height of the attacking front.

**Figure 14.**The invading fluid front at the top of the channel, ${X}_{t}$, over time for different n.

**Figure 15.**The invading fluid front at the bottom of the channel, ${X}_{b}$, over time for different n.

**Figure 16.**The mean thickness of the static layer at the top of the channel, ${H}_{\mathrm{t}}$, over time for different n.

**Figure 17.**The mean thickness of the static layer at the bottom of the channel, ${H}_{\mathrm{b}}$, over time for different n.

**Figure 25.**The location of the advancing front at the bottom of the channel over time for different m.

**Figure 27.**The mean thickness of the static layer over time at the top of the channel for different m.

**Figure 28.**The mean thickness of the static layer over time at the bottom of the channel for different m.

m | Re | Ri | At | Ca | Inclination Angle | Pr | T |
---|---|---|---|---|---|---|---|

20 | 100 | 1 | 0.2 | $\infty $ | 45 | 1 | 50 |

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**MDPI and ACS Style**

Esmaeilpour, M.; Gholami Korzani, M.
Analyzing Impacts of Interfacial Instabilities on the Sweeping Power of Newtonian Fluids to Immiscibly Displace Power-Law Materials. *Processes* **2021**, *9*, 742.
https://doi.org/10.3390/pr9050742

**AMA Style**

Esmaeilpour M, Gholami Korzani M.
Analyzing Impacts of Interfacial Instabilities on the Sweeping Power of Newtonian Fluids to Immiscibly Displace Power-Law Materials. *Processes*. 2021; 9(5):742.
https://doi.org/10.3390/pr9050742

**Chicago/Turabian Style**

Esmaeilpour, Morteza, and Maziar Gholami Korzani.
2021. "Analyzing Impacts of Interfacial Instabilities on the Sweeping Power of Newtonian Fluids to Immiscibly Displace Power-Law Materials" *Processes* 9, no. 5: 742.
https://doi.org/10.3390/pr9050742