# Identification and Mapping of Three Distinct Breakup Morphologies in the Turbulent Inertial Regime of Emulsification—Effect of Weber Number and Viscosity Ratio

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{max}, can be accurately predicted by the viscosity-corrected Kolmogorov –Hinze theory [8,11]. This theoretical framework is based on a balance where disruptive (turbulent) stress is compared to the total resistance to deformation resulting from Laplace pressure and viscous resistance. Breakup events occur in one of two regimes: in the turbulent inertial regime, if the drop diameter is larger than the Kolmogorov length scale, D > η,

_{C}denotes continuous phase viscosity, ρ

_{C}denotes continuous phase viscosity and ε denotes dissipation rate of turbulent kinetic energy) and in the turbulent viscous regime if the drop is smaller, D < η. For the turbulent inertial regime, the viscosity-corrected Kolmogorov–Hinze theory predicts [8,11]:

_{1}and c

_{2}are empirical constants (c

_{1}= 0.86 and c

_{2}= 0.37, [8]), γ is interfacial tension and μ

_{D}is the disperse phase viscosity. This semi-empirical relationship can be transformed into a dimensionless form using a Weber number,

_{D}/μ

_{C}, and the ratio between drop diameter and Kolmogorov length scale, D/η, resulting in [12]:

_{2}in Equation (2)). Thus, in terms of how it influences the largest surviving drop diameter, increasing the interfacial tension or increasing the disperse phase viscosity (ratio) are two interchangeable effects, according to the theoretical framework.

_{D}/μ

_{C}= 10–24) translate into small variations in the morphology at the initial breakup (e.g., extending the length of the thin neck before breakup and delaying the deformation sequence). These results also comply rather well with emulsification experiments [28].

_{D}/μ

_{C}) in these previous studies is small in comparison to the industrially relevant conditions for emulsification. Little is still known on at what We the transition between different breakup morphologies occur, and it is still unknown if larger viscosity differences can lead to not only a slight modification of breakup morphology [28] but also transitions between breakup morphologies.

## 2. Materials and Methods

#### 2.1. Numerical Drop Breakup Experiments

_{0})

^{3}, where D

_{0}is the size of the initial injected drop. This corresponds to a volume fraction of 1.7%, which can be compared to the 1–20% typically used in emulsification using high-pressure homogenizers in industrially relevant settings. Turbulence is continuously injected into the domain using ABC-forcing [41,42], ensuring a Taylor-scale Reynolds number of 33.

_{η}[12]. A drop that does not reach a critically deformed state [43] during this time is considered to survive passage.

_{η}.

#### 2.2. Investigated Cases

_{0}, and the drop diameter to Kolmogorov length scale ratio, D

_{0}/η, were kept constant throughout all simulations in order to ensure that the spatial resolution (in terms of computational cells per drop diameter), as well as the breakup regime as described by the viscosity-corrected Kolmogorov–Hinze framework did not vary between condition (i.e., remained in the turbulent inertial regime, D

_{0}/η = 22). Furthermore, note that the viscosity-corrected Kolmogorov–Hinze theory suggests that the outcome of an emulsification experiment is determined by only three dimensionless numbers: the Weber number, the viscosity ratio and the drop diameter to Kolmogorov length scale (Equation (4)). Keeping the latter constant suggests that all remaining variation could be described in terms of the first two.

#### 2.3. Dissipation Rate of Turbulent Kinetic Energy

_{i}denotes velocity in dimension x

_{i}), based on the continuous phase DNS data and averaged across the computational domain.

#### 2.4. Quantifying Interfacial Area

## 3. Results and Discussion

_{D}/μ

_{C}= 1), Section 3.2 the intermediary-viscosity case (μ

_{D}/μ

_{C}= 22) and Section 3.3 the high-viscosity case (μ

_{D}/μ

_{C}= 40). Section 3.4 discusses the phenomenon of non-monotonic transitions between breakup morphologies and its relation to turbulence modulation. Section 3.5 constructs an initial breakup morphology map based on We and the viscosity ratio, and Section 3.6 discusses implications for emulsification processing.

#### 3.1. Effect of We at a Low Viscosity Ratio (μ_{D}/μ_{C} = 1.0)

_{D}/μ

_{C}= 1) deforming in the turbulent flow (iso-surface at α = 0.5). Results comply with previous investigations [12,22,26]: The spherical shape is lost almost instantly, and the drop deforms in several directions. As time progresses, a sheet structure develops. When sufficiently thin, holes appear and deform further into narrow threads from which the first fragment detaches. The time of initial breakup—defined from when the first fragment detached [37]—is 7.2 τ

_{η}. However, note that the initial breakup time is difficult to characterize under this morphology since the limiting spatial resolution makes it difficult to fully distinguish detachment from rupture without fragmentation (i.e., hole formation) (see Figure 1).

_{D}/μ

_{C}= 1). As the external stress becomes smaller in relation to stabilization (i.e., as We decreases from 100 to 11), the morphology gradually shifts from a case where the drop is breaking as a sheet that almost appears to be ‘dissolving’ due to its poor ability resist external stress (We = 100), to a case with a sheet breaking into a coherent network of threads (We = 11). However, throughout this range (We = 11–100), a breakup occurs via a highly deformed sheet. This will be referred to as a ‘sheet breakup morphology’ in the present study. Sheet breakup, with its high multi-directional deformation and with rupture occurring almost simultaneously in several locations, is similar to what Komrakova [22] refers to as ‘burst breakup’ and has elements of what is referred to as ‘bag breakup’ under conditions of liquid drops breaking in gaseous jets [49].

_{D}/μ

_{C}= 1). Within the sheet breakup regime (We = 11–100), there is a trend of increased breakup time with decreasing Weber number. The trend is comparable but less steep than the breakup time prediction suggested by Rivière et al. [24]; see line in Figure 3.

_{B}> 90 τ

_{η}). These drops go through several sequences of deformation–relaxation but never reach a critically deformed state. Thus, the critical Weber number for the low viscosity ratio (at the investigated flow realization) is in the range of 1.4–1.6.

#### 3.2. Effect of We at an Intermediate Viscosity Ratio (μ_{D}/μ_{C} = 22)

_{D}/μ

_{C}= 22) as a function of Weber number. In a previous investigation, we identified sheet breakup at a high Weber number and bulb breakup at lower values for this viscosity ratio [12]. Figure 5 suggests that the sheet regime extends down to We = 30, and the bulb breakup regime starts from We = 11. The corresponding initial breakup times can be seen in Figure 6.

_{η}= 10–13, Figure 7). When the thread-neck has become sufficiently thin, the structure is critically deformed and will eventually release a small fragment from the top of the thread (Figure 7, t/τ

_{η}= 15). This breakup morphology is referred to as ‘thread breakup’ in the present study.

_{η}. This suggests that the critical Weber number for the intermediary viscosity ratio is in the range 2.6–2.85, i.e., somewhat higher than for the low viscosity case (which is expected due to the extra resistance to deformation offered by the viscous resistance, cf. Equation (4)). However, as seen in Figure 5, if We is decreased further (from We = 2.55 to 2.4), bulb breakup re-appears in a narrow range (We = 2.2–2.4). Lowering the Weber number below We = 2.2, again, results in no breakup taking place during the investigated time span. This non-monotonous effect is, again, surprising (suggesting a more stabilized drop breaking when a less stabilized do not) and relates to the non-monotonous trend in breakup time observed for the low-viscosity case (red box in Figure 3). This phenomenon is discussed further in Section 3.4.

#### 3.3. Effect of We at a High Viscosity Ratio (μ_{D}/μ_{C} = 40)

_{D}/μ

_{C}= 40), as a function of Weber number. The corresponding breakup time can be seen in Figure 9. As seen in Figure 8, the drop is always stabilized enough to avoid it from thinning into a sheet (in the Weber number range investigated). The cases with a high Weber number (We = 100 and We = 60), result in thread breakup. At lower Weber numbers (We from 30 to 2.85), the combination of Laplace pressure and viscous stabilization hinders the thread from being pulled out from the drop, and deformation proceeds onwards to bulb breakup. A drop with a Weber number of two or below passes through a sequence of deformation–relaxation cycles but does not reach critical deformation under the investigated time span (i.e., t

_{B}> 90 τ

_{η}). Thus, the critical Weber number is in the range 2.0–2.85 at this viscosity ratio.

#### 3.4. Non-Monotonous Breakup and Turbulence Modulation

_{D}/μ

_{C}= 1.0), We = 2.85 breaks before We = 4.85, despite being more stabilized. Moreover, for the intermediary viscosity ratio case (μ

_{D}/μ

_{C}= 22), a We = 2.4 drop breaks, whereas at We = 2.6, the drop does not (see Figure 5). Both phenomena appear in relatively narrow We-intervals, indicating that they are secondary effects—the main effect is still an increased breakup tendency with increasing Weber number (as seen from breakup times in Figure 3, Figure 6 and Figure 9). However, the cause of this deviatory behavior must be understood in order to understand what controls the transition between breakup regimes.

_{D}denotes disperse phase density), describing the relative effect of viscous stabilization, is monotonically increasing across Weber numbers. Thus, there is no reason to expect drops with a lower Weber number to be more susceptible to breakup than higher Weber number drops simply due to viscous stabilization.

_{η}. This appears to correlate to the time when the total interfacial area of the deforming drop starts to increase rapidly (indicating fast deformation globally; see Figure 10B). As seen in Figure 10A, there is a systematic effect of the turbulence being slightly more suppressed the higher the Weber number. Thus, the more stabilized drop encounters somewhat stronger turbulent stresses. This trend of increased turbulence suppression with increasing Weber number is consistent with previous investigations of turbulence modulation in isotropic homogenous turbulence and has been hypothesized to relate to the increased extent of energy exchange for more deformable interfaces [39].

_{D}/µ

_{C}= 22). From Figure 11B, showing the evolution of interfacial area with time, there appears to be two major periods of intense deformation (~8 τ

_{η}and ~35 τ

_{η}). Especially in the later deformation period, a clear and monotonic effect can be seen where the higher the Weber number, the greater the dissipation rate of TKE. The effect is still minor. However, when comparing drops with a relatively small difference in Weber number, it explains why more stabilized drops can sometimes break when less stabilized ones do not. From Figure 5, we see that We = 2.60 (less stabilized) does not break, whereas We = 2.40 (more stabilized) does break. The Laplace pressure stabilization is 7.4% higher for We = 2.40 compared to We = 2.6. This can be compared to the turbulence modulation effect, which gives a 3.1% lower dissipation rate of TKE for the high We drop. Since the turbulent disruptive stress scales with a dissipation rate of TKE to a power of 2/3 (Equation (3)), this results in a 9.9% higher disruptive stress for the We = 2.4 drop. Consequently, the turbulent decreases faster than the Laplace stabilization in this region of Weber numbers and the We = 2.4 drop breaks, despite being more stabilized, as observed in Figure 5.

_{D}/μ

_{C}= 40). The same effect of increased turbulence attenuation with increasing We can be seen for the lower viscosity cases, at least for some intervals (i.e., at the deformation occurring at t/τ

_{η}= 20 and between t/τ

_{η}= 30–40; see Figure 12B). However, non-monotonic effects were not observed, either in terms of critical Weber number nor in terms of breakup time (see Figure 8 and Figure 9). This might be due to the increased stabilizing effect of viscosity (Oh = 0.5 at the critical Weber number for the high-viscosity case, as compared to Oh = 0.25 for the intermediary viscosity case).

#### 3.5. A Parameter Map for Initial Breakup

_{1}= 0.86 and c

_{2}= 0.37 corresponds to We

_{Cr}= 0.61 at μ

_{D}/μ

_{C}= 1). To allow for a quantitative comparison of how the critical Weber number scales with viscosity according to Kolmogorov–Hinze theory, however, the semi-empirical prediction of critical Weber number has been introduced as a dashed line in Figure 13, scaled so as to give a critical Weber number of 1.8 at μ

_{D}/μ

_{C}= 1. As seen in the figure, similar trends of how viscosity influences the critical Weber number are obtained after rescaling.

_{D}/μ

_{C}~5–10).

#### 3.6. Practical Implications for Emulsification Processes

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Parameter settings (densities and viscosities) corresponding to the simulated cases (see Table 1) in the dimensionless units used in the code. The dissipation rate of TKE (averaged over space and time) is 3.07 (

**-**), and the initial drop diameter is 2 (

**-**). The length of the simulation domain is 2π.

μ_{D}/μ_{C} | ρ_{C} (-) | ρ_{C} (-) | μ_{C} (-) | μ_{D} (-) |
---|---|---|---|---|

1.0 | 1.0 | 1.0 | 0.06 | 0.06 |

5.0 | 1.0 | 1.0 | 0.06 | 0.30 |

10 | 1.0 | 1.0 | 0.06 | 0.60 |

22 | 1.0 | 1.0 | 0.06 | 1.32 |

40 | 1.0 | 1.0 | 0.06 | 2.40 |

**Table A2.**Parameter settings (interfacial tension) corresponding to the simulated cases (see Table 1) in the dimensionless units used in the code. The dissipation rate of TKE (averaged over space and time) is 3.07 (-), and the initial drop diameter is 2 (-). The length of the simulation domain is 2π.

We | 100 | 60 | 30 | 20 | 13 | 11 | 9.0 | 7.0 | 4.85 | 2.85 | 2.60 | 2.55 | 2.40 | 2.20 | 2.00 | 1.80 | 1.60 | 1.40 | 1.20 | 0.96 |

γ (-) | 0.13 | 0.23 | 0.469 | 0.67 | 1.00 | 1.22 | 1.49 | 1.92 | 2.80 | 4.70 | 5.16 | 5.26 | 5.54 | 6.09 | 6.70 | 7.45 | 8.38 | 9.58 | 11.18 | 14.0 |

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**Figure 1.**Breakup sequence for We = 100, μ

_{D}/μ

_{C}= 1. Illustrating the sheet breakup morphology. (Iso-surfaces at α = 0.5).

**Figure 2.**Morphology at the initial breakup as a function of Weber number for a low viscosity drop (μ

_{D}/μ

_{C}= 1). Green arrows show the point of initial detachment.

**Figure 3.**Time of initial breakup, t

_{B}, as a function of Weber number, We, for a low viscosity drop (μ

_{D}/μ

_{C}= 1). Solid line displays average breakup time as suggested by Rivière et al. [24]. Arrows indicate cases with t

_{B}> 90 τ

_{η}. Colored fields show breakup morphology; see Figure 2. Red box displays the non-monotonic breakup time trend; see Section 3.4.

**Figure 4.**Breakup sequence for We = 7.0, μ

_{D}/μ

_{C}= 1, illustrating the bulb breakup morphology. (Iso-surfaces at α = 0.5).

**Figure 5.**Morphology at initial breakup as a function of Weber number for in intermediary viscosity drop (μ

_{D}/μ

_{C}= 22). Green arrows show the point of initial detachment.

**Figure 6.**Time of initial breakup, t

_{B}, as a function of Weber number, We, for an intermediary viscosity drop (μ

_{D}/μ

_{C}= 22). Solid line displays average breakup time as suggested by Rivière et al. [24]. Arrows indicate cases with t

_{B}> 90 τ

_{η}. Colored fields show breakup morphology; see Figure 2.

**Figure 7.**Breakup sequence for We = 7.0, μ

_{D}/μ

_{C}= 1. Illustrating the thread breakup morphology. (Iso-surfaces at α = 0.5).

**Figure 8.**Morphology at initial breakup as a function of Weber number for a high viscosity drop (μ

_{D}/μ

_{C}= 40). Green arrows show the point of initial detachment.

**Figure 10.**Suppression of turbulence for the low viscosity drop (μ

_{D}/μ

_{C}= 1.0). (

**A**) Dissipation rate of TKE, ε, normalized to the one-phase flow time-averaged value, ε*, as a function of time, t. (

**B**) Total interfacial area, A, normalized with its initial value, A

_{0}.

**Figure 11.**Suppression of turbulence for an intermediary viscosity drop (μ

_{D}/μ

_{C}= 22). (

**A**) Dissipation rate of TKE, ε, normalized to the one-phase flow time-averaged value, ε*, as a function of time, t. (

**B**) Total interfacial area, A, normalized with its initial value, A

_{0}.

**Figure 12.**Suppression of turbulence for a high viscosity drop (μ

_{D}/μ

_{C}= 40). (

**A**) Dissipation rate of TKE, ε, normalized to the one-phase flow time-averaged value, ε*, as a function of time, t. (

**B**) Total interfacial area, A, normalized with its initial value, A

_{0}.

**Figure 13.**Parameter map, showing the initial breakup morphology (S: Sheet, T: Thread, B: bulb; N: none) as a function of Weber number, We, and viscosity ratio, μ

_{D}/μ

_{C}. Solid lines and colored fields are free-hand attempts to illustrate demarcations between morphologies. Dashed line shows the effect of the viscosity ratio on critical Weber number according to the viscosity-corrected Kolmogorov–Hinze theory [8] (rescaled with a proportionality constant complying with the numerical simulations; see text).

We | μ_{D}/μ_{C} = 1.0 | μ_{D}/μ_{C} = 5.0 | μ_{D}/μ_{C} = 10 | μ_{D}/μ_{C} = 22 | μ_{D}/μ_{C} = 40 |
---|---|---|---|---|---|

100 | X | X | X | X | X |

60 | X | X | X | X | X |

30 | X | X | X | X | X |

20 | X | X | |||

13 | X | X | X | X | X |

11 | X | X | X | X | X |

9.0 | X | X | X | X | X |

7.0 | X | X | X | X | X |

4.85 | X | X | X | X | X |

2.85 | X | X | X | X | X |

2.60 | X | ||||

2.55 | X | ||||

2.40 | X | ||||

2.20 | X | ||||

2.00 | X | X | X | X | X |

1.80 | X | ||||

1.60 | X | ||||

1.40 | X | ||||

1.20 | X | ||||

0.96 | X | X | X | X |

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## Share and Cite

**MDPI and ACS Style**

Håkansson, A.; Olad, P.; Innings, F.
Identification and Mapping of Three Distinct Breakup Morphologies in the Turbulent Inertial Regime of Emulsification—Effect of Weber Number and Viscosity Ratio. *Processes* **2022**, *10*, 2204.
https://doi.org/10.3390/pr10112204

**AMA Style**

Håkansson A, Olad P, Innings F.
Identification and Mapping of Three Distinct Breakup Morphologies in the Turbulent Inertial Regime of Emulsification—Effect of Weber Number and Viscosity Ratio. *Processes*. 2022; 10(11):2204.
https://doi.org/10.3390/pr10112204

**Chicago/Turabian Style**

Håkansson, Andreas, Peyman Olad, and Fredrik Innings.
2022. "Identification and Mapping of Three Distinct Breakup Morphologies in the Turbulent Inertial Regime of Emulsification—Effect of Weber Number and Viscosity Ratio" *Processes* 10, no. 11: 2204.
https://doi.org/10.3390/pr10112204