# Modeling the Excess Velocity of Low-Viscous Taylor Droplets in Square Microchannels

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## Abstract

**:**

## 1. Introduction

## 2. Hydrodynamic Fundamentals of Taylor Flows

## 3. Concept of Excess Velocity

## 4. Model Specification

## 5. Model Calibration

## 6. Model Validation

## 7. Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

Acronyms/Abbreviations | |

GA | Genetic Algorithm |

PIV | Particle Image Velocimetry |

PSA | Pattern Search Algorithm |

Dimensionless Quantities | |

$\lambda $ | Viscosity ratio [−] |

$Ca$ | Capillary number [−] |

$Oh$ | Ohnesorge number [−] |

$Re$ | Reynolds number [−] |

Greek Symbols | |

$\beta $ | geometric coefficient of resistance [−] |

$\delta $ | wall film-thickness [$\mathsf{\mu}$m] |

$\eta $ | dynamic viscosity [Pa · s] |

$\Gamma $ | control surface [-] |

$\Omega $ | flow resistance [Pa ·s · m${}^{-3}$] |

$\omega $ | weight factor [−] |

$\psi $ | dimensionless equilibrium function $[-]$ |

$\rho $ | density [kg · m${}^{-3}$] |

$\sigma $ | interfacial tension [N · m${}^{-1}$] |

Roman Symbols | |

A | cross-sectional area [$\mathsf{\mu}$m${}^{2}$] |

a | semi-minor axis of ellipse [$\mathsf{\mu}$m] |

$ar$ | aspect ratio $[-]$ |

b | semi-major axis of ellipse [$\mathsf{\mu}$m] |

c | fitting coefficient [−] |

D | hydraulic diameter [$\mathsf{\mu}$m] |

d | characteristic length [m] |

H | channel height [$\mathsf{\mu}$m] |

h | droplet height [$\mathsf{\mu}$m] |

k | dimensionless ratio [−] |

l | length [$\mathsf{\mu}$m] |

m | fitting coefficient [−] |

n | fitting coefficient [−] |

Q | volume flow rate [$\mathsf{\mu}$l · min${}^{-1}$] |

R | radius [$\mathsf{\mu}$m] |

u | velocity [mm · s${}^{-1}$] |

W | channel width [$\mathsf{\mu}$m] |

x | cap length till gutter entrance [$\mathsf{\mu}$m] |

Superscripts | |

$\mathcal{M}$ | model calibration data |

$meas$ | measurement data |

$mod$ | model results |

$rel$ | relative |

$stat$ | stationary |

Subscripts | |

0 | superficial/total/offset |

$\beta $ | concerning resistance factor |

c | continuous phase |

$c,b$ | back droplet cap |

$c,f$ | front droplet cap |

$c,i$ | front or back droplet cap |

$cap$ | concerning droplet cap curvature |

$ch$ | channel |

d | disperse/droplet phase |

$ex$ | excess scale concept |

f | wall film |

$fb$ | from front to back |

g | gutter |

$g,b$ | gutter at back of droplet |

$g,f$ | gutter at front of droplet |

$g,i$ | i-th droplet gutter |

$LP$ | Laplace pressure |

$mean$ | arithmetic mean |

$rel$ | relative value |

s | slug |

$slip$ | slipping scale concept |

Other Symbols | |

$\overline{...}$ | averaged value |

## Appendix A. Considerations for Mixed Terms

**Figure A1.**Dimensionless film-area and excess velocity for $\frac{{l}_{d}}{W}=3$ and $\beta =3.892$ for the proposed model.

**Figure A2.**Measured values and retrieved correlation for the droplet cap deformation ratios ${k}_{c,f}$ and ${k}_{c,b}$ for different $Ca$ numbers used in the proposed model. Additionally, the calculated dimensionless gutter-radii ${k}_{g,f}$, ${k}_{g,b}$ based on the model of Mießner et al. [24] are shown.

## Appendix B. Criterion for Neglecting the Viscous Heating

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**Figure 1.**Comparison of different concepts of calculating the relative droplet velocity: excess velocity from this work (black solid line), slip velocity (grey solid line) and difference of both concepts (black dashed line).

**Figure 2.**Prominent averaged and local velocity for a flowing droplet, where overlined entities represent area-averaged velocities. The film flow is not shown in this drawing. (

**a**) Velocities for a fixed point-of-view (fixed frame); (

**b**) flow balance at a steady control surface; (

**c**) velocities for a moving point-of-view with the velocity ${u}_{d}$ (moving frame); and (

**d**) flow balance at a moving control surface.

**Figure 3.**Declaration of relevant geometry for the model of a Taylor droplet flowing through a rectangular microchannel with the droplet velocity ${u}_{d}$. (

**a**) Top-view of x-y-plane, characterizing the droplet with the front (${a}_{f}$, ${b}_{f}$) and back cap (${a}_{b}f$, ${b}_{b}$), as well as the channel width W and droplet width w. (

**b**) Droplet front-view in y-z-plane with the droplet area ${A}_{d}$, gutter area ${A}_{g}$, film area ${A}_{f}$, channel height H and the droplet height h. Only one representation of each area is shown. (

**c**) Close-up of the droplet corner region with gutter radius (${R}_{g}$) and the film-thickness $\delta $.

**Figure 5.**Calibration function $\psi $ for measured (triangles) and modeled data ${\psi}^{\mathcal{M}}$ for the measured droplet lengths ${l}_{d}$ and correlated $\beta $ (crosses) over the $Ca$ number. The black solid line depicts the calibration function for an averaged $\overline{{l}_{d}}$ and $\overline{\beta}$ to clarify the underlying systematic.

**Figure 6.**Comparison of calculated resistance factor $\beta $ (squares) and correlation (solid and dashed line) of our model. The data of the simulation from Shams et al. [27] is shown as squares. Both datasets are corrected by the offset ${\beta}_{0}$ to compensate for the influence of the flow form (co-current or counter-current).

**Figure 7.**Comparison of our model (stars) and measurements (circles) with the measurements (triangles) and correlation from Jose and Cubaud [22] for a Taylor droplet in co-flow (axis scaling and normalization kept for comparability). The inclination for low $Ca$ numbers (hatched area) is discussed in the text. Since the influence of ${l}_{d}$ correlates linearly with $\Delta {p}_{LP}$ instead of $Ca$ in our model and $\beta $ is not included in the x-axis normalization, we additionally show the borders of our model for the minimum/maximum ${l}_{d}$ and $\beta $ of our measurements.

Parameter | Lower Boundary | Upper Boundary |
---|---|---|

${m}_{cap,f}$ | 1.00 | 9.90 |

${c}_{cap,f}$ | 0.40 | 1.50 |

${n}_{cap,f}$ | 1.00 | 1.005 |

${m}_{cap,b}$ | −2.50 | $-1.00$ |

${c}_{cap,b}$ | 0.30 | 0.75 |

${n}_{cap,b}$ | 0.995 | 1.00 |

${m}_{\beta}$ | 0.00 | 10.00 |

${c}_{\beta}$ | 0.50 | 1.5 |

${n}_{\beta}$ | 0.50 | 20 |

${\mathit{\omega}}_{1}$ | ${\mathit{\omega}}_{2}$ | ${\mathit{\omega}}_{3}$ |
---|---|---|

3.0 | 5.0 | 3.7 |

Genetic Algorithm | |

population size | 200 individuals |

creation function | random feasible population |

scaling function | ranking |

selection function | stochastic uniform |

mutation function | adaptive feasible |

crossover function | scattered |

Pattern Search Algorithm | |

search method | Latin Hypercube |

poll method | complete poll |

Target Value | ${\mathit{m}}_{\mathit{i}}$ | ${\mathit{c}}_{\mathit{i}}$ | ${\mathit{n}}_{\mathit{i}}$ |
---|---|---|---|

${k}_{f}$ | 4.8761 | 0.7465 | 1.0021 |

${k}_{b}$ | −1.6967 | 0.4745 | 0.9980 |

$\beta $ | 6.1280 | 1.2105 | 1.4541 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Helmers, T.; Kemper, P.; Thöming, J.; Mießner, U.
Modeling the Excess Velocity of Low-Viscous Taylor Droplets in Square Microchannels. *Fluids* **2019**, *4*, 162.
https://doi.org/10.3390/fluids4030162

**AMA Style**

Helmers T, Kemper P, Thöming J, Mießner U.
Modeling the Excess Velocity of Low-Viscous Taylor Droplets in Square Microchannels. *Fluids*. 2019; 4(3):162.
https://doi.org/10.3390/fluids4030162

**Chicago/Turabian Style**

Helmers, Thorben, Philip Kemper, Jorg Thöming, and Ulrich Mießner.
2019. "Modeling the Excess Velocity of Low-Viscous Taylor Droplets in Square Microchannels" *Fluids* 4, no. 3: 162.
https://doi.org/10.3390/fluids4030162