# Simulation of Pressure-Driven and Channel-Based Microfluidics on Different Abstract Levels: A Case Study

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- Simulations at high abstraction levels come with less computational costs than simulations at lower abstraction levels. While not surprising at all, this is, to the best of our knowledge, the first case study that investigates the performance of simulations and quantifies the computational cost for microfluidic devices at different abstraction levels.
- Applying more computational efforts does not always yield a better simulation quality. In fact, we show that for certain use cases, applying a (computationally more expensive) lower abstraction does not significantly improve the accuracy of the results. Insights like those enable the designers or researcher to properly trade-off the available approaches and to save a lot of simulation time without a severe less of quality.
- The choice of the applied simulator should reflect the need of the designer or researcher. If, e.g., droplets are considered and the end-user is only interested in the position of them, the 1D abstraction level might be sufficient (providing a very fast solution). If, instead, also the integrity or the deformation of the droplet is of interest, the more accurate but also much slower 2D/3D approach might be needed. Again, insights which equip the designers or researcher to choose the right solution for the task.
- For some use cases, simulations provide inconclusive results. This shows that, although simulations really can help in many use cases, they might not always be the “ground-truth”. After all, they do not reflect the real world in a perfect fashion but heavily rely on the underlying models and assumptions. Being aware of that and the complementary approaches helps designers and researchers to constantly reflect when a certain simulation can be trusted and, in case of contradicting results, which simulation approach most likely covered the real world best.

## 2. Considered Use Cases

#### 2.1. Fluid Flow through a Channel

**Example**

**1.**

#### 2.2. Non-Newtonian Fluid Flow

**Example**

**2.**

#### 2.3. Fluid Mixing

**Example**

**3.**

#### 2.4. Droplet Microfluidics

**Example**

**4.**

## 3. Applied Simulation Methods

#### 3.1. 1D Models

#### 3.1.1. Non-Newtonian Fluid Flow

#### 3.1.2. Fluid Mixing

#### 3.1.3. Droplet Microfluidics

#### 3.2. Finite Volume Method

#### 3.3. Lattice-Boltzmann Method

## 4. Conducted Simulations

#### 4.1. Non-Newtonian Fluid Flow

#### 4.1.1. Setup of the Use Case

#### 4.1.2. Setup of the Simulation

^{−8}. To get the shear strain rate $\dot{\gamma}$ in the FVM, the gradient of the velocity field is solved for explicitly every iteration step.

^{−5}for both solvers. In the LBM, the shear strain rate is available locally and no finite difference scheme is necessary [46].

#### 4.2. Fluid Mixing

#### 4.2.1. Setup of the Use Case

#### 4.2.2. Setup of the Simulation

#### 4.3. Droplet Microfluidics

#### 4.3.1. Problem Setup

#### 4.3.2. Implementation

## 5. Obtained Results and Discussion

#### 5.1. Non-Newtonian Fluid Flow

#### 5.2. Fluid Mixing

#### 5.3. Droplet Microfluidics

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Narayanamurthy, V.; Jeroish, Z.; Bhuvaneshwari, K.; Bayat, P.; Premkumar, R.; Samsuri, F.; Yusoff, M.M. Advances in passively driven microfluidics and lab-on-chip devices: A comprehensive literature review and patent analysis. RSC Adv.
**2020**, 10, 11652–11680. [Google Scholar] [CrossRef] [PubMed] - Carrell, C.; Kava, A.; Nguyen, M.; Menger, R.; Munshi, Z.; Call, Z.; Nussbaum, M.; Henry, C. Beyond the lateral flow assay: A review of paper-based microfluidics. Microelectron. Eng.
**2019**, 206, 45–54. [Google Scholar] [CrossRef] - Chung, S.K.; Rhee, K.; Cho, S.K. Bubble actuation by electrowetting-on-dielectric (EWOD) and its applications: A review. Int. J. Precis. Eng. Manuf.
**2010**, 11, 991–1006. [Google Scholar] [CrossRef] - Squires, T.M.; Quake, S.R. Microfluidics: Fluid physics at the nanoliter scale. Rev. Mod. Phys.
**2005**, 77, 977–1026. [Google Scholar] [CrossRef] [Green Version] - Oh, K.W.; Lee, K.; Ahn, B.; Furlani, E.P. Design of pressure-driven microfluidic networks using electric circuit analogy. Lab A Chip
**2012**, 12, 515–545. [Google Scholar] [CrossRef] - Erickson, D. Towards numerical prototyping of labs-on-chip: Modeling for integrated microfluidic devices. Microfluid. Nanofluidics
**2005**, 1, 301–318. [Google Scholar] [CrossRef] - Zhang, J. Lattice Boltzmann method for microfluidics: Models and applications. Microfluid. Nanofluidics
**2011**, 10, 1–28. [Google Scholar] [CrossRef] - Grimmer, A.; Chen, X.; Hamidović, M.; Haselmayr, W.; Ren, C.L.; Wille, R. Simulation before fabrication: A case study on the utilization of simulators for the design of droplet microfluidic networks. RSC Adv.
**2018**, 8, 34733–34742. [Google Scholar] [CrossRef] [Green Version] - Ferziger, J.H.; Perić, M.; Street, R.L. Computational Methods for Fluid Dynamics; Springer: Berlin/Heidelberg, Germany, 2002; Volume 3. [Google Scholar]
- LeVeque, R.J. Finite Volume Methods for Hyperbolic Problems; Cambridge University Press: Cambridge, UK, 2002; Volume 31. [Google Scholar]
- McNamara, G.R.; Zanetti, G. Use of the Boltzmann Equation to Simulate Lattice-Gas Automata. Phys. Rev. Lett.
**1988**, 61, 2332–2335. [Google Scholar] [CrossRef] - Krüger, T.; Kusumaatmaja, H.; Kuzmin, A.; Shardt, O.; Silva, G.; Viggen, E.M. The Lattice Boltzmann Method; Springer: Berlin/Heidelberg, Germany, 2017; Volume 10, pp. 4–15. [Google Scholar]
- Wörner, M. Numerical modeling of multiphase flows in microfluidics and micro process engineering: A review of methods and applications. Microfluid. Nanofluidics
**2012**, 12, 841–886. [Google Scholar] [CrossRef] - Glatzel, T.; Litterst, C.; Cupelli, C.; Lindemann, T.; Moosmann, C.; Niekrawietz, R.; Streule, W.; Zengerle, R.; Koltay, P. Computational fluid dynamics (CFD) software tools for microfluidic applications–A case study. Comput. Fluids
**2008**, 37, 218–235. [Google Scholar] [CrossRef] - Di Carlo, D.; Edd, J.F.; Humphry, K.J.; Stone, H.A.; Toner, M. Particle segregation and dynamics in confined flows. Phys. Rev. Lett.
**2009**, 102, 094503. [Google Scholar] [CrossRef] [Green Version] - Mora, A.E.M. Numerical study of the dynamics of a droplet in a T-junction microchannel using OpenFOAM. Chem. Eng. Sci.
**2019**, 196, 514–526. [Google Scholar] [CrossRef] - Chun, B.; Ladd, A. Inertial migration of neutrally buoyant particles in a square duct: An investigation of multiple equilibrium positions. Phys. Fluids
**2006**, 18, 031704. [Google Scholar] [CrossRef] [Green Version] - Kobel, S.; Valero, A.; Latt, J.; Renaud, P.; Lutolf, M. Optimization of microfluidic single cell trapping for long-term on-chip culture. Lab Chip
**2010**, 10, 857–863. [Google Scholar] [CrossRef] [Green Version] - Bazaz, S.R.; Mashhadian, A.; Ehsani, A.; Saha, S.C.; Krüger, T.; Warkiani, M.E. Computational inertial microfluidics: A review. Lab Chip
**2020**, 20, 1023–1048. [Google Scholar] [CrossRef] - Boyd, J.; Buick, J.; Green, S. A second-order accurate lattice Boltzmann non-Newtonian flow model. J. Phys. A Math. Gen.
**2006**, 39, 14241. [Google Scholar] [CrossRef] - Cornish, R. Flow in a pipe of rectangular cross-section. Proc. R. Soc. London. Ser. A Contain. Pap. A Math. Phys. Character
**1928**, 120, 691–700. [Google Scholar] - Shang, L.; Cheng, Y.; Zhao, Y. Emerging droplet microfluidics. Chem. Rev.
**2017**, 117, 7964–8040. [Google Scholar] [CrossRef] - Quarteroni, A.; Tuveri, M.; Veneziani, A. Computational vascular fluid dynamics: Problems, models and methods. Comput. Vis. Sci.
**2000**, 2, 163–197. [Google Scholar] [CrossRef] - Perktold, K.; Resch, M.; Florian, H. Pulsatile non-Newtonian flow characteristics in a three-dimensional human carotid bifurcation model. J. Biomech. Eng. Nov.
**1991**, 113, 464–475. [Google Scholar] [CrossRef] - Carreau, P.J. Rheological equations from molecular network theories. Trans. Soc. Rheol.
**1972**, 16, 99–127. [Google Scholar] [CrossRef] - Bird, R.B.; Armstrong, R.C.; Hassager, O. Dynamics of Polymeric Liquids. Vol. 1: Fluid Mechanics; Wiley: Hoboken, NJ, USA, 1987. [Google Scholar]
- Gijsen, F.J.; van de Vosse, F.N.; Janssen, J. The influence of the non-Newtonian properties of blood on the flow in large arteries: Steady flow in a carotid bifurcation model. J. Biomech.
**1999**, 32, 601–608. [Google Scholar] [CrossRef] - Fink, G.; Mitteramskogler, T.; Hintermüller, M.A.; Jakoby, B.; Wille, R. Automatic Design of Microfluidic Gradient Generators. IEEE Access
**2022**, 10, 28155–28164. [Google Scholar] [CrossRef] - Herold, K.E.; Herold, K.E.; Rasooly, A. Lab on a Chip Technology: Biomolecular Separation and Analysis; Caister Academic: Poole, UK, 2009; Volume 2. [Google Scholar]
- Vestad, T.; Marr, D.; Oakey, J. Flow control for capillary-pumped microfluidic systems. J. Micromechanics Microengineering
**2004**, 14, 1503. [Google Scholar] [CrossRef] - Chou, H.P.; Unger, M.A.; Quake, S.R. A microfabricated rotary pump. Biomed. Microdevices
**2001**, 3, 323–330. [Google Scholar] [CrossRef] - Studer, V.; Pépin, A.; Chen, Y.; Ajdari, A. An integrated AC electrokinetic pump in a microfluidic loop for fast and tunable flow control. Analyst
**2004**, 129, 944–949. [Google Scholar] [CrossRef] - Convery, N.; Gadegaard, N. 30 years of microfluidics. Micro Nano Eng.
**2019**, 2, 76–91. [Google Scholar] [CrossRef] - Moukhtari, F.E.; Lecampion, B. A semi-infinite hydraulic fracture driven by a shear-thinning fluid. J. Fluid Mech.
**2018**, 838, 573–605. [Google Scholar] [CrossRef] [Green Version] - Sochi, T. Analytical solutions for the flow of Carreau and Cross fluids in circular pipes and thin slits. Rheol. Acta
**2015**, 54, 745–756. [Google Scholar] [CrossRef] [Green Version] - Cussler, E.L.; Cussler, E.L. Diffusion: Mass Transfer in Fluid Systems; Cambridge University Press: Cambridge, UK, 2009. [Google Scholar]
- Wu, Z.; Nguyen, N.T.; Huang, X. Nonlinear diffusive mixing in microchannels: Theory and experiments. J. Micromechanics Microengineering
**2004**, 14, 604–611. [Google Scholar] [CrossRef] - Grimmer, A.; Wille, R. Designing Droplet Microfluidic Networks; Springer: Berlin/Heidelberg, Germany, 2020. [Google Scholar]
- OpenFOAM. Available online: https://openfoam.org/ (accessed on 22 March 2022).
- Hardy, J.; Pomeau, Y.; De Pazzis, O. Time evolution of a two-dimensional model system. I. Invariant states and time correlation functions. J. Math. Phys.
**1973**, 14, 1746–1759. [Google Scholar] [CrossRef] - Enskog, D. Kinetische Theorie der Vorgänge in Mässig Verdünnten Gasen…; Almquist & Wiksell: Stockholm, Sweden, 1917; Volume 1. [Google Scholar]
- Chapman, S. VI. On the law of distribution of molecular velocities, and on the theory of viscosity and thermal conduction, in a non-uniform simple monatomic gas. Philos. Trans. R. Soc. London. Ser. A Contain. Pap. A Math. Phys. Character
**1916**, 216, 279–348. [Google Scholar] - Latt, J.; Malaspinas, O.; Kontaxakis, D.; Parmigiani, A.; Lagrava, D.; Brogi, F.; Belgacem, M.B.; Thorimbert, Y.; Leclaire, S.; Li, S.; et al. Palabos: Parallel Lattice Boltzmann Solver. Comput. Math. Appl.
**2021**, 81, 334–350. [Google Scholar] [CrossRef] - Takken, M. Microfluidics-Abstraction-Levels. Available online: https://github.com/micheltakken/Microfluidics-Abstraction-Levels (accessed on 31 May 2022).
- Shmukler, M. Density of Blood-The Physics Factbook. 2004. Available online: https://hypertextbook.com/facts/2004/MichaelShmukler.shtml, (accessed on 4 April 2022).
- Boyd, J.; Buick, J.M.; Green, S. Analysis of the Casson and Carreau-Yasuda non-Newtonian blood models in steady and oscillatory flows using the lattice Boltzmann method. Phys. Fluids
**2007**, 19, 093103. [Google Scholar] [CrossRef] [Green Version] - Hirt, C.W.; Nichols, B.D. Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys.
**1981**, 39, 201–225. [Google Scholar] [CrossRef] - Dauyeshova, B.; Rojas-Solórzano, L.R.; Monaco, E. Numerical simulation of diffusion process in T-shaped micromixer using Shan-Chen Lattice Boltzmann Method. Comput. Fluids
**2018**, 167, 229–240. [Google Scholar] [CrossRef] - Shan, X.; Chen, H. Lattice Boltzmann model for simulating flows with multiple phases and components. Phys. Rev. E
**1993**, 47, 1815. [Google Scholar] [CrossRef] [Green Version] - Vanapalli, S.A.; Banpurkar, A.G.; Van Den Ende, D.; Duits, M.H.; Mugele, F. Hydrodynamic resistance of single confined moving drops in rectangular microchannels. Lab Chip
**2009**, 9, 982–990. [Google Scholar] [CrossRef] - Hashmi, A.; Xu, J. On the quantification of mixing in microfluidics. J. Lab. Autom.
**2014**, 19, 488–491. [Google Scholar] [CrossRef] [Green Version]

**Figure 2.**A set of flow profiles through a channel of length l with width w, where the walls on the top and bottom are assumed to be rigid. The flow profile of a Newtonian fluid is shown by the solid line and the flow profiles for the shear thinning and thickening fluids are given by the dashed lines.

**Figure 3.**Flow mixing in a meandering channel [28].

**Figure 5.**The lattices that are used for the LBM simulations. (

**a**) The D2Q9 lattice for 2D LBM simulations; (

**b**) The D3Q19 lattice for 3D LBM simulations.

**Figure 8.**The concentration distribution of fluid A for the 1D approach, the 2D FVM approach, and the 2D LBM approach. For the 2D approach, the concentration distribution was taken at the measurement line m in Figure 6b.

**Figure 10.**Results of the simulations for droplet microfluidics. (

**a**) Droplet position at different timestamps as simulated with the 1D approach; (

**b**) Droplet position and deformation at different timestamps as simulated with the 2D FVM approach; (

**c**) Droplet position and deformation at different timestamps as simulated with the 2D LBM approach; (

**d**) Droplet position and deformation at different timestamps as simulated with the 3D FVM approach. (

**e**) Droplet position and deformation at different timestamps as simulated with the 3D LBM approach.

2D FVM | 2D LBM | 3D FVM | 3D LBM | |
---|---|---|---|---|

Time [hh:mm:ss] | 00:02:34 | 00:05:31 | 07:02:27 | 06:31:32 |

Required Memory [MB] | 132.8 | 14.6 | 5819.1 | 1410.1 |

**Table 2.**Absolute Mixing Index (AMI) for all mixing simulations, the runtime in CPU time, and required memory in MB for the 2D and 3D simulations.

1D | 2D FVM | 2D LBM | 3D FVM | 3D LBM | |
---|---|---|---|---|---|

AMI | 0.654 | 0.665 | 0.680 | 0.654 | 0.519 |

Time [hh:mm:ss] | - | 00:06:38 | 00:08:18 | 11:44:36 | 13:52:23 |

Required Memory [MB] | - | 109.6 | 20.0 | 1530.0 | 515.1 |

2D FVM | 2D LBM | 3D FVM | 3D LBM | |
---|---|---|---|---|

Time [hh:mm:ss] | 00:03:29 | 00:02:23 | 06:06:13 | 04:26:45 |

Required Memory [MB] | 96.1 | 32.4 | 898.1 | 711.6 |

Level | Non-Newtonian Fluid Flow | Fluid Mixing | Droplet Microfluidics | ||||
---|---|---|---|---|---|---|---|

Rectangular Channel | Other | Straight Channel | Meander | Position | Split | Deformation | |

1D | |||||||

2D | * | * | |||||

3D | * | * |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Takken, M.; Wille, R.
Simulation of Pressure-Driven and Channel-Based Microfluidics on Different Abstract Levels: A Case Study. *Sensors* **2022**, *22*, 5392.
https://doi.org/10.3390/s22145392

**AMA Style**

Takken M, Wille R.
Simulation of Pressure-Driven and Channel-Based Microfluidics on Different Abstract Levels: A Case Study. *Sensors*. 2022; 22(14):5392.
https://doi.org/10.3390/s22145392

**Chicago/Turabian Style**

Takken, Michel, and Robert Wille.
2022. "Simulation of Pressure-Driven and Channel-Based Microfluidics on Different Abstract Levels: A Case Study" *Sensors* 22, no. 14: 5392.
https://doi.org/10.3390/s22145392