# Investigation of Magneto Hydro-Dynamics Effects on a Polymer Chain Transfer in Micro-Channel Using Dissipative Particle Dynamics Method

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Simulation

#### 2.1. Magneto-Hydrodynamics

#### 2.1.1. Analytical Solution for MHD in Simple Channel

#### 2.2. Dissipative Particle Dynamics Method

#### 2.3. Polymer Chain

_{c}should be modified for this type of simulation. In this paper, the harmonic spring force, ${\overrightarrow{f}}_{i,i\pm 1}^{\mathrm{S}}$, or ${\overrightarrow{F}}_{\mathrm{p},i}$ is used between beads which is added to F

_{c}in DPD formulation. For the beads consisting a polymer chain, the following conservative harmonic force presents the intermolecular bonds [32,34,53].

## 3. Results

#### 3.1. Validation of MHD-DPD Results with Analytical Solution

#### 3.2. Short Polymer Chain Transfer in MHD Flow

#### 3.3. Long Polymer Chain Transfer in MHD Flow

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Li, X.; Pivkin, I.V.; Liang, H. Hydrodynamic effects on flow-induced polymer translocation through a microfluidic channel. Polymer
**2013**, 54, 4309–4317. [Google Scholar] [CrossRef] - Xu, Z.; Yang, Y.; Zhu, G.; Chen, P.; Huang, Z.; Dai, X.; Hou, C.; Yan, L.T. Simulating Transport of Soft Matter in Micro/Nano Channel Flows with Dissipative Particle Dynamics. Adv. Theory Simul.
**2019**, 2, 1800160. [Google Scholar] [CrossRef] - Li, P.C. Microfluidic Lab-on-A-Chip for Chemical and Biological Analysis and Discovery; CRC Press: Boca Raton, FL, USA, 2005. [Google Scholar]
- Darbandi, M.; Zakeri, R.; Schneider, G.E. Simulation of Polymer Chain Driven by DPD Solvent Particles in Nanoscale Flows. In Proceedings of the ASME 2010 8th International Conference on Nanochannels, Microchannels, and Minichannels Collocated with 3rd Joint US-European Fluids Engineering Summer Meeting, Montreal, QC, Canada, 1–5 August 2010; pp. 1035–1040. [Google Scholar]
- Chen, C.-W.; Jiang, Y. Computational Fluid Dynamics Study of Magnus Force on an Axis-Symmetric, Disk-Type AUV with Symmetric Propulsion. Symmetry
**2019**, 11, 397. [Google Scholar] [CrossRef] [Green Version] - Irandoost Shahrestani, M.; Maleki, A.; Safdari Shadloo, M.; Tlili, I. Numerical Investigation of Forced Convective Heat Transfer and Performance Evaluation Criterion of Al
_{2}O_{3}/Water Nanofluid Flow inside an Axisymmetric Microchannel. Symmetry**2020**, 12, 120. [Google Scholar] [CrossRef] [Green Version] - Maleki, A.; Elahi, M.; Assad, M.E.H.; Nazari, M.A.; Shadloo, M.S.; Nabipour, N. Thermal conductivity modeling of nanofluids with ZnO particles by using approaches based on artificial neural network and MARS. J. Therm. Anal. Calorim.
**2020**. [Google Scholar] [CrossRef] - Guillouzic, S.; Slater, G.W. Polymer translocation in the presence of excluded volume and explicit hydrodynamic interactions. Phys. Lett. A
**2006**, 359, 261–264. [Google Scholar] [CrossRef] - Muthukumar, M.; Kong, C. Simulation of polymer translocation through protein channels. Proc. Natl. Acad. Sci. USA
**2006**, 103, 5273–5278. [Google Scholar] [CrossRef] [Green Version] - Liu, S.; Ban, X.; Wang, B.; Wang, X. A Symmetric Particle-Based Simulation Scheme towards Large Scale Diffuse Fluids. Symmetry
**2018**, 10, 86. [Google Scholar] [CrossRef] [Green Version] - Ikonen, T.; Bhattacharya, A.; Ala-Nissila, T.; Sung, W. Unifying model of driven polymer translocation. Phys. Rev. E
**2012**, 85, 051803. [Google Scholar] [CrossRef] [Green Version] - Jin, H.; Chen, B.; Zhao, X.; Cao, C. Molecular dynamic simulation of hydrogen production by catalytic gasification of key intermediates of biomass in supercritical water. J. Energy Resour. Technol.
**2018**, 140, 041801. [Google Scholar] [CrossRef] - Xu, B.; Jin, H.; Li, H.; Guo, Y.; Fan, J. Investigation on the evolution of the coal macromolecule in the process of combustion with Molecular dynamics method. J. Energy Resour. Technol.
**2020**, 142. [Google Scholar] [CrossRef] - Hoogerbrugge, P.; Koelman, J. Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics. EPL (Europhys. Lett.)
**1992**, 19, 155. [Google Scholar] [CrossRef] - Zakeri, R.; Lee, E.S. Similar Region in Electroosmotic Flow Rate for Newtonian and non-Newtonian Fluids using dissipative particle dynamics (DPD). In Proceedings of the ASME 2014 International Mechanical Engineering Congress and Exposition, Montreal, QC, Canada, 14–20 November 2014. [Google Scholar]
- Safdari Shadloo, M. Numerical simulation of compressible flows by lattice Boltzmann method. Numer. Heat Transf. Part A Appl.
**2019**, 75, 167–182. [Google Scholar] [CrossRef] - Vasheghani Farahani, M.; Foroughi, S.; Norouzi, S.; Jamshidi, S. Mechanistic Study of Fines Migration in Porous Media Using Lattice Boltzmann Method Coupled With Rigid Body Physics Engine. J. Energy Resour. Technol.
**2019**, 141. [Google Scholar] [CrossRef] - Almasi, F.; Shadloo, M.; Hadjadj, A.; Ozbulut, M.; Tofighi, N.; Yildiz, M. Numerical simulations of multi-phase electro-hydrodynamics flows using a simple incompressible smoothed particle hydrodynamics method. Comput. Math. Appl.
**2019**. [Google Scholar] [CrossRef] - Fatehi, R.; Rahmat, A.; Tofighi, N.; Yildiz, M.; Shadloo, M.S. Density-based smoothed particle hydrodynamics methods for incompressible flows. Comput. Fluids
**2019**, 185, 22–33. [Google Scholar] [CrossRef] - Hopp-Hirschler, M.; Shadloo, M.S.; Nieken, U. A smoothed particle hydrodynamics approach for thermo-capillary flows. Comput. Fluids
**2018**, 176, 1–19. [Google Scholar] [CrossRef] - Shadloo, M.S.; Oger, G.; Le Touzé, D. Smoothed particle hydrodynamics method for fluid flows, towards industrial applications: Motivations, current state, and challenges. Comput. Fluids
**2016**, 136, 11–34. [Google Scholar] [CrossRef] - Hopp-Hirschler, M.; Shadloo, M.S.; Nieken, U. Viscous fingering phenomena in the early stage of polymer membrane formation. J. Fluid Mech.
**2019**, 864, 97–140. [Google Scholar] [CrossRef] - Zhang, K.; Manke, C.W. Simulation of polymer solutions by dissipative particle dynamics. Mol. Simul.
**2000**, 25, 157–166. [Google Scholar] [CrossRef] - Willemsen, S.; Hoefsloot, H.; Iedema, P. Mesoscopic simulation of polymers in fluid dynamics problems. J. Stat. Phys.
**2002**, 107, 53–65. [Google Scholar] [CrossRef] - Pastorino, C.; Kreer, T.; Müller, M.; Binder, K. Comparison of dissipative particle dynamics and Langevin thermostats for out-of-equilibrium simulations of polymeric systems. Phys. Rev. E
**2007**, 76, 026706. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Duong-Hong, D.; Han, J.; Wang, J.S.; Hadjiconstantinou, N.G.; Chen, Y.Z.; Liu, G.R. Realistic simulations of combined DNA electrophoretic flow and EOF in nano-fluidic devices. Electrophoresis
**2008**, 29, 4880–4886. [Google Scholar] [CrossRef] [PubMed] - Pan, H.; Ng, T.; Li, H.; Moeendarbary, E. Dissipative particle dynamics simulation of entropic trapping for DNA separation. Sens. Actuators A Phys.
**2010**, 157, 328–335. [Google Scholar] [CrossRef] - Masoud, H.; Alexeev, A. Selective control of surface properties using hydrodynamic interactions. Chem. Commun.
**2011**, 47, 472–474. [Google Scholar] [CrossRef] - Guo, J.; Li, X.; Liu, Y.; Liang, H. Flow-induced translocation of polymers through a fluidic channel: A dissipative particle dynamics simulation study. J. Chem. Phys.
**2011**, 134, 134906. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Yang, K.; Vishnyakov, A.; Neimark, A.V. Polymer translocation through a nanopore: DPD study. J. Phys. Chem. B
**2013**, 117, 3648–3658. [Google Scholar] [CrossRef] - Ranjith, S.K.; Patnaik, B.; Vedantam, S. Transport of DNA in hydrophobic microchannels: A dissipative particle dynamics simulation. Soft Matter
**2014**, 10, 4184–4191. [Google Scholar] [CrossRef] - Zakeri, R. Dissipative particle dynamics simulation of the soft micro actuator using polymer chain displacement in electro-osmotic flow. Mol. Simul.
**2019**, 45, 1488–1497. [Google Scholar] [CrossRef] - Mao, J.; Yao, Y.; Zhou, Z.; Hu, G. Polymer translocation through nanopore under external electric field: Dissipative particle dynamics study. Appl. Math. Mech.
**2015**, 36, 1581–1592. [Google Scholar] [CrossRef] - Zakeri, R.; Lee, E.S. Simulation of nano polymer chain sensor in electroosmotic flow using dissipative particle dynamics (DPD) method. In Proceedings of the ASME 2014 International Mechanical Engineering Congress and Exposition, Montreal, QC, Canada, 14–20 November 2014. [Google Scholar]
- Alarifi, I.M.; Abokhalil, A.G.; Osman, M.; Lund, L.A.; Ayed, M.B.; Belmabrouk, H.; Tlili, I. MHD flow and heat transfer over vertical stretching sheet with heat sink or source effect. Symmetry
**2019**, 11, 297. [Google Scholar] [CrossRef] [Green Version] - Khan, I.; Alqahtani, A.M. MHD Nanofluids in a Permeable Channel with Porosity. Symmetry
**2019**, 11, 378. [Google Scholar] [CrossRef] [Green Version] - Laser, D.J.; Santiago, J.G. A review of micropumps. J. Micromech. Microeng.
**2004**, 14, R35. [Google Scholar] [CrossRef] - Lim, S.; Choi, B. A study on the MHD (magnetohydrodynamic) micropump with side-walled electrodes. J. Mech. Sci. Technol.
**2009**, 23, 739–749. [Google Scholar] [CrossRef] - Kang, H.-J.; Choi, B. Development of the MHD micropump with mixing function. Sens. Actuators A Phys.
**2011**, 165, 439–445. [Google Scholar] [CrossRef] - Ito, K.; Takahashi, T.; Fujino, T.; Ishikawa, M. Influences of channel size and operating conditions on fluid behavior in a MHD micro pump for micro total analysis system. J. Int. Counc. Electr. Eng.
**2014**, 4, 220–226. [Google Scholar] [CrossRef] [Green Version] - Khan, M.A.; Hristovski, I.R.; Marinaro, G.; Kosel, J. Magnetic Composite Hydrodynamic Pump With Laser-Induced Graphene Electrodes. IEEE Trans. Magn.
**2017**, 53, 1–4. [Google Scholar] [CrossRef] [Green Version] - Zhou, X.; Gao, M.; Gui, L. A Liquid-Metal Based Spiral Magnetohydrodynamic Micropump. Micromachines
**2017**, 8, 365. [Google Scholar] [CrossRef] [Green Version] - Kefayati, G.R.; Gorji-Bandpy, M.; Sajjadi, H.; Ganji, D. Lattice Boltzmann simulation of MHD mixed convection in a lid-driven square cavity with linearly heated wall. Sci. Iran.
**2012**, 19, 1053–1065. [Google Scholar] [CrossRef] [Green Version] - Ghahderijani, M.J.; Esmaeili, M.; Afrand, M.; Karimipour, A. Numerical simulation of MHD fluid flow inside constricted channels using lattice Boltzmann method. J. Appl. Fluid Mech.
**2017**, 10, 1639–1648. [Google Scholar] [CrossRef] - Javaherdeh, K.; Najjarnezami, A. Lattice Boltzmann simulation of MHD natural convection in a cavity with porous media and sinusoidal temperature distribution. Appl. Math. Mech.
**2018**, 39, 1187–1200. [Google Scholar] [CrossRef] - Chaabane, R.; Jemni, A. Lattice Boltzmann approach for MagnetoHydroDynamic convective heat transfer. Energy Procedia
**2019**, 162, 181–190. [Google Scholar] [CrossRef] - Jafari, S.; Zakeri, R.; Darbandi, M. DPD simulation of non-Newtonian electroosmotic fluid flow in nanochannel. Mol. Simul.
**2018**, 44, 1444–1453. [Google Scholar] [CrossRef] - Elmars, B.; Yu, M.; Ozols, R. Heat and Mass Transfer in MHD Flows; World Scientific: Singapore, 1987; Volume 3. [Google Scholar]
- Gold, R.R. Magnetohydrodynamic pipe flow. Part 1. J. Fluid Mech.
**1962**, 13, 505–512. [Google Scholar] [CrossRef] - Asma, M.; Othman, W.; Muhammad, T.; Mallawi, F.; Wong, B. Numerical Study for Magnetohydrodynamic Flow of Nanofluid Due to a Rotating Disk with Binary Chemical Reaction and Arrhenius Activation Energy. Symmetry
**2019**, 11, 1282. [Google Scholar] [CrossRef] [Green Version] - Karniadakis, G.; Beskok, A.; Aluru, N. Microflows and Nanoflows: Fundamentals and Simulation; Springer Science & Business Media: Berlin, Germany, 2006; Volume 29. [Google Scholar]
- Jehser, M.; Zifferer, G.; Likos, C.N. Scaling and Interactions of Linear and Ring Polymer Brushes via DPD Simulations. Polymers
**2019**, 11, 541. [Google Scholar] [CrossRef] [Green Version] - Nikunen, P.; Karttunen, M.; Vattulainen, I. How would you integrate the equations of motion in dissipative particle dynamics simulations? Comput. Phys. Commun.
**2003**, 153, 407–423. [Google Scholar] [CrossRef] [Green Version] - Duong-Hong, D.; Phan-Thien, N.; Fan, X. An implementation of no-slip boundary conditions in DPD. Comput. Mech.
**2004**, 35, 24–29. [Google Scholar] [CrossRef] - Sheikholeslami, M.; Shehzad, S. Magnetohydrodynamic nanofluid convection in a porous enclosure considering heat flux boundary condition. Int. J. Heat Mass Transf.
**2017**, 106, 1261–1269. [Google Scholar] [CrossRef] - Yapici, M.K.; Al Nabulsi, A.; Rizk, N.; Boularaoui, S.M.; Christoforou, N.; Lee, S. Alternating magnetic field plate for enhanced magnetofection of iron oxide nanoparticle conjugated nucleic acids. J. Magn. Magn. Mater.
**2019**, 469, 598–605. [Google Scholar] [CrossRef]

**Figure 2.**Comparison of dimensionless velocity profiles of DPD particles (

**left**) and dimensionless average velocities (

**right**) with analytical results under influence of MHD in simple channel by changing values of Ha.

**Figure 3.**Motion of polymer chain with different values for Ha parameter and harmonic bond constant (K) from number of time steps 1 to 15000 for 20 beads (

**left**) and a chain polymer through DPD particles at number of time step 14000 with K = 5000 and Ha = 20 (

**right**).

**Figure 4.**Dimensionless velocity of polymer mass center for different Ha and K values for a polymer chain consisting of 20 beads (

**left**) and temporal variation of average kinetic temperature (

**right**).

**Figure 5.**Temporal evolution of radius of gyration squared (

**left**) and end-end distance (

**right**) for different Ha and K values for a polymer chain consisting of 20 beads.

**Figure 6.**Motion of polymer chain with different Ha parameter and harmonic bond constant (K) from time step 1 to time step 15000 for a 50-beads polymer chain (

**left**) and a snapshot of polymer chain among DPD particles at time step14000 for K = 5000 and Ha = 50 (

**right**).

**Figure 7.**Dimensionless velocity of polymer mass center for different Ha and K values for a 50-beads polymer chain (

**left**) and the temporal evolution of average kinetic temperature (

**right**).

**Figure 8.**Radius of gyration squared (

**left**) and end-end distance (

**right**) for different Ha and K values for a 50-beads polymer chain.

**Figure 9.**Short (N = 20) and long (N = 50) polymer chain transfer in microchannel for Ha = 20 and K = 5000 from number of time steps from 1 to 20000.

Variables | a_{ij}Different Particles | a_{jj}Same Particles | Number of Particles | Simulation Box (Channel Size) | Time Step | $\mathit{\sigma}$ | $\mathit{\gamma}$ | Cut Off Radious | Periodic Boundary Condition |
---|---|---|---|---|---|---|---|---|---|

Value | 3 | 10 | 4000 | 20 (length) × 50 (height) | 0.001 | 3 | 4.5 | 1 | x-direction |

Variables | Spring Constant | Number of Beads |
---|---|---|

Value | 500, 5000 | 20, 50 |

Variables | $\mathit{L}$ | $\mathit{\mu}\mathit{L}{\mathbf{\left(}\raisebox{1ex}{$\mathit{\sigma}$}\!\left/ \!\raisebox{-1ex}{$\mathit{\rho}\mathit{\nu}$}\right.\mathbf{\right)}}^{\mathbf{0.5}}$ | Ha Number |
---|---|---|---|

Value | 25 | 1 | 1, 2, 7, 10, 20 |

**Table 4.**Average values of test cases calculation from number of time steps from 1 to 15000 with different effective input parameters.

Spring Constant | Number of Beads | Ha Number | $\overline{\mathit{R}\mathit{g}\mathbf{2}}$ | $\overline{\mathit{V}\mathit{c}\mathit{m}}$ | $\overline{\mathit{T}}$ | $\overline{\mathit{N}}$ |
---|---|---|---|---|---|---|

500 | 20 | 1 | 0.54919 | 0.99985 | 0.87459 | 1.5659 |

5000 | 20 | 1 | 0.11650 | 1.00033 | 0.86930 | 0.7998 |

500 | 20 | 20 | 0.53191 | 1.00027 | 0.86012 | 1.96924 |

5000 | 20 | 20 | 0.11838 | 0.99982 | 0.87195 | 0.78627 |

500 | 50 | 1 | 6.7853 | 1.00161 | 0.860313 | 2.9855 |

5000 | 50 | 1 | 0.90651 | 1.00477 | 0.86932 | 3.3996 |

500 | 50 | 20 | 4.3087 | 0.99984 | 0.86096 | 2.7783 |

5000 | 50 | 20 | 0.85215 | 0.99985 | 0.87459 | 1.8455 |

© 2020 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**

Zakeri, R.; Sabouri, M.; Maleki, A.; Abdelmalek, Z.
Investigation of Magneto Hydro-Dynamics Effects on a Polymer Chain Transfer in Micro-Channel Using Dissipative Particle Dynamics Method. *Symmetry* **2020**, *12*, 397.
https://doi.org/10.3390/sym12030397

**AMA Style**

Zakeri R, Sabouri M, Maleki A, Abdelmalek Z.
Investigation of Magneto Hydro-Dynamics Effects on a Polymer Chain Transfer in Micro-Channel Using Dissipative Particle Dynamics Method. *Symmetry*. 2020; 12(3):397.
https://doi.org/10.3390/sym12030397

**Chicago/Turabian Style**

Zakeri, Ramin, Moslem Sabouri, Akbar Maleki, and Zahra Abdelmalek.
2020. "Investigation of Magneto Hydro-Dynamics Effects on a Polymer Chain Transfer in Micro-Channel Using Dissipative Particle Dynamics Method" *Symmetry* 12, no. 3: 397.
https://doi.org/10.3390/sym12030397