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

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## 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

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**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 |

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**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