**3. Results**

In this section, the results of numerical simulations of MHD flow in micro channel are presented regarding the polymer chain motion under the influence of magnetic field (MHD flow) using DPD method. We developed our DPD code for current simulation. First, the DPD results for the simple channel flow with the results of the analytical solution are compared to evaluate the accuracy of suggested method for this type of simulation. Then, the study by embedding the polymer chain in a simple channel is extended to study the effect of MHD flow on polymer chain and the related motion characteristics. Different conditions have been considered including changing the magnetic field or *Ha*-value, spring constant (harmonic bond constant of polymer), or *K* parameter and the number of polymer beads in the polymer chain.

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

In order to evaluate and validate the results, the simple microchannel flow under external force of magnetic field is simulated. The simulation parameters are presented in Tables 1–3. The analytical velocity profiles are also calculated in accordance with Equation (6) in Section 2.1.1 with the assumption of *Vv* = 1. As can be seen in Figure 2 (left), by increasing the value of parameter Ha Number, dimensionless velocity profiles not only increase to a certain extent in terms of value, but they also flatten more in plug like shape and tend to remain constant thereafter. For example, there is a dramatic change of about 80% for maximum value of dimensionless velocity when the value *Ha* Number is increased from 1.0 to 2.0. However, the changes are much less evident from *Ha* = 10 to *Ha* = 20. For a better study of variation of MHD flow, the dimensionless average velocity of particles in channel considering different values of *Ha* is compared with the analytical calculation in Figure 2 (right). As it is expected, increasing the value of *Ha*, the differences would decrease remarkably and would approach the constant value of unity. Also, the results show a proper agreemen<sup>t</sup> with the analytical results. The range of discrepancy between DPD solution and analytical solution is between 3% to 7%. The level of accuracy depends on time steps and number of iterations. Therefore, the parameter of *Ha* number is a suitable criterion for studying the velocity profile behavior in micro-channels and DPD method is a proper method for this type of fluid simulation.



**Table 2.** Setting parameters for polymer chain.



**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*.

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

**Table 3.** Setting parameters for magnetic parameter.

In order to extend the results to the case of polymer translocation in microchannel, the movement of a short no charge polymer chain consisting of 20 beads in a microchannel has been analyzed under influence of MHD flow. Figure 3 illustrates the movement of the polymer chain under different magnetic field strengths (or *Ha*-values) as well as different polymer harmonic bond coefficients

(or spring constants), considering periodic boundary condition at inlet/outlet boundaries. As can be observed, by increasing the *Ha* values from 1 to *Ha* = 20, the relative movement of the short polymer would be more with respect to the hardness coefficient (spring constant) of 500. It has been simulated that conspicuous differences in *x*-direction for the cases of *Ha* = 1 and 20 with the harmonic bond constant of *K* = 500 (red lines with circular symbol and blue lines with diamond symbol) is occurred compared to the cases of *K* = 5000 (green lines with triangle symbols and black lines with gradient symbols). By increasing the harmonic bond constant from 500 to 5000 (10 times), the length of polymers is decreased almost to the half. Results show that choosing the higher *Ha* values and higher harmonic bond constants provides proper polymer chain transfer for low length of polymer cases.

**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**).

In the following, the effects of magnetic field and hardness of polymers on the properties of polymer chain are investigated. The variation of dimensionless velocity of mass center of the polymer during the time is depicted in the Figure 4 (left). As can be seen, the oscillations decrease over time and, in the case of more severe magnetic field and higher spring coefficient value, the oscillation is greatly reduced. Therefore, in the condition of *Ha* = 20 and *K* = 5000 causes the polymer to move with less velocity changes. As is evident from Figure 4 (right), temporal evolutions of the average kinetic temperature for all cases reach to unity and fluid condition approaches the equilibrium. Also, by examining the radius of gyration squared for the polymer chain in Figure 5 (left), it is observed that the amount of distortion and perturbation is greatly reduced around the 80% by increasing the harmonic bond constant from 500 to 5000. Changing the magnetic field has little effect on the polymer chain perturbations. Such behavior can be expected by studying the end-to-end distance of the polymer chain as shown in Figure 5 (right). Results indicate that in the non-equilibrium situation, the amount of perturbation for a short polymer chain is high for a short period of time, and in the case of higher *Ha* and *K* values, low oscillation is resulted for those mentioned parameters. Therefore, again, having a chain with a higher harmonic bond constant has about 80% lower oscillation in this study.

**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.

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

One of the significant factors in polymer chain transfer is the length of polymer chain. In this study, the effect of MHD is investigated on no charge polymer chain motion and consequently, the length of chain would be influenced from magnetic field. By consideration of previous results, the motion of a polymer chain consisting of 50 beads having different harmonic bond constant values is depicted in Figure 6. As can be expected, higher magnetic field has more effect on the motion in the case of the higher spring constant or *K* = 5000 compared to *K* = 500. Also, by increasing the harmonic bond constant, the polymer prefers to collapse and in higher magnetic field, compression is enhanced compared to other conditions. In the maximum circumstance, 40% compression is observed in the case of *Ha* = 20 and *K* = 5000. It can be concluded that the proper selection of parameters for transfer of polymer chain are the higher *Ha* and *K* values.

**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**).

It can also be concluded from the study of the dimensionless velocity of mass center the rate of velocity variation would greatly decrease as the harmonic bond constant and chain length increase. As Figure 7 indicates, more reduction would be resulted for the magnetic field with *Ha* = 20 and *K* = 5000. Similar to the previous results, the average kinetic temperature would converge to unity over the time and fluid would move towards the equilibrium condition. In these cases, for high length of polymer chain, the higher magnetic field and the higher polymer hardness has a proper result.

**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**).

By examining the radius of gyration squared and the end-to-end distance, it can be concluded from Figure 8 that a significant decrease of about 75% would occur by increasing the harmonic bond constant from 500 to 5000. It should be noted that higher length of polymer chain delays the equilibrium condition. In this case, higher *Ha* value along with a higher harmonic bond constant present proper polymer chain transfer. In Figure 9, these transfers for both cases of short and long polymer chain with *Ha* = 20 and *K* = 5000 as a high *Ha* and *K* values are shown as a proper polymer chain transfer in this study. For more information from quantity aspect, average properties of radius of gyration squared, dimensionless velocity of polymer mass center, temporal evolution of average kinetic temperature, and the length of polymer chain by consideration of Figure 2 to 8 and different input DPD variable are presented in Table 4. As can be seen again, test case of *Ha* = 20 and *K*=5000 show less perturbation and less strength in polymer chain through transfer in the micro channel.

**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.


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