**1. Introduction**

Understanding the behavior of polymer chains in micro/nano-scale flows is an important issue, as it is the gateway to di fferent scientific and technological research activities in di fferent fields of study such as biology, genetics, etc. The translocation of polymers through nano/micro-scale passages is encountered in many biological processes in living cells or chemical processes such as DNA (Deoxyribonucleic acid) motion through narrow pores, protein translocation through cell membranes, and penetration of viruses into the cell nucleus. Knowledge of such processes can be beneficial in developing some technical analysis procedures concerning genomic partitioning and rapid DNA sequencing [1–3]. There are several ways to transfer polymer through narrow pores or micro channel including electroosmotic micro pump, magneto hydrodynamic method, and pressure driven flow [3]. Due to high controllability on fluid flow using electrical field or magnetic field, they are proper methods for fluid pumping [3,4].

The di fficulties and costs associated with the experimental studies promote the researchers to use the computational simulation methods as the preliminary design and analysis tools to narrow down the design parameters' envelopes before getting into the actual manufacturing process. Regarding the small scale of simulated systems, molecular simulation methods would be the best choices. Such simulation methods have been successfully applied to di fferent nano-scale flow problems. Researches have used di fferent numerical simulation methods such as computational fluid dynamics [5–7], molecular dynamics [8], Langevin dynamics [9,10], and Brownian dynamics [11] to study prediction of polymer chains behavior in fluid flows. Based on the Lagrangian methods such as Molecular Dynamics [12–15], lattice Boltzmann method [16,17] or smoothed particle hydrodynamics method [18–21], the dissipative particle dynamics (DPD) method is a mesoscopic method [12,13,22] which has been vastly used in micro/nano-scale simulations. It benefits from the lower computational cost compared with the molecular dynamics method via using the clusters of molecules, known as beads, instead of considering all actual molecules.

There are several studies in relation of DPD and polymer chain motion in nano/micro flows. Zhang and Manke [23] used the DPD method to study the motion of polymers and polymer solutions rheology in the spherical particles with adsorbed polymers. They found that Newtonian behavior is governing on polymer solutions or polymer in sphere suspensions at low shear rates, but shear-thinning behavior is formed at higher shear rates. Willemsen et al. [24] used the DPD method to investigate the motion of a polymer within a square capillary and the e ffect of polymers on melting process in a shear flow. Pastorino et al. [25] compared the dynamics of Langevin and DPD as a thermostat term in non-equilibrium simulations of polymeric systems. They studied polymer brushes in di fferent systems including the relative sliding motion, Poiseuille and Couette flows of polymeric liquids, and brush-melt interfaces to compare these two di fferent thermostats. Based on the DPD method, Duong-Hong et al. [26] introduced an electrophoresis model for DNA, which they simulated the coupled DNA electro-osmotic and electrophoretic motion in micro/nano-scale passages. Using this model, they were able to capture the free-draining mobility of DNA while avoiding the expensive electrostatic interactions in the molecular simulations. They also computed DNA mobilities in realistic geometries with a good accuracy. Their results indicated that the Ti-channel has a better separating performance than the Tp-channel. Pan et al. [27] used the DPD method to study the DNA separation in a micro-device using an entropic trapping mechanism. They showed that longer DNA strands have a higher speed than shorter ones. They concluded that the entropic trapping is the consequence of delayed entrance. Moreover, they concluded that corner trapping does not contribute to DNA separation. Masoud and Alexeev [28] used the DPD method to design nano-structured surfaces capable of selective regulation of collision between microchannel walls and polymer, which is suspended in fluid though the microchannel. By utilizing di fferent geometries for nanoscopic posts attached to the internal channel surfaces, they could attract the suspended nanoparticles and polymeric chains to the walls or repel them. Guo et al. [29] studied the translocation of polymers in fluid through a microchannel using the DPD method. They predicted the relation between shear stress and length of polymer chain for the average translocation time. Moreover, they observed two di fferent mechanisms for translocation including single-file and double-folded translocation. They also mentioned the possibility of clogging at the entrance of the channel for polymers longer than a critical length.

All of the mentioned studies used pressure driven flow for transfer of polymer through microchannel. However, lack of studies in the area of di fferent body forces are observed. Yang et al. [30] studied the motion of a polymer chain through a hole using DPD method. They considered two di fferent driving forces, namely the uniform hydrostatic force implemented to whole solvent particles and polymer chains, and also uniform electrostatic force which is employed as a body force, applied to selected charged particles in the chain and some ions in the solvent which were charged oppositely. They found that the power-law correlations should be used for coil-like chains and it is not proper for globular chains. Ranjith et al. [31] used the DPD method to investigate the e ffect of finite slip at hydrophobic microchannel walls on the hydrodynamics and the dynamics of the DNA chain. They showed that an asymmetric velocity profile caused by hydrophobic and hydrophilic walls can affect the location of the DNA molecules. They used this e ffect to propose a simple arrangemen<sup>t</sup> for separation of short and long DNA chains. Zakeri [32] used the DPD method to simulate the performance of a soft polymer micro-actuator in electro-osmotic flow in a simple micro-channel and a convergent–divergent one. The results indicated that the amplitude of reciprocating motion of polymer increases as the electric field is enhanced, the number of beads is decreased, the spring constant is increased, or more length of a polymer chain is exposed against the fluid flow motion.

A thorough review on the literature shows that there were studies in which the electrical field is used as the transport driving force for polymer chains, e.g., [26,32,33]. Base on the Zakeri [32,34], electroosmotic is a proper external force to move DPD particles in a micro channel, However, the transport of polymer chains using the magnetic forces is not regarded frequently. Magnetic force [35,36] is also another body force capable of inducing fluid flow. This driving force plays an important role in micro pumps (e.g., see [37–42]). Kefayati [43] employed LBM method to simulate the e ffect of MHD flow in a lid-driven cavity problem for various Hartman numbers. Ghahderijani [44] used LBM method to simulate MHD flow in simple micro channels. Javaherdeh and Najjarnezami [45] used LBM to investigate natural convection in a porous cavity with sinusoidally heated walls considering the e ffects of magnetic field. They investigated the e ffects of Hartmann number, porosity and Darcy number on the fluid flow, and heat transfer. Chaabane and Jemni [46] used the LBM to study the convection heat transfer in a 2D enclosure containing a conductive fluid. They examined the effects of Hartmann number, Rayleigh number, Prandtl number on the flow, and temperature fields. Although LBM is a proper simulation method in microscale, the freedom degree of DPD method is higher and it treats more realistically based on the real physics of particles interaction [17,47]. Based on the several references [32,34], the e ffect of polymer chain from electroosmotic flow was presented, but polymer transfer from in MHD flow requires more investigation.

In this paper, the DPD simulation method is employed to simulate MHD flow in simple symmetric micro channel, also the motion of a polymer chain through micro channel is studied to investigate the various physical properties of polymer chain influenced from MHD flow.

## **2. Numerical Simulation**

To present the simulation of MHD in micro channels which a ffects the polymer chain transfer through fluid particles, we discuss in the three main subjects in the Sections 2.1–2.3, including the magneto-hydrodynamics equations, the details of DPD method, and the molecular model of polymer chain.
