J. Cent. South Univ. Technol. (2008) 15(s1): 234-238

DOI: 10.1007/s11771-008-353-8

Dynamics simulation of electrorheological suspensions in

poiseuille flow field

ZHU Shi-sha(朱石沙)^{1, 2}, LUO Cheng(罗 成)¹, ZHOU Jie(周 杰)¹, CHEN Na(陈 娜)¹

（1. College of Mechanical Engineer, Xiangtan University, Xiangtan 411105, China

2. Changshu Institute of Technology, Changshu 215500, China)

Abstract: Based on a modified Maxwell-Wagner model, molecular dynamics is carried out to simulate the structural changes of ER (electrorheological) suspensions in a poiseuille flow field. The simulation results show that the flow assists in the collection of particles at the electrodes under a low pressure gradient，and the negative ER effect will show under a high pressure gradient. By analyzing the relationship curves of the shear stress and the pressure gradient in different relaxation time, it is found that for the same kind of ER suspensions materials, there is an optimal dielectric relaxation frequency.

Key words: electrorheological suspensions; molecular dynamics; pressure gradient value; relaxation time.

1 Introduction

Concerning the researches on the structural characteristic of ER suspensions in an external electric field, the most representative one is the study of HOMARD and MASAO^[1] who investigated the accumulation characteristic of  ER fluids by applying molecular dynamics. In addition, TAO^[2] simulated the effect of temperature on the stabilization structure of ER fluids by Monte Carlo method. In his work, it is considered that the formation of final configuration has been achieved by Brownian motion under a certain temperature, while electro-sensitive particles have been gathered into two different chains. ZHU^[3] put forward a capture effect in a poiseuille flow field in the process of his experimental study, and analyzed the effect on microstructure of ER fluids. However, their research works are based on a electrostatic polarization model, which forecasted some ER effects successfully, but can not be explained in some experiments the phenomenon of shear strength dependent on electric field less than square^[4]. Based on the modified Maxwell-Wagner^[5], the authors have modified the force equations of electro-sensitive particles, and simulated the accumulation changes of ER fluids in the poiseuille flow field by adopting molecular dynamics to seek the array configurations characteristics of electro-sensitive particles following time changes in the micro-mechanism caused by the electric field.

2 Dynamics model

Molecule dynamics method is adopted to simulate the movement characteristic of electro-sensitive particles in the poiseuille flow field. First of all, Newton’s movement equations should be established to identify the force expression of particles in a compound field. The modified Maxwell-Wagner model is figured that an electro-sensitive particle will be polarized and turned into a dipole in an external electric field, besides, the interaction forces between these dipole particles depend mainly on the dielectric mismatch of electro-sensitive particles and basic medium fluid except the intensity of a electric field. It is noticeable, in which the dielectric constant is a function of shear rate.

In the poiseuille flow field, the movement equations of particle i is given by

      (1)

where  F_i is the electrostatic resultant force on the particle i; m is the mass of particle, ?p/?x is the pressure gradient. z is the particle diameter, and h is the height between two electrode plates.

               ₍₂₎

where  f_ij′ is the electrostatic resultant force on i particle by particle j and all its images(induced by the periodicity-boundary condition); f_ij^rep is the short-ranged

repulsive force to prevent particle i and particle j overlapping one another (the particles as rigid sphere);  f_i ^rep is the electrostatic resultant force on i particle by all its images(induced by the periodicity-boundary condition); f_i ^wall is the the short-ranged repulsive force to prevent particle i coming into the electrodes(the electrodes as rigid).

In a shear flow field or a pressure gradient field, based on the modified Maxwell-Wagner model, the polarization force on j particle by particle i is given by 

    ₍₃₎

where _R is a particle radius_,  is a tilted anger of dipole moment, and is a modified polarization frequency.

                         (4)

                 (5)

                       (6)

where  H_i is the distance from i particle to the electrode plates. The flow resistance force acting on i particle can be given by stockes friction.

          (7)

The above forces have been summed for n-1 particles respectively, and then adding all forces, the inertia of particles has been neglected, so the movement equations of particles can be expressed as:

                    (8)

where  r* is the particle’s displacement, the asterisk indicates a dimensionless idea. S* is known as the relative pressure gradient given by

             (9)

The yielding shear stress of ER suspensions at a certain moment is calculated by

                       (10)

where V* is a systemic volume (area for two- dimensional simulation), is the part forces of total forces acting on i particle along x direction, and Z_i* is a non-dimensional value of particle i in z coordinates.

_{3 simulation results}

In this study, the adoptive simulation implement is a PDETOOL software package provided by MATLAB. simulation is based on periodic boundary conditions, and the effect of electric field frequency on ER effect is not considered in the DC field. Inside these square boxes, the boundary length is 10, the uniform radius of particles is 0.5, time step is given by ?t*=0.01, Step number is 500, dimensionless time unit is 0.058 s. So the total time of simulation is 0.29.

Related parameters: R=50 μm, η_c=1.6 Pa?s, ε₀=8.85×10^{-12 F/m}, ε_p=20, ε_c=2, σ_p =1×10^-8 S/m, σ_c=0, E₀=

1 kV/mm, F₀=0.000 002 6,=0.058, l₀=2R =0.0001, S=0.003 6×, =30.

First, the configurations of ER fluids following time changes have been simulated in a poiseuille flow field between the two electrode plates, when  is 0.

In Fig.1, it can be seen that in the fifty-step time, particles of ER fluids at the middle form a bit chains. At the 100-step moment, the chains number can be seen. However, the arrays of chains are out of order, from 100 steps to 500 steps, the particle movement of ER fluids becomes slow, and the evolvement speed of configurations also gradually become slow. At the same time, the structures of chains begin to stabilize. To 500 steps, the stable chain structures have already shaped, and these chains can be seen clearly.

The evolution configurations of ER fluids particles have been simulated when the pressure gradient is 0.15. Compared with Fig.1, when the step number is 50 uniformly, the pressure gradient field speeds up the accumulation of particles, and it can be found that the effect of the pressure gradient field on displacements of particles is the most obvious in Fig.2. When step number is 500, the structures of particles in Fig.2 are more focused, compared particles in Fig.1, under no pressure gradient. Because of the pressure gradient field, the movement speeds of particles in Fig.2 are more quickly than those in Fig.1.

In addition, the relationship between the shear rate and the pressure gradient can be given by.

Fig.1 Particle structures of ER fluids in process from 0 s to 0.29 s when electric field applied: (a) Initial distribution of particles; (b) Distribution of particles at step number of 50 (The time interval is 0.029 s); (c) Distribution of particles at step number of 100 (The time interval is 0.058 s); (d) Distribution of particles at step number of 200 (The time interval is 0.116 s); (e) Distribution of particles at step number of 350 (The time interval is 0.203 s); (f) Distribution of particles at step number of 500 (The time interval is 0.290 s)

The relationship between non-dimensional shear stress and non-dimensional pressure gradient is described in a group of different relaxation time to verify the effect of dielectric relaxation time on ER effect.

Fig.3 describes that the average shear stress changes with pressure gradient, when the dipole is at the relaxation time of 0.02, 0.2 and 2 s, respectively, where the range of the pressure gradient is from 0 to 9.6, and the size of the relative shear stress is from 0 to 30. According to a parameter () representing the ratio

Fig.2 Configuration of ER fluids in process from 0 s to 0.29 s when electric field and pressure gradient applied: (a) Initial distribution of particles; (b) Distribution of particles at step number of 50 (The time interval is 0.029 s); (c) Distribution of particles at step number of 100 (The time interval is 0.058 s); (d) Distribution of particles at step number of 300 (The time interval is 0.174 s); (e) Distribution of particles at step number of 400 (The time interval is 0.232 s); (f) Distribution of particles at step number of 500 (The time interval is 0.290 s)

of electrostatic force and hydrodynamic force, it is known that the size of Mn is from 0 to 0.013 5. As shown in Fig.3(a), the ER effect is positive, which is illuminated when Mn is 0.02, the actual polarization-relaxation time that is 0.001 6 s has a positive ER effect. In Fig.3(b), when τ_MW is 0.2 s and the actual relaxation time is 0.016, a strong ER effect begins to happen, and the maximum value will appear subsequently, then the corresponding pressure gradient is 2.8 and the actual pressure gradient is 0.072 8. When the pressure gradient achieves 8.5, the

 

actual pressure gradient is 0.22, the negative ER effect will appear. The above phenomenon explains that the high pressure gradient goes against arousing ER effect. The possible cause is that the chains of ER fluids particles have been swept away by the high pressure gradient flow. In Fig.3(c), when the relaxation time is 2 (the real time is 0.16 s), it can be seen that the intensity of ER effect is very weak, besides there is also a maximum value, the corresponding relative pressure gradient is 1.5(the real pressure gradient is 0.039). In addition, when the relative pressure gradient is 3 (the real value is 0.078), the negative ER effect will appear. The reason for the above phenomenon is that the relaxation time is too long and not conducive to the formation of ER effect.

4 Conclusions

Simulation results show that under a low pressure gradient, ER effect is the most obvious because of the flow assisting in the collection of particles at the electrodes. But under a high pressure gradient, the particles are swept away by the flow. For appearing a best ER effect, a optimal dielectric relaxation frequency that is about from 10² to 10⁵ exists in theory, that is a reasonable interpretation that the negative ER effect will appear because of too long relaxation time. The analytical results show that if the relaxation time of particles is altered, the micro-chain structure of ER fluids will change and its mechanical properties will be affected.

References

[1] HOWARD S, MASAO D. Shear Resistance of ER fluids under time varying electric fields [J]. J Rheol, 1992, 36(6): 1143-1163.

[2] TAO R. Structure and dynamics of dipolar fluids under strong shear [J]. J Chem Eng Sci, 2006, 61(7): 2186-2190.

[3] ZHU Shi-sha, SUN Hong-li. Capture effect of electrorheological suspensions in flow field [J]. Chinese Journal of Mechanical Engineering, 2007, 20(2): 97-100. (in Chinese)

[4] WINSLOW W M. Induced fibrilation of suspensions [J]. J Appl phys, 1949, 20: 1137-1140.

[5] KHUSID B, ACRIVOS A. Effects of conductivity in electric-field- induced aggregation in electrorheological fluids [J]. Phys Rev E, 1995, 52(2): 1669-1693.

(Edited by HE Xue-feng)

Foundation item: Project (50771089) supported by the National Natural Science Foundation of China

Received date: 2008-06-25; Accepted date: 2008-08-05

Corresponding author: ZHU Shi-sha, Ph D; Tel: +86-732-8292547; E-mail: zssxtdx@xtu.edu.cn