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Particle mass effect of electrorheological fluids in Poiseuille flow
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Physica B 367 (2005) 229–236 www.elsevier.com/locate/physb
Particle mass effect of electrorheological fluids in Poiseuille flow Zhao Xiao-Peng, Gao Xiu-Min, Gao Dan-Jun, Luo Chun-Rong Department of Applied Physics, Institute of Electrorheological Technology 141#, Northwestern Polytechnical University, Xi’an 710072, PR China Received 23 January 2005; received in revised form 14 June 2005; accepted 14 June 2005
Abstract Considering the mass effect and short-range interaction of multi-particles in electrorheological (ER) fluids, the structure evolution in Poiseuille flow was simulated by means of the equivalent plate conduction model and the molecular dynamics method. It was seen that the velocity peak value in the transition zone decreases obviously with particle’s mass, time and width to form the plug zone increase, which was compared with those not considering the particle’s mass. The experiment to observe the structure evolution of ER fluids was also devised. The structure evolution observed in experiment complies with the simulation fundamentally. Moreover, the hypothesis that there is no slip between particles and electrodes in contact zone was also testified by our experiment. r 2005 Elsevier B.V. All rights reserved. PACS: 45.70.Mg; 47.50.+d; 45.50.j Keywords: ER fluids; Multi-interaction; Poiseuille flow; Structure evolution
1. Introduction An electrorheological (ER) fluid, a kind of typical soft matter, is usually a suspension of high dielectric constant particles dispersed in insulating oil of low dielectric constant [1–3]. Its mechanical properties (viscosity, shear stress) [4–6], optical properties (diffraction, optical rotation) [7–9] and Corresponding author. Tel.: +086 29 8495950.
E-mail address: [email protected] (Z. Xiao-Peng).
microwave behavior (scattering, attenuation) [10,11] can be adjusted under external electric field. Therefore, in the science and technology field in general, interest has been generated [12–14]. Structure evolution of ER fluids, to which an external electric field is applied, not only reflects the interaction between particles and host liquid, but also is important for designing ER components. As one of the main work status, Poiseuille flow contributes greatly to many mechanic–electric components, for example, ER fluid valves, mounts
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and dampers [15]. In early theoretical researches, the continuum model was applied to study the Poiseuille flow in ER fluids [16,17]. The ER fluid was regarded as homogeneous and continuous liquid, namely, the mean velocity gradient had no relation with position. The velocity profile between electrodes evolved from a parabolic distribution into a plug zone under an external electric field [18]. However, the continuum model takes no consideration of intrinsic structure changes in ER fluids under an external electric field. Actually, ER fluids can form chains and columns under external electric field [19–22]. Tamura and Doi [23] used molecular dynamics to simulate the Poiseuille flow of ER fluids. They treated the polarized particles as electric dipoles and obtained some results in accordance with experimental findings. Taking the edge effect into consideration, one of the authors of this paper and co-workers used the dipole model to simulate Poiseuille flow. They not only found the plug zone but also a breathing transition zone in the contact zone [24]. The dipole model is one of the basal models, through which many results in accordance with experimental findings were obtained [20–23,25]. However, this model shows its own limitations to some extent. For example, according to the dipole model, the static shear stress of ER fluids is square proportional to the external electric field, but that conflicted with the experimental result in some cases [26]. In particular, the model took no consideration of the short-range interaction of multi-particles. Considering the short-range interaction, Foulc and Felici [27] proposed a conduction model of ER fluids, and obtained a relationship between the static shear stress and the external electric field. Their calculated results agreed with experimental results . However, the conduction model considering the nonlinear characteristic made it difficult to calculate the problem of interaction of multi-particles. The Zhao research group has proposed an equivalent plate conduction model [28]. They simplified the nonlinear problem of interaction of particles to linear superposition and made it easier to calculate the interaction of multi-particles. Recently, we used the equivalent plate conduction model and molecular dynamics to simulate the Poiseuille flow of
ER fluids. The results were compared with those of the dipole model. It was found that the time taken to form the plug zone is greatly shortened and the velocity peak value of the breathing transition zone increases markedly. The three-dimensional velocity profile of particles in ER fluids and the structure evolution were also simulated [29]. However, in order to calculate easily, the mass of the particles was neglected completely in all previous researches, so the conclusions only suit those conditions that volume fraction of particles and Re number are very small. In the actual electrorheological system, the mass of the particles plays a very important role. In this paper, we first considered the mass effect of particles in ER fluids, namely, the effect of an inertia term occurring in the equation of motion of the particles was considered. The equivalent plate conduction model and molecular dynamics were employed to simulate the features of the structure evolution in Poiseuille flow. The results were compared with those obtained without taking the mass effect into account. At the same time, we made an experimental device and observed the structure evolution of ER fluids in Poiseuille flow. In addition, our experiment proved the hypothesis that particles close to the electrodes are immobile.
2. Equation of Poiseuille flow and simulation 2.1. Simulation method It is postulated that particles, which are regarded as spheres, scatter in the insulating host liquid homogeneously. d is the average particle diameter, p is the dielectric constant of the particles, f is the dielectric constant of the host fluids, m is the mass of one particle and Z is the viscosity of the host liquid. ER fluids are filled between two parallel electrodes. A homogeneous shear flow g_ ~ ex is applied to ER fluids, where g_ represents the shear rate. Generally, particles scatter randomly in ER fluids without external electric field. When the external electric field is applied, there will be electrostatic interactions among the particles. The equation of motion of the particles under the external electric field and
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the shear flow is expressed as follows [24]: 0 * 1 * * d2 Ri X ~ dR ~el 3pdZ@ i g_ zi~ Fi ¼ F m 2 ¼ ex A þ B i , i dt dt (1) *
where Ri represents the position of particle i at time ~el is the electrostatic force acting t. The first term F i on particle i, the second term is *the Stokes viscous shear force, and the third term Bi is the Brownian force. The latter two forces are applied to particle i via continuum fluids. When the electrostatic forces, the collision repulsion forces, the short-range repulsion forces, and the image forces of particles are all considered, Eq. (1) turns into
~i dR g_ zi~ 3pdZ ex dt
! þ Bi ,
where Dx ¼ xj xi , Dz ¼ zj zi , a is the average radius of the particles, ~ ip ; ~ j; ~ iy are the unit vectors of the x, y and z-axes, respectively, and s is the conductivity of the particles. Because the short-range repulsion force among particles decreases sharply with increasing distant between particles, we use an exponential form as follows [30]: * * ~rep ðR ~ij Þ ¼ F ij ðRij Þ exp ½100ðRij 1Þ~ F er . ij d The total image force applied on particle i is
~self ¼ F ~ij ð0; 2 zi Þ þ F i
ð2Þ
*
~ij ðR ~ij Þ where Rij is the relative position vector, F represents the electrostatic force exerted on particle rep ~ij Þ is the shorti by particle j at position rj , F~ij ðR range repulsion force applied upon particle i by self particle j at position rj , F~ is the total image force i
wall acting on the particle, and F~i is the short-range repulsion force between particle i and the electrodes. Based on the equivalent plate conduction model, the electrostatic interaction between two particles is [28]
~ij ðR ~ij Þ ¼ f~ðDx ; Dz þ 2 req Þ F þ f~ðDx ; Dz 2 req Þ 2 f~ðDx ; Dz Þ, ð3Þ where req is the thickness of a plate * in the equivalent plate conduction model and f ðDx ; Dz Þ is given by ffiffiffiffiffiffiffiffi2 Z Z 1 Z p1x 1 s2 a2 1 ~ dx1 dx2 pffiffiffiffiffiffiffiffi dy1 f ðDx ; Dz Þ ¼ 4pf 1 1x21 1 ffiffiffiffiffiffiffiffi p Z 1x2 ~ ~ 2 ðD þ x x Þ~ x 1 2 ip þ ðy1 y2 Þj þ Dz iy
pffiffiffiffiffiffiffiffi dy2 , 2 2 2 3=2 2 ½ðD þ x x Þ þ ðy y Þ þ D 1x2 x 1 2 1 2 z
ð4Þ
(5)
þ1 X ~ij ð0; 2 zi 2 s l z Þ ½F s¼1
~ij ð0; 2 zi þ 2 s l z Þ, F
X ~i d2 R ~ij Þ þ F ~rep ðR ~ij Þ m 2 ¼ ½F~ij ðR ij dt jai ~self þ F~wall þF i i
231
ð6Þ
where l z is the distance between two electrodes. The Brownian force is generally so weak that it can be neglected. Then the equation of motion of particle i under both pressure p0 and electric field E is given by X ~ij ðR ~ij Þ þ F~rep ðR ~ij Þ ½F ij jai 0 ~ ~self 3pdZ½dRi p z ðh zÞ ~ þF ex i dt 2Z *
d2 Ri ¼m 2 ð7Þ dt After eliminating its dimension, the scaling factors for time, velocity, and length are 3pZd2 =F 0 , F 0 =3pZd, and d respectively. We get *
X ~ d2 Ri dR i ~ ðR ~ ÞþF ~rep ðR ~ Þ þ ¼ ½F G ij ij ij ij dt 2 dt jai n
~self þ S zi ðh zi Þ ~ ex . þF i 2 ð8Þ
Here S ¼
3pZd d2 0 12p2 f d 0 p ¼ p; F0 Z s2
F0 ¼
s2 d2 , 4pf
Gn ¼
mF 0 , 9p2 Z2 d3 ð9Þ
where i; j ¼ 1; 2; :::; M (M is the number of particles), h is the distance between the top and
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bottom contact zones, and S is the non-dimensional pressure gradient or relative pressure gradient. Gn represents the coefficient of the mass term, which reveals the mass effect of particles or their volume fraction. From the above processes, we can infer that Gn is the equivalent particle mass or the mass of the particles in unit volume, so G n increases with the volume fraction of particles in ER fluids. The simulation unit cell Lx Lz ¼ 2 30 is assumed, and the number of particles in the total chain, which includes an entire chain under the effect of an electric field, is 30. Five chains periodically distributed left or right are used to simulate infinite chains formed in the horizontal direction. The same set-up was used in the literature [24,28,29]. Based on observing experiments, the chain-like or column-like structure of the particles has been assumed. 2.2. Simulation results and discussion If the volume fraction is so high that the coefficient of the mass term is approximately equivalent to those of other terms in the equation of motion, the mass of particles should be considered. Since particles in ER fluids form chains and columns under the external electric field, the effects induced by the particle mass should not be neglected during the motion of particles. In addition, in flow researches focused on the problem of dynamic responses, neglecting the mass term would result in a certain inaccuracy. Strictly speaking, the mass term can be neglected just under the condition that the volume fraction and Re number are very small. In this paper, Eq. (8) was resolved by the method of four-power Runge–Kutta. Choosing G* ¼ 0.1–0.4 and S ¼ 0:002, we calculated the two-dimensional velocity and position profile, and the three-dimensional velocity evolution profile, respectively. 2.2.1. Two-dimensional velocity evolution profile The evolution process of the two-dimensional velocity profile is studied for G* ¼ 0.1–0.4 . Fig. 1 shows the velocity sections with G* ¼ 0.1 at different times. The velocity section is depicted as curve a in this figure at t ¼ 0. When the electric field is applied, the velocities of the particles
30 25 20 N
232
a
b
15 10 5 0 0.0
0.2
0.4
0.6 V
0.8
1.0
1.2
Fig. 1. Velocity evolving profile with G* ¼ 0.1.
decrease with time, and the velocity section changes monotonically with the envelope of curve a. The central particles, which flow up more rapidly than the side particles, tend to be at the same speed. The velocity section reaches curve b at the moment t ¼ 0:747, when the volume flux reaches a minimum. The velocity of the middle particles, which move forwards as a whole, is almost the same. These phenomena are similar to the results from the continuous medium mechanics method, where it is referred to as the ‘‘plug zone’’. After the ER fluids are in the state of the plug zone, the so-called ‘transition zone’ appears in the velocity profile. The speed of particles close to the two electrodes changes sharply, increasing and then decreasing rapidly in a short time. During that period, the velocity peaks appear at both ends of the profile. Fig. 2 shows the velocity sections of G* ¼ 0.4 at different times. We can conclude from the simulation that the velocity peak value of the transition zone decreases and the width of the plug zone broadens with the increasing mass term. Table 1 shows the results with different coefficients of mass terms. It can be seen that the time to form the plug zone for G 40 is longer than that for G ¼ 0. All these phenomena are attributed to the mass effect of particles in high volume fraction ER fluids, in which particles form the compact structure of chains and columns under the external electric field. With increasing volume fraction, the interactions among particles become stronger and
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30
25
25
20
20
a
b
15
N
N
233
15
10
10
5
5
0 0.0
a1
a2
a3
a4
0
0.2
0.4
0.6 V
0.8
1.0
1.2
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 X
Fig. 2. Velocity evolving profile with G* ¼ 0.4.
Fig. 3. Position profiles of particles with Gn ¼ 0:3.
Table 1 Peak value and plug time with different coefficients of mass terms
2.2.3. Three-dimensional velocity evolving profile The structure of the ER fluid under an applied electric field is not limited to a two-dimensional multi-chain structure. Tao and Sun [3] verified that it is a BCT structure. The column in an ER fluid is usually rather thick; sometime it includes several hundred chains. For example, one of the columns observed in the experiments by Chen et al. [31] is composed of approximately 50 12 chains . It has been determined that the yield stress of a five-chain BCT structure is 39% greater than that of a single chain and the yield stress of a nine-chain BCT structure is 72% greater [28]. Thus, the threedimensional velocity evolving profiles need to be further studied and the results will approach reality more closely. The three-dimensional velocity evolving process in ER fluids is simulated at Gn ¼ 0:3. When a pressure gradient is applied to ER fluids, the velocity profile turns out to be a parabolic distribution. After the chain got into plug states, the velocity decreases, and the middle segment of the profile becomes relatively smooth. The peak value in the velocity profile comes out when the chain gets into the transitional state (Fig. 4(a)). Subsequently, fluids come back to the plug state again with a wider plug zone amid the profile (Fig. 4(b)). We conclude that the peak value in velocity profile decreases obviously and the width of the plug zone increases clearly compared with the simulation without considering the mass term [29].
Coefficients
0
0.1
0.2
0.3
0.4
Peak value Plug time
1.03 7.8
0.747 9.3
0.635 9.4
0.540 9.4
0.459 9.4
the chains become denser, so the mass term effect gets more significant. 2.2.2. Two-dimension position profiles Fig. 3 shows the positions of a single chain at different time with Gn ¼ 0:3. The contact zones are located at the top and bottom of the chains. The thickness is the diameter of a particle. There is no slip between particles and the electrodes in the contact zone. The initial position of the chain is shown as state a1 with t ¼ 0 in Fig. 3. The chain, which is forced to move parallel to the x-axis, takes 9.3-unit time to get into plug state a2. Compared with the simulation without considering the mass effect, the curvature of the chain is smaller. With increasing time, the chain gets into the transitional state (see a3), forward curving appears at each end of the chain like that when the mass effect is neglected [29], but the curvature is smaller. Finally, the chain moves forward and comes back into plug states again. At the same time, the middle segment of the chain becomes smoother.
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234 1.4 1.2
Velocity
1.0 0.8 0.6 0.4
Fig. 5. Schematic diagram of experimental setup. 0.2 0.0 0
1
(a)
2 Y
3
10
5
4 0
25 30 15 20 s Particle
1.4 1.2
Velocity
1.0 0.8 0.6 0.4 0.2 0.0
(b)
0
1
2 Y
3
4
0
5
10
25 15 20 s Particle
30
Fig. 4. Evolution of three-dimensional velocity profile with mass effect.
3. Experiment of Poiseuille flow in ER fluids 3.1. Equipment and method Fig. 5 is the schematic figure of the experimental equipment designed to observe Poiseuille flow. The ER fluid valve, in which we can observe Poiseuille flow, was composed of two parallel electrodes made of transparent glass. The inner surfaces were coated with conductive films as electrodes. We can observe the fluid flow in any direction because the electrodes are transparent. The length of the ER fluid valve is 80.0 mm, the width is 11.5 mm, and the height is 6.0 mm.The distance between the electrodes is 11.5 mm. Two pipes are connected with each end of the valve so that ER fluids can flow through easily. The electrodes are connected to a high-voltage power source (0–5 kV). The ER
fluid used in our experiment is composed of starch powder and voltage transformer oil, and the volume fraction is 10% and 20%, respectively. During experiment, ER fluids were injected into the valve at a certain speed, and the electric field was applied to ER fluids. The CCD picked up pictures of the structure evolution process of ER fluids. At the same time, we also can take the picture of the process. The video recorder (VRC) recorded the process by tape. A video card passes the graphic information into the computer through video ports and transforms the graphs into digital video files. The application program (Video Capture) of video card (EZ Capture) captures the graphs by frame. Comparing the difference between images in different frames, the computer can give us the structure evolution profile. In our experiment, the following experimental procedures are adopted. (1) Put ER fluids into ER fluids valve slowly. (2) Keep the external pressure unchanged, and make the ER fluids flow naturally. (3) Increase the external electric field slowly, and observe the flow states of ER fluids. In another state, keep the external pressure unchanged, increase the external electric field slowly, and observe the states of ER fluids in the valve. The two-dimensional position profiles and the change of the angle formed by the end segment and the electrodes are observed experimentally. The hypothesis that the particles clinging to the electrodes are not movable is also observed in experiment. 3.2. Results and discussion (1) When the volume fraction of the ER fluid is 10%, and 200 V/mm external electric field is
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applied, the particles in the ER fluid form chains quickly. Under the effect of pressure, these chains curve forward, but the middle segments of the chains become relatively smooth (as in Fig. 6). The experimental result complies with that of the simulation (as in Fig. 3). (2) With increasing the pressure gradient, the particle chains break. The middle segments of the chains move forward, while the end segments located in contact zones tilt (shown in Fig. 6(a)). The end segment formed an angle with the electrodes, and this angle increases with the pressure gradient. If the pressure keeps constant, the angle decreases as the external electric field increases. When the pressure gradient P keeps increasing, the leaned endsegment chains slide or drift along the electrodes. The chains then were called drifting chains. These drifting chains would move forward and
Fig. 6. Structure evolution process of ER fluids in experiment.
235
combine with other middle segment chains to become new chains. (3) Keeping the external electric field at 200 V/mm and increasing the pressure gradient P, when P is smaller than a certain constant Pmax, we observe that the particles near the electrodes do not move and form contact zones. If P reaches Pmax, particles clinging to the electrodes begin to move. If the electric field increases to 400 V/mm, even though P reaches Pmax, the particles clinging to the electrodes stay still stationary. We can conclude that the particles near the electrodes do not move when P is smaller than Pmax (shown as Fig. 6(b)). Therefore, the hypothesis in reference [32,33] that there is no slip between particles and electrodes in contact zone was testified in our experiment. (4) For a volume fraction of ER fluid of 20% at an electric field of 200 V/mm, the chain forming process is faster than that of the 10% volume fraction ER fluid. Under the electric field of 400 V/mm, particles would form chains and columns at once, and the structure almost does not change under a low-pressure gradient. Chains and columns curve forward and then deposit, obviously, with increasing pressure gradient. Based on the experiment observation, we found that there are two stages of structure evolution in ER fluids: initial stage and yield stage. In the initial stage, the polarized particles attract each other and form chains parallel to the electric field between electrodes; then the particles gather and deposit in the valve under the external electric field. Because the time to reconstruct the chains and columns is much shorter than the characteristic time of flow in ER fluids, it is easy to observe the chains and their branches, formed by gathered particles. In the yield stage, the valve was filled with ER fluids, and the deposit of particles makes the flow path narrow. Then the valve is divided into two parts: static deposit part and flow path. The flow path always forms at the place where the flow resistance is minimum. In our research, theoretical and experimental results accord only qualitatively. There are two main relations: (1) In the theoretical Section 2.2.2,
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the simulation shows that the chains curve forward and middle segments of chains become relatively smooth (see Fig. 3) under an external electric field and pressure. In experimental Fig. 6(a), we can see that chains curve forward and middle segments of chains become relatively smooth, in accordance with the simulation result. (2) In the literature and in our simulation, there is a hypothesis that the particles clinging to the electrodes are not movable. In our experiment, this hypothesis is testified. Shown as Fig. 6, there is no slip between particles and electrodes in the contact zone. 4. Conclusions The velocity peak value of the transitional zone in ER fluids decreases obviously, but the time to form the plug zone prolongs and the width of the plug zone increases with the mass term. In the evolution process of ER fluids under a pressure gradient, the middle segments of the chains curve forward, and form a relatively smooth zone. The angle between the electrodes and the end segments increases with increasing pressure gradient, and decreases with the external electric field. When P is smaller than the critical value Pmax, the particles near the electrodes are movable and Pmax also increases with the external electric field.
Acknowledgments This work was supported by the National Nature Science Foundation of China for Distinguished Young Scholar under Grant no. 50025207 and the Key Research Plan of the National Science Foundation of China under Grant no. 90101005. References [1] T.C. Halsey, Science 258 (1992) 761. [2] J.T. Woestmena, Phys. Rev. E, 47 (1993) 2942.
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Physica B 367 (2005) 229–236 www.elsevier.com/locate/physb
Particle mass effect of electrorheological fluids in Poiseuille flow Zhao Xiao-Peng, Gao Xiu-Min, Gao Dan-Jun, Luo Chun-Rong Department of Applied Physics, Institute of Electrorheological Technology 141#, Northwestern Polytechnical University, Xi’an 710072, PR China Received 23 January 2005; received in revised form 14 June 2005; accepted 14 June 2005
Abstract Considering the mass effect and short-range interaction of multi-particles in electrorheological (ER) fluids, the structure evolution in Poiseuille flow was simulated by means of the equivalent plate conduction model and the molecular dynamics method. It was seen that the velocity peak value in the transition zone decreases obviously with particle’s mass, time and width to form the plug zone increase, which was compared with those not considering the particle’s mass. The experiment to observe the structure evolution of ER fluids was also devised. The structure evolution observed in experiment complies with the simulation fundamentally. Moreover, the hypothesis that there is no slip between particles and electrodes in contact zone was also testified by our experiment. r 2005 Elsevier B.V. All rights reserved. PACS: 45.70.Mg; 47.50.+d; 45.50.j Keywords: ER fluids; Multi-interaction; Poiseuille flow; Structure evolution
1. Introduction An electrorheological (ER) fluid, a kind of typical soft matter, is usually a suspension of high dielectric constant particles dispersed in insulating oil of low dielectric constant [1–3]. Its mechanical properties (viscosity, shear stress) [4–6], optical properties (diffraction, optical rotation) [7–9] and Corresponding author. Tel.: +086 29 8495950.
E-mail address: [email protected] (Z. Xiao-Peng).
microwave behavior (scattering, attenuation) [10,11] can be adjusted under external electric field. Therefore, in the science and technology field in general, interest has been generated [12–14]. Structure evolution of ER fluids, to which an external electric field is applied, not only reflects the interaction between particles and host liquid, but also is important for designing ER components. As one of the main work status, Poiseuille flow contributes greatly to many mechanic–electric components, for example, ER fluid valves, mounts
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.06.023
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and dampers [15]. In early theoretical researches, the continuum model was applied to study the Poiseuille flow in ER fluids [16,17]. The ER fluid was regarded as homogeneous and continuous liquid, namely, the mean velocity gradient had no relation with position. The velocity profile between electrodes evolved from a parabolic distribution into a plug zone under an external electric field [18]. However, the continuum model takes no consideration of intrinsic structure changes in ER fluids under an external electric field. Actually, ER fluids can form chains and columns under external electric field [19–22]. Tamura and Doi [23] used molecular dynamics to simulate the Poiseuille flow of ER fluids. They treated the polarized particles as electric dipoles and obtained some results in accordance with experimental findings. Taking the edge effect into consideration, one of the authors of this paper and co-workers used the dipole model to simulate Poiseuille flow. They not only found the plug zone but also a breathing transition zone in the contact zone [24]. The dipole model is one of the basal models, through which many results in accordance with experimental findings were obtained [20–23,25]. However, this model shows its own limitations to some extent. For example, according to the dipole model, the static shear stress of ER fluids is square proportional to the external electric field, but that conflicted with the experimental result in some cases [26]. In particular, the model took no consideration of the short-range interaction of multi-particles. Considering the short-range interaction, Foulc and Felici [27] proposed a conduction model of ER fluids, and obtained a relationship between the static shear stress and the external electric field. Their calculated results agreed with experimental results . However, the conduction model considering the nonlinear characteristic made it difficult to calculate the problem of interaction of multi-particles. The Zhao research group has proposed an equivalent plate conduction model [28]. They simplified the nonlinear problem of interaction of particles to linear superposition and made it easier to calculate the interaction of multi-particles. Recently, we used the equivalent plate conduction model and molecular dynamics to simulate the Poiseuille flow of
ER fluids. The results were compared with those of the dipole model. It was found that the time taken to form the plug zone is greatly shortened and the velocity peak value of the breathing transition zone increases markedly. The three-dimensional velocity profile of particles in ER fluids and the structure evolution were also simulated [29]. However, in order to calculate easily, the mass of the particles was neglected completely in all previous researches, so the conclusions only suit those conditions that volume fraction of particles and Re number are very small. In the actual electrorheological system, the mass of the particles plays a very important role. In this paper, we first considered the mass effect of particles in ER fluids, namely, the effect of an inertia term occurring in the equation of motion of the particles was considered. The equivalent plate conduction model and molecular dynamics were employed to simulate the features of the structure evolution in Poiseuille flow. The results were compared with those obtained without taking the mass effect into account. At the same time, we made an experimental device and observed the structure evolution of ER fluids in Poiseuille flow. In addition, our experiment proved the hypothesis that particles close to the electrodes are immobile.
2. Equation of Poiseuille flow and simulation 2.1. Simulation method It is postulated that particles, which are regarded as spheres, scatter in the insulating host liquid homogeneously. d is the average particle diameter, p is the dielectric constant of the particles, f is the dielectric constant of the host fluids, m is the mass of one particle and Z is the viscosity of the host liquid. ER fluids are filled between two parallel electrodes. A homogeneous shear flow g_ ~ ex is applied to ER fluids, where g_ represents the shear rate. Generally, particles scatter randomly in ER fluids without external electric field. When the external electric field is applied, there will be electrostatic interactions among the particles. The equation of motion of the particles under the external electric field and
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the shear flow is expressed as follows [24]: 0 * 1 * * d2 Ri X ~ dR ~el 3pdZ@ i g_ zi~ Fi ¼ F m 2 ¼ ex A þ B i , i dt dt (1) *
where Ri represents the position of particle i at time ~el is the electrostatic force acting t. The first term F i on particle i, the second term is *the Stokes viscous shear force, and the third term Bi is the Brownian force. The latter two forces are applied to particle i via continuum fluids. When the electrostatic forces, the collision repulsion forces, the short-range repulsion forces, and the image forces of particles are all considered, Eq. (1) turns into
~i dR g_ zi~ 3pdZ ex dt
! þ Bi ,
where Dx ¼ xj xi , Dz ¼ zj zi , a is the average radius of the particles, ~ ip ; ~ j; ~ iy are the unit vectors of the x, y and z-axes, respectively, and s is the conductivity of the particles. Because the short-range repulsion force among particles decreases sharply with increasing distant between particles, we use an exponential form as follows [30]: * * ~rep ðR ~ij Þ ¼ F ij ðRij Þ exp ½100ðRij 1Þ~ F er . ij d The total image force applied on particle i is
~self ¼ F ~ij ð0; 2 zi Þ þ F i
ð2Þ
*
~ij ðR ~ij Þ where Rij is the relative position vector, F represents the electrostatic force exerted on particle rep ~ij Þ is the shorti by particle j at position rj , F~ij ðR range repulsion force applied upon particle i by self particle j at position rj , F~ is the total image force i
wall acting on the particle, and F~i is the short-range repulsion force between particle i and the electrodes. Based on the equivalent plate conduction model, the electrostatic interaction between two particles is [28]
~ij ðR ~ij Þ ¼ f~ðDx ; Dz þ 2 req Þ F þ f~ðDx ; Dz 2 req Þ 2 f~ðDx ; Dz Þ, ð3Þ where req is the thickness of a plate * in the equivalent plate conduction model and f ðDx ; Dz Þ is given by ffiffiffiffiffiffiffiffi2 Z Z 1 Z p1x 1 s2 a2 1 ~ dx1 dx2 pffiffiffiffiffiffiffiffi dy1 f ðDx ; Dz Þ ¼ 4pf 1 1x21 1 ffiffiffiffiffiffiffiffi p Z 1x2 ~ ~ 2 ðD þ x x Þ~ x 1 2 ip þ ðy1 y2 Þj þ Dz iy
pffiffiffiffiffiffiffiffi dy2 , 2 2 2 3=2 2 ½ðD þ x x Þ þ ðy y Þ þ D 1x2 x 1 2 1 2 z
ð4Þ
(5)
þ1 X ~ij ð0; 2 zi 2 s l z Þ ½F s¼1
~ij ð0; 2 zi þ 2 s l z Þ, F
X ~i d2 R ~ij Þ þ F ~rep ðR ~ij Þ m 2 ¼ ½F~ij ðR ij dt jai ~self þ F~wall þF i i
231
ð6Þ
where l z is the distance between two electrodes. The Brownian force is generally so weak that it can be neglected. Then the equation of motion of particle i under both pressure p0 and electric field E is given by X ~ij ðR ~ij Þ þ F~rep ðR ~ij Þ ½F ij jai 0 ~ ~self 3pdZ½dRi p z ðh zÞ ~ þF ex i dt 2Z *
d2 Ri ¼m 2 ð7Þ dt After eliminating its dimension, the scaling factors for time, velocity, and length are 3pZd2 =F 0 , F 0 =3pZd, and d respectively. We get *
X ~ d2 Ri dR i ~ ðR ~ ÞþF ~rep ðR ~ Þ þ ¼ ½F G ij ij ij ij dt 2 dt jai n
~self þ S zi ðh zi Þ ~ ex . þF i 2 ð8Þ
Here S ¼
3pZd d2 0 12p2 f d 0 p ¼ p; F0 Z s2
F0 ¼
s2 d2 , 4pf
Gn ¼
mF 0 , 9p2 Z2 d3 ð9Þ
where i; j ¼ 1; 2; :::; M (M is the number of particles), h is the distance between the top and
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bottom contact zones, and S is the non-dimensional pressure gradient or relative pressure gradient. Gn represents the coefficient of the mass term, which reveals the mass effect of particles or their volume fraction. From the above processes, we can infer that Gn is the equivalent particle mass or the mass of the particles in unit volume, so G n increases with the volume fraction of particles in ER fluids. The simulation unit cell Lx Lz ¼ 2 30 is assumed, and the number of particles in the total chain, which includes an entire chain under the effect of an electric field, is 30. Five chains periodically distributed left or right are used to simulate infinite chains formed in the horizontal direction. The same set-up was used in the literature [24,28,29]. Based on observing experiments, the chain-like or column-like structure of the particles has been assumed. 2.2. Simulation results and discussion If the volume fraction is so high that the coefficient of the mass term is approximately equivalent to those of other terms in the equation of motion, the mass of particles should be considered. Since particles in ER fluids form chains and columns under the external electric field, the effects induced by the particle mass should not be neglected during the motion of particles. In addition, in flow researches focused on the problem of dynamic responses, neglecting the mass term would result in a certain inaccuracy. Strictly speaking, the mass term can be neglected just under the condition that the volume fraction and Re number are very small. In this paper, Eq. (8) was resolved by the method of four-power Runge–Kutta. Choosing G* ¼ 0.1–0.4 and S ¼ 0:002, we calculated the two-dimensional velocity and position profile, and the three-dimensional velocity evolution profile, respectively. 2.2.1. Two-dimensional velocity evolution profile The evolution process of the two-dimensional velocity profile is studied for G* ¼ 0.1–0.4 . Fig. 1 shows the velocity sections with G* ¼ 0.1 at different times. The velocity section is depicted as curve a in this figure at t ¼ 0. When the electric field is applied, the velocities of the particles
30 25 20 N
232
a
b
15 10 5 0 0.0
0.2
0.4
0.6 V
0.8
1.0
1.2
Fig. 1. Velocity evolving profile with G* ¼ 0.1.
decrease with time, and the velocity section changes monotonically with the envelope of curve a. The central particles, which flow up more rapidly than the side particles, tend to be at the same speed. The velocity section reaches curve b at the moment t ¼ 0:747, when the volume flux reaches a minimum. The velocity of the middle particles, which move forwards as a whole, is almost the same. These phenomena are similar to the results from the continuous medium mechanics method, where it is referred to as the ‘‘plug zone’’. After the ER fluids are in the state of the plug zone, the so-called ‘transition zone’ appears in the velocity profile. The speed of particles close to the two electrodes changes sharply, increasing and then decreasing rapidly in a short time. During that period, the velocity peaks appear at both ends of the profile. Fig. 2 shows the velocity sections of G* ¼ 0.4 at different times. We can conclude from the simulation that the velocity peak value of the transition zone decreases and the width of the plug zone broadens with the increasing mass term. Table 1 shows the results with different coefficients of mass terms. It can be seen that the time to form the plug zone for G 40 is longer than that for G ¼ 0. All these phenomena are attributed to the mass effect of particles in high volume fraction ER fluids, in which particles form the compact structure of chains and columns under the external electric field. With increasing volume fraction, the interactions among particles become stronger and
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30
25
25
20
20
a
b
15
N
N
233
15
10
10
5
5
0 0.0
a1
a2
a3
a4
0
0.2
0.4
0.6 V
0.8
1.0
1.2
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 X
Fig. 2. Velocity evolving profile with G* ¼ 0.4.
Fig. 3. Position profiles of particles with Gn ¼ 0:3.
Table 1 Peak value and plug time with different coefficients of mass terms
2.2.3. Three-dimensional velocity evolving profile The structure of the ER fluid under an applied electric field is not limited to a two-dimensional multi-chain structure. Tao and Sun [3] verified that it is a BCT structure. The column in an ER fluid is usually rather thick; sometime it includes several hundred chains. For example, one of the columns observed in the experiments by Chen et al. [31] is composed of approximately 50 12 chains . It has been determined that the yield stress of a five-chain BCT structure is 39% greater than that of a single chain and the yield stress of a nine-chain BCT structure is 72% greater [28]. Thus, the threedimensional velocity evolving profiles need to be further studied and the results will approach reality more closely. The three-dimensional velocity evolving process in ER fluids is simulated at Gn ¼ 0:3. When a pressure gradient is applied to ER fluids, the velocity profile turns out to be a parabolic distribution. After the chain got into plug states, the velocity decreases, and the middle segment of the profile becomes relatively smooth. The peak value in the velocity profile comes out when the chain gets into the transitional state (Fig. 4(a)). Subsequently, fluids come back to the plug state again with a wider plug zone amid the profile (Fig. 4(b)). We conclude that the peak value in velocity profile decreases obviously and the width of the plug zone increases clearly compared with the simulation without considering the mass term [29].
Coefficients
0
0.1
0.2
0.3
0.4
Peak value Plug time
1.03 7.8
0.747 9.3
0.635 9.4
0.540 9.4
0.459 9.4
the chains become denser, so the mass term effect gets more significant. 2.2.2. Two-dimension position profiles Fig. 3 shows the positions of a single chain at different time with Gn ¼ 0:3. The contact zones are located at the top and bottom of the chains. The thickness is the diameter of a particle. There is no slip between particles and the electrodes in the contact zone. The initial position of the chain is shown as state a1 with t ¼ 0 in Fig. 3. The chain, which is forced to move parallel to the x-axis, takes 9.3-unit time to get into plug state a2. Compared with the simulation without considering the mass effect, the curvature of the chain is smaller. With increasing time, the chain gets into the transitional state (see a3), forward curving appears at each end of the chain like that when the mass effect is neglected [29], but the curvature is smaller. Finally, the chain moves forward and comes back into plug states again. At the same time, the middle segment of the chain becomes smoother.
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234 1.4 1.2
Velocity
1.0 0.8 0.6 0.4
Fig. 5. Schematic diagram of experimental setup. 0.2 0.0 0
1
(a)
2 Y
3
10
5
4 0
25 30 15 20 s Particle
1.4 1.2
Velocity
1.0 0.8 0.6 0.4 0.2 0.0
(b)
0
1
2 Y
3
4
0
5
10
25 15 20 s Particle
30
Fig. 4. Evolution of three-dimensional velocity profile with mass effect.
3. Experiment of Poiseuille flow in ER fluids 3.1. Equipment and method Fig. 5 is the schematic figure of the experimental equipment designed to observe Poiseuille flow. The ER fluid valve, in which we can observe Poiseuille flow, was composed of two parallel electrodes made of transparent glass. The inner surfaces were coated with conductive films as electrodes. We can observe the fluid flow in any direction because the electrodes are transparent. The length of the ER fluid valve is 80.0 mm, the width is 11.5 mm, and the height is 6.0 mm.The distance between the electrodes is 11.5 mm. Two pipes are connected with each end of the valve so that ER fluids can flow through easily. The electrodes are connected to a high-voltage power source (0–5 kV). The ER
fluid used in our experiment is composed of starch powder and voltage transformer oil, and the volume fraction is 10% and 20%, respectively. During experiment, ER fluids were injected into the valve at a certain speed, and the electric field was applied to ER fluids. The CCD picked up pictures of the structure evolution process of ER fluids. At the same time, we also can take the picture of the process. The video recorder (VRC) recorded the process by tape. A video card passes the graphic information into the computer through video ports and transforms the graphs into digital video files. The application program (Video Capture) of video card (EZ Capture) captures the graphs by frame. Comparing the difference between images in different frames, the computer can give us the structure evolution profile. In our experiment, the following experimental procedures are adopted. (1) Put ER fluids into ER fluids valve slowly. (2) Keep the external pressure unchanged, and make the ER fluids flow naturally. (3) Increase the external electric field slowly, and observe the flow states of ER fluids. In another state, keep the external pressure unchanged, increase the external electric field slowly, and observe the states of ER fluids in the valve. The two-dimensional position profiles and the change of the angle formed by the end segment and the electrodes are observed experimentally. The hypothesis that the particles clinging to the electrodes are not movable is also observed in experiment. 3.2. Results and discussion (1) When the volume fraction of the ER fluid is 10%, and 200 V/mm external electric field is
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applied, the particles in the ER fluid form chains quickly. Under the effect of pressure, these chains curve forward, but the middle segments of the chains become relatively smooth (as in Fig. 6). The experimental result complies with that of the simulation (as in Fig. 3). (2) With increasing the pressure gradient, the particle chains break. The middle segments of the chains move forward, while the end segments located in contact zones tilt (shown in Fig. 6(a)). The end segment formed an angle with the electrodes, and this angle increases with the pressure gradient. If the pressure keeps constant, the angle decreases as the external electric field increases. When the pressure gradient P keeps increasing, the leaned endsegment chains slide or drift along the electrodes. The chains then were called drifting chains. These drifting chains would move forward and
Fig. 6. Structure evolution process of ER fluids in experiment.
235
combine with other middle segment chains to become new chains. (3) Keeping the external electric field at 200 V/mm and increasing the pressure gradient P, when P is smaller than a certain constant Pmax, we observe that the particles near the electrodes do not move and form contact zones. If P reaches Pmax, particles clinging to the electrodes begin to move. If the electric field increases to 400 V/mm, even though P reaches Pmax, the particles clinging to the electrodes stay still stationary. We can conclude that the particles near the electrodes do not move when P is smaller than Pmax (shown as Fig. 6(b)). Therefore, the hypothesis in reference [32,33] that there is no slip between particles and electrodes in contact zone was testified in our experiment. (4) For a volume fraction of ER fluid of 20% at an electric field of 200 V/mm, the chain forming process is faster than that of the 10% volume fraction ER fluid. Under the electric field of 400 V/mm, particles would form chains and columns at once, and the structure almost does not change under a low-pressure gradient. Chains and columns curve forward and then deposit, obviously, with increasing pressure gradient. Based on the experiment observation, we found that there are two stages of structure evolution in ER fluids: initial stage and yield stage. In the initial stage, the polarized particles attract each other and form chains parallel to the electric field between electrodes; then the particles gather and deposit in the valve under the external electric field. Because the time to reconstruct the chains and columns is much shorter than the characteristic time of flow in ER fluids, it is easy to observe the chains and their branches, formed by gathered particles. In the yield stage, the valve was filled with ER fluids, and the deposit of particles makes the flow path narrow. Then the valve is divided into two parts: static deposit part and flow path. The flow path always forms at the place where the flow resistance is minimum. In our research, theoretical and experimental results accord only qualitatively. There are two main relations: (1) In the theoretical Section 2.2.2,
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the simulation shows that the chains curve forward and middle segments of chains become relatively smooth (see Fig. 3) under an external electric field and pressure. In experimental Fig. 6(a), we can see that chains curve forward and middle segments of chains become relatively smooth, in accordance with the simulation result. (2) In the literature and in our simulation, there is a hypothesis that the particles clinging to the electrodes are not movable. In our experiment, this hypothesis is testified. Shown as Fig. 6, there is no slip between particles and electrodes in the contact zone. 4. Conclusions The velocity peak value of the transitional zone in ER fluids decreases obviously, but the time to form the plug zone prolongs and the width of the plug zone increases with the mass term. In the evolution process of ER fluids under a pressure gradient, the middle segments of the chains curve forward, and form a relatively smooth zone. The angle between the electrodes and the end segments increases with increasing pressure gradient, and decreases with the external electric field. When P is smaller than the critical value Pmax, the particles near the electrodes are movable and Pmax also increases with the external electric field.
Acknowledgments This work was supported by the National Nature Science Foundation of China for Distinguished Young Scholar under Grant no. 50025207 and the Key Research Plan of the National Science Foundation of China under Grant no. 90101005. References [1] T.C. Halsey, Science 258 (1992) 761. [2] J.T. Woestmena, Phys. Rev. E, 47 (1993) 2942.
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