Lattice Boltzmann Simulation of One Particle Migrating in a Pulsating Flow in Microvessel
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1 Commun. Theor. Phys. 56 (2011) Vol. 56, No. 4, October 15, 2011 Lattice Boltzmann Simulation of One Particle Migrating in a Pulsating Flow in Microvessel QIU Bing ( ), 1, TAN Hui-Li ( Û), 2 and LI Hua-Bing (ÓÙÏ) 1 1 School of Material Science and Engineering, Guilin University of Electronic Technology, Guilin , China 2 School of Physical Science and Technology, Guangxi Normal University, Guilin , China (Received April 15, 2011) Abstract A lattice Boltzmann model of two dimensions is used to simulate the movement of a single rigid particle suspended in a pulsating flow in micro vessel. The particle is as big as a red blood cell, and the micro vessel is four times as wide as the diameter of the particle. It is found that Segré Silberberg effect will not respond to the pulsation of the flow when the Reynolds number is relatively high. However, when the Reynolds number is low enough, Segré Silberberg effect disappears. In the steady flow, different initial position leads to different equilibrium positions. In a pulsating flow, different frequencies of pulsation also cause different equilibrium positions. Particularly, when the frequency of pulsation is closed to the human heart rate, Segré Silberberg effect presents again. The evolutions of velocity, rotation, and trajectory of the particle are investigated to find the dynamics of such abnormal phenomenon. PACS numbers: ef, mf Key words: Segré Silberberg effect, lattice Boltzmann method, red blood cell, pulsating flow The movements of suspended particles in flows are sometimes abnormal. For example, it was observed that suspended rigid particles in laminar flow tend to concentrate into a region about half-way between the centerline and the wall of tube. [1] This remarkable phenomenon is called Segré Silberberg effect, and lots of researches have been done to explain its dynamics. [2 4] However, most of these researches are based on a steady flow and under a macro scale. The movements of suspended particles in micro flows and pulsating flows are supposed to be abnormal. In this article we built a lattice Boltzmann model to find weather Segré Silberberg effect could also be presented in flows of very low Reynolds number and flows of pulsating. In our researches, a rigid suspended particle as big as a human red blood cell, is set in a two-dimensional long straight vessel with its width four times the diameter of the particle. It has been found that Segré Silberberg effect is distinct under this proportion of system size. [5] The suspended particle is released in the pulsating flow from at rest and then migrates with the fluid. The pressure drop is adjusted to keep the velocity of the particle in a reasonable value. For the size of system is very small, and the reasonable velocity of a red blood cell in micro vessel is slow, the Reynolds number (Re) is very low accordingly. The trajectories of the particle from different initial positions or under different frequency of pulsation are compared with that in a relatively high Re flow. Our study is based on the lattice Boltzmann method, which has been widely used nowadays in complex flow researches. [6 8] Recently, we have used this method to simulate the deformation of red blood cell in shear flow. [9] We have also proposed a scheme [10] to simplify the calculation and improved the stability of simulation as well, which has been proved to be accurate when Reynolds Numbers is small. We will use this scheme in our researches to evaluate the hydrodynamic forces exerting on the boundary of the particle. For simplicity, D2Q9 lattice Boltzmann model [11] is adopted in our researches. The evolution of distribution function f i ( x, t) is computed by the following formula. [12] f i ( x + e i δt, t + δt) f i ( x, t) = 1 τ (f i f (eq) i ), (1) where i represents the direction of velocity, δt is a timestep. τ is the single relaxation time, which is related to the dimensionless viscous coefficients of the fluid υ and in our model their relationship is: f (eq) i υ = 2τ 1 6, (2) is the equilibrium distribution function. It is computed by the following formula: [11] [ = α i ρ e i u ( e i u) u2], (3) f (eq) i where α 0 = 4/9, α 1 = α 2 = α 3 = α 4 = 1/9, α 5 = α 6 = α 7 = α 8 = 1/36. The density of fluid ρ and the flow velocity u are calculated by the following formula. ρ = f i, (4) Supported by the National Natural Science Foundation of China under Grant Nos , , and To whom correspondence should be addressed, qbingbing@sina.com c 2011 Chinese Physical Society and IOP Publishing Ltd i=0
2 No. 4 Communications in Theoretical Physics 757 u = 1 ρ f i e i. (5) i=0 Half-way bounceback boundary condition [13] is imposed at the walls of the vessel. And periodic boundary condition [14] is imposed at the inlet and outlet of the simulating region. Despite the space in fluid is discrete, the particle can move across the lattices under the action of hydrodynamic force. The hydrodynamic force and the torque acting on the particle are calculated by integrating the stress tensor on the boundary of it using the following formula: [14] F = [ˆσ ρ u( u u S )] ds, (6) S T = r {[ˆσ ρ u( u u S )] ds}, (7) S σ xy = 1 2τ Pδ xy τ 1/2 2τ f i (e ix u x )(e iy u y ), (8) i 1 P = 1 3 ρ, (9) P here is the pressure and ˆσ is the stress tensor in the fluid. S is the boundary of the particle. The distribution function on the boundary f i is replaced by the distribution function on the fluid node nearest to the boundary. [10] The boundary condition on the surface of the particle is solved with the scheme which is first presented by Filippova and Hanel [15] and later improved by Mei et al. [16] where T is the period of the pulsation. According to this formula, the maximum of P is 2 P m, and the minimum is zero, which indicates that the blood will never flow back. D s = 8 µm is the diameter of the particle. It is approximate to the diameter of a human red blood cell. And in our lattice model, it is described as 14 lattice spacing. D = 32 µm is the diameter of the vessel. The initial lateral displacement y c of the red blood cell varies from 0.2D to 0.4D. The Reynolds number of the particle is defined as: Re = u pd s, (11) ν where u p is the velocity of the particle and ν is the viscosity of fluid. In our simulation ν = 0.01 cm 2 /s which is approximate to the viscosity of human blood plasma in micro-artery. The relationship of ν and υ determines the value of δt. So when ν is certain, we can change τ to adjust the value of δt. In our simulation τ = 0.8, which ensures the stability of the simulation and makes the computation to be acceptable. In the case that the flow is pulsant, we calculate the mean Reynolds numbers by averaging the Re in one complete period. Fig. 1 Schematic diagram of the model. The black circle represents the particle and the two parallel lines represent the walls of the infinitely long micro vessel. Periodical pressure drop is imposed at the inlet and outlet of the simulating region. Figure 1 is the schematic diagram of our model. The black circle represents the particle and the two parallel lines represent the walls of the infinitely long micro vessel. However, the simulation is conducted in a region with its length 4 times the width of the vessel, thus the particle is kept in the middle of the simulation region through a so called remesh procedure. Pressure drop P is imposed at the inlet and the outlet of the region to drive the fluid and the particle. In case of steady flow, the pressure drop is P m. When the flow reaches a steady state, the red blood cell is released from at rest. In case of pulsating flow, the pressure drop is calculated by the following formula: ( P(t) = P m + P m sin 2π t ), (10) T Fig. 2 Trajectories of the particle released from different initial position in steady flow. (a) corresponds to the case that the Reynolds number is 0.84, (b) corresponds to very low Reynolds numbers, which is reasonable for real human micro-vessel. Figure 2 is the trajectories of the particle released from different initial positions in a steady flow. Figure 2(a) corresponds to the case that the Reynolds number is
3 758 Communications in Theoretical Physics Vol It is seen that particles released from different initial position finally move to a same equilibrium position. Segré Silberberg effect is presented perfectly. However, the speed of particle reaches about 10.5 cm/s, which is too high for real blood flow in micro-vessel. Figure 2(b) corresponds to the case that the Reynolds number is much lower. In this case the velocity of the particle is about 1.45 cm/s, which is a reasonable value for real case. It is seen that the equilibrium position divides into several. Different initial position leads to different equilibrium position. For the final velocity of the particle is not exactly the same, the final Reynolds numbers is slightly different too. Comparing these two pictures we can see that the migration of particle in micro flow is very different from that in macro flow. Fig. 3 Trajectories of the particle in pulsating flow of different frequencies. (a) corresponds to the case that the mean Reynolds number is 0.84, and particle is released at y c = 0.2D; (b), (c), and (d) correspond to the cases that the mean Reynolds numbers are much lower, and the particle is released at y c = 0.25D for (b), y c = 0.31D for (c) and y c = 0.35D for (d). Figure 3 is the trajectories of the particle in pulsating flow of different frequencies from the same initial position. Figure 3(a) corresponds to the case that the mean Reynolds number is It is seen that the trajectories of particle respond little to the pulsation of the flow. Figures 3(b), 3(c), and 3(d) correspond to the very low Reynolds numbers, which are coincident with the real micro blood flow. For the case of Figs. 3(b) and 3(d), different frequencies of pulsation lead to different equilibrium positions. For the case of Fig. 3(c), the particle is released just at the Segré Silberberg equilibrium position, thus it will keep its position under various frequencies. Significant difference is presented again between the migration in micro flow and that in macro flow. Figure 4 is the trajectories of particle released from different initial position in pulsating flow. The mean Reynolds Number here is Re = The period of pulsation is T = s for Fig. 4(a), T = s for Fig. 4(b) and T = 0.8 s for Fig. 4(c). It is seen that the equilibrium positions tends to gather up when the period of pulsation is getting longer. Particularly, when the period reaches the level of human heart rate, that is T = 0.8 s, particles from different initial positions finally move to the Segr e Silberberg equilibrium position. This is very interesting. It seems that appropriate pulsation makes the suspended particle in micro flow act like that in the macro flow. And human body just chooses this frequency. It is hardly to say whether it is accidental or inevitable. To find the dynamics of these differences, we put down the velocity evolution of the particle. Figure 6 presents the evolution of the velocity and angular velocity of the particle in pulsating flow. Figures 5(a), 5(c), and 5(e) correspond to the case that the mean Reynolds number is Figures 5(b), 5(d), and 5(f) correspond to the
4 No. 4 Communications in Theoretical Physics 759 case that the mean Reynolds number is Comparing Fig. 5(a) with Fig. 5(b), we can see that the evolutions of u px are basically in step with the pulsation of the flow. Only that in Fig. 5(b), the maximum of u px in every period increasing gradually at first before it reaches a steady value. This is because that it takes long time for the particle to move to the equilibrium position when Reynolds Number is very low. And before that, restricted by the maximum velocity of fluid around, the maximum u px of the particle is lower than that at the equilibrium position. Fig. 4 Trajectories of the particle released from different initial positions in pulsating flow. The period of pulsation is T = s for (a), T = 0.08 s for (b) and T = 0.8 s for (c). The mean Reynolds Number here is Re = In Fig. 5(e) and Fig. 5(f) we can see that the evolution of angular velocity is basically in step with the evolution of u px. That is because that no-slip boundary condition is conducted on the surface of particle. So the faster it rolls the faster it goes. Figures 5(c) and 5(d) are the evolutions of u px. Figure 5(c) indicates that the particle reaches the equilibrium position quickly before the fluid complete one period of pulsation. After that, the pulsation of flow will never affect u px any more. But in Fig. 5(d), for it takes long time for the particle to reach the equilibrium position, u px fluctuates at first with the pulsation of the flow, and the amplitude of it declines quickly. After the particle reaches the equilibrium position, infinitesimal disturbance can still be observed, with its period coincident with that of the pulsation of flow. From here we suppose that why there are so many differences between migration in micro flow and macro flow is because that the sensitivity of the particle to the variation of the fluid is different. When Reynolds number is very low, the particle has enough time to respond to the slight variations of the fluid while that in macro flow has not. And the accumulation of these responses would finally result in significant difference in their migrations. Using the Lattice Boltzmann simulation, we have studied the migration of one rigid particle suspended in pulsation flow in micro vessel. Simulation results show significant differences between migration in macro flow and in micro flow. Our researches indicate that the particle in micro flow have enough time to respond to the slight variation of fluid, while that in macro flow has not. So the presentation of Segré Silberberg effect has much to do with the inertia of the particle, and the migration of particle in very low Reynolds number flow could better reflect the characteristic of flow. Our researches also suggest that human heart rate may be an inevitable choice of human body. When the period of pulsation approaches to human heart rate, the migration in micro flow is much like that in macro flow. These phenomena are strange and have never been reported as we know. Nevertheless, our model is too idealistic. The reliability of such phenomena should be conformed again by other research methods.
5 Communications in Theoretical Physics 760 Vol. 56 Fig. 5 Velocity evolution of the particle in pulsating flow. (a), (c), (e) correspond to the case that the mean Reynolds number is 0.84, and (b), (d), (f) correspond to the case that the mean Reynolds number is References [1] G. Segre and A. Silberberg, Nature (London) 189 (1961) 209. [2] H.H. Yi, L.J. Fan, X.F. Yang, and H.B. Li, Chin. Phys. Lett. 26 (2009) [3] H. Basag aog lu, P. Meakin, S. Succi, G.R. Redden, and T.R. Ginn, Phys. Rev. E 77 (2008) [4] C.R. Choi and C.N. Kim, Korean J. Chem. Eng. 27 (2010) [5] M. Tachibana, Rheol. Acta 12 (1973) 58. [6] H.B. Yang, Y. Liu, Y.S. Xu, and J.L. Kou, Commun. Theor. Phys. 54 (2010) 886. [7] H.B. Li, L. Jin, and B. Qiu, Chin. Phys. Lett. 25 (2008) [8] Y.Y. Chen, H.H. Yi, H.B. Li, and H.P. Fang, Commun. Theor. Phys. 51 (2009) 331. [9] J. Shi, B. Qiu, and H.L. Tan, Commun. Theor. Phys. 51 (2009) [10] H.B. Li, C.Y. Zhang, X.Y. Lu, and H.P. Fang, Chin. Phys. Lett. 24 (2007) [11] Y.H. Qian, D. D Humires, and P. Lallemand, Europhys. Lett. 17 (1992) 479. [12] H. Chen, S. Chen, and W.H. Matthaeus, Phys. Rev. A 45 (1992) R5339. [13] D.P. Ziegler, J. Stat. Phys. 71 (1993) [14] T. Inamuro, K. Meaba, and F. Ogino, Int. J. Multiphase Flow 26 (2000) [15] O. Filippova and D. Hanel, Comput. Fluids 26 (1997) 697. [16] R. Mei, L. Lou, and W. Shyy, J. Comput. Phys. 155 (1999) 307.
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