A NUMERICAL ALGORITHM BASED ON FICTITIOUS DOMAIN METHOD FOR THE SIMULATION OF MICROORGANISMS SWIMMING IN A VISCOELASTIC FLUID
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摘要: 微生物是自然生态系统的重要组成部分, 掌握微生物在复杂流体中的运动特性可以为微型器件的设计制造提供理论指导. 壁面效应是微生物游动研究中的重要问题之一, 已有研究表明微生物在壁面附近存在复杂的行为特征. 然而已有研究大多集中于微生物在牛顿流体中的游动模拟, 仅有少数涉及黏弹性流体等非牛顿流体. 本文采用直接力虚拟区域法与乔列斯基分解相结合的数值方法, 引入Squirmer微生物游动模型, 研究了微生物在黏弹性流体中的游动问题. 首先给出求解黏弹性流体本构方程的数值格式; 并将该方法应用于研究微生物游动中的壁面效应. 研究结果表明, 游动方向是影响微生物颗粒壁面效应的重要因素. 流体弹性应力会对微生物产生一个反向转矩, 影响微生物的游动方向, 从而阻碍微生物逃离壁面. 微生物颗粒在黏弹性流体中与壁面作用时间较长, 几乎达到牛顿流体的两倍以上.Abstract: Microorganisms are one of the important parts of natural ecosystem, understanding the kinematic behaviors of microorganisms swimming in complex fluids could provide guidance for the design and manufacturing of MEMS. Wall effects are one of the most important scientific problems of the research of microorganism swimming, and recent work reveals that microorganisms show complicated swimming behaviors near the wall. However, most of the work reported in the literatures focused on microorganism swimming in Newtonian fluid, less attention is paid on microorganism swimming in viscoelastic fluid or other non-Newtonian fluids. A direct-forcing fictitious domain method combined with Cholesky decomposition for the simulation of microorganisms swimming in a viscoelastic fluid is reported in this paper. The squirmer model is applied to represent the swimming of microorganisms. The numerical schemes for the discretization of Giesekus constitutive equation are first presented and validated. The newly developed simulation model is then applied to investigate the effect of planar wall on swimming dynamics of current squirmer in viscoelastic flow, i.e., Giesekus fluid. The results show that the swimming direction of squirmer is a critical factor of the wall-trapping effect. The fluid elasticity affects the swimmer motion near solid wall by generating an elastic torque which reorient the swimming direction. The time for the squirmer to contact planar wall in viscoelastic fluid is almost twice of that in Newtonian fluid.
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Key words:
- fictitious domain method /
- viscoelastic fluid /
- microorganisms /
- Cholesky decomposition
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图 3 两种类型粒子游动平均速度
${U_{{\rm{GK}}}}$ 随$ We $ 变化规律: (a)$ \beta = 5 $ , (b)$ \beta = - 5 $ . (b)插图为$ \beta = - 5,We = 2 $ 工况下粒子游动速度$ U $ 随时间的变化曲线; 黑色实线为粗网格$h = 1/16,\Delta t = 0.000\;5$ 计算结果, 红色实线为细网格$h = 1/32,\Delta t = 0.000\;25$ 计算结果Figure 3. The average swimming speed
${U_{{\rm{GK}}}}$ of squirmers: (a)$ \beta = 5 $ , (b)$ \beta = - 5 $ . Inset: squirmer swimming speed computed when using two different meshes at$ \beta = - 5,We = 2 $ ; black solid line represents the result of coarse grid$h = 1/16,\Delta t = 0.000\;5$ , and the red represents the result of fine grid$h = 1/32,\Delta t = 0.000\;25$ 
图 4 微生物颗粒(
$ \beta = 3 $ )与壁面作用轨迹图, 粒子与壁面初始距离$ d = 2 $ , 游动方向与$ x $ 轴夹角$\gamma = - \text{π} /4$ Figure 4. Trajectory of puller (
$ \beta = 3 $ ) contact with wall, the initial distance between particles and wall$ d = 2 $ , the angle between swimming direction and$ x $ -axis$\gamma = - \text{π} /4$ 
图 5 微生物颗粒自由游动流场结构图. 坐标系为随体坐标; 黑色箭头表示颗粒游动方向, 背景云图表示构型张量的迹
${\rm{Tr}}\left( {{{\boldsymbol{C}}}} \right)$ . 左半部分图为$ We = 0 $ 结果; 右半部分图为$ We = 4 $ 结果Figure 5. Flow field around squirmer particles free swimming in viscoelastic fluids with comoving frame; the black arrow represents the swimming direction of particles, and the background cloud chart represents the trace of the configuration tensor
${\rm{Tr}}\left( {\boldsymbol{C}} \right)$ . The left part of figure shows the result at$ We = 0 $ ; the right part shows the result at$ We = 4 $ 
图 6 不同类型微生物颗粒与壁面间隔距离
$ d $ 随时间变化曲线. 黏弹性流体的参数为:$ Re = 0.01 $ ,$ We = 0\sim 4 $ ,$ \alpha = 0.2 $ ,$ {\mu _r} = 0.5 $ ,$ {\rho _r} = 1.0 $ Figure 6. The distance between squirmer particles and the wall in the y direction as a function of time. Parameters of viscoelastic fluid:
$ Re = 0.01 $ ,$ We = 0\sim 4 $ ,$ \alpha = 0.2 $ ,$ {\mu _r} = 0.5 $ ,$ {\rho _r} = 1.0 $ 
图 7 微生物颗粒游动方向与
$ x $ 轴夹角随时间变化曲线. 黏弹性流体的参数为:$ Re = 0.01 $ ,$ We = 0\sim 4 $ ,$ \alpha = 0.2 $ ,$ {\mu _r} = 0.5 $ ,$ {\rho _r} = 1.0 $ Figure 7. Angle between squirmer particles swimming direction and the
$ x $ -axis as a function of time. Parameters of the viscoelastic fluid:$ Re = 0.01 $ ,$ We = 0\sim 4 $ ,$ \alpha = 0.2 $ ,$ {\mu _r} = 0.5 $ ,$ {\rho _r} = 1.0 $ 
图 8 微生物颗粒在
$ x - y $ 平面内受到的水动力转矩$ {T_z} $ 随时间变化曲线. 其中,$ {T_{v,z}} $ 表示黏性转矩$ z $ 方向分量,$ {T_{e,z}} $ 表示弹性转矩$ z $ 方向分量, 计算公式见式(33). 黏弹性流体的参数为:$ Re = 0.01 $ ,$ We = 0,4 $ ,$ \alpha = 0.2 $ ,$ {\mu _r} = 0.5 $ ,$ {\rho _r} = 1.0 $ Figure 8. Hydrodynamic torque in the z direction computed using Eq. (33).
$ {T_{v,z}} $ represents the viscous torque and$ {T_{e,z}} $ the elastic torque. Parameters of the viscoelastic fluid:$ Re = 0.01 $ ,$ We = 0,4 $ ,$ \alpha = 0.2 $ ,$ {\mu _r} = 0.5 $ ,$ {\rho _r} = $ 1.0
图 9 微生物颗粒在壁面方向受到的水动力
$ {F_y} $ 随时间变化曲线. 其中,$ {F_{v,y}} $ 表示黏性应力$ y $ 方向分量,$ {F_{e,y}} $ 表示弹性应力$ y $ 方向分量, 计算公式见式(33). 黏弹性流体的参数为:$ Re = 0.01 $ ,$ We = 0,4 $ ,$ \alpha = 0.2 $ ,$ {\mu _r} = 0.5 $ ,$ {\rho _r} = 1.0 $ Figure 9. Hydrodynamic forces in the y direction computed using Eq. (33).
$ {F_{v,y}} $ represents the viscous force and$ {F_{e,y}} $ the elastic force. Parameters of viscoelastic fluid:$ Re = 0.01 $ ,$ We = 0,4 $ ,$ \alpha = 0.2 $ ,$ {\mu _r} = 0.5 $ ,$ {\rho _r} = 1.0 $ -
[1] Guasto JS, Rusconi R, Stocker R. Fluid mechanics of planktonic microorganisms. Annual Review of Fluid Mechanics, 2012, 44: 373-400 doi: 10.1146/annurev-fluid-120710-101156 [2] Yazdi S, Ardekani AM. Bacterial aggregation and biofilm formation in a vortical flow. Biomicrofluidics, 2012, 6(4): 044114 doi: 10.1063/1.4771407 [3] Karshalev E, Zhang Y, Esteban-Fernández de Ávila B, et al. Micromotors for active delivery of minerals toward the treatment of iron deficiency anemia. Nano Letters, 2019, 19(11): 7816-7826 doi: 10.1021/acs.nanolett.9b02832 [4] Zhang Y, Yuan K, Zhang L. Micro/nanomachines: from functionalization to sensing and removal. Advanced Materials Technologies, 2019, 4(4): 1800636 doi: 10.1002/admt.201800636 [5] Vyskocil J, Mayorga-Martinez CC, Jablonska E, et al. Cancer cells microsurgery via asymmetric bent surface Au/Ag/Ni microrobotic scalpels through a transversal rotating magnetic field. ACS Nano, 2020, 14(7): 8247-8256 doi: 10.1021/acsnano.0c01705 [6] Drescher K, Leptos KC, Tuval I, et al. Dancing volvox: hydrodynamic bound states of swimming algae. Physical Review Letters, 2009, 102(16): 168101 doi: 10.1103/PhysRevLett.102.168101 [7] Kantsler V, Dunkel J, Polin M, et al. Ciliary contact interactions dominate surface scattering of swimming eukaryotes. Proceedings of the National Academy of Sciences, 2013, 110(4): 1187-1192 doi: 10.1073/pnas.1210548110 [8] Lauga E, DiLuzio WR, Whitesides GM, et al. Swimming in circles: motion of bacteria near solid boundaries. Biophysical Journal, 2006, 90(2): 400-412 doi: 10.1529/biophysj.105.069401 [9] Kuron M, Stärk P, Holm C, et al. Hydrodynamic mobility reversal of squirmers near flat and curved surfaces. Soft Matter, 2019, 15(29): 5908-5920 doi: 10.1039/C9SM00692C [10] Chaithanya KV, Thampi SP. Wall-curvature driven dynamics of a microswimmer. Physical Review Fluids, 2021, 6(8): 083101 doi: 10.1103/PhysRevFluids.6.083101 [11] Ishimoto K, Gaffney EA, Smith DJ. Squirmer hydrodynamics near a periodic surface topography. arXiv Preprint, 2109.05528, 2021 [12] Narinder N, Gomez-Solano JR, Bechinger C. Active particles in geometrically confined viscoelastic fluids. New Journal of Physics, 2019, 21(9): 093058 doi: 10.1088/1367-2630/ab40e0 [13] Narinder N, Zhu W, Bechinger C. Active colloids under geometrical constraints in viscoelastic media. The European Physical Journal E, 2021, 44(3): 1-6 [14] Li GJ, Karimi A, Ardekani AM. Effect of solid boundaries on swimming dynamics of microorganisms in a viscoelastic fluid. Rheologica Acta, 2014, 53(12): 911-926 doi: 10.1007/s00397-014-0796-9 [15] Ouyang Z, Lin J, Ku X. Hydrodynamic properties of squirmer swimming in power-law fluid near a wall. Rheologica Acta, 2018, 57(10): 655-671 doi: 10.1007/s00397-018-1107-7 [16] Ahana P, Thampi SP. Confinement induced trajectory of a squirmer in a two dimensional channel. Fluid Dynamics Research, 2019, 51(6): 065504 doi: 10.1088/1873-7005/ab4d08 [17] Leoni M, Paoluzzi M, Eldeen S, et al. Surfing and crawling macroscopic active particles under strong confinement: Inertial dynamics. Physical Review Research, 2020, 2(4): 043299 doi: 10.1103/PhysRevResearch.2.043299 [18] Yazdi S, Ardekani AM, Borhan A. Locomotion of microorganisms near a no-slip boundary in a viscoelastic fluid. Physical Review E, 2014, 90(4): 043002 doi: 10.1103/PhysRevE.90.043002 [19] Alves MA, Oliveira PJ, Pinho FT. Numerical methods for viscoelastic fluid flows. Annual Review of Fluid Mechanics, 2021, 53: 509-541 doi: 10.1146/annurev-fluid-010719-060107 [20] King RC, Apelian MR, Armstrong RC, et al. Numerically stable finite element techniques for viscoelastic calculations in smooth and singular geometries. Journal of Non-Newtonian Fluid Mechanics, 1988, 29: 147-216 doi: 10.1016/0377-0257(88)85054-7 [21] Oliveira PJ. Method for time-dependent simulations of viscoelastic flows: vortex shedding behind cylinder. Journal of Non-newtonian Fluid Mechanics, 2001, 101(1-3): 113-137 doi: 10.1016/S0377-0257(01)00146-X [22] Thomases B. An analysis of the effect of stress diffusion on the dynamics of creeping viscoelastic flow. Journal of Non-Newtonian Fluid Mechanics, 2011, 166(21-22): 1221-1228 doi: 10.1016/j.jnnfm.2011.07.009 [23] Vaithianathan T, Collins LR. Numerical approach to simulating turbulent flow of a viscoelastic polymer solution. Journal of Computational Physics, 2003, 187(1): 1-21 doi: 10.1016/S0021-9991(03)00028-7 [24] Harten A. High resolution schemes for hyperbolic conservation laws. Journal of Computational Physics, 1997, 135(2): 260-278 doi: 10.1006/jcph.1997.5713 [25] Van Leer B. Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method. Journal of Computational Physics, 1979, 32(1): 101-136 [26] Liu XD, Osher S, Chan T. Weighted essentially non-oscillatory schemes. Journal of Computational Physics, 1994, 115(1): 200-212 doi: 10.1006/jcph.1994.1187 [27] Fattal R, Kupferman R. Constitutive laws for the matrix-logarithm of the conformation tensor. Journal of Non-Newtonian Fluid Mechanics, 2004, 123(2-3): 281-285 doi: 10.1016/j.jnnfm.2004.08.008 [28] Lozinski A, Owens RG. An energy estimate for the Oldroyd B model: theory and applications. Journal of Non-newtonian Fluid Mechanics, 2003, 112(2-3): 161-176 doi: 10.1016/S0377-0257(03)00096-X [29] Lighthill MJ. On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers. Communications on Pure and Applied Mathematics, 1952, 5(2): 109-118 doi: 10.1002/cpa.3160050201 [30] Blake JR. A spherical envelope approach to ciliary propulsion. Journal of Fluid Mechanics, 1971, 46(1): 199-208 doi: 10.1017/S002211207100048X [31] Li G, Ostace A, Ardekani AM. Hydrodynamic interaction of swimming organisms in an inertial regime. Physical Review E, 2016, 94(5): 053104 doi: 10.1103/PhysRevE.94.053104 [32] Li GJ, Ardekani AM. Hydrodynamic interaction of microswimmers near a wall. Physical Review E, 2014, 90(1): 013010 doi: 10.1103/PhysRevE.90.013010 [33] Lin Z, Gao T. Direct-forcing fictitious domain method for simulating non-Brownian active particles. Physical Review E, 2019, 100(1): 013304 doi: 10.1103/PhysRevE.100.013304 [34] Giesekus H. A simple constitutive equation for polymer fluids based on the concept of deformation-dependent tensorial mobility. Journal of Non-Newtonian Fluid Mechanics, 1982, 11(1-2): 69-109 doi: 10.1016/0377-0257(82)85016-7 [35] Glowinski R, Pan TW, Hesla TI, et al. A distributed Lagrange multiplier/fictitious domain method for particulate flows. International Journal of Multiphase Flow, 1999, 25(5): 755-794 doi: 10.1016/S0301-9322(98)00048-2 [36] Yu Z, Shao X. A direct-forcing fictitious domain method for particulate flows. Journal of Computational Physics, 2007, 227(1): 292-314 doi: 10.1016/j.jcp.2007.07.027 [37] Yu Z, Lin Z, Shao X, et al. A parallel fictitious domain method for the interface-resolved simulation of particle-laden flows and its application to the turbulent channel flow. Engineering Applications of Computational Fluid Mechanics, 2016, 10(1): 160-170 doi: 10.1080/19942060.2015.1092268 [38] Zhu C, Yu Z, Pan D, et al. Interface-resolved direct numerical simulations of the interactions between spheroidal particles and upward vertical turbulent channel flows. Journal of Fluid Mechanics, 2020, 891: A6 [39] Xia Y, Lin Z, Pan D, et al. Turbulence modulation by finite-size heavy particles in a downward turbulent channel flow. Physics of Fluids, 2021, 33(6): 063321 doi: 10.1063/5.0053540 [40] Zhu C, Qian L, Lin Z, et al. Turbulent channel flow of a binary mixture of neutrally buoyant ellipsoidal particles. Physics of Fluids, 2022, 34(5): 053609 doi: 10.1063/5.0089088 [41] Zhu L, Lauga E, Brandt L. Self-propulsion in viscoelastic fluids: Pushers vs pullers. Physics of Fluids, 2012, 24(5): 051902 doi: 10.1063/1.4718446 -
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