Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T05:29:57.666Z Has data issue: false hasContentIssue false
  • English
  • Français

A Lattice Boltzmann-Direct Forcing/Fictitious Domain Model for Brownian Particles in Fluctuating Fluids

Published online by Cambridge University Press:  20 August 2015

Deming Nie  and
Jianzhong Lin
Deming Nie*
Affiliation:
State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China College of Metrology and Technology Engineering, China Jiliang University, Hangzhou 310018, China
Jianzhong Lin*
Affiliation:
State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China College of Metrology and Technology Engineering, China Jiliang University, Hangzhou 310018, China
*
Email:nieinhz@cjlu.edu.cn
Corresponding author.Email:linjz@cjlu.edu.cn, mecjzlin@zju.edu.cn
Get access

Abstract

The previously developed LB-DF/FD method derived from the lattice Boltzmann method and Direct Forcing/Fictitious Domain method is extended to deal with 3D particle’s Brownian motion. In the model the thermal fluctuations are introduced as random forces and torques acting on the Brownian particle. The hydrodynamic interaction is introduced by directly resolving the fluid motions. A sphere fluctuating in a cubic box with the periodic boundary is considered to validate the present model. By examining the velocity autocorrelation function (VCF) and rotational velocity autocorrelation function (RVCF), it has been found that in addition to the two relaxation times, the mass density ratio should be taken into consideration to check the accuracy and effectiveness of the present model. Furthermore, the fluctuation-dissipation theorem and equipartition theorem have been investigated for a single spherical particle. Finally, a Brownian particle trapped in a harmonic potential has been simulated to further demonstrate the ability of the LB-DF/FD model.

Keywords

68U2076T2076M2882B80Lattice Boltzmann methodfictitious domainBrownian motion
Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 4 , April 2011 , pp. 959 - 973
Copyright
Copyright © Global Science Press Limited 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Park, J. S., Choi, C. K. and Kihm, K. D., Temperature measurement for a nanoparticle suspension by detecting the Brownian motion using optical serial sectioning microscopy (OSSM), Meas. Sci. Technol., 16 (2005), 1418–1429.Google Scholar
[2]Chung, K., Cho, J. K., Park, E. S., Breedveld, V. and Lu, H., Three-dimensional in situ temperature measurement in microsystems using Brownian motion of nanoparticles, Anal. Chem., 81 (2009), 991–999.Google Scholar
[3]MacKintosh, F. C. and Schmidt, C. F., Microrheology, Curr. Opin. Coll. Interface. Sci., 4(1999), 300–307.CrossRefGoogle Scholar
[4]Tischer, C., Altmann, S., Fisinger, S., Hoäber, J. K. H., Stelzer, E. H. K. and Florin, E.-L., Three-dimensional thermal noise imaging, Appl. Phys. Lett., 79 (2001), 3878–3880.CrossRefGoogle Scholar
[5]Jeney, S., Stelzer, E. H. K., Grübmuller, H. and Florin, E.-L., Mechanical properties of single motor molecules studies by three-dimensional thermal force probing in optical tweezers, Chem. Phys. Chem., 5 (2004), 1150–1158.Google Scholar
[6]Dean Astumian, R., Thermodynamics and kinetics of a Brownian motor, Science, 276 (1997), 917–922.Google Scholar
[7]Dere’nyi, I. and Astumian, R. D., AC separation of particles by biased Brownian motion in a two-dimensional sieve, Phys. Rev. E., 58 (1998), 7781–7784.CrossRefGoogle Scholar
[8]Keblinski, P., Phillpot, S. R., Choi, S. U. S. and Eastman, J. A., Mechanisms of heat flow in suspensions of nano-sized particles (nanofluids), Int. J. Heat. Mass. Trans., 45 (2002), 855–863.CrossRefGoogle Scholar
[9]Lee, S., Choi, S. U. S., Li, S. and Eastman, J. A., Measuring thermal conductivity of fluids containing oxide nanopartieles, J. Heat. Trans., 121 (1999), 280–289.Google Scholar
[10]Prasher, R., Bhattacharya, P. and Phelan, P. E., Thermal conductivity of nanoscale colloidal solutions (nanofluids), Phys. Rev. Lett., 94 (2005), 025901.Google Scholar
[11]Jang, S. P. and Choi, S. U. S., Role of Brownian motion in the enhanced thermal conductivity of nanofluids, Appl. Phys. Lett., 84 (2004), 4316–4318.CrossRefGoogle Scholar
[12]Bhattacharya, P., Saha, S. K., Yadav, A. and Phelan, P. E., Brownian dynamics simulation to determine the effective thermal conductivity of nanofluids, J. Appl. Phys., 95 (2004), 6492–6494.CrossRefGoogle Scholar
[13]Sonechkin, D. M., Climate dynamics as a nonlinear Brownian motion, Int. J. Bifurcation. Chaos. Appl. Sci. Eng., 8 (1998), 799–803.Google Scholar
[14]Sottinen, T., Fractional Brownian motion, random walks and binary market models, Finance. Stochast., 5 (2001), 343–355.Google Scholar
[15]Shiftan, Y., Button, K. and Nijkamp, P., Transportation Planning, Ediward Elgar Publishing, Inc., U.K., 2007.Google Scholar
[16]Ermak, D. L. and McCammon, J. A., Brownian dynamics with hydrodynamic interactions, J. Chem. Phys., 69 (1978), 1352–1360.CrossRefGoogle Scholar
[17]Brady, J. F. and Bossis, G., Self-diffusion of Brownian particles in concentrated suspensions under shear, J. Chem. Phys., 87 (1987), 5437–5448.Google Scholar
[18]Brady, J. F., Bossis, G., Stokesian dynamics, Annu. Rev. Fluid. Mech., 20 (1988), 111–157.Google Scholar
[19]Ladd, A. J. C., Short-time motion of colloidal particles: numerical simulation via a fluctuating Lattice-Boltzmann equation, Phys. Rev. Lett., 70 (1993), 1339–1342.Google Scholar
[20]Landau, L. D. and Lifshitz, E. M., Fluid Mechanics, London, Pergamon Press, 1959.Google Scholar
[21]Sharma, N. and Patankar, N. A., Direct numerical simulation of the Brownian motion of particles by using fluctuating hydrodynamic equations, J. Comput. Phys., 201 (2004), 466–486.CrossRefGoogle Scholar
[22]Ladd, A. J. C., Numerical simulations of particulate suspensions via a discretized Boltzmann equation part I: theoretical foundation, J. Fluid. Mech., 271 (1994), 285–310.Google Scholar
[23]Ladd, A. J. C., Numerical simulations of particulate suspensions via a discretized Boltzmann equation part II: numerical results, J. Fluid. Mech., 271 (1994), 311–339.Google Scholar
[24]Ahlrichs, P. and Dünweg, B., Lattice-Boltzmann simulation of polymer-solvent systems, Int. J. Mod. Phys. C., 9 (1998), 1429–1438.CrossRefGoogle Scholar
[25]Ahlrichs, P. and Dünweg, B., Simulation of a single polymer chain in solution by combining lattice Boltzmann and molecular dynamics, J. Chem. Phys., 111 (1999), 8225–8239.Google Scholar
[26]Dünweg, B., Schiller, U. D. and Ladd, A. J. C., Statistical mechanics of the fluctuating lattice Boltzmann equation, Phys. Rev. E., 76 (2007), 036704.CrossRefGoogle ScholarPubMed
[27]Dünweg, B. and Ladd, A. J. C., Lattice Boltzmann simulations of soft matter systems, Adv. Poly. Sci., 221 (2009), 89–166.Google Scholar
[28]Yu, Z. S., Shao, X. M. and Wachs, A., A fictitious domain method for particulate flows with heat transfer, J. Comput. Phys., 217 (2005), 424–452.Google Scholar
[29]Iwashita, T., Nakayama, Y. and Yamamoto, R., A numerical model for Brownian particles fluctuating in incompressible fluids, J. Phys. Soc. Jpn., 7 (2008), 074007.Google Scholar
[30]Benzi, R., Succi, S. and Vergassola, M. R., The lattice Boltzmann equation: theory and applications, Phys. Rep., 222 (1992), 145–197.Google Scholar
[31]Chen, S. Y. and Doolen, G. D., Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid. Mech., 30 (1998), 329–364.CrossRefGoogle Scholar
[32]Wang, M. and Pan, N., Predictions of effective physical properties of complex multiphase materials, Mater. Sci. Eng. Rep., 63 (2008), 1–30.Google Scholar
[33]Qi, D., Lattice-Boltzmann simulations of particles in non-zero-Reynolds-number flows, J. Fluid. Mech., 385 (1999), 41–62.Google Scholar
[34]Ladd, A. J. C. and Verberg, R., Lattice-Boltzmann simulations of particle-fluid suspensions, J. Stat. Phys., 104 (2001), 1191–1251.Google Scholar
[35]Aidun, C. K. and Ding, E.-J., Dynamics of particle sedimentation in a vertical channel: peri-oddoubling bifurcation and chaotic state, Phys. Fluids., 15 (2003), 1612–1621.Google Scholar
[36]Lin, J. Z., Shi, X. and You, Z. J., Effects of the aspect ratio on the sedimentation of a fiber in Newtonian fluids, J. Aerosol. Sci., 34 (2003), 909–921.Google Scholar
[37]Lin, J. Z., Wang, Y. L. and James, A. O., Sedimentation of rigid cylindrical particles with mechanical contacts, Chin. Phys. Lett., 22 (2005), 628–631.Google Scholar
[38]Shi, X., Lin, J. Z. and Yu, Z. S., Discontinuous Galerkin spectral element lattice Boltzmann method on triangular element, Int. J. Numer. Methods. Fluids, 42 (2003), 1249–1261.CrossRefGoogle Scholar
[39]Ku, X. K. and Lin, J. Z., Effect of two bounding walls on the rotational motion of a fiber in the simple shear flow, Fiber. Polym., 10 (2009), 302–309.Google Scholar
[40]Xia, Z. H., Connington, K. W., Rapaka, S., Yue, P. T., Feng, J. J. and Chen, S. Y., Flow patterns in the sedimentation of an elliptical particle, J. Fluid. Mech., 625 (2009), 249–272.CrossRefGoogle Scholar
[41]Feng, Z. G. and Michaelides, E. E., The immersed boundary-lattice Boltzmann method for solving fluid-particles interaction problems, J. Comput. Phys., 195 (2004), 602–628.Google Scholar
[42]Feng, Z. G. and Michaelides, E. E., Proteus: a direct forcing method in the simulations of particulate flows, J. Comput. Phys., 202 (2005), 20–51.CrossRefGoogle Scholar
[43]Shi, X. and Phan-Thien, N., Distributed Lagrange multiplier/fictitious domain method in the framework of lattice Boltzmann method for fluid-structure interactions, J. Comput. Phys., 206 (2005), 81–94.CrossRefGoogle Scholar
[44]Shi, X. and Lim, S. P., A LBM-DLM/FD method for 3D fluid-structure interactions, J. Comput. Phys., 226 (2007), 2028–2043.Google Scholar
[45]Nie, D. M. and Lin, J. Z., A LB-DF/FD method for particle suspensions, Commun. Comput. Phys., 7 (2010), 544–563.Google Scholar
[46]Widom, A., Velocity fluctuations of a hard-core Brownian particle, Phys. Rev. A., 3 (1971), 1394–1396.Google Scholar
[47]Hauge, E. H. and Martin-L, A.öf, Fluctuating hydrodynamics and Brownian motion, J. Stat. Phys., 7 (1973), 259–281.Google Scholar
[48]Hinch, E. J., Application of the Langevin equation to fluid suspensions, J. Fluid. Mech., 72 (1975), 499–511.Google Scholar
[49]Yu, Z. S. and Shao, X. M., A direct-forcing fictitious domain method for particulate flows, J. Comput. Phys., 227 (2007), 292–314.CrossRefGoogle Scholar
[50]Uhlmann, M., An immersed boundary method with direct forcing for the simulation of par-ticulate flows, J. Comput. Phys., 209 (2005), 448–476.Google Scholar
[51]Clercx, H. J. H. and Schram, P. P. J. M, Brownian particles in shear flow and harmonic potentials: a study of long-time tails, Phys. Rev. A., 46 (1992), 1942–1950.Google Scholar