Hostname: page-component-cb9f654ff-5kfdg Total loading time: 0 Render date: 2025-08-29T23:22:10.550Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamics of a spherical particle in a low-Reynolds-number fluid confined between two concentric spherical walls

Published online by Cambridge University Press:  27 August 2025

Zhuang Sun ,
Juan J. de Pablo  and
Xikai Jiang Open the ORCID record for Xikai Jiang [Opens in a new window]
Zhuang Sun
Affiliation:
State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, PR China School of Engineering Science, University of Chinese Academy of Sciences, Beijing 100049, PR China
Juan J. de Pablo
Affiliation:
Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL 60637, USA Materials Science Division, Argonne National Laboratory, Lemont, IL 60439, USA Tandon School of Engineering, New York University, Brooklyn, NY 11201, USA
Xikai Jiang*
Affiliation:
State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, PR China School of Engineering Science, University of Chinese Academy of Sciences, Beijing 100049, PR China
*
Corresponding author: Xikai Jiang, xikaij@imech.ac.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

Dynamics of a spherical particle and the suspending low-Reynolds-number fluid confined between two concentric spherical walls were studied numerically. We calculated the particle’s hydrodynamic mobilities at various locations in the confined space. It was observed that the mobility is largest near the middle of confined space along the radial direction, and decays as the particle becomes closer to no-slip walls. At a certain confinement level, the maximal mobility occurs near the inner wall. We also calculated the drift velocity of the particle perpendicular to the external force. The magnitude of the drift velocity normalised by the velocity along the external force was found to depend on particle location and the confinement level; it is observed that the maximal drift velocity occurs near the wall. Fluid vortices in the confined space induced by particle motion were observed and analysed. In addition, we studied particle trajectories in the flow when the walls rotate at constant angular velocities. The externally applied force, rotation-induced flow and centrifugal/centripetal force, and particle–wall interaction lead to various modes of particle motion. This work lays the foundation to understand and manipulate particulate transport in microfluidic applications such as intracellular transport and encapsulation technologies.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Micro-/Nano-fluid dynamics: Microscale transport Multiphase and Particle-laden Flows: Particle/fluid flow

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1018 , 10 September 2025 , A4
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press

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.)

Article purchase

Temporarily unavailable

References

Aderogba, K. 2009 Theory of images in spherical-layered axisymmetric viscous hydrodynamics. J. Appl. Mech. 76 (6), 061007.10.1115/1.3130450CrossRefGoogle Scholar
Ando, T., Chow, E. & Skolnick, J. 2013 Dynamic simulation of concentrated macromolecular solutions with screened long-range hydrodynamic interactions: algorithm and limitations. J. Chem. Phys. 139 (12), 121922.10.1063/1.4817660CrossRefGoogle ScholarPubMed
Aponte-Rivera, C., Su, Y. & Zia, R.N. 2018 Equilibrium structure and diffusion in concentrated hydrodynamically interacting suspensions confined by a spherical cavity. J. Fluid Mech. 836, 413–450.10.1017/jfm.2017.801CrossRefGoogle Scholar
Aponte-Rivera, C. & Zia, R.N. 2016 Simulation of hydrodynamically interacting particles confined by a spherical cavity. Phys. Rev. Fluids 1, 023301.10.1103/PhysRevFluids.1.023301CrossRefGoogle Scholar
Bonaccurso, E., Kappl, M. & Butt, H.-J. 2002 Hydrodynamic force measurements: boundary slip of water on hydrophilic surfaces and electrokinetic effects. Phys. Rev. Lett. 88, 076103.10.1103/PhysRevLett.88.076103CrossRefGoogle ScholarPubMed
Cervantes-Martínez, A.E., Ramírez-Saito, A., Armenta-Calderón, R., Ojeda-López, M.A. & Arauz-Lara, J.L. 2011 Colloidal diffusion inside a spherical cell. Phys. Rev. E 83, 030402.10.1103/PhysRevE.83.030402CrossRefGoogle ScholarPubMed
Chen, G. & Jiang, X. 2022 Motion of a sphere and the suspending low-Reynolds-number fluid confined in a cubic cavity. Theor. Appl. Mech. Lett. 12 (4), 100352.10.1016/j.taml.2022.100352CrossRefGoogle Scholar
Chen, G. & Jiang, X. 2023 Single-particle dynamics in a low-Reynolds-number fluid under spherical confinement. J. Fluid Mech. 969, A15.10.1017/jfm.2023.572CrossRefGoogle Scholar
Chow, E. & Skolnick, J. 2015 Effects of confinement on models of intracellular macromolecular dynamics. Proc. Natl Acad. Sci. USA 112 (48), 14846–14851.10.1073/pnas.1514757112CrossRefGoogle ScholarPubMed
Cohen-Fix, O. 2021 Cell biology: how does the nucleus get its membrane? Curr. Biol. 31 (18), R1077–R1079.10.1016/j.cub.2021.08.014CrossRefGoogle ScholarPubMed
Crowdy, D. 2011 Treadmilling swimmers near a no-slip wall at low Reynolds number. Intl J. Nonlinear Mech. 46 (4), 577–585.10.1016/j.ijnonlinmec.2010.12.010CrossRefGoogle Scholar
Crowdy, D. & Samson, O. 2011 Hydrodynamic bound states of a low-Reynolds-number swimmer near a gap in a wall. J. Fluid Mech. 667, 309–335.10.1017/S0022112010004465CrossRefGoogle Scholar
Daddi-Moussa-Ider, A., Löwen, H. & Gekle, S. 2018 Creeping motion of a solid particle inside a spherical elastic cavity. Eur. Phys. J. E 41 (104), 1–14.10.1140/epje/i2018-11715-7CrossRefGoogle ScholarPubMed
Dagastine, R.R., Webber, G.B., Manica, R., Stevens, G.W., Grieser, F. & Chan, D.Y.C. 2010 Viscosity effects on hydrodynamic drainage force measurements involving deformable bodies. Langmuir 26 (14), 11921–11927.10.1021/la1012473CrossRefGoogle ScholarPubMed
Davis, A.M.J. & Crowdy, D.G. 2012 Stresslet asymptotics for a treadmilling swimmer near a two-dimensional corner: hydrodynamic bound states. Proc. R. Soc. A: Math. Phys. Engng Sci. 468 (2148), 3765–3783.10.1098/rspa.2012.0237CrossRefGoogle Scholar
Delmotte, B. 2019 Hydrodynamically bound states of a pair of microrollers: a dynamical system insight. Phys. Rev. Fluids 4, 044302.10.1103/PhysRevFluids.4.044302CrossRefGoogle Scholar
Drescher, K., Leptos, K.C., Tuval, I., Ishikawa, T., Pedley, T.J. & Goldstein, R.E. 2009 Dancing Volvox: hydrodynamic bound states of swimming algae. Phys. Rev. Lett. 102, 168101.10.1103/PhysRevLett.102.168101CrossRefGoogle ScholarPubMed
Driscoll, M., Delmotte, B., Youssef, M., Sacanna, S., Donev, A. & Chaikin, P. 2017 Unstable fronts and motile structures formed by microrollers. Nat. Phys. 13, 375–379.10.1038/nphys3970CrossRefGoogle Scholar
Faltas, M.S. & Saad, E.I. 2011 Stokes flow past an assemblage of slip eccentric spherical particle-in-cell models. Math. Meth. Appl. Sci. 34 (13), 1594–1605.10.1002/mma.1465CrossRefGoogle Scholar
Felderhof, B.U. & Sellier, A. 2012 Mobility matrix of a spherical particle translating and rotating in a viscous fluid confined in a spherical cell, and the rate of escape from the cell. J. Chem. Phys. 136 (5), 054703.10.1063/1.3681368CrossRefGoogle Scholar
Gibson, L.J., Zhang, S., Stilgoe, A.B., Nieminen, T.A. & Rubinsztein-Dunlop, H. 2019 Machine learning wall effects of eccentric spheres for convenient computation. Phys. Rev. E 99, 043304.10.1103/PhysRevE.99.043304CrossRefGoogle ScholarPubMed
Gonzalez, E., Aponte-Rivera, C. & Zia, R.N. 2021 Impact of polydispersity and confinement on diffusion in hydrodynamically interacting colloidal suspensions. J. Fluid Mech. 925, A35.10.1017/jfm.2021.563CrossRefGoogle Scholar
Goto, K., Sakata, S., Moritani, K. & Inui, N. 2017 Mean distance of two Brownian particles trapped in a suspended droplet and its dependence on the Debye length. Physica A: Stat. Mech. Applics. 466, 511–520.10.1016/j.physa.2016.09.037CrossRefGoogle Scholar
Grelier, M., Sivak, D.A. & Ehrich, J. 2024 Unlocking the potential of information flow: maximizing free-energy transduction in a model of an autonomous rotary molecular motor. Phys. Rev. E 109, 034115.10.1103/PhysRevE.109.034115CrossRefGoogle Scholar
Guo, H., Man, Y. & Zhu, H. 2024 Hydrodynamic bound states of rotating microcylinders in a confining geometry. Phys. Rev. Fluids 9, 014102.10.1103/PhysRevFluids.9.014102CrossRefGoogle Scholar
Guriyanova, S., Semin, B., Rodrigues, T.S., Butt, H.-J. & Bonaccurso, E. 2009 Hydrodynamic drainage force in a highly confined geometry: role of surface roughness on different length scales. Microfluid Nanofluid 8, 653–663.10.1007/s10404-009-0498-2CrossRefGoogle Scholar
Hamilton, J.K., Gilbert, A.D., Petrov, P.G. & Ogrin, F.Y. 2018 Torque driven ferromagnetic swimmers. Phys. Fluids 30 (9), 092001.10.1063/1.5046360CrossRefGoogle Scholar
Hernández-Ortiz, J.P., de Pablo, J.J., Graham, M.D. 2007 Fast computation of many-particle hydrodynamic and electrostatic interactions in a confined geometry. Phys. Rev. Lett. 98, 140602.10.1103/PhysRevLett.98.140602CrossRefGoogle Scholar
Hoell, C., Löwen, H., Menzel, A.M. & Daddi-Moussa-Ider, A. 2019 Creeping motion of a solid particle inside a spherical elastic cavity: II. Asymmetric motion. Eur. Phys. J. E 42 (89), 1–14.10.1140/epje/i2019-11853-4CrossRefGoogle ScholarPubMed
Hu, Y., Zhang, X. & Wang, W. 2010 Boundary conditions at the liquid–liquid interface in the presence of surfactants. Langmuir 26 (13), 10693–10702.10.1021/la101025hCrossRefGoogle ScholarPubMed
Ioannou, I.A., Monck, C., Ceroni, F., Brooks, N.J., Kuimova, M.K. & Elani, Y. 2024 Nucleated synthetic cells with genetically driven intercompartment communication. Proc. Natl Acad. Sci. USA 121 (36), e2404790121.10.1073/pnas.2404790121CrossRefGoogle ScholarPubMed
Kabacaoğlu, G. & Biros, G. 2019 Sorting same-size red blood cells in deep deterministic lateral displacement devices. J. Fluid Mech. 859, 433–475.10.1017/jfm.2018.829CrossRefGoogle Scholar
Keh, H.J. & Lee, T.C. 2010 Axisymmetric creeping motion of a slip spherical particle in a nonconcentric spherical cavity. Theor. Comput. Fluid Dyn. 24 (5), 497–510.10.1007/s00162-010-0181-yCrossRefGoogle Scholar
Konopka, M.C., Shkel, I.A., Cayley, S., Record, M.T. & Weisshaar, J.C. 2006 Crowding and confinement effects on protein diffusion in vivo. J. Bacteriol. 188 (17), 6115–6123.10.1128/JB.01982-05CrossRefGoogle ScholarPubMed
Kramer, S.C., Davies, D.R. & Wilson, C.R. 2021 Analytical solutions for mantle flow in cylindrical and spherical shells. Geosci. Model Dev. 14 (4), 1899–1919.10.5194/gmd-14-1899-2021CrossRefGoogle Scholar
Lavrenteva, O., Prakash, J. & Nir, A. 2016 Effect of added mass on the interaction of bubbles in a low-Reynolds-number shear flow. Phys. Rev. E 93, 023105.10.1103/PhysRevE.93.023105CrossRefGoogle Scholar
Lee, J. & Ladd, A.J.C. 2007 Particle dynamics and pattern formation in a rotating suspension. J. Fluid Mech. 577, 183–209.10.1017/S002211200700465XCrossRefGoogle Scholar
Lee, T.C. & Keh, H.J. 2013 Slow motion of a spherical particle in a spherical cavity with slip surfaces. Intl J. Engng Sci. 69, 1–15.10.1016/j.ijengsci.2013.03.010CrossRefGoogle Scholar
Li, J., Jiang, X., Singh, A., Heinonen, O.G., Hernández-Ortiz, J.P. & de Pablo, J.J. 2020 Structure and dynamics of hydrodynamically interacting finite-size Brownian particles in a spherical cavity: spheres and cylinders. J. Chem. Phys. 152 (20), 204109.10.1063/1.5139431CrossRefGoogle Scholar
Liu, L., Wu, F., Ju, X.-J., Xie, R., Wang, W., Niu, C.H. & Chu, L.-Y. 2013 Preparation of monodisperse calcium alginate microcapsules via internal gelation in microfluidic-generated double emulsions. J. Colloid Interface Sci. 404, 85–90.10.1016/j.jcis.2013.04.044CrossRefGoogle ScholarPubMed
Luo, H.-Y., Jiang, C., Dou, S.-X., Wang, P.-Y. & Li, H. 2024 Quantum dot-based three-dimensional single-particle tracking characterizes the evolution of spatiotemporal heterogeneity in necrotic cells. Anal. Chem. 96 (29), 11682–11689.10.1021/acs.analchem.4c00097CrossRefGoogle ScholarPubMed
Magrinya, P., Palacios-Alonso, P., Llombart, P., Delgado-Buscalioni, R., Alexander-Katz, A., Arriaga, L.R. & Aragones, J.L. 2025 Rolling vesicles: from confined rotational flows to surface-enabled motion. Proc. Natl Acad. Sci. USA 122 (13), e2424236122.10.1073/pnas.2424236122CrossRefGoogle ScholarPubMed
Maheshwari, A.J., Sunol, A.M., Gonzalez, E., Endy, D. & Zia, R.N. 2019 Colloidal hydrodynamics of biological cells: a frontier spanning two fields. Phys. Rev. Fluids 4, 110506.10.1103/PhysRevFluids.4.110506CrossRefGoogle Scholar
Martinez-Pedrero, F., Navarro-Argemí, E., Ortiz-Ambriz, A., Pagonabarraga, I. & Tierno, P. 2018 Emergent hydrodynamic bound states between magnetically powered micropropellers. Sci. Adv. 4 (1), eaap9379.10.1126/sciadv.aap9379CrossRefGoogle ScholarPubMed
Maxey, M.R. & Riley, J.J. 1983 Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26 (4), 883–889.10.1063/1.864230CrossRefGoogle Scholar
van de Meent, J.-W., A.J., Gladden, L.F. & Goldstein, R.E. 2010 Measurement of cytoplasmic streaming in single plant cells by magnetic resonance velocimetry. J. Fluid Mech. 642, 5–14.10.1017/S0022112009992187CrossRefGoogle Scholar
Michaelides, E.E. 1997 Review – the transient equation of motion for particles, bubbles, and droplets. J. Fluids Engng 119 (2), 233–247.10.1115/1.2819127CrossRefGoogle Scholar
Nan, L., Zhang, H., Weitz, D.A. & Shum, H.C. 2024 Development and future of droplet microfluidics. Lab on a Chip 24, 1135–1153.10.1039/D3LC00729DCrossRefGoogle ScholarPubMed
O’Neill, M.E. & Majumdar, S.R. 1970 a Asymmetrical slow viscous fluid motions caused by the translation or rotation of two spheres. Part I: The determination of exact solutions for any values of the ratio of radii and separation parameters. Z. Angew. Math. Phys. 21, 164–179.10.1007/BF01590641CrossRefGoogle Scholar
O’Neill, M.E. & Majumdar, S.R. 1970 b Asymmetrical slow viscous fluid motions caused by the translation or rotation of two spheres. Part II: Asymptotic forms of the solutions when the minimum clearance between the spheres approaches zero. Z. Angew. Math. Phys. 21, 180–187.10.1007/BF01590642CrossRefGoogle Scholar
Oseen, C.W. 1927 Neuere Methoden Und Ergebnisse in der Hydrodynamik, vol. 1. Akademische Verlagsgesellschaft MBH.Google Scholar
Pozrikidis, C. 1992 Boundary integral and singularity methods for linearized viscous flow. Cambridge Texts in Applied Mathematics, vol. 8. Cambridge University Press.Google Scholar
Prindle, J.R., de Cuba, O.I.C. & Gahlmann, A. 2023 Single-molecule tracking to determine the abundances and stoichiometries of freely-diffusing protein complexes in living cells: past applications and future prospects. J. Chem. Phys. 159 (7), 071002.10.1063/5.0155638CrossRefGoogle ScholarPubMed
Rallabandi, B. 2021 Inertial forces in the Maxey–Riley equation in nonuniform flows. Phys. Rev. Fluids 6, L012302.10.1103/PhysRevFluids.6.L012302CrossRefGoogle Scholar
Sandoval-Jiménez, I.M., Jacinto-Méndez, D., Toscano-Flores, L.G. & Carbajal-Tinoco, M.D. 2020 Brownian-particle motion used to characterize mechanical properties of lipid vesicles. J. Chem. Phys. 152 (1), 014901.10.1063/1.5133092CrossRefGoogle ScholarPubMed
Sellier, A. 2008 Slow viscous motion of a solid particle in a spherical cavity. Comput. Model. Engng Sci. 25 (3), 165–180.Google Scholar
Sheng, J. et al. 2024 Formylation boosts the performance of light-driven overcrowded alkene-derived rotary molecular motors. Nat. Chem. 16, 1330–1338.10.1038/s41557-024-01521-0CrossRefGoogle ScholarPubMed
Shi, A. & Schwartz, D.K. 2024 Bridging macroscopic diffusion and microscopic cavity escape of Brownian and active particles in irregular porous networks. ACS Nano 18 (34), 22864–22873.10.1021/acsnano.4c02873CrossRefGoogle ScholarPubMed
Singh, A., Li, J., Jiang, X., Hernández-Ortiz, J.P., Jaeger, H.M. & de Pablo, J.J. 2020 Shape induced segregation and anomalous particle transport under spherical confinement. Phys. Fluids 32 (5), 053307.10.1063/5.0002906CrossRefGoogle Scholar
Skolnick, J. 2016 Perspective: on the importance of hydrodynamic interactions in the subcellular dynamics of macromolecules. J. Chem. Phys. 145 (10), 100901.10.1063/1.4962258CrossRefGoogle ScholarPubMed
Stein, D.B., de Caino, G., Lauga, E., Shelley, M.J. & Goldstein, R.E. 2021 Swirling instability of the microtubule cytoskeleton. Phys. Rev. Lett. 126, 028103.10.1103/PhysRevLett.126.028103CrossRefGoogle ScholarPubMed
Sun, Z., Chen, G., de Pablo, J.J. & Jiang, X. 2024 Particulate transport in the low-Reynolds-number fluid confined in a spherical cavity. Chin. J. Theor. Appl. Mech. 56 (5), 1284–1296.Google Scholar
Sunol, A.M. & Zia, R.N. 2023 Confined Brownian suspensions: equilibrium diffusion, thermodynamics, and rheology. J. Rheol. 67 (2), 433–460.10.1122/8.0000520CrossRefGoogle Scholar
Swan, J.W. & Brady, J.F. 2010 Particle motion between parallel walls: hydrodynamics and simulation. Phys. Fluids 22 (10), 103301.10.1063/1.3487748CrossRefGoogle Scholar
Tallapragada, P. & Sudarsanam, S. 2019 Chaotic advection and mixing by a pair of microrotors in a circular domain. Phys. Rev. E 100, 062207.10.1103/PhysRevE.100.062207CrossRefGoogle Scholar
Tang, T., Zhao, H., Shen, S., Yang, L. & Lim, C.T. 2024 Enhancing single-cell encapsulation in droplet microfluidics with fine-tunable on-chip sample enrichment. Microsyst. Nanoengng 10 (3), 1–12.Google ScholarPubMed
Tsai, S.-T. 2022 Sedimentation motion of sand particles in moving water (I): the resistance on a small sphere moving in non-uniform flow. Theor. Appl. Mech. Lett. 12 (6), 100392.10.1016/j.taml.2022.100392CrossRefGoogle Scholar
Vollenbroek, J.C., Nieuwelink, A.-E., Bomer, J.G., Tiggelaar, R.M., van den Berg, A., Weckhuysen, B.M. & Odijk, M. 2023 Droplet microreactor for high-throughput fluorescence-based measurements of single catalyst particle acidity. Microsyst. Nanoengng 9 (39), 1–11.Google ScholarPubMed
Wang, H. & Hu, G.H. 2017 P granules phase transition induced by cytoplasmic streaming in Caenorhabditis elegans embryo. Sci. China Phys. Mech. Astron. 60 (1), 018711.10.1007/s11433-016-0388-6CrossRefGoogle Scholar
Wang, X., Hou, Y., Yao, L., Gao, M. & Ge, M. 2016 Generation, characterization, and application of hierarchically structured self-assembly induced by the combined effect of self-emulsification and phase separation. J. Am. Chem. Soc. 138 (7), 2090–2093.10.1021/jacs.5b12149CrossRefGoogle ScholarPubMed
Xiang, L., Chen, K., Yan, R., Li, W. & Xu, K. 2020 Single-molecule displacement mapping unveils nanoscale heterogeneities in intracellular diffusivity. Nat. Meth. 17 (5), 524–530.10.1038/s41592-020-0793-0CrossRefGoogle ScholarPubMed
Supplementary material: File

Sun et al. supplementary material

Sun et al. supplementary material
Download Sun et al. supplementary material(File)
File 403.4 KB