- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Fluids

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Weakly deformable poroelastic particle in an unbounded Stokes flow

Simon M. Finney1,*, Matthew G. Hennessy2, Andreas Münch1, and Sarah L. Waters1,†

  • *Contact author: finney@maths.ox.ac.uk
  • †Contact author: sarah.waters@maths.ox.ac.uk
PDF

Phys. Rev. Fluids 10, 093603 – Published 23 September, 2025

DOI: https://doi.org/10.1103/p3g6-gkww

Abstract

A framework is developed to study the deformation of a spherical poroelastic particle immersed in an unbounded, three-dimensional Stokes flow. The flow is driven by imposing a steady far-field condition, and the particle is modeled using a two-phase approach, where a deformable solid skeleton is fully saturated by the surrounding viscous fluid. Slip is permitted on the interface between the poroelastic particle and the surrounding Stokes flow. We consider the regime in which the ratio of typical viscous fluid stress to elastic stiffness is small, leading to small deformations and a decoupling of the fluid and solid problems. The traction exerted by the fluid on the particle and the Darcy pressure within are computed and used to formulate a purely solid-mechanics problem for the equilibrium particle deformation. To demonstrate the method, two example far-field flow profiles (shear flow and Poiseuille flow) are considered. Closed-form solutions for the translational velocity, rotation, and surface deformation of the particle are presented and analyzed as functions of the particle's permeability, slip, and Poisson's ratio. We show that the rotation of the particle is not impacted by its poroelasticity and that the Poisson's ratio plays a key role in selecting the dominant mechanism of particle deformation. For incompressible particles, the shear traction exerted by the fluid on the particle drives the deformation, causing the deformation to decrease with slip. For compressible particles, the Darcy pressure in the particle drives the deformation, and the deformation increases with slip.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (55)

  1. D. Caccavo, An overview on the mathematical modeling of hydrogels' behavior for drug delivery systems, Int. J. Pharm. 560, 175 (2019).
  2. M. Penn and M. Hennessy, Optimal loading of hydrogel-based drug-delivery systems, Appl. Math. Modell. 112, 649 (2022).
  3. J. Li and D. J. Mooney, Designing hydrogels for controlled drug delivery, Nat. Rev. Mater. 1, 16071 (2016).
  4. M. Villone and P. Maffettone, Dynamics, rheology, and applications of elastic deformable particle suspensions: A review, Rheol. Acta 58, 109 (2019).
  5. M. M. Villone, J. K. Nunes, Y. Li, H. A. Stone, and P. L. Maffettone, Design of a microfluidic device for the measurement of the elastic modulus of deformable particles, Soft Matter 15, 880 (2019).
  6. Y. Chen, K. Guo, L. Jiang, S. Zhu, Z. Ni, and N. Xiang, Microfluidic deformability cytometry: A review, Talanta 251, 123815 (2023).
  7. A. K. Haudenschild, A. H. Hsieh, S. Kapila, and J. C. Lotz, Pressure and distortion regulate human mesenchymal stem cell gene expression, Ann. Biomed. Eng. 37, 492 (2009).
  8. H. Zhang, A. Kay, N. R. Forsyth, K. K. Liu, and A. J. El Haj, Gene expression of single human mesenchymal stem cell in response to fluid shear, J. Tissue Eng. 3, 2041731412451988 (2012).
  9. H. Li, A. Muhammad, and P. Norbert, Mechanosensitive channels and their functions in stem cell differentiation, Exp. Cell Res. 374, 259 (2019).
  10. B. Martinac, Y. A. Nikolaev, G. Silvani, N. Bavi, V. Romanov, Y. Nakayama, A. D. Martinac, P. Rohde, O. Bavi, and C. D. Cox, Cell membrane mechanics and mechanosensory transduction, Curr. Top. Membr. 86, 83 (2020).
  11. F. M. Eroshkin and A. G. Zaraisky, Mechano-sensitive regulation of gene expression during the embryonic development, Genesis 55, e23026 (2017).
  12. M. C. Arno, Engineering the mammalian cell surface with synthetic polymers: Strategies and applications, Macromol. Rapid Commun. 41, 2000302 (2020).
  13. T. T. Pham, P. L. Tran, C. D. Phung, H. T. Nguyen, C. H. Nguyen, C. S. Yong, J. O. Kim, S. Yook, and J.-H. Jeong, Surface-triggered in situ gelation for tunable conformal hydrogel coating of therapeutic cells and biomedical devices, Adv. Funct. Mater. 31, 2010169 (2021).
  14. M. Biot, General theory of three dimensional consolidation, J. Appl. Phys. 12, 155 (1941).
  15. M. Biot, Nonlinear and semilinear rheology of porous solids, J. Geophys. Res. 78, 4924 (1973).
  16. S. Whitaker, Flow in porous media I: A theoretical derivation of Darcy's law, Transp. Porous Media 1, 3 (1986).
  17. L. Durlofsky and J. Brady, Analysis of the Brinkman equation as a model for flow in porous media, Phys. Fluids 30, 3329 (1987).
  18. B. Padmavathi, T. Amaranath, and S. Nigam, Stokes flow past a porous sphere using Brinkman's model, Z. Angew. Math. Phys. 44, 929 (1993).
  19. S.-M. Yang and W.-H. Hong, Motions of a porous particle in Stokes flow: Part 1. Unbounded single-fluid domain problem, Korean J. Chem. Eng. 5, 23 (1988).
  20. S.-M. Yang and W.-H. Hong, Motions of a porous particle in Stokes flow part 2. Linear flows near a fluid interface, Korean J. Chem. Eng. 6, 234 (1989).
  21. S.-M. Yang and L. G. Leal, Motions of a porous particle in Stokes-flow near a plane-fluid interface, Physicochemical Hydrodynamics 11, 543 (1989).
  22. E. Saad and M. Faltas, Slow motion of a porous sphere translating along the axis of a circular cylindrical pore subject to a stress jump condition, Transp. Porous Media 102, 91 (2014).
  23. J. Prakash and G. Sekhar, Slow motion of a porous spherical particle with a rigid core in a spherical fluid cavity, Meccanica 52, 91 (2017).
  24. X. Yao, C. Ng, J. Teo, Marcos, and T. Wong, Slow viscous flow of two porous spherical particles translating along the axis of a cylinder, J. Fluid Mech. 861, 643 (2019).
  25. C. Peskin, The immersed boundary method, Acta Numerica 11, 479 (2002).
  26. W. Strychalski, C. Copos, O. Lewis, and R. Guy, A poroelastic immersed boundary method with applications to cell biology, J. Comput. Phys. 282, 77 (2015).
  27. C. Copos and R. Guy, A porous viscoelastic model for the cell cytoskeleton, ANZIAM J. 59, 472 (2018).
  28. Y. Leng, M. de Lucio, and H. Gomez, Using poro-elasticity to model the large deformation of tissue during subcutaneous injection, Comput. Methods Appl. Mech. Eng. 384, 113919 (2021).
  29. L. Li, J. Zhang, Z. Xu, Y.-N. Young, J. Feng, and P. Yue, An arbitrary Lagrangian-Eulerian method for simulating interfacial dynamics between a hydrogel and a fluid, J. Comput. Phys. 451, 110851 (2022).
  30. T. Murata, Deformation of an elastic particle suspended in an arbitrary flow field, J. Phys. Soc. Jpn. 50, 1009 (1981).
  31. B. Nasouri, A. Khot, and G. Elfring, Elastic two-sphere swimmer in Stokes flow, Phys. Rev. Fluids 2, 043101 (2017).
  32. Y.-N. Young, Y. Mori, and M. J. Miksis, Slightly deformable Darcy drop in linear flows, Phys. Rev. Fluids 4, 063601 (2019).
  33. M. M. Villone, F. Greco, M. A. Hulsen, and P. L. Maffettone, Numerical simulations of deformable particle lateral migration in tube flow of Newtonian and viscoelastic media, J. Non-Newtonian Fluid Mech. 234, 105 (2016).
  34. S. M. Finney, M. G. Hennessy, A. Münch, and S. L. Waters, The impact of confinement on the deformation of an elastic particle under axisymmetric tube flow, IMA J. Appl. Math. 89, 498 (2024).
  35. A. Mietke, O. Otto, S. Girardo, P. Rosendahl, A. Taubenberger, S. Golfier, E. Ulbricht, S. Aland, J. Guck, and E. Fischer-Friedrich, Extracting cell stiffness from real-time deformability cytometry: Theory and experiment, Biophys. J. 109, 2023 (2015).
  36. H. Brenner and J. Happel, Slow viscous flow past a sphere in a cylindrical tube, J. Fluid Mech. 4, 195 (1958).
  37. G. S. Beavers and D. Joseph, Boundary conditions at a naturally permeable wall, J. Fluid Mech. 30, 197 (1967).
  38. P. Saffman, On the boundary condition at the surface of a porous medium, Stud. Appl. Math. 50, 93 (1971).
  39. T. Levy and E. Sanchez-Palencia, On boundary conditions for fluid flow in porous media, Int. J. Eng. Sci. 13, 923 (1975).
  40. J. Ochoa-Tapia and S. Whitaker, Momentum transfer at the boundary between a porous medium and a homogeneous fluid—I. Theoretical development, Int. J. Heat Mass Transf. 38, 2635 (1995).
  41. M. Minale, Momentum transfer within a porous medium. II. Stress boundary condition, Phys. Fluids 26, 123102 (2014).
  42. J. J. Feng and Y.-N. Young, Boundary conditions at a gel-fluid interface, Phys. Rev. Fluids 5, 124304 (2020).
  43. Z. Xu, J. Zhang, Y.-N. Young, P. Yue, and J. Feng, Comparison of four boundary conditions for the fluid-hydrogel interface, Phys. Rev. Fluids 7, 093301 (2022).
  44. R. Ruiz-Baier, M. Taffetani, H. D. Westermeyer, and I. Yotov, The Biot–Stokes coupling using total pressure: Formulation, analysis and application to interfacial flow in the eye, Comput. Methods Appl. Mech. Eng. 389, 114384 (2022).
  45. S. Badia, A. Quaini, and A. Quarteroni, Coupling Biot and Navier–Stokes equations for modeling fluid–poroelastic media interaction, J. Comput. Phys. 228, 7986 (2009).
  46. C. W. MacMinn, E. R. Dufresne, and J. S. Wettlaufer, Large deformations of a soft porous material, Phys. Rev. Appl. 5, 044020 (2016).
  47. See Supplemental Material at http://link.aps.org/supplemental/10.1103/p3g6-gkww for the asymptotic reduction of the nonlinear problem, as well as the reformulation of the interfacial conditions for the fluid and solid problems, which includes Refs. [30, 37, 46, 54, 55].
  48. H. Lamb, Hydrodynamics, 6th ed. (Dover, New York, 1945), pp. 595–596.
  49. P. Howell, G. Kozyreff, and J. Ockendon, Applied Solid Mechanics, Cambridge Texts in Applied Mathematics (Cambridge University Press, Cambridge, 2008).
  50. T. Tran-Cong, On the completeness of the Papkovich-Neuber solution, Q. Appl. Math. 47, 645 (1989).
  51. I. Noichl and C. Schönecker, Dynamics of elastic, nonheavy spheres sedimenting in a rectangular duct, Soft Matter 18, 2462 (2022).
  52. M. Moradi, W. Shi, and E. Nazockdast, General solutions of linear poro-viscoelastic materials in spherical coordinates, J. Fluid Mech. 946, A22 (2022).
  53. S. M. Finney, Poroelastic sphere unbounded, GitHub repository (2024), https://github.com/finneysimon/PoroelasticSphereUnbounded.
  54. S. N. Atluri and A. Cazzani, Rotations in computational solid mechanics, Arch. Comput. Methods Eng. 2, 49 (1995).
  55. J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics, 1st ed., Mechanics of Fluids and Transport Processes (Springer Dordrecht, 1983).

Outline

Information

Phys. Rev. Fluids 10, 093603– Published 23 September, 2025
  • Received 6 September 2024
  • Accepted 21 July 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation