- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Access by Xinjiang University

Flow rate-pressure drop relation for deformable channels via fluidic and elastic reciprocal theorems

Evgeniy Boyko1,2,3,*, Howard A. Stone1,†, and Ivan C. Christov2,‡

  • 1Department of Mechanical and Aerospace Engineering, Princeton University, New Jersey 08544, USA
  • 2School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47907, USA
  • 3Davidson School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907, USA

  • *eboyko@princeton.edu; eboyko@purdue.edu
  • †hastone@princeton.edu
  • ‡christov@purdue.edu
PDF

Phys. Rev. Fluids 7, L092201 – Published 2 September, 2022

DOI: https://doi.org/10.1103/PhysRevFluids.7.L092201

Abstract

Viscous flows through configurations manufactured or naturally assembled from soft materials apply both pressure and shear stress at the solid-liquid interface, leading to deformation of the fluidic conduit's cross section, which, in turn, affects the flow rate-pressure drop relation. Conventionally, calculating this flow rate-pressure drop relation requires solving the complete elastohydrodynamic problem, which couples the fluid flow and elastic deformation. In this Letter, we use the reciprocal theorems for Stokes flow and linear elasticity to derive a closed-form expression for the flow rate-pressure drop relation in deformable microchannels, bypassing the detailed calculation of the solution to the fluid-structure-interaction problem. For small deformations (under a domain perturbation scheme), our theory provides the leading-order effect of the interplay between the fluid stresses and the compliance of the channel on the flow rate-pressure drop relation. Our approach uses solely the fluid flow solution and the elastic deformation due to the corresponding fluid stress distribution in an undeformed channel, eliminating the need to solve the coupled elastohydrodynamic problem. Unlike previous theoretical studies that neglected the presence of lateral sidewalls (and considered shallow geometries of effectively infinite width), our approach allows us to determine the influence of confining sidewalls on the flow rate-pressure drop relation. In particular, for the flow-rate-controlled situation and the Kirchhoff-Love plate-bending theory for the elastic deformation, we show a trade-off between the effect of compliance of the deforming top wall and the drag due to sidewalls on the pressure drop. Whereas compliance decreases the pressure drop, the drag due to sidewalls increases it. Our theoretical framework may provide insight into existing experimental data and pave the way for the design of novel optimized soft microfluidic configurations of different cross-sectional shapes.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (46)

  1. T. Gervais, J. El-Ali, A. Günther, and K. F. Jensen, Flow-induced deformation of shallow microfluidic channels, Lab Chip 6, 500 (2006).
  2. E. Seker, D. C. Leslie, H. Haj-Hariri, J. P. Landers, M. Utz, and M. R. Begley, Nonlinear pressure-flow relationships for passive microfluidic valves, Lab Chip 9, 2691 (2009).
  3. I. C. Christov, Soft hydraulics: from Newtonian to complex fluid flows through compliant conduits, J. Phys.: Condens. Matter 34, 063001 (2022).
  4. T. Thorsen, S. J. Maerkl, and S. R. Quake, Microfluidic large-scale integration, Science 298, 580 (2002).
  5. M. Gomez, D. E. Moulton, and D. Vella, Passive Control of Viscous Flow via Elastic Snap-Through, Phys. Rev. Lett. 119, 144502 (2017).
  6. K. Hosokawa, K. Hanada, and R. Maeda, A polydimethylsiloxane (PDMS) deformable diffraction grating for monitoring of local pressure in microfluidic devices, J. Microelectromechan. Syst. 12, 1 (2002).
  7. O. Ozsun, V. Yakhot, and K. L. Ekinci, Non-invasive measurement of the pressure distribution in a deformable micro-channel, J. Fluid Mech. 734, R1 (2013).
  8. 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).
  9. Y. Matia, T. Elimelech, and A. D. Gat, Leveraging internal viscous flow to extend the capabilities of beam-shaped soft robotic actuators, Soft Robotics 4, 126 (2017).
  10. A. Vurgaft, S. B. Elbaz, and A. D. Gat, Forced motion of a cylinder within a liquid-filled elastic tube–a model of minimally invasive medical procedures, J. Fluid Mech. 881, 1048 (2019).
  11. D. Chakraborty, J. R. Prakash, J. Friend, and L. Yeo, Fluid-structure interaction in deformable microchannels, Phys. Fluids 24, 102002 (2012).
  12. S. I. Rubinow and J. B. Keller, Flow of a viscous fluid through an elastic tube with applications to blood flow, J. Theor. Biol. 35, 299 (1972).
  13. A. H. Shapiro, Steady flow in collapsible tubes, ASME J. Biomech. Eng. 99, 126 (1977).
  14. 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).
  15. T. Shidhore and I. C. Christov, Static response of deformable microchannels: a comparative modelling study, J. Phys.: Condens. Matter 30, 054002 (2018).
  16. V. Anand, J. D. J. Rathinaraj, and I. C. Christov, Non-Newtonian fluid–structure interactions: Static response of a microchannel due to internal flow of a power-law fluid, J. Non-Newtonian Fluid Mech. 264, 62 (2019).
  17. A. Mehboudi and J. Yeom, Experimental and theoretical investigation of a low-Reynolds-number flow through deformable shallow microchannels with ultra-low height-to-width aspect ratios, Microfluid. Nanofluid. 23, 66 (2019).
  18. X. Wang and I. C. Christov, Theory of the flow-induced deformation of shallow compliant microchannels with thick walls, Proc. R. Soc. London, Ser. A 475, 20190513 (2019).
  19. L. Ramos-Arzola and O. Bautista, Fluid structure-interaction in a deformable microchannel conveying a viscoelastic fluid, J. Non-Newtonian Fluid Mech. 296, 104634 (2021).
  20. H. A. Lorentz, A general theorem concerning the motion of a viscous fluid and a few consequences derived from it (in Dutch), Versl. Konigl. Akad. Wetensch. Amst. 5, 168 (1896).
  21. J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics (Martinus Nijhoff Publishers, The Hague, 1983)
  22. H. Masoud and H. A. Stone, The reciprocal theorem in fluid dynamics and transport phenomena, J. Fluid Mech. 879, P1 (2019).
  23. C. Berdan II and L. G. Leal, Motion of a sphere in the presence of a deformable interface, I: Perturbation of the interface from flat: the effects on drag and torque, J. Colloid Interface Sci. 87, 62 (1982).
  24. V. A. Shaik and A. M. Ardekani, Motion of a model swimmer near a weakly deforming interface, J. Fluid Mech. 824, 42 (2017).
  25. B. Rallabandi, B. Saintyves, T. Jules, T. Salez, C. Schönecker, L. Mahadevan, and H. A. Stone, Rotation of an immersed cylinder sliding near a thin elastic coating, Phys. Rev. Fluids 2, 074102 (2017).
  26. B. Rallabandi, N. Oppenheimer, M. Y. Ben Zion, and H. A. Stone, Membrane-induced hydroelastic migration of a particle surfing its own wave, Nat. Phys. 14, 1211 (2018).
  27. A. Daddi-Moussa-Ider, B. Rallabandi, S. Gekle, and H. A. Stone, Reciprocal theorem for the prediction of the normal force induced on a particle translating parallel to an elastic membrane, Phys. Rev. Fluids 3, 084101 (2018).
  28. A. Kargar-Estahbanati and B. Rallabandi, Lift forces on three-dimensional elastic and viscoelastic lubricated contacts, Phys. Rev. Fluids 6, 034003 (2021).
  29. V. Bertin, Y. Amarouchene, E. Raphael, and T. Salez, Soft-lubrication interactions between a rigid sphere and an elastic wall, J. Fluid Mech. 933, A23 (2022).
  30. L. G. Leal, The motion of small particles in non-Newtonian fluids, J. Non-Newtonian Fluid Mech. 5, 33 (1979).
  31. L. G. Leal, Particle motions in a viscous fluid, Annu. Rev. Fluid Mech. 12, 435 (1980).
  32. R. F. Day and H. A. Stone, Lubrication analysis and boundary integral simulations of a viscous micropump, J. Fluid Mech. 416, 197 (2000).
  33. S. Michelin and E. Lauga, A reciprocal theorem for boundary-driven channel flows, Phys. Fluids 27, 111701 (2015).
  34. E. Boyko and H. A. Stone, Reciprocal theorem for calculating the flow rate–pressure drop relation for complex fluids in narrow geometries, Phys. Rev. Fluids 6, L081301 (2021).
  35. E. Boyko and H. A. Stone, Pressure-driven flow of the viscoelastic Oldroyd-B fluid in narrow non-uniform geometries: analytical results and comparison with simulations, J. Fluid Mech. 936, A23 (2022).
  36. J. C. Maxwell, On the calculation of the equilibrium and stiffness of frames, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 27, 294 (1864).
  37. E. Betti, Teoria della elasticitá, Il Nuovo Cimento 7-8, 69 (1872).
  38. J. D. Achenbach, Reciprocity in Elastodynamics (Cambridge University Press, Cambridge, UK, 2003).
  39. A. Martínez-Calvo, A. Sevilla, G. G. Peng, and H. A. Stone, Start-up flow in shallow deformable microchannels, J. Fluid Mech. 885, A25 (2020).
  40. S. Timoshenko and S. Woinkowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1987).
  41. P. Howell, G. Kozyreff, and J. Ockendon, Applied Solid Mechanics, Cambridge Texts in Applied Mathematics No. 43 (Cambridge University Press, Cambridge, UK, 2009).
  42. L. G. Leal, Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes (Cambridge University Press, Cambridge, UK, 2007).
  43. H. Bruus, Theoretical Microfluidics (Oxford University Press, Oxford, 2008).
  44. A. E. H. Love, The small free vibrations and deformation of a thin elastic shell, Philos. Trans. R. Soc., London, Ser. A 179, 491 (1888).
  45. D. G. Duffy, Green's Functions with Applications (Chapman and Hall/CRC, Boca Raton, FL, 2015).
  46. S. P. Sutera and R. Skalak, The history of Poiseuille's law, Annu. Rev. Fluid Mech. 25, 1 (1993).

Outline

Information

Phys. Rev. Fluids 7, L092201– Published 2 September, 2022
  • Received 22 March 2022
  • Accepted 19 August 2022
Article part of CHORUS
Crossmark - Check for updates button

©2022 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation