Hostname: page-component-6dbcb7884d-64ff6 Total loading time: 0 Render date: 2025-02-13T15:09:51.790Z Has data issue: false hasContentIssue false
  • English
  • Français

Flow rate–pressure drop relation for deformable shallow microfluidic channels

Published online by Cambridge University Press:  21 February 2018

Ivan C. Christov Open the ORCID record for Ivan C. Christov [Opens in a new window] ,
Vincent Cognet ,
Tanmay C. Shidhore  and
Howard A. Stone
Ivan C. Christov*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA
Vincent Cognet
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA École Normale Supérieure de Cachan, Cachan, CEDEX, France
Tanmay C. Shidhore
Affiliation:
School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA
Howard A. Stone*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
*
†Email addresses for correspondence: christov@purdue.edu, hastone@princeton.edu
†Email addresses for correspondence: christov@purdue.edu, hastone@princeton.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Laminar flow in devices fabricated from soft materials causes deformation of the passage geometry, which affects the flow rate–pressure drop relation. For a given pressure drop, in channels with narrow rectangular cross-section, the flow rate varies as the cube of the channel height, so deformation can produce significant quantitative effects, including nonlinear dependence on the pressure drop (Gervais et al., Lab on a Chip, vol. 6, 2006, pp. 500–507). Gervais et al. proposed a successful model of the deformation-induced change in the flow rate by heuristically coupling a Hookean elastic response with the lubrication approximation for Stokes flow. However, their model contains a fitting parameter that must be found for each channel shape by performing an experiment. We present a perturbation approach for the flow rate–pressure drop relation in a shallow deformable microchannel using the theory of isotropic quasi-static plate bending and the Stokes equations under a lubrication approximation (specifically, the ratio of the channel’s height to its width and of the channel’s height to its length are both assumed small). Our result contains no free parameters and confirms Gervais et al.’s observation that the flow rate is a quartic polynomial of the pressure drop. The derived flow rate–pressure drop relation compares favourably with experimental measurements.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Low-Reynolds-number flows: Lubrication theory Micro-/Nano-fluid dynamics: Microfluidics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 841 , 25 April 2018 , pp. 267 - 286
Copyright
© 2018 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.)

References

Abramowitz, M. & Stegun, I. A.(Eds) 1972 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series, vol. 55. National Bureau of Standards, http://www.nr.com/aands/.Google Scholar
Ando, K., Sanada, T., Inaba, K., Damazo, J. S., Shepherd, J. E., Colonius, T. & Brennen, C. E. 2011 Shock propagation through a bubbly liquid in a deformable tube. J. Fluid Mech. 671, 339–363.Google Scholar
Anoop, R. & Sen, A. K. 2015 Capillary flow enhancement in rectangular polymer microchannels with a deformable wall. Phys. Rev. E 92, 013024.Google Scholar
Bahga, S. S., Bercovici, M. & Santiago, J. G. 2012 Robust and high-resolution simulations of nonlinear electrokinetic processes invariable cross-section channels. Electrophoresis 33, 3036–3051.Google Scholar
Bodnár, T., Galdi, G. P. & Nečasová, Š.(Eds) 2014 Fluid–Structure Interaction and Biomedical Applications. Advances in Mathematical Fluid Mechanics, Birkhäuser.Google Scholar
Bruus, H. 2008 Theoretical Microfluidics. Oxford University Press.Google Scholar
Chakraborty, D., Prakash, J. R., Friend, J. & Yeo, L. 2012 Fluid–structure interaction in deformable microchannels. Phys. Fluids 24, 102002.Google Scholar
Cheung, P., Toda-Peters, K. & Shen, A. Q. 2012 In situ pressure measurement within deformable rectangular polydimethylsiloxane microfluidic devices. Biomicrofluidics 6, 026501.CrossRefGoogle ScholarPubMed
Conrad, W. A. 1969 Pressure–flow relationships in collapsible tubes. IEEE Trans. Biomed. Engng BME‐16, 284–295.Google Scholar
Dendukuri, D., Gu, S. S., Pregibon, D. C., Hatton, T. A. & Doyle, P. S. 2007 Stop-flow lithography in a microfluidic device. Lab on a Chip 7, 818–828.CrossRefGoogle Scholar
Ducloué, L., Hazel, A. L., Thompson, A. B. & Juel, A. 2017 Reopening modes of a collapsed elasto-rigid channel. J. Fluid Mech. 819, 121–146.CrossRefGoogle Scholar
Duprat, C. & Stone, H. A. 2016 Fluid–Structure Interactions in Low-Reynolds-Number Flows. The Royal Society of Chemistry.Google Scholar
Elbaz, S. B. & Gat, A. D. 2014 Dynamics of viscous liquid within a closed elastic cylinder subject to external forces with application to soft robotics. J. Fluid Mech. 758, 221–237.Google Scholar
Elbaz, S. B. & Gat, A. D. 2016 Axial creeping flow in the gap between a rigid cylinder and a concentric elastic tube. J. Fluid Mech. 806, 580–602.Google Scholar
Fung, Y. C. 1997 Biomechanics: Circulation, 2nd edn. Springer.Google Scholar
Gervais, T., El-Ali, J., Günther, A. & Jensen, K. F. 2006 Flow-induced deformation of shallow microfluidic channels. Lab on a Chip 6, 500–507.Google Scholar
Ghosal, S. 2002 Lubrication theory for electro-osmotic flow in a microfluidic channel of slowly varying cross-section and wall charge. J. Fluid Mech. 459, 103–128.CrossRefGoogle Scholar
Gomez, M., Moulton, D. E. & Vella, D. 2017 Passive control of viscous flow via elastic snap-through. Phys. Rev. Lett. 119, 144502.Google Scholar
Grotberg, J. B. & Jensen, O. E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 36, 121–147.Google Scholar
Happel, J. R. & Brenner, H. 1983 Low Reynolds Number Hydrodynamics, 2nd edn. Martinus Nijhoff Publishers.CrossRefGoogle Scholar
Hardy, B. S., Uechi, K., Zhen, J. & Kavehpour, H. P. 2009 The deformation of flexible PDMS microchannels under a pressure driven flow. Lab on a Chip 9, 935–938.Google Scholar
Holden, M. A., Kumar, S., Beskok, A. & Cremer, P. S. 2003 Microfluidic diffusion diluter: bulging of PDMS microchannels under pressure-driven flow. J. Micromech. Microengng 13, 412–418.Google Scholar
Iliev, O., Mikelić, A. & Popov, P. 2008 On upscaling certain flows in deformable porous media. Multiscale Model. Simul. 7, 93–123.Google Scholar
Johnson, K. L. 1985 Contact Mechanics. Cambridge University Press.CrossRefGoogle Scholar
Johnston, I. D., McCluskey, D. K., Tan, C. K. L. & Tracey, M. C. 2014 Mechanical characterization of bulk Sylgard 184 for microfluidics and microengineering. J. Micromech. Microengng 24, 035017.Google Scholar
Katz, A. I., Chen, Y. & Moreno, A. H. 1969 Flow through a collapsible tube: experimental analysis and mathematical model. Biophys. J. 9, 1261–1279.CrossRefGoogle ScholarPubMed
Kizilova, N., Hamadiche, M. & Gad-El-Hak, M. 2012 Mathematical models of biofluid flows in compliant ducts. Arch. Mech. 64, 65–94.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1986 Theory of Elasticity, 3rd edn. Butterworth-Heinemann.Google Scholar
Lauga, E., Stroock, A. D. & Stone, H. A. 2004 Three-dimensional flows in slowly varying planar geometries. Phys. Fluids 16, 3051–3062.CrossRefGoogle Scholar
Lebovitz, N. R. 1982 Perturbation expansions on perturbed domains. SIAM Rev. 24, 381–400.CrossRefGoogle Scholar
Lötters, J. C., Olthuis, W., Veltink, P. H. & Bergveld, P. 1997 The mechanical properties of the rubber elastic polymer polydimethylsiloxane for sensor applications. J. Micromech. Microengng 7, 145–147.Google Scholar
Love, A. E. H. 1888 The small free vibrations and deformation of a thin elastic shell. Phil. Trans. R. Soc. Lond. A 179, 491–546.Google Scholar
Mukherjee, U., Chakraborty, J. & Chakraborty, S. 2013 Relaxation characteristics of a compliant microfluidic channel under electroosmotic flow. Soft Matt. 9, 1562–1569.CrossRefGoogle Scholar
Niu, P., Nablo, B. J., Bhadriraju, K. & Reyes, D. R. 2017 Uncovering the contribution of microchannel deformation to impedance-based flow rate measurements. Anal. Chem. 89, 11372–11377.Google Scholar
Ozsun, O., Yakhot, V. & Ekinci, K. L. 2013 Non-invasive measurement of the pressure distribution in a deformable micro-channel. J. Fluid Mech. 734, R1.Google Scholar
Panda, P., Yuet, K. P., Dendukuri, D., Hatton, T. A. & Doyle, P. S. 2009 Temporal response of an initially deflected PDMS channel. New J. Phys. 11, 115001.CrossRefGoogle Scholar
Pedley, T. J. 1980 The Fluid Mechanics of Large Blood Vessels. Cambridge University Press.Google Scholar
Raj, A. & Sen, A. K. 2016 Flow-induced deformation of compliant microchannels and its effect on pressure–flow characteristics. Microfluid. Nanofluid. 20, 31.CrossRefGoogle Scholar
Raj, M. K., DasGupta, S. & Chakraborty, S. 2017 Hydrodynamics in deformable microchannels. Microfluid. Nanofluid. 21, 70.Google Scholar
Rubinow, S. I. & Keller, J. B. 1972 Flow of a viscous fluid through an elastic tube with applications to blood flow. J. Theor. Biol. 34, 299–313.Google Scholar
Schomburg, W. K. 2011 Introduction to Microsystem Design. Springer.CrossRefGoogle Scholar
Seker, E., Leslie, D. C., Haj-Hariri, H., Landers, J. P., Utz, M. & Begley, M. R. 2009 Nonlinear pressure–flow relationships for passive microfluidic valves. Lab on a Chip 9, 2691–2697.Google Scholar
Small, M. K. & Nix, W. D. 1992 Analysis of the accuracy of the bulge test in determining the mechanical properties of thin films. J. Mat. Res. 7, 1553–1563.CrossRefGoogle Scholar
Sollier, E., Murray, C., Maoddi, P. & Di Carlo, D. 2011 Rapid prototyping polymers for microfluidic devices and high pressure injections. Lab on a Chip 11, 3752–3765.CrossRefGoogle ScholarPubMed
Squires, T. M. & Quake, S. R. 2005 Microfluidics: fluid physics at the nanoliter scale. Rev. Mod. Phys. 77, 977–1026.Google Scholar
Stone, H. A., Stroock, A. D. & Ajdari, A. 2004 Engineering flows in small devices: Microfluidics toward a lab-on-a-chip. Annu. Rev. Fluid Mech. 36, 381–411.Google Scholar
Sutera, S. P. & Skalak, R. 1993 The history of Poiseuille’s law. Annu. Rev. Fluid Mech. 25, 1–19.Google Scholar
Tavakol, B., Froehlicher, G., Holmes, D. P. & Stone, H. A. 2017 Extended lubrication theory: improved estimates of flow in channels with variable geometry. Proc. R. Soc. Lond. A 473, 20170234.Google Scholar
Timoshenko, S. & Woinowsky-Krieger, S. 1959 Theory of Plates and Shells, 2nd edn. McGraw-Hill.Google Scholar
Van Dyke, M. D. 1975 Perturbation Methods in Fluid Mechanics. Parabolic Press.Google Scholar
Whittaker, R. J., Heil, M., Jensen, O. E. & Waters, S. L. 2010 A rational derivation of a tube law from shell theory. Q. J. Mech. Appl. Maths 63, 465–496.Google Scholar
Xia, Y. & Whitesides, G. M. 1998 Soft lithography. Annu. Rev. Mater. Sci. 28, 153–184.Google Scholar