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Start-up flow in shallow deformable microchannels

Published online by Cambridge University Press:  27 December 2019

Alejandro Martínez-Calvo*
Affiliation:
Grupo de Mecánica de Fluidos, Departamento de Ingeniería Térmica y de Fluidos, Universidad Carlos III de Madrid, Av. Universidad 30, 28911 Leganés (Madrid), Spain
Alejandro Sevilla
Affiliation:
Grupo de Mecánica de Fluidos, Departamento de Ingeniería Térmica y de Fluidos, Universidad Carlos III de Madrid, Av. Universidad 30, 28911 Leganés (Madrid), Spain
Gunnar G. Peng
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, CambridgeCB3 0WA, UK
Howard A. Stone
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ08544, USA
*Corresponding
Email address for correspondence: amcalvo@ing.uc3m.es

Abstract

Microfluidic systems are usually fabricated with soft materials that deform due to the fluid stresses. Recent experimental and theoretical studies on the steady flow in shallow deformable microchannels have shown that the flow rate is a nonlinear function of the pressure drop due to the deformation of the upper soft wall. Here, we extend the steady theory of Christov et al. (J. Fluid Mech., vol. 841, 2018, pp. 267–286) by considering the start-up flow from rest, both in pressure-controlled and in flow-rate-controlled configurations. The characteristic scales and relevant parameters governing the transient flow are first identified, followed by the development of an unsteady lubrication theory assuming that the inertia of the fluid is negligible, and that the upper wall can be modelled as an elastic plate under pure bending satisfying the Kirchhoff–Love equation. The model is governed by two non-geometrical dimensionless numbers: a compliance parameter $\unicode[STIX]{x1D6FD}$, which compares the characteristic displacement of the upper wall with the undeformed channel height, and a parameter $\unicode[STIX]{x1D6FE}$ that compares the inertia of the solid with its flexural rigidity. In the limit of negligible solid inertia, $\unicode[STIX]{x1D6FE}\rightarrow 0$, a quasi-steady model is developed, whereby the fluid pressure satisfies a nonlinear diffusion equation, with $\unicode[STIX]{x1D6FD}$ as the only parameter, which admits a self-similar solution under pressure-controlled conditions. This simplified lubrication description is validated with coupled three-dimensional numerical simulations of the Navier equations for the elastic solid and the Navier–Stokes equations for the fluid. The agreement is very good when the hypotheses behind the model are satisfied. Unexpectedly, we find fair agreement even in cases where the solid and liquid inertia cannot be neglected.

Type
JFM Papers
Copyright
© 2019 Cambridge University Press

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References

Anna, S. L. 2016 Droplets and bubbles in microfluidic devices. Annu. Rev. Fluid Mech. 48, 285309.CrossRefGoogle Scholar
Bico, J., Reyssat, É. & Roman, B. 2018 Elastocapillarity: when surface tension deforms elastic solids. Annu. Rev. Fluid Mech. 50, 629659.CrossRefGoogle Scholar
Bisplinghoff, R. L., Ashley, H. & Halfman, R. L. 2013 Aeroelasticity. Courier Corporation.Google Scholar
Bruus, H. 2008 Theoretical Microfluidics. Oxford University Press.Google Scholar
Cancelli, C. & Pedley, T. J. 1985 A separated-flow model for collapsible-tube oscillations. J. Fluid Mech. 157, 375404.CrossRefGoogle Scholar
Cassot, F., Lauwers, F., Fouard, C., Prohaska, S. & Lauwers-Cances, V. 2006 A novel three-dimensional computer-assisted method for a quantitative study of microvascular networks of the human cerebral cortex. Microcirculation 13 (1), 118.CrossRefGoogle ScholarPubMed
Cheung, P., Toda-Peters, K. & Shen, A. Q. 2012 In situ pressure measurement within deformable rectangular polydimethylsiloxane microfluidic devices. Biomicrofluidics 6 (2), 026501.CrossRefGoogle ScholarPubMed
Christov, I. C., Cognet, V., Shidhore, T. C. & Stone, H. A. 2018 Flow rate–pressure drop relation for deformable shallow microfluidic channels. J. Fluid Mech. 841, 267286.CrossRefGoogle Scholar
Conrad, W. A. 1969 Pressure–flow relationships in collapsible tubes. IEEE Trans. Biomed. Engng 16 (4), 284295.CrossRefGoogle ScholarPubMed
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 (7), 818828.CrossRefGoogle Scholar
Duprat, C. & Stone, H. A.(Eds) 2016 Fluid–structure interactions in low-Reynolds-number flows. In Soft Matter Series. The Royal Society of Chemistry.Google Scholar
El-Ali, J., Sorger, P. K. & Jensen, K. F. 2006 Cells on chips. Nature 442 (7101), 403.CrossRefGoogle 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, 221237.CrossRefGoogle 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, 580602.CrossRefGoogle Scholar
Elbaz, S. B., Jacob, H. & Gat, A. D. 2018 Transient gas flow in elastic microchannels. J. Fluid Mech. 846, 460481.CrossRefGoogle Scholar
Fung, Y.-C. 1993a Biomechanics: Circulation. Springer.CrossRefGoogle Scholar
Fung, Y.-C. 1993b Biomechanics: Mechanical Properties of Living Tissues. Springer.CrossRefGoogle Scholar
Fung, Y.-C. 1993c Biomechanics: Motion, Flow, Stress, and Growth. Springer.CrossRefGoogle Scholar
Gervais, T., El-Ali, J., Günther, A. & Jensen, K. F. 2006 Flow-induced deformation of shallow microfluidic channels. Lab on a Chip 6 (4), 500507.CrossRefGoogle ScholarPubMed
Goldsmith, H. L. & Skalak, R. 1975 Hemodynamics. Annu. Rev. Fluid Mech. 7 (1), 213247.CrossRefGoogle Scholar
Grotberg, J. B. 1994 Pulmonary flow and transport phenomena. Annu. Rev. Fluid Mech. 26 (1), 529571.CrossRefGoogle Scholar
Grotberg, J. B. 2001 Respiratory fluid mechanics and transport processes. Annu. Rev. Fluid Mech. 3 (1), 421457.Google ScholarPubMed
Grotberg, J. B. & Jensen, O. E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 36, 121147.CrossRefGoogle Scholar
Happel, J. & Brenner, H. 2012 Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media. Springer Science & Business Media.Google 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 (7), 935938.CrossRefGoogle Scholar
Heil, M. 1997 Stokes flow in collapsible tubes: computation and experiment. J. Fluid Mech. 353, 285312.CrossRefGoogle Scholar
Heil, M. & Hazel, A. L. 2011 Fluid–structure interaction in internal physiological flows. Annu. Rev. Fluid Mech. 43, 141162.CrossRefGoogle Scholar
Heil, M., Jensen, O. E., Pedley, T. J. & Carpenter, P. W. 2003 Flow in collapsible tubes and past other highly compliant boundaries. In Proceedings of the IUTAM Symposium held at the University of Warwick, United Kingdom, 26–30 March 2001 (ed. Carpenter, P. W. & Pedley, T. J.). Springer.Google Scholar
Heil, M. & Pedley, T. J. 1995 Large axisymmetric deformation of a cylindrical shell conveying a viscous flow. J. Fluids Struct. 9 (3), 237256.CrossRefGoogle Scholar
Howell, P., Kozyreff, G. & Ockendon, J. 2009 Applied Solid Mechanics. Cambridge University Press.Google Scholar
Huppert, H. E. 1982 The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 4358.CrossRefGoogle Scholar
Ilievski, F., Mazzeo, A. D., Shepherd, R. F., Chen, X. & Whitesides, G. M. 2011 Soft robotics for chemists. Angew. Chem. Intl Ed. Engl. 123 (8), 19301935.CrossRefGoogle Scholar
Juel, A., Pihler-Puzović, D. & Heil, M. 2018 Instabilities in blistering. Annu. Rev. Fluid Mech. 50, 691714.CrossRefGoogle Scholar
Kerr, A. D. 1964 Elastic and viscoelastic foundation models. J. Appl. Mech. 31 (3), 491498.CrossRefGoogle Scholar
Lasheras, J. C. 2007 The biomechanics of arterial aneurysms. Annu. Rev. Fluid Mech. 39, 293319.CrossRefGoogle Scholar
Lauga, E. 2016 Bacterial hydrodynamics. Annu. Rev. Fluid Mech. 48, 105130.CrossRefGoogle Scholar
Lister, J. R., Peng, G. G. & Neufeld, J. A. 2013 Viscous control of peeling an elastic sheet by bending and pulling. Phy. Rev. Lett. 111 (15), 154501.CrossRefGoogle Scholar
Love, A. E. H. 1888 The small free vibrations and deformation of a thin elastic shell. Phil. Trans. R. Soc. Lond. A 179, 491546.CrossRefGoogle Scholar
Majidi, C. 2014 Soft robotics: a perspective – current trends and prospects for the future. Soft Robot. 1 (1), 511.CrossRefGoogle 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.CrossRefGoogle Scholar
Païdoussis, M. P., Price, S. J. & De Langre, E. 2010 Fluid–Structure Interactions: Cross-Flow-Induced Instabilities. Cambridge University Press.CrossRefGoogle 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 (11), 115001.CrossRefGoogle Scholar
Pattle, R. E. 1959 Diffusion from an instantaneous point source with a concentration-dependent coefficient. Q. J. Mech. Appl. Maths 12 (4), 407409.CrossRefGoogle Scholar
Pedley, T. J. 1980 The Fluid Mechanics of Large Blood Vessels. Cambridge University Press.CrossRefGoogle Scholar
Pedley, T. J. & Luo, X. Y. 1998 Modelling flow and oscillations in collapsible tubes. Theor. Comput. Fluid Dyn. 10 (1–4), 277294.CrossRefGoogle Scholar
Polygerinos, P., Correll, N., Morin, S. A., Mosadegh, B., Onal, C. D., Petersen, K., Cianchetti, M., Tolley, M. T. & Shepherd, R. F. 2017 Soft robotics: review of fluid-driven intrinsically soft devices; manufacturing, sensing, control, and applications in human–robot interaction. Adv. Engng Mater. 19 (12), 1700016.CrossRefGoogle Scholar
Popel, A. S. & Johnson, P. C. 2005 Microcirculation and hemorheology. Annu. Rev. Fluid Mech. 37, 4369.CrossRefGoogle ScholarPubMed
Raj, A. & Sen, A. K. 2016 Flow-induced deformation of compliant microchannels and its effect on pressure–flow characteristics. Microfluid. Nanofluid. 20 (2), 31.CrossRefGoogle Scholar
Raj, M. K., DasGupta, S. & Chakraborty, S. 2017 Hydrodynamics in deformable microchannels. Microfluid. Nanofluid. 21 (4), 70.CrossRefGoogle Scholar
Rivero-Rodríguez, J. & Scheid, B. 2018 Bubble dynamics in microchannels: inertial and capillary migration forces. J. Fluid Mech. 842, 215247.CrossRefGoogle Scholar
Rivero-Rodriguez, J. & Scheid, B. 2019 Mass transfer around bubbles flowing in cylindrical microchannels. J. Fluid Mech. 869, 110142.CrossRefGoogle Scholar
Rodríguez-Rodríguez, J., Sevilla, A., Martínez-Bazán, C. & Gordillo, J. M. 2015 Generation of microbubbles with applications to industry and medicine. Annu. Rev. Fluid Mech. 47, 405429.CrossRefGoogle Scholar
Rus, D. & Tolley, M. T. 2015 Design, fabrication and control of soft robots. Nature 521 (7553), 467.CrossRefGoogle ScholarPubMed
Sackmann, E. K., Fulton, A. L. & Beebe, D. J. 2014 The present and future role of microfluidics in biomedical research. Nature 507 (7491), 181.CrossRefGoogle ScholarPubMed
Secomb, T. W., Hsu, R. & Pries, A. R. 2002 Blood flow and red blood cell deformation in nonuniform capillaries: effects of the endothelial surface layer. Microcirculation 9 (3), 189196.CrossRefGoogle ScholarPubMed
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 (18), 26912697.CrossRefGoogle ScholarPubMed
Sforza, D. M., Putman, C. M. & Cebral, J. R. 2009 Hemodynamics of cerebral aneurysms. Annu. Rev. Fluid Mech. 41, 91107.CrossRefGoogle ScholarPubMed
Shapiro, A. H. 1977 Steady flow in collapsible tubes. Trans. ASME: J. Biomech. Engng 99 (3), 126147.Google Scholar
Shepherd, R. F., Ilievski, F., Choi, W., Morin, S. A., Stokes, A. A., Mazzeo, A. D., Chen, X., Wang, M. & Whitesides, G. M. 2011 Multigait soft robot. Proc. Natl Acad. Sci. USA 108 (51), 2040020403.CrossRefGoogle ScholarPubMed
Shidhore, T. C. & Christov, I. C. 2018 Static response of deformable microchannels: a comparative modelling study. J. Phys.: Condens. Matter 30 (5), 054002.Google ScholarPubMed
Squires, T. M. & Quake, S. R. 2005 Microfluidics: fluid physics at the nanoliter scale. Rev. Mod. Phys. 77 (3), 977.CrossRefGoogle 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, 381411.CrossRefGoogle Scholar
Tabeling, P. 2005 Introduction to Microfluidics. Oxford University Press.Google Scholar
Taylor, C. A. & Draney, M. T. 2004 Experimental and computational methods in cardiovascular fluid mechanics. Annu. Rev. Fluid Mech. 36, 197231.CrossRefGoogle Scholar
Weibel, D. B., Siegel, A. C., Lee, A., George, A. H. & Whitesides, G. M. 2007 Pumping fluids in microfluidic systems using the elastic deformation of poly(dimethylsiloxane). Lab on a Chip 7 (12), 18321836.CrossRefGoogle Scholar
Whitesides, G. M. 2006 The origins and the future of microfluidics. Nature 442 (7101), 368373.CrossRefGoogle ScholarPubMed
Xia, Y. & Whitesides, G. M. 1998 Soft lithography. Angew. Chem. Intl Ed. Engl. 37 (5), 550575.3.0.CO;2-G>CrossRefGoogle ScholarPubMed
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