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Transient gas flow in elastic microchannels

  • Shai B. Elbaz (a1), Hila Jacob (a1) and Amir D. Gat (a1)


We study pressure-driven propagation of gas into a two-dimensional microchannel bounded by linearly elastic substrates. Relevant fields of application include lab-on-a-chip devices, soft robotics and respiratory flows. Applying the lubrication approximation, the flow field is governed by the interaction between elasticity and viscosity, as well as weak rarefaction and low-Mach-number compressibility effects, characteristic of gaseous microflows. A governing equation describing the evolution of channel height is derived for the problem. Several physical limits allow simplification of the governing equation and solution by self-similarity. These limits, representing different physical regimes and their corresponding time scales, include compressibility–elasticity–viscosity, compressibility–viscosity and elasticity–viscosity dominant balances. Transition of the flow field between these regimes and corresponding exact solutions is illustrated for the case of an impulsive mass insertion in which the order of magnitude of the deflection evolves in time. For an initial channel thickness which is similar to the elastic deformation generated by the background pressure, a symmetry between compressibility and elasticity allows us to obtain a self-similar solution which includes weak rarefaction effects. The presented results are validated by numerical solutions of the evolution equation.


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Arkilic, E. B., Breuer, K. S. & Schmidt, M. A. 2001 Mass flow and tangential momentum accommodation in silicon micromachined channels. J. Fluid Mech. 437, 2943.
Arkilic, E. B., Schmidt, M. A. & Breuer, K. S. 1997 Gaseous slip flow in long microchannels. J. Microelectromech. Syst. 6, 167178.
Aubert, C. & Colin, S. 2001 High-order boundary conditions for gaseous flows in rectangular microchannels. Microscale Therm. Engng 5, 4154.
Barenblatt, G. I. 1952 On some unsteady fluid and gas motions in a porous medium. Prikl. Mat. Mekh. 16 (1), 6778 (in Russian).
Cercignani, C. 2000 Rarefied Gas Dynamics. Macmillan.
Chapman, S. & Cowling, T. G. 1952 The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion of Gases, Notes Added in 1951. Cambridge University Press.
Cheung, P., Toda-Peters, K. & Shen, A. Q. 2012 In situ pressure measurement within deformable rectangular polydimethylsiloxane microfluidic devices. Biomicrofluidics 6 (2), 026501.
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.
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.
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.
Gad-el-Hak, M. 1999 The fluid mechanics of microdevices. Trans. ASME J. Fluids Engng 121, 533.
Gat, A. D., Frankel, I. & Weihs, D. 2008 Gas flows through constricted shallow micro-channels. J. Fluid Mech. 602, 427442.
Gat, A. D., Frankel, I. & Weihs, D. 2009 A higher-order Hele-Shaw approximation with application to gas flows through shallow micro-channels. J. Fluid Mech. 638, 141160.
Gat, A. D., Frankel, I. & Weihs, D. 2010a Compressible flows through micro-channels with sharp edged turns and bifurcations. Microfluid. Nanofluid. 8 (5), 619629.
Gat, A. D., Frankel, I. & Weihs, D. 2010b Gas flows through shallow T-junctions and parallel microchannel networks. Phys. Fluids 22 (9), 092001.
Gaver, D. P., Halpern, D., Jensen, O. E. & Grotberg, J. B. 1996 The steady motion of a semi-infinite bubble through a flexible-walled channel. J. Fluid Mech. 319, 2565.
George, D., Anoop, R. & Sen, A. K. 2015 Elastocapillary powered manipulation of liquid plug in microchannels. Appl. Phys. Lett. 107 (26), 261601.
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.
Grotberg, J. B. & Jensen, O. E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 36, 121147.
Günther, A., Khan, S. A., Thalmann, M., Trachsel, F. & Jensen, K. F. 2004 Transport and reaction in microscale segmented gas–liquid flow. Lab on a Chip 4 (4), 278286.
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.
Hewitt, I. J., Balmforth, N. J. & De Bruyn, J. R. 2015 Elastic-plated gravity currents. Eur. J. Appl. Maths 26 (01), 131.
Ho, C. M. & Tai, Y. C. 1998 Micro-electro-mechanical-systems (MEMS) and fluid flows. Annu. Rev. Fluid Mech. 30, 579612.
Hodges, S. R. & Jensen, O. E. 2002 Spreading and peeling dynamics in a model of cell adhesion. J. Fluid Mech. 460, 381409.
Hosoi, A. E. & Mahadevan, L. 2004 Peeling, healing, and bursting in a lubricated elastic sheet. Phys. Rev. Lett. 93 (13), 137802.
Huppert, H. E. 1982 The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 4358.
Ilievski, F., Mazzeo, A. D., Shepherd, R. F., Chen, X. & Whitesides, G. M. 2011 Soft robotics for chemists. Angew. Chem. 123 (8), 19301935.
Jang, J. & Wereley, S. 2004 Pressure distributions of gaseous slip flow in straight and uniform rectangular microchannels. Microfluid. Nanofluid. 1, 4151.
Kang, C., Roh, C. & Overfelt, R. A. 2014 Pressure-driven deformation with soft polydimethylsiloxane (PDMS) by a regular syringe pump: challenge to the classical fluid dynamics by comparison of experimental and theoretical results. RSC Adv. 4 (7), 31023112.
Lee, W. Y., Wong, M. & Zohar, Y. 2001 Gas flow in microchannels with bends. J. Micromesh. Microeng. 11, 635644.
Lee, W. Y., Wong, M. & Zohar, Y. 2002 Pressure loss in constriction microchannels. J. Microelectromech. Syst. 11, 236244.
Leibenzon, L. S. 1930 The Motion of a Gas in a Porous Medium. Complete Works, vol. 2. Acad. Sciences URSS (Russian), 63.
Leslie, D. C., Easley, C. J., Seker, E., Karlinsey, J. M., Utz, M., Begley, M. R. & Landers, J. P. 2009 Frequency-specific flow control in microfluidic circuits with passive elastomeric features. Nat. Phys. 5 (3), 231235.
Li, M. & Brasseur, J. G. 1993 Non-steady peristaltic transport in finite-length tubes. J. Fluid Mech. 248, 129151.
Lister, J. R., Peng, G. G. & Neufeld, J. A. 2013 Viscous control of peeling an elastic sheet by bending and pulling. Phys. Rev. Lett. 111 (15), 154501.
Liu, J., Tai, Y.-C. & Ho, C.-M. 1995 MEMS for pressure distribution studies of gaseous flows in microchannels. In Proceedings of the IEEE Micro Electro Mechanical Systems, pp. 209215.
McEwan, A. D. & Taylor, G. I. 1966 The peeling of a flexible strip attached by a viscous adhesive. J. Fluid Mech. 26 (1), 115.
Muskat, M. 1937 The Flow of Homogeneous Fluids Through Porous Media. McGraw-Hill.
Onal, C. D. 2016 System-level challenges in pressure-operated soft robotics. In SPIE Defense+ Security, 983627. International Society for Optics and Photonics.
Onal, C. D., Chen, X., Whitesides, G. M. & Rus, D. 2017 Soft mobile robots with on-board chemical pressure generation. In Robotics Research, pp. 525540. Springer.
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69 (3), 931.
Pong, K.-C., Ho, C.-M., Liu, J. & Tai, Y.-C. 1994 Non-linear pressure distribution in uniform microchannels. ASME-Publications-Fed. 197, 5151.
Shidhore, T. C. & Christov, I. C. 2018 Static response of deformable microchannels: a comparative modelling study. J. Phys.: Condens. Matter 30 (5), 054002.
Srinivas, S. S. & Kumaran, V. 2017 Effect of viscoelasticity on the soft-wall transition and turbulence in a microchannel. J. Fluid Mech. 812, 10761118.
Takagi, D. & Balmforth, N. J. 2011 Peristaltic pumping of viscous fluid in an elastic tube. J. Fluid Mech. 672, 196218.
Taylor, G. I. & Saffman, P. G. 1957 Effects of compressibility at low Reynolds number. J. Phys. Chem. Ref. Data 24, 553562.
Thorey, C. & Michaut, C. 2016 Elastic-plated gravity currents with a temperature-dependent viscosity. J. Fluid Mech. 805, 88117.
Young, Y.-N. & Stone, H. A. 2017 Long-wave dynamics of an elastic sheet lubricated by a thin liquid film on a wetting substrate. Phys. Rev. Fluids 2 (6), 064001.
Yu, Z. T. F., Lee, Y.-K., Wong, M. & Zohar, Y. 2005 Fluid flows in microchannels with cavities. J. Microelectromech. Syst. 14, 13861398.
Zaouter, T., Lasseux, D. & Prat, M. 2018 Gas slip flow in a fracture: local Reynolds equation and upscaled macroscopic model. J. Fluid Mech. 837, 413442.
Zeng, W., Jacobi, I., Beck, D. J., Li, S. & Stone, H. A. 2015 Characterization of syringe-pump-driven induced pressure fluctuations in elastic microchannels. Lab on a Chip 15 (4), 11101115.
Zohar, Y., Lee, S. Y. K., Lee, W. Y., Jiang, L. & Tong, P. 2002 Subsonic gas flow in a straight and uniform microchannel. J. Fluid Mech. 472, 125151.
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Transient gas flow in elastic microchannels

  • Shai B. Elbaz (a1), Hila Jacob (a1) and Amir D. Gat (a1)


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