Acta Mechanica

, Volume 229, Issue 10, pp 4113–4129 | Cite as

Slip flow in a microchannel driven by rhythmic wall contractions

  • Krishnashis ChatterjeeEmail author
  • Anne Staples
Original Paper


We adopt a recent minimal mathematical model of a pumping mechanism in entomological respiratory systems and consider the model’s behavior in the slip flow regime, which occurs naturally in the distalmost portions of insect respiratory systems. In the model, a phase lag in the timing of two neighboring wall contractions in a rectangular microchannel produces a unidirectional flow. The current study investigates the results of incorporating slip effects into the model by introducing first-order accurate slip boundary conditions to investigate the method’s performance for slip flows at the microscale in the slip flow regime. The two-dimensional Navier–Stokes equations are solved with microscale and lubrication theory assumptions, and the tangential momentum accommodation coefficient is assumed to be one, so that the slip flow parameter \(\beta \) is identically equivalent to the Knudsen number, Kn. The variations of the axial velocity, pressure gradient, and total pressure along the channel are determined for three representative Knudsen numbers that span the continuum and slip flow regimes. It was observed that for the shear-driven flow investigated here, the overall effect of increasing the amount of slip is to decrease the volumetric flow rate and that the phase lag for producing maximum flow is in the range of \(63^\circ \)\(67^\circ \), while in the no-slip case the optimum phase lag is approximately \(63^\circ \). The results suggest that shear-driven flows at the microscale in the slip flow regime may see a reduction in flow rate in contrast to pressure-driven microscale gas flows in the slip flow regime.


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The authors would like to thank the National Science Foundation (Grant No. 1437387) for providing the funding support for this study and Dr. Yasser Aboelkassem for providing valuable insights.


  1. 1.
    Socha, J.J., Lee, W.K., Harrison, J.F., Waters, J.S., Fezzaa, K., Westneat, M.W.: Correlated patterns of tracheal compression and convective gas exchange in a carabid beetle. J. Exp. Biol. 211, 3409–3420 (2008)CrossRefGoogle Scholar
  2. 2.
    Uchida, A., Aoki, H.: Unsteady flows in a semi-infinite contracting or expanding pipe. J. Fluid Mech. 82, 371–387 (1977)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Secomb, T.W.: Flow in a channel with pulsating walls. J. Fluid Mech. 88, 273–288 (1978)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Zhang, T., Jia, L., Wang, Z.: Validation of Navier–Stokes equations for slip flow analysis within transition region. Int. J. Heat Mass Transf. 51, 6323–6327 (2008)CrossRefGoogle Scholar
  5. 5.
    Arkilic, E.B., Schmidt, M.A.: Gaseous slip flow in long microchannels. J. Microelectromech. Syst. 6(2), 167–178 (1997)CrossRefGoogle Scholar
  6. 6.
    Agrawal, A.: A comprehensive review on gas flow in microchannels. Int. J. Micro-Nano Scale Transp. 2(1), 1–40 (2011)CrossRefGoogle Scholar
  7. 7.
    Aboelkassem, Y., Staples, A.E.: Flow transport in a microchannel induced by moving wall contractions: a novel micropumping mechanism. Acta Mech. 223, 463–480 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Meijing, L., Brasseur, J.G.: Non-steady peristaltic transport in finite-length tubes. J. Fluid Mech. 248, 129–151 (1993)CrossRefGoogle Scholar
  9. 9.
    Aboelkassem, Y., Staples, A.E., Socha, J.J.: Microscale flow pumping inspired by rhythmic tracheal compressions in insects. In: Proc. ASME Press. Vessels Piping Conf., PVP2011, 57061 (2011)Google Scholar
  10. 10.
    Singh, P., Radhakrishnan, V., Narayan, K.A.: Squeezing flow between parallel plates. Ingenious Arch. 60, 274–281 (1990)CrossRefGoogle Scholar
  11. 11.
    Kwang-Hua Chu, A.: Transport control within a microtube. Phys. Rev. E 70, 061902 (2004)CrossRefGoogle Scholar
  12. 12.
    Ralph, M.E., Pedley, T.J.: Flow in a channel with moving indentation. J. Fluid Mech. 190, 87–112 (1988)CrossRefGoogle Scholar
  13. 13.
    Ramachandra Rao, A., Mishra, M.: Nonlinear and curvature effects on peristaltic flow of a viscous fluid in an asymmetric channel. Acta Mech. 168, 35–59 (2004)CrossRefGoogle Scholar
  14. 14.
    Macagno, E.O., Christensen, J.: Fluid mechanics of the duodenum. Ann. Rev. Fluid Mech. 12, 139–158 (1980)CrossRefGoogle Scholar
  15. 15.
    Macagno, E.O., Christensen, J., Lee, L.: Modeling the effect of wall movement on absorption in the intestine. Am. J. Physiol. 243, G541–G550 (1982)Google Scholar
  16. 16.
    Dongari, N., Agarwal, A.: Analytical solution of gaseous slip flow in long microchannels. Int. J. Heat Mass Transf. 50, 3411–3421 (2007)CrossRefGoogle Scholar
  17. 17.
    Ebaid, A.: Effects of magnetic field and wall slip conditions on the peristaltic transport of a newtonian fluid in an asymmetric microchannel. Phys. Lett. A. 372, 4493–4499 (2008)CrossRefGoogle Scholar
  18. 18.
    Westneat, M.W., Betz, O., Blob, R.W., Fezzaa, K., Cooper, W.J., Lee, W.K.: Tracheal respiration in insects visualized with synchrotron X-ray imaging. Science 299, 558–560 (2003)CrossRefGoogle Scholar
  19. 19.
    Aboelkassem, Y., Staples, A.E.: Selective pumping in a network: insect-style microscale flow transport. Bioinspir. Biomim. 8, 026004 (2013)CrossRefGoogle Scholar
  20. 20.
    Aboelkassem, Y., Staples, A.E.: Bioinspired pumping model for flow in a microtube with rhythmic wall contractions. J. Fluids Struct. 42, 187–204 (2013)CrossRefGoogle Scholar
  21. 21.
    San, O., Staples, A.E.: Dynamics of pulsatile flows through elastic microtubes. Int. J. Appl. Mech. 04, 1250006 (2012)CrossRefGoogle Scholar
  22. 22.
    Skalak, F.M., Wang, C.Y.: On the unsteady squeezing of a viscous fluid from a tube. J. Aust. Math. Soc. (Ser. B) 21, 65–74 (1979)CrossRefGoogle Scholar
  23. 23.
    Shapiro, A.H., Jaffrin, M.Y., Weinberg, S.L.: Peristaltic pumping with long wavelengths at low Reynolds number. J. Fluid. Mech. 37, 799–825 (1969)CrossRefGoogle Scholar
  24. 24.
    Tretheway, D.C., Liu, X., Meinhart, C.D.: Analysis of slip flow in microchannels. In: Proc. 11th Inter. Symp. Appl. Laser Tech. to Fluid Mech., Lisbon, pp. 8–11 (2002)Google Scholar
  25. 25.
    Davey, A.: On the stability of flow in an elliptic pipe which is nearly circular. J. Fluid Mech. 87(2), 233–241 (1978)CrossRefGoogle Scholar
  26. 26.
    Prusa, V.: On the influence of boundary condition on stability of Hagen–Poiseuille flow. Comput. Math. Appl. 57(5), 763–771 (2009)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Aboelkassem, Y., Staples, A.E.: Stokeslets-meshfree computations and theory for flow in a collapsible microchannel. Theor. Comput. Fluid Dyn. 27(5), 681–700 (2013)CrossRefGoogle Scholar
  28. 28.
    Aboelkassem, Y., Staples, A.E.: A three-dimensional model for flow pumping in a microchannel inspired by insect respiration. Acta Mech. 225(2), 493–507 (2014)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Huh, D., Kim, H.J., Fraser, J.P., Shea, D.E., Khan, M., Bahinski, A., Hamilton, G.A., Ingber, D.E.: Microfabrication of human organs-on-chips. Nat. Am. Inc. 8(11), 2135–2157 (2013)Google Scholar
  30. 30.
    Dittrich, P.S., Manz, A.: Lab-on-a-chip: microfluidics in drug discovery. Nature 5, 210–218 (2006)CrossRefGoogle Scholar
  31. 31.
    Grinias, J.P., Kennedy, R.T.: Advances in and prospects of microchip liquid chromatography. Trends Analyt. Chem. 81, 110–117 (2016)CrossRefGoogle Scholar
  32. 32.
    Unger, M.A., Chou, H.P., Thorsen, T., Scherer, A., Quake, S.R.: Monolithic microfabricated valves and pumps by multilayer soft lithography. Science 288(5463), 113–116 (2000)CrossRefGoogle Scholar
  33. 33.
    Thorsen, T., Maerkl, S.J., Quake, S.R.: Microfluidic large-scale integration. Science 298(5593), 580–584 (2002)CrossRefGoogle Scholar
  34. 34.
    Agrawal, A., Prabhu, S.V.: Survey on measurement of tangential momentum accommodation coefficient. J. Vac. Sci. Technol. A 26(4), 634–645 (2008)CrossRefGoogle Scholar
  35. 35.
    Socha, J.J., Forster, T.D., Greenlee, K.J.: Issues of convection in insect respiration: insights from synchrotron X-ray imaging and beyond. Resp. Physiol. Neurobiol. 173, S65–S73 (2010)CrossRefGoogle Scholar
  36. 36.
    Lauga, E., Stone, H.A.: Effective slip in pressure-driven Stokes flow. J. Fluid Mech. 489, 55–77 (2003)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Biomedical Engineering and MechanicsVirginia TechBlacksburgUSA

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