Acta Biotheoretica

, Volume 65, Issue 3, pp 211–231 | Cite as

Microscale Gaseous Slip Flow in the Insect Trachea and Tracheoles

Regular Article


An analytical investigation into compressible gas flow with slight rarefactions through the insect trachea and tracheoles during the closed spiracle phase is undertaken, and a complete set of asymptotic analytical solutions is presented. We first obtain estimates of the Reynolds and Mach numbers at the channel terminal ends where the tracheoles directly deliver respiratory gases to the cells, by comparing the magnitude of the different forces in the compressible gas flow. The 2D Navier–Stokes equations with a slip boundary condition are used to investigate compressibility and rarefied effects in the trachea and tracheoles. Expressions for the velocity components, pressure gradients and net flow inside the trachea are then presented. Numerical simulations of the tracheal compressible flow are performed to validate the analytical results from this study. This work extends previous work of Arkilic et al. (J Microelectromech Syst 6(2):167–178, 1997) on compressible flows through a microchannel. Novel devices for microfluidic compressible flow transport may be invented from results obtained in this study.


Microchannels Compressible flow Navier Stokes equations Microfluidic devices 



Professors Shirley Abelman and Frances Duncan thank the NRF South Africa for financial support. The reviewers are thanked for their careful reading of our manuscript. Their useful comments have resulted in an improved manuscript.

Compliance with ethical standards

Conflict of interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


  1. Aboelkassem Y, Staples AE (2012) Flow transport in a microchannel induced by moving wall contractions: a novel micropumping mechanism. Acta Mech 223:463–480CrossRefGoogle Scholar
  2. Aboelkassem Y, Staples AE (2013) Selective pumping in a network: insect-style microscale flow transport. Bioinspir Biomim 8:026004CrossRefGoogle Scholar
  3. Arkilic EB, Schmidt MA, Kenneth SB (1997) Gaseous slip flow in long microchannels. J Microelectromech Syst 6(2):167–178CrossRefGoogle Scholar
  4. Beskok A, Karniadakis G (1993) Simulation of heat and momentum transfer in micro-geometries. AIAA Paper, 93-3269Google Scholar
  5. Cai C, Boyd I, Fan J, Candler GV (2000) Direct simulation methods for low-speed microchannel flows. J Thermophys Heat Transf 14:368–378CrossRefGoogle Scholar
  6. Cai C, Liu DD, Boyd ID (2007) Compressible gas flow inside a two dimensional uniform microchannel. AIAA Aerospace Sciences Meeting, Reno, NevadaGoogle Scholar
  7. Duncan FD, Byrne MJ (2002) Respiratory airflow in a wingless beetle. J Exp Biol 205:2489–2497Google Scholar
  8. Glowinski R, Lichnewsky A (1990) Computing methods in applied sciences and engineering. SIAM, Philadelphia, pp 162–164Google Scholar
  9. Gullan PJ, Cranston PS (2014) The insects: an outline of entomology, 5th edn. Wiley, New YorkGoogle Scholar
  10. Hetz SK, Wasserthal LT, Hermann S, Kaden H, Oelbner W (1994) Direct oxygen measurements in the tracheal system of lepidopterous pupae using miniaturized amperometric sensors. Bioelectrochem Bioenerg 33(2):165–170CrossRefGoogle Scholar
  11. Li M, Brasseur JG (1993) Non-steady peristaltic transport in finite-length tubes. J Fluid Mech 248:129–151CrossRefGoogle Scholar
  12. Lord R (1977) Tangential momentum accommodation coefficients of rare gases on polycrystalline metal surfaces. In: Potter J (ed) Rarefied gas dynamics. American Institute of Aeronautics and Astronautics, New YorkGoogle Scholar
  13. Massey B (1989) Mechanics of fluids, 6th edn. Van Nostrand, LondonCrossRefGoogle Scholar
  14. Nation JL (2002) Insect physiology and biochemistry. CRC Press, Boca Raton, pp 327–347Google Scholar
  15. Oran ES, Oh CK, Cybyk BZ (1998) Direct simulation Monte Carlo: recent advances and applications. Annu Rev Fluid Mech 30:403–441CrossRefGoogle Scholar
  16. Piekos E, Breuer K (1995) DSMC modeling of micromechanical devices. AIAA Paper, 95-2089Google Scholar
  17. Pong K, Ho C, Liu J, Tai Y (1994) Non-linear pressure distribution in uniform microchannels. In Application of microfabrication to fluid mechanics, ASME Winter Annual Meeting, Chicago, IL, pp 51–56Google Scholar
  18. Schmitz A, Perry SF (1999) Stereological determination of tracheal volume and diffusing capacity of the tracheal walls in the stick insect Carausius morosus (Phasmatodea, Lonchodidae). J Physiol Biochem Zool 72(2):205–218CrossRefGoogle Scholar
  19. Simelane SM, Abelman S, Duncan FD (2016) Gas exchange models for a flexible tracheal system. J Acta Biotheor 64(2):161–196CrossRefGoogle Scholar
  20. Socha JJ, Lee WK, Harrison JF, Waters JS, Fezzaa K, Westneat MW (2008) Correlated patterns of tracheal compression and convective gas exchange in a carabid beetle. J Exp Biol 211:3409–3420CrossRefGoogle Scholar
  21. Socha JJ, Forster TD, Greenlee KJ (2010) Issues of convection in insect respiration: insights from synchrotron X-ray imaging and beyond. Respir Physiol Neurobiol 173:65–73CrossRefGoogle Scholar
  22. Wang X, Cheng C, Wang S, Liu S (2009) Electroosmotic pumps and their applications in microfluidic systems. Microfluid Nanofluidics 6(2):145CrossRefGoogle Scholar
  23. Wasserthal LT (2014) Periodic heartbeat reversals cause cardiogenic inspiration and expiration with coupled spiracle leakage in resting bowflies, Calliphora vicina. J Exp Biol 217:1543–1554CrossRefGoogle Scholar
  24. White FM (1991) Viscous fluid flow. McGraw-Hill, BostonGoogle Scholar
  25. Withers CP (1992) Comparative animal physiology. Sanders College Publishing, OrlandoGoogle Scholar
  26. Wolfram Research, Inc (2012) Mathematica, Version 9.0, Champaign, ILGoogle Scholar
  27. Xu K, Li Z (2004) Microchannel flow in the slip regime: gas-kinetic BGK-Burnett solutions. J Fluid Mech 513:87–110CrossRefGoogle Scholar
  28. Zohar Y, Lee SY, Lee WY, Jiang L, Tong P (2004) Subsonic gas flow in a straight and uniform microchannel. J Fluid Mech 472:125–151Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.School of Computer Science and Applied MathematicsUniversity of the WitwatersrandJohannesburgSouth Africa
  2. 2.School of Animal, Plant and Environmental SciencesUniversity of the WitwatersrandJohannesburgSouth Africa

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