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Microscale Gaseous Slip Flow in the Insect Trachea and Tracheoles

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Abstract

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.

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Notes

  1. We define the aspect ratio as the ratio of the longest side to the shortest side, i.e. we define the aspect ratio as the length (L) to breadth (B) ratio (\(\delta ^{-1} = L/B \gg 1\)).

  2. A value of \(\sigma = 1\) has been used for most engineering problems although values for \(\sigma > 1\) (corresponding to \(\sigma _m < 1\)) have been reported under controlled test conditions (Arkilic et al. 1997; Lord 1977).

  3. The \(\delta ^0\)-term for \(\tilde{v}\) is equal to zero and is not included in the expansion of \(\tilde{v}\) because it is clear from the non-dimensional continuity equation (Eq. 2.8) that the highest order \(\tilde{v}\) term that is required to satisfy the differential continuity equation is \(\mathcal {O} (\delta )\).

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Acknowledgements

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.

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Authors and Affiliations

  1. School of Computer Science and Applied Mathematics, University of the Witwatersrand, Private Bag 3 Wits, Johannesburg, 2050, South Africa

    S. M. Simelane & S. Abelman

  2. School of Animal, Plant and Environmental Sciences, University of the Witwatersrand, Private Bag 3 Wits, Johannesburg, 2050, South Africa

    F. D. Duncan

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  1. S. M. Simelane
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  2. S. Abelman
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  3. F. D. Duncan
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Correspondence to S. Abelman.

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Simelane, S.M., Abelman, S. & Duncan, F.D. Microscale Gaseous Slip Flow in the Insect Trachea and Tracheoles. Acta Biotheor 65, 211–231 (2017). https://doi.org/10.1007/s10441-017-9312-9

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  • DOI: https://doi.org/10.1007/s10441-017-9312-9

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