Hostname: page-component-7dd5485656-frp75 Total loading time: 0 Render date: 2025-10-30T02:42:48.165Z Has data issue: false hasContentIssue false
  • English
  • Français

Extended Thermodynamic Approach for Non-Equilibrium Gas Flow

Published online by Cambridge University Press:  03 June 2015

G. H. Tang ,
G. X. Zhai ,
W. Q. Tao ,
X. J. Gu  and
D. R. Emerson
G. H. Tang*
Affiliation:
MOE Key Laboratory of Thermo-Fluid Science and Engineering, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, P.R. China
G. X. Zhai
Affiliation:
MOE Key Laboratory of Thermo-Fluid Science and Engineering, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, P.R. China
W. Q. Tao
Affiliation:
MOE Key Laboratory of Thermo-Fluid Science and Engineering, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, P.R. China
X. J. Gu
Affiliation:
Computational Science and Engineering Department, STFC Daresbury Laboratory, Warrington WA4 4AD, UK
D. R. Emerson
Affiliation:
Computational Science and Engineering Department, STFC Daresbury Laboratory, Warrington WA4 4AD, UK
*
Email:ghtang@mail.xjtu.edu.cn
Get access

Abstract

Gases in microfluidic structures or devices are often in a non-equilibrium state. The conventional thermodynamic models for fluids and heat transfer break down and the Navier-Stokes-Fourier equations are no longer accurate or valid. In this paper, the extended thermodynamic approach is employed to study the rarefied gas flow in microstructures, including the heat transfer between a parallel channel andpressure-driven Poiseuille flows through a parallel microchannel andcircular microtube. The gas flow characteristics are studied and it is shown that the heat transfer in the non-equilibrium state no longer obeys the Fourier gradient transport law. In addition, the bimodal distribution of streamwise and spanwise velocity and temperature through a long circular microtube is captured for the first time.

Keywords

76P0582B40Non-equilibriummoment methodmicrochannelfriction coefficient

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 13 , Issue 5 , May 2013 , pp. 1330 - 1356
Copyright
Copyright © Global Science Press Limited 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Gad-el-Hak, M.The Fluid mechanics of microdevices-the Freeman Scholar lecture. ASME J. Fluids Eng., 121: 5–33,1999.Google Scholar
[2]Beskok, A.Validation of a new velocity-slip model for separated gas microflows, Num. Heat Transfer, Part B, 40: 451–471,2001.Google Scholar
[3]Cercignani, C.The Boltzmann Equation and Its Applications. Springer-Verlag, New York, 1988.CrossRefGoogle Scholar
[4]Bird, G.Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon Press, Oxford, 1994.Google Scholar
[5]Tang, G. H., Tao, W. Q. and He, Y. L.Lattice Boltzmann method for gaseous microflows using kinetic theory boundary conditions. Phys. Fluids, 17: 058101, 2005.CrossRefGoogle Scholar
[6]Tang, G. H., Zhang, Y. H., Gu, X. J. and Emerson, D. R.Lattice Boltzmann modelling Knudsen layer effect in non-equilibrium flows. Europhys. Lett., 83: 40008, 2008.Google Scholar
[7]Mieussens, L.Discrete-velocity models and numerical schemes for the Boltzmann-BGK equation in plane and axisymmetric geometries. J. Comput. Phys., 162: 429–466, 2000.Google Scholar
[8]Sharipov, F.Non-isothermal gas flow through rectangular microchannels. J. Micromech. Microeng., 9: 394–401, 1999.CrossRefGoogle Scholar
[9]Titarev, V. A.Implicit unstructured-mesh method for calculating Poiseuille flows of rarefied gas. Commun.Comput. Phys., 8: 427–444,2010.Google Scholar
[10]Xu, K. and Huang, J. C.A unified gas-kinetic scheme for continuum and rarefied flows. J. Comput. Phys., 229: 7747–7764, 2010.Google Scholar
[11]Gu, X. J. and Emerson, D. R.A high-order moment approach for capturing non-equilibrium phenomena in the transition regime. J. Fluid Mech., 636: 177–216,2009.Google Scholar
[12]Taheri, P., Torrilhon, M. and Struchtrup, H.Couette and Poiseuille microflows: Analytical solutions for regularised 13-moment equations. Phys. Fluids, 21: 017102, 2009.Google Scholar
[13]Gu, X. J., Emerson, D. R. and Tang, G. H.Kramers’ problem and the Knudsen minimum: A theoretical analysis using a linearised 26-moment approach. Contin. Mech. Thermodyn., 21: 345–360, 2009.Google Scholar
[14]Taheri, P. and Struchtrup, H.An extended macroscopic transport model for rarefied gas flows in long capillaries with circular cross section. Phys. Fluids, 22: 112004, 2010.Google Scholar
[15]Gu, X. J. and Emerson, D. R.Modeling oscillatory flows in the transition regime with a high-order moment method. Microfluid. Nanofluid., 10: 389–401,2011.Google Scholar
[16]Grad, H.On the kinetic theory of rarefied gases. Comm. Pure Appl. Math., 2: 331–407,1949.CrossRefGoogle Scholar
[17]Struchtrup, H. and Torrilhon, M.Regularization of Grad’s 13 moment equations: Derivation and linear analysis. Phys. Fluids, 15: 2668–2680,2003.Google Scholar
[18]Struchtrup, H.Macroscopic Transport Equations for Rarefied Gas Flows. Springer, Berlin Heidelberg, 2005.Google Scholar
[19]Lockerby, D. A., Reese, J. M. and Gallis, M. A.The usefulness of higher-order constitutive relations for describing the Knudsen layer. Phys. Fluids, 17: 100609, 2005.CrossRefGoogle Scholar
[20]Lockerby, D. A. and Reese, J. M.On the modelling of isothermal gas flows at the microscale. J. Fluid Mech., 604: 235–261,2008.CrossRefGoogle Scholar
[21]Young, J. B.Calculation of Knudsen layers and jump conditions using the linearized G13 and R13 moment methods. Int. J. Heat Mass Transfer, 54: 2902–2912,2011.Google Scholar
[22]Struchtrup, H. and Torrilhon, M.Higher order effects in rarefied channel flows. Phys. Rev. E, 78: 046301, 2008Google Scholar
[23]Maxwell, J. C.On stresses in rarefied gases a rising from inequalities of temperature. Phil. Trans. Roy. Soc., 17: 231–256,1879.Google Scholar
[24]Struchtrup, H. and Weiss, W.Temperature jump and velocity slip in the moment method. Contin. Mech. Thermodyn., 12: 1–18, 2000.Google Scholar
[25]Hadjiconstantinou, N. G.Oscillatory shear-driven gas flows in the transition and free molecular flow regimes. Phys. Fluids, 17: 100611, 2005.Google Scholar
[26]Patankar, S. V.Numerical Heat Transfer and Fluid Flow. Hemisphere Publishing Corporation, 1980.Google Scholar
[27]Rhie, C. M. and Chow, W. L.Numerical study of turbulent flow past an airfoil with trailing edge separation. AIAA J., 21: 1525–1532, 1983.CrossRefGoogle Scholar
[28]Gallis, M. A., Rader, D. J. and Torczynski, J. R.Calculations of the near-wall thermophoretic force in rarefied gas flow. Phys. Fluids, 14: 4290–4301, 2002.Google Scholar
[29]Li, Q., He, Y. L., Tang, G. H. and Tao, W. Q.Lattice Boltzmann modeling of microchannel flows in the transition flow regime. Microfluid. Nanofluid., 10: 607–618,2011.CrossRefGoogle Scholar
[30]Shen, C., Tian, D. B., Xie, C. and Fan, J.Examination of the LBM in simulation of microchannel flow in transitional regime. Microscale Thermal Eng., 8: 423–432, 2010.Google Scholar
[31]Arkilic, E. B., Schmidt, M. A. and Breuer, K. S.Slip flow in microchannel. J. Microelectromech. Syst., 6: 167–174, 1997.Google Scholar
[32]Tang, G. H., He, Y. L. and Tao, W. Q.Comparison of gas slip models with solutions of lin-earized Boltzmann equation and direct simulation of Monte Carlo method. Int. J. Mod. Phys. C, 18: 203–216, 2007.Google Scholar
[33]Hadjiconstantinou, N. G. and O, SimekConstant-wall-temperature Nusselt number in micro and nano-channels. ASME J. Heat Transfer 124(2): 356–364, 2002.Google Scholar
[34]Ohwada, T., Sone, Y. and Aoki, K.Numerical analysis of the Poiseuille and thermal transpi-ration flows between two parallel plates on the basis of the Boltzmann equation for hardsphere molecules. Phys. Fluids A, 1: 2042–2049, 1989.CrossRefGoogle Scholar
[35]Loyalka, S. K. and Hamoodi, S. A.Poiseuille flow of a rarefied gas in a cylindrical tube: Solution of linearized Boltzmann equation. Phys. Fluids A, 2: 2061–2065, 1990.Google Scholar