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

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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