Skip to main content Accessibility help

Extended Thermodynamic Approach for Non-Equilibrium Gas Flow

  • G. H. Tang (a1), G. X. Zhai (a1), W. Q. Tao (a1), X. J. Gu (a2) and D. R. Emerson (a2)...


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.


Corresponding author


Hide All
[1]Gad-el-Hak, M.The Fluid mechanics of microdevices-the Freeman Scholar lecture. ASME J. Fluids Eng., 121: 533,1999.
[2]Beskok, A.Validation of a new velocity-slip model for separated gas microflows, Num. Heat Transfer, Part B, 40: 451471,2001.
[3]Cercignani, C.The Boltzmann Equation and Its Applications. Springer-Verlag, New York, 1988.
[4]Bird, G.Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon Press, Oxford, 1994.
[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.
[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.
[7]Mieussens, L.Discrete-velocity models and numerical schemes for the Boltzmann-BGK equation in plane and axisymmetric geometries. J. Comput. Phys., 162: 429466, 2000.
[8]Sharipov, F.Non-isothermal gas flow through rectangular microchannels. J. Micromech. Microeng., 9: 394401, 1999.
[9]Titarev, V. A.Implicit unstructured-mesh method for calculating Poiseuille flows of rarefied gas. Commun.Comput. Phys., 8: 427444,2010.
[10]Xu, K. and Huang, J. C.A unified gas-kinetic scheme for continuum and rarefied flows. J. Comput. Phys., 229: 77477764, 2010.
[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: 177216,2009.
[12]Taheri, P., Torrilhon, M. and Struchtrup, H.Couette and Poiseuille microflows: Analytical solutions for regularised 13-moment equations. Phys. Fluids, 21: 017102, 2009.
[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: 345360, 2009.
[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.
[15]Gu, X. J. and Emerson, D. R.Modeling oscillatory flows in the transition regime with a high-order moment method. Microfluid. Nanofluid., 10: 389401,2011.
[16]Grad, H.On the kinetic theory of rarefied gases. Comm. Pure Appl. Math., 2: 331407,1949.
[17]Struchtrup, H. and Torrilhon, M.Regularization of Grad’s 13 moment equations: Derivation and linear analysis. Phys. Fluids, 15: 26682680,2003.
[18]Struchtrup, H.Macroscopic Transport Equations for Rarefied Gas Flows. Springer, Berlin Heidelberg, 2005.
[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.
[20]Lockerby, D. A. and Reese, J. M.On the modelling of isothermal gas flows at the microscale. J. Fluid Mech., 604: 235261,2008.
[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: 29022912,2011.
[22]Struchtrup, H. and Torrilhon, M.Higher order effects in rarefied channel flows. Phys. Rev. E, 78: 046301, 2008
[23]Maxwell, J. C.On stresses in rarefied gases a rising from inequalities of temperature. Phil. Trans. Roy. Soc., 17: 231256,1879.
[24]Struchtrup, H. and Weiss, W.Temperature jump and velocity slip in the moment method. Contin. Mech. Thermodyn., 12: 118, 2000.
[25]Hadjiconstantinou, N. G.Oscillatory shear-driven gas flows in the transition and free molecular flow regimes. Phys. Fluids, 17: 100611, 2005.
[26]Patankar, S. V.Numerical Heat Transfer and Fluid Flow. Hemisphere Publishing Corporation, 1980.
[27]Rhie, C. M. and Chow, W. L.Numerical study of turbulent flow past an airfoil with trailing edge separation. AIAA J., 21: 15251532, 1983.
[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: 42904301, 2002.
[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: 607618,2011.
[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: 423432, 2010.
[31]Arkilic, E. B., Schmidt, M. A. and Breuer, K. S.Slip flow in microchannel. J. Microelectromech. Syst., 6: 167174, 1997.
[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: 203216, 2007.
[33]Hadjiconstantinou, N. G. and O, SimekConstant-wall-temperature Nusselt number in micro and nano-channels. ASME J. Heat Transfer 124(2): 356364, 2002.
[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: 20422049, 1989.
[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: 20612065, 1990.


Extended Thermodynamic Approach for Non-Equilibrium Gas Flow

  • G. H. Tang (a1), G. X. Zhai (a1), W. Q. Tao (a1), X. J. Gu (a2) and D. R. Emerson (a2)...


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed