Non-Newtonian Couette-Poiseuille flow of a dilute gas

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March 2011, 4(1): 361-384. doi: 10.3934/krm.2011.4.361

Non-Newtonian Couette-Poiseuille flow of a dilute gas

1.	Département de Physique, Université Moulay Ismaïl, Meknès, Morocco
2.	Departamento de Física, Universidad de Extremadura, E-06071 Badajoz, Spain

Received September 2010 Revised October 2010 Published January 2011

Abstract
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The steady state of a dilute gas enclosed between two infinite parallel plates in relative motion and under the action of a uniform body force parallel to the plates is considered. The Bhatnagar-Gross-Krook model kinetic equation is analytically solved for this Couette-Poiseuille flow to first order in the force and for arbitrary values of the Knudsen number associated with the shear rate. This allows us to investigate the influence of the external force on the non-Newtonian properties of the Couette flow. Moreover, the Couette-Poiseuille flow is analyzed when the shear-rate Knudsen number and the scaled force are of the same order and terms up to second order are retained. In this way, the transition from the bimodal temperature profile characteristic of the pure force-driven Poiseuille flow to the parabolic profile characteristic of the pure Couette flow through several intermediate stages in the Couette-Poiseuille flow are described. A critical comparison with the Navier-Stokes solution of the problem is carried out.

Keywords: Poiseuille flow, Couette flow, non-Newtonian properties., Bhatnagar-Gross-Krook kinetic model.

Mathematics Subject Classification: Primary: 76P05, 82B40; Secondary: 82C40, 82C0.

Citation: Mohamed Tij, Andrés Santos. Non-Newtonian Couette-Poiseuille flow of a dilute gas. Kinetic & Related Models, 2011, 4 (1) : 361-384. doi: 10.3934/krm.2011.4.361

References:

[1]	M. Alam and V. K. Chikkadi, Velocity distribution function and correlations in a granular Poiseuille flow,, J. Fluid Mech., 653 (2010), 175. doi: 10.1017/S0022112010000200.
[2]	M. Alaoui and A. Santos, Poiseuille flow driven by an external force,, Phys. Fluids A, 4 (1992), 1273. doi: 10.1063/1.858245.
[3]	K. Aoki, S. Takata and T. Nakanishi, A Poiseuille-type flow of a rarefied gas between two parallel plates driven by a uniform external force,, Phys. Rev. E, 65 (2002). doi: 10.1103/PhysRevE.65.026315.
[4]	E. Asmolov, N. K. Makashev and V. I. Nosik, Heat transfer between parallel plates in a gas of Maxwellian molecules,, Sov. Phys. Dokl., 24 (1979), 892.
[5]	P. L. Bhatnagar, E. P. Gross and M. Krook, A model collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems,, Phy. Rev., 94 (1954), 511. doi: 10.1103/PhysRev.94.511.
[6]	J. J. Brey, A. Santos and J. W. Dufty, Heat and momentum transport far from equilibrium,, Phys. Rev. A, 36 (1987), 2842. doi: 10.1103/PhysRevA.36.2842.
[7]	C. Cercignani, "The Boltzmann Equation and Its Applications,'', Springer-Verlag, (1988).
[8]	C. Cercignani, "Mathematical Methods in Kinetic Theory,'', Plenum Press, (1990).
[9]	C. Cercignani, M. Lampis and S. Lorenzani, Plane Poiseuille-Couette problem in micro-electro-mechanical systems applications with gas-rarefaction effects,, Phy. Fluids, 18 (2006). doi: 10.1063/1.2335847.
[10]	C. Cercignani and F. Sernagiotto, Cylindrical Poiseuille flow of a rarefied gas,, Phys. Fluids, 9 (1966), 40. doi: 10.1063/1.1761530.
[11]	S. Chapman and T. G. Cowling, "The Mathematical Theory of Nonuniform Gases,'', Cambridge University Press, (1970).
[12]	V. Chikkadi and M. Alam, Slip velocity and stresses in granular Poiseuille flow via event-driven simulation,, Phys. Rev. E, 80 (2009). doi: 10.1103/PhysRevE.80.021303.
[13]	J. R. Dorfman and H. van Beijeren, The kinetic theory of gases,, in, (1977), 65.
[14]	A. I. Erofeev and O. G. Friedlander, Macroscopic models for non-equilibrium flows of monatomic gas and normal solutions,, in, (2007), 117.
[15]	R. Esposito, J. L. Lebowitz and R. Marra, A hydrodynamic limit of the stationary Boltzmann equation in a slab,, Commun. Math. Phys., 160 (1994), 49. doi: 10.1007/BF02099789.
[16]	M. A. Gallis, J. R. Torczynski, D. J. Rader, M. Tij and A. Santos, Normal solutions of the Boltzmann equation for highly nonequilibrium Fourier flow and Couette flow,, Phys. Fluids, 18 (2006).
[17]	M. A. Gallis, J. R. Torczynski, D. J. Rader, M. Tij and A. Santos, Analytical and numerical normal solutions of the Boltzmann equation for highly nonequilibrium Fourier and Couette flows,, in, (2007), 251.
[18]	L. S. García-Colín, R. M. Velasco and F. J. Uribe, Beyond the Navier-Stokes equations: Burnett hydrodynamics,, Phys. Rep., 465 (2008), 149. doi: 10.1016/j.physrep.2008.04.010.
[19]	V. Garzó and M. López de Haro, Nonlinear transport for a dilute gas in steady Couette flow,, Phys. Fluids, 9 (1997), 776. doi: 10.1063/1.869232.
[20]	V. Garzó and A. Santos, "Kinetic Theory of Gases in Shear Flows. Nonlinear Transport,'', Kluwer Academic Publishers, (2003).
[21]	S. Hess and M. Malek Mansour, Temperature profile of a dilute gas undergoing a plane Poiseuille flow,, Physica A, 272 (1999), 481. doi: 10.1016/S0378-4371(99)00254-X.
[22]	L. P. Kadanoff, G. R. McNamara and G. Zanetti, A Poiseuille viscometer for lattice gas automata,, Complex Syst., 1 (1987), 791.
[23]	L. P. Kadanoff, G. R. McNamara and G. Zanetti, From automata to fluid flow: Comparisons of simulation and theory,, Phys. Rev. A, 40 (1989), 4527. doi: 10.1103/PhysRevA.40.4527.
[24]	C. S. Kim, J. W. Dufty, A. Santos and J. J. Brey, Hilbert-class or "normal'' solutions for stationary heat flow,, Phys. Rev. A, 39 (1989), 328. doi: 10.1103/PhysRevA.39.328.
[25]	C. S. Kim, J. W. Dufty, A. Santos and J. J. Brey, Analysis of nonlinear transport in Couette flow,, Phys. Rev A, 40 (1989), 7165. doi: 10.1103/PhysRevA.40.7165.
[26]	G. M. Kremer, "An Introduction to the Boltzmann Equation and Transport Processes in Gases,'', Springer, (2010). doi: 10.1007/978-3-642-11696-4.
[27]	M. Malek Mansour, F. Baras and A. L. Garcia, On the validity of hydrodynamics in plane Poiseuille flows,, Physica A, 240 (1997), 255. doi: 10.1016/S0378-4371(97)00149-0.
[28]	N. K. Makashev and V. I. Nosik, Steady Couette flow (with heat transfer) of a gas of Maxwellian molecules,, Sov. Phys. Dokl., 25 (1981), 589.
[29]	J. M. Montanero, M. Alaoui, A. Santos and V. Garzó, Monte Carlo simulation of the Boltzmann equation for steady Fourier flow,, Phys. Rev. E, 49 (1994), 367. doi: 10.1103/PhysRevE.49.367.
[30]	J. M. Montanero and V. Garzó, Nonlinear Couette flow in a dilute gas: Comparison between theory and molecular dynamics simulation,, Phys. Rev. E, 58 (1998), 1836. doi: 10.1103/PhysRevE.58.1836.
[31]	J. M. Montanero, A. Santos and V. Garzó, Monte Carlo simulation of nonlinear Couette flow in a dilute gas,, Phys. Fluids, 12 (2000), 3060. doi: 10.1063/1.1313563.
[32]	R. S. Myong, Coupled nonlinear constitutive models for rarefied and microscale gas flows: subtle interplay of kinematics and dissipation effects,, Cont. Mech. Thermodyn., 21 (2009), 389. doi: 10.1007/s00161-009-0112-6.
[33]	V. I. Nosik, Heat transfer between parallel plates in a mixture of gases of Maxwellian molecules,, Sov. Phys. Dokl., 25 (1981), 495.
[34]	V. I. Nosik, Degeneration of the Chapman-Enskog expansion in one-dimensional motions of Maxwellian molecule gases,, in, 13 (1983), 237.
[35]	T. Ohwada, Y. Sone and K. Aoki, Numerical analysis of the Poiseuille and thermal transpiration flows between two parallel plates on the basis of the Boltzmann equation for hard-sphere molecules,, Phys. Fluids A, 1 (1989), 2042. doi: 10.1063/1.857478.
[36]	M. C. Potter, Stability of plane Couette-Poiseuille flow,, J. Fluid Mech., 24 (1966), 609. doi: 10.1017/S0022112066000855.
[37]	D. Risso and P. Cordero, Dilute gas Couette flow: Theory and molecular dynamics simulation,, Phys. Rev. E, 56 (1997), 489. doi: 10.1103/PhysRevE.56.489.
[38]	D. Risso and P. Cordero, Generalized hydrodynamics for a Poiseuille flow: theory and simulations,, Phys. Rev. E, 58 (1998), 546. doi: 10.1103/PhysRevE.58.546.
[39]	M. Sabbane, M. Tij and A. Santos, Maxwellian gas undergoing a stationary Poiseuille flow in a pipe,, Physica A, 327 (2003), 264. doi: 10.1016/S0378-4371(03)00513-2.
[40]	A. Santos, Solutions of the moment hierarchy in the kinetic theory of Maxwell models,, Cont. Mech. Thermodyn., 21 (2009), 361. doi: 10.1007/s00161-009-0113-5.
[41]	A. Santos, J. J. Brey and V. Garzó, Kinetic model for steady heat flow,, Phys. Rev. A, 34 (1986), 5047. doi: 10.1103/PhysRevA.34.5047.
[42]	A. Santos, J. J. Brey, C. S. Kim and J. W. Dufty, Velocity distribution for a gas with steady heat flow,, Phys. Rev. A, 39 (1989), 320. doi: 10.1103/PhysRevA.39.320.
[43]	A. Santos, V. Garzó and J. J. Brey, Comparison between the homogeneous-shear and the sliding-boundary methods to produce shear flow,, Phys. Rev. A, 46 (1992), 8018. doi: 10.1103/PhysRevA.46.8018.
[44]	A. Santos and M. Tij, Gravity-driven Poiseuille flow in dilute gases. Elastic and inelastic collisions,, in, (2006), 53.
[45]	Y. Sone, Asymptotic theory of a steady flow of a rarefied gas past bodies for small Knudsen numbers,, in, (1991), 19.
[46]	Y. Sone, Flows induced by temperature fields in a rarefied gas and their ghost effect on the behavior of a gas in the continuum limit,, Annu. Rev. Fluid Mech., 32 (2000), 779. doi: 10.1146/annurev.fluid.32.1.779.
[47]	Y. Sone, "Kinetic Theory and Fluid Dynamics,'', Birkhäuser, (2002).
[48]	H. Struchtrup, "Macroscopic Transport Equations for Rarefied Gas Flows. Approximation Methods in Kinetic Theory,'', Springer, (2005).
[49]	H. Struchtrup and M Torrilhon, Higher-order effects in rarefied channel flows,, Phys. Rev. E, 78 (2008). doi: 10.1103/PhysRevE.78.046301.
[50]	S. A. Suslov and T. D. Tran, Revisiting plane Couette-Poiseuille flows of a piezo-viscous fluid,, J. Non-Newton. Fluid Mech., 154 (2006), 170.
[51]	P. Taheri, M. Torrilhon and H. Struchtrup, Couette and Poiseuille microflows: analytical solutions for regularized 13-moment equations,, Phys. Fluids, 21 (2009). doi: 10.1063/1.3064123.
[52]	R. Tehver, F. Toigo, J. Koplik and J. R. Banavar, Thermal walls in computer simulations,, Phys. Rev. E, 57 (1998). doi: 10.1103/PhysRevE.57.R17.
[53]	E. M. Thurlow and J. C. Klewicki, Experimental study of turbulent Poiseuille-Couette flow,, Phys. Fluids, 12 (2000), 865. doi: 10.1063/1.870341.
[54]	M. Tij, V. Garzó and A. Santos, Influence of gravity on nonlinear transport in the planar Couette flow,, Phys. Fluids, 11 (1999), 893. doi: 10.1063/1.869960.
[55]	M. Tij, M. Sabbane and A. Santos, Nonlinear Poiseuille flow in a gas,, Phys. Fluids, 10 (1998), 1021. doi: 10.1063/1.869621.
[56]	M. Tij and A. Santos, Perturbation analysis of a stationary nonequilibrium flow generated by an external force,, J. Stat. Phys., 76 (1994), 1399. doi: 10.1007/BF02187068.
[57]	M. Tij and A. Santos, Combined heat and momentum transport in a dilute gas,, Phys. Fluids, 7 (1995), 2858. doi: 10.1063/1.868662.
[58]	M. Tij and A. Santos, Non-Newtonian Poiseuille flow of a gas in a pipe,, Physica A, 289 (2001), 336. doi: 10.1016/S0378-4371(00)00405-2.
[59]	M. Tij and A. Santos, Poiseuille flow in a heated granular gas,, J. Stat. Phys., 117 (2004), 901. doi: 10.1007/s10955-004-5710-x.
[60]	M. Tij, E. E. Tahiri, J. M. Montanero, V. Garzó, A. Santos and J. W. Dufty, Nonlinear Couette flow in a low density granular gas,, J. Stat. Phys., 103 (2001), 1035. doi: 10.1023/A:1010317207358.
[61]	B. D. Todd and D. J. Evans, Temperature profile for Poiseuille flow,, Phys. Rev. E, 55 (1997), 2800. doi: 10.1103/PhysRevE.55.2800.
[62]	K. P. Travis, B. D. Todd and D. J. Evans, Poiseuille flow of molecular fluids,, Physica A, 240 (1997), 315. doi: 10.1016/S0378-4371(97)00155-6.
[63]	C. Truesdell and R. G. Muncaster, "Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas,'', Academic Press, (1980).
[64]	F. J. Uribe and A. L. Garcia, Burnett description for plane Poiseuille flow,, Phys. Rev. E, 60 (1999), 4063. doi: 10.1103/PhysRevE.60.4063.
[65]	F. Vega Reyes, A. Santos and V. Garzó, Non-Newtonian granular hydrodynamics. What do the inelastic simple shear flow and the elastic Fourier flow have in common?,, Phys. Rev. Lett., 104 (2010). doi: 10.1103/PhysRevLett.104.028001.
[66]	P. Welander, On the temperature jump in a rarefied gas,, Akiv för Fysik, 7 (1954), 507.
[67]	K. Xu, Super-Burnett solutions for Poiseuille flow,, Phys. Fluids, 15 (2003), 2077. doi: 10.1063/1.1577564.
[68]	Y. Zheng, A. L. Garcia and B. J. Alder, Comparison of kinetic theory and hydrodynamics for Poiseuille flow,, J. Stat. Phys., 109 (2002), 495. doi: 10.1023/A:1020498111819.

show all references

References:

[1]	M. Alam and V. K. Chikkadi, Velocity distribution function and correlations in a granular Poiseuille flow,, J. Fluid Mech., 653 (2010), 175. doi: 10.1017/S0022112010000200.
[2]	M. Alaoui and A. Santos, Poiseuille flow driven by an external force,, Phys. Fluids A, 4 (1992), 1273. doi: 10.1063/1.858245.
[3]	K. Aoki, S. Takata and T. Nakanishi, A Poiseuille-type flow of a rarefied gas between two parallel plates driven by a uniform external force,, Phys. Rev. E, 65 (2002). doi: 10.1103/PhysRevE.65.026315.
[4]	E. Asmolov, N. K. Makashev and V. I. Nosik, Heat transfer between parallel plates in a gas of Maxwellian molecules,, Sov. Phys. Dokl., 24 (1979), 892.
[5]	P. L. Bhatnagar, E. P. Gross and M. Krook, A model collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems,, Phy. Rev., 94 (1954), 511. doi: 10.1103/PhysRev.94.511.
[6]	J. J. Brey, A. Santos and J. W. Dufty, Heat and momentum transport far from equilibrium,, Phys. Rev. A, 36 (1987), 2842. doi: 10.1103/PhysRevA.36.2842.
[7]	C. Cercignani, "The Boltzmann Equation and Its Applications,'', Springer-Verlag, (1988).
[8]	C. Cercignani, "Mathematical Methods in Kinetic Theory,'', Plenum Press, (1990).
[9]	C. Cercignani, M. Lampis and S. Lorenzani, Plane Poiseuille-Couette problem in micro-electro-mechanical systems applications with gas-rarefaction effects,, Phy. Fluids, 18 (2006). doi: 10.1063/1.2335847.
[10]	C. Cercignani and F. Sernagiotto, Cylindrical Poiseuille flow of a rarefied gas,, Phys. Fluids, 9 (1966), 40. doi: 10.1063/1.1761530.
[11]	S. Chapman and T. G. Cowling, "The Mathematical Theory of Nonuniform Gases,'', Cambridge University Press, (1970).
[12]	V. Chikkadi and M. Alam, Slip velocity and stresses in granular Poiseuille flow via event-driven simulation,, Phys. Rev. E, 80 (2009). doi: 10.1103/PhysRevE.80.021303.
[13]	J. R. Dorfman and H. van Beijeren, The kinetic theory of gases,, in, (1977), 65.
[14]	A. I. Erofeev and O. G. Friedlander, Macroscopic models for non-equilibrium flows of monatomic gas and normal solutions,, in, (2007), 117.
[15]	R. Esposito, J. L. Lebowitz and R. Marra, A hydrodynamic limit of the stationary Boltzmann equation in a slab,, Commun. Math. Phys., 160 (1994), 49. doi: 10.1007/BF02099789.
[16]	M. A. Gallis, J. R. Torczynski, D. J. Rader, M. Tij and A. Santos, Normal solutions of the Boltzmann equation for highly nonequilibrium Fourier flow and Couette flow,, Phys. Fluids, 18 (2006).
[17]	M. A. Gallis, J. R. Torczynski, D. J. Rader, M. Tij and A. Santos, Analytical and numerical normal solutions of the Boltzmann equation for highly nonequilibrium Fourier and Couette flows,, in, (2007), 251.
[18]	L. S. García-Colín, R. M. Velasco and F. J. Uribe, Beyond the Navier-Stokes equations: Burnett hydrodynamics,, Phys. Rep., 465 (2008), 149. doi: 10.1016/j.physrep.2008.04.010.
[19]	V. Garzó and M. López de Haro, Nonlinear transport for a dilute gas in steady Couette flow,, Phys. Fluids, 9 (1997), 776. doi: 10.1063/1.869232.
[20]	V. Garzó and A. Santos, "Kinetic Theory of Gases in Shear Flows. Nonlinear Transport,'', Kluwer Academic Publishers, (2003).
[21]	S. Hess and M. Malek Mansour, Temperature profile of a dilute gas undergoing a plane Poiseuille flow,, Physica A, 272 (1999), 481. doi: 10.1016/S0378-4371(99)00254-X.
[22]	L. P. Kadanoff, G. R. McNamara and G. Zanetti, A Poiseuille viscometer for lattice gas automata,, Complex Syst., 1 (1987), 791.
[23]	L. P. Kadanoff, G. R. McNamara and G. Zanetti, From automata to fluid flow: Comparisons of simulation and theory,, Phys. Rev. A, 40 (1989), 4527. doi: 10.1103/PhysRevA.40.4527.
[24]	C. S. Kim, J. W. Dufty, A. Santos and J. J. Brey, Hilbert-class or "normal'' solutions for stationary heat flow,, Phys. Rev. A, 39 (1989), 328. doi: 10.1103/PhysRevA.39.328.
[25]	C. S. Kim, J. W. Dufty, A. Santos and J. J. Brey, Analysis of nonlinear transport in Couette flow,, Phys. Rev A, 40 (1989), 7165. doi: 10.1103/PhysRevA.40.7165.
[26]	G. M. Kremer, "An Introduction to the Boltzmann Equation and Transport Processes in Gases,'', Springer, (2010). doi: 10.1007/978-3-642-11696-4.
[27]	M. Malek Mansour, F. Baras and A. L. Garcia, On the validity of hydrodynamics in plane Poiseuille flows,, Physica A, 240 (1997), 255. doi: 10.1016/S0378-4371(97)00149-0.
[28]	N. K. Makashev and V. I. Nosik, Steady Couette flow (with heat transfer) of a gas of Maxwellian molecules,, Sov. Phys. Dokl., 25 (1981), 589.
[29]	J. M. Montanero, M. Alaoui, A. Santos and V. Garzó, Monte Carlo simulation of the Boltzmann equation for steady Fourier flow,, Phys. Rev. E, 49 (1994), 367. doi: 10.1103/PhysRevE.49.367.
[30]	J. M. Montanero and V. Garzó, Nonlinear Couette flow in a dilute gas: Comparison between theory and molecular dynamics simulation,, Phys. Rev. E, 58 (1998), 1836. doi: 10.1103/PhysRevE.58.1836.
[31]	J. M. Montanero, A. Santos and V. Garzó, Monte Carlo simulation of nonlinear Couette flow in a dilute gas,, Phys. Fluids, 12 (2000), 3060. doi: 10.1063/1.1313563.
[32]	R. S. Myong, Coupled nonlinear constitutive models for rarefied and microscale gas flows: subtle interplay of kinematics and dissipation effects,, Cont. Mech. Thermodyn., 21 (2009), 389. doi: 10.1007/s00161-009-0112-6.
[33]	V. I. Nosik, Heat transfer between parallel plates in a mixture of gases of Maxwellian molecules,, Sov. Phys. Dokl., 25 (1981), 495.
[34]	V. I. Nosik, Degeneration of the Chapman-Enskog expansion in one-dimensional motions of Maxwellian molecule gases,, in, 13 (1983), 237.
[35]	T. Ohwada, Y. Sone and K. Aoki, Numerical analysis of the Poiseuille and thermal transpiration flows between two parallel plates on the basis of the Boltzmann equation for hard-sphere molecules,, Phys. Fluids A, 1 (1989), 2042. doi: 10.1063/1.857478.
[36]	M. C. Potter, Stability of plane Couette-Poiseuille flow,, J. Fluid Mech., 24 (1966), 609. doi: 10.1017/S0022112066000855.
[37]	D. Risso and P. Cordero, Dilute gas Couette flow: Theory and molecular dynamics simulation,, Phys. Rev. E, 56 (1997), 489. doi: 10.1103/PhysRevE.56.489.
[38]	D. Risso and P. Cordero, Generalized hydrodynamics for a Poiseuille flow: theory and simulations,, Phys. Rev. E, 58 (1998), 546. doi: 10.1103/PhysRevE.58.546.
[39]	M. Sabbane, M. Tij and A. Santos, Maxwellian gas undergoing a stationary Poiseuille flow in a pipe,, Physica A, 327 (2003), 264. doi: 10.1016/S0378-4371(03)00513-2.
[40]	A. Santos, Solutions of the moment hierarchy in the kinetic theory of Maxwell models,, Cont. Mech. Thermodyn., 21 (2009), 361. doi: 10.1007/s00161-009-0113-5.
[41]	A. Santos, J. J. Brey and V. Garzó, Kinetic model for steady heat flow,, Phys. Rev. A, 34 (1986), 5047. doi: 10.1103/PhysRevA.34.5047.
[42]	A. Santos, J. J. Brey, C. S. Kim and J. W. Dufty, Velocity distribution for a gas with steady heat flow,, Phys. Rev. A, 39 (1989), 320. doi: 10.1103/PhysRevA.39.320.
[43]	A. Santos, V. Garzó and J. J. Brey, Comparison between the homogeneous-shear and the sliding-boundary methods to produce shear flow,, Phys. Rev. A, 46 (1992), 8018. doi: 10.1103/PhysRevA.46.8018.
[44]	A. Santos and M. Tij, Gravity-driven Poiseuille flow in dilute gases. Elastic and inelastic collisions,, in, (2006), 53.
[45]	Y. Sone, Asymptotic theory of a steady flow of a rarefied gas past bodies for small Knudsen numbers,, in, (1991), 19.
[46]	Y. Sone, Flows induced by temperature fields in a rarefied gas and their ghost effect on the behavior of a gas in the continuum limit,, Annu. Rev. Fluid Mech., 32 (2000), 779. doi: 10.1146/annurev.fluid.32.1.779.
[47]	Y. Sone, "Kinetic Theory and Fluid Dynamics,'', Birkhäuser, (2002).
[48]	H. Struchtrup, "Macroscopic Transport Equations for Rarefied Gas Flows. Approximation Methods in Kinetic Theory,'', Springer, (2005).
[49]	H. Struchtrup and M Torrilhon, Higher-order effects in rarefied channel flows,, Phys. Rev. E, 78 (2008). doi: 10.1103/PhysRevE.78.046301.
[50]	S. A. Suslov and T. D. Tran, Revisiting plane Couette-Poiseuille flows of a piezo-viscous fluid,, J. Non-Newton. Fluid Mech., 154 (2006), 170.
[51]	P. Taheri, M. Torrilhon and H. Struchtrup, Couette and Poiseuille microflows: analytical solutions for regularized 13-moment equations,, Phys. Fluids, 21 (2009). doi: 10.1063/1.3064123.
[52]	R. Tehver, F. Toigo, J. Koplik and J. R. Banavar, Thermal walls in computer simulations,, Phys. Rev. E, 57 (1998). doi: 10.1103/PhysRevE.57.R17.
[53]	E. M. Thurlow and J. C. Klewicki, Experimental study of turbulent Poiseuille-Couette flow,, Phys. Fluids, 12 (2000), 865. doi: 10.1063/1.870341.
[54]	M. Tij, V. Garzó and A. Santos, Influence of gravity on nonlinear transport in the planar Couette flow,, Phys. Fluids, 11 (1999), 893. doi: 10.1063/1.869960.
[55]	M. Tij, M. Sabbane and A. Santos, Nonlinear Poiseuille flow in a gas,, Phys. Fluids, 10 (1998), 1021. doi: 10.1063/1.869621.
[56]	M. Tij and A. Santos, Perturbation analysis of a stationary nonequilibrium flow generated by an external force,, J. Stat. Phys., 76 (1994), 1399. doi: 10.1007/BF02187068.
[57]	M. Tij and A. Santos, Combined heat and momentum transport in a dilute gas,, Phys. Fluids, 7 (1995), 2858. doi: 10.1063/1.868662.
[58]	M. Tij and A. Santos, Non-Newtonian Poiseuille flow of a gas in a pipe,, Physica A, 289 (2001), 336. doi: 10.1016/S0378-4371(00)00405-2.
[59]	M. Tij and A. Santos, Poiseuille flow in a heated granular gas,, J. Stat. Phys., 117 (2004), 901. doi: 10.1007/s10955-004-5710-x.
[60]	M. Tij, E. E. Tahiri, J. M. Montanero, V. Garzó, A. Santos and J. W. Dufty, Nonlinear Couette flow in a low density granular gas,, J. Stat. Phys., 103 (2001), 1035. doi: 10.1023/A:1010317207358.
[61]	B. D. Todd and D. J. Evans, Temperature profile for Poiseuille flow,, Phys. Rev. E, 55 (1997), 2800. doi: 10.1103/PhysRevE.55.2800.
[62]	K. P. Travis, B. D. Todd and D. J. Evans, Poiseuille flow of molecular fluids,, Physica A, 240 (1997), 315. doi: 10.1016/S0378-4371(97)00155-6.
[63]	C. Truesdell and R. G. Muncaster, "Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas,'', Academic Press, (1980).
[64]	F. J. Uribe and A. L. Garcia, Burnett description for plane Poiseuille flow,, Phys. Rev. E, 60 (1999), 4063. doi: 10.1103/PhysRevE.60.4063.
[65]	F. Vega Reyes, A. Santos and V. Garzó, Non-Newtonian granular hydrodynamics. What do the inelastic simple shear flow and the elastic Fourier flow have in common?,, Phys. Rev. Lett., 104 (2010). doi: 10.1103/PhysRevLett.104.028001.
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