American Institute of Mathematical Sciences

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

Non-Newtonian Couette-Poiseuille flow of a dilute gas

Mohamed Tij 1, and  Andrés Santos 2,

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

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. Google Scholar

[2]

M. Alaoui and A. Santos, Poiseuille flow driven by an external force,, Phys. Fluids A, 4 (1992), 1273. doi: 10.1063/1.858245. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

C. Cercignani, "The Boltzmann Equation and Its Applications,'', Springer-Verlag, (1988). Google Scholar

[8]

C. Cercignani, "Mathematical Methods in Kinetic Theory,'', Plenum Press, (1990). Google Scholar

[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. Google Scholar

[10]

C. Cercignani and F. Sernagiotto, Cylindrical Poiseuille flow of a rarefied gas,, Phys. Fluids, 9 (1966), 40. doi: 10.1063/1.1761530. Google Scholar

[11]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Nonuniform Gases,'', Cambridge University Press, (1970). Google Scholar

[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. Google Scholar

[13]

J. R. Dorfman and H. van Beijeren, The kinetic theory of gases,, in, (1977), 65. Google Scholar

[14]

A. I. Erofeev and O. G. Friedlander, Macroscopic models for non-equilibrium flows of monatomic gas and normal solutions,, in, (2007), 117. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

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[20]

V. Garzó and A. Santos, "Kinetic Theory of Gases in Shear Flows. Nonlinear Transport,'', Kluwer Academic Publishers, (2003). Google Scholar

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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. Google Scholar

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L. P. Kadanoff, G. R. McNamara and G. Zanetti, A Poiseuille viscometer for lattice gas automata,, Complex Syst., 1 (1987), 791. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

V. I. Nosik, Heat transfer between parallel plates in a mixture of gases of Maxwellian molecules,, Sov. Phys. Dokl., 25 (1981), 495. Google Scholar

[34]

V. I. Nosik, Degeneration of the Chapman-Enskog expansion in one-dimensional motions of Maxwellian molecule gases,, in, 13 (1983), 237. Google Scholar

[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. Google Scholar

[36]

M. C. Potter, Stability of plane Couette-Poiseuille flow,, J. Fluid Mech., 24 (1966), 609. doi: 10.1017/S0022112066000855. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[44]

A. Santos and M. Tij, Gravity-driven Poiseuille flow in dilute gases. Elastic and inelastic collisions,, in, (2006), 53. Google Scholar

[45]

Y. Sone, Asymptotic theory of a steady flow of a rarefied gas past bodies for small Knudsen numbers,, in, (1991), 19. Google Scholar

[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. Google Scholar

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[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. Google Scholar

[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. Google Scholar

[53]

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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. Google Scholar

[55]

M. Tij, M. Sabbane and A. Santos, Nonlinear Poiseuille flow in a gas,, Phys. Fluids, 10 (1998), 1021. doi: 10.1063/1.869621. Google Scholar

[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. Google Scholar

[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. Google Scholar

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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. Google Scholar

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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. Google Scholar

[2]

M. Alaoui and A. Santos, Poiseuille flow driven by an external force,, Phys. Fluids A, 4 (1992), 1273. doi: 10.1063/1.858245. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

C. Cercignani, "The Boltzmann Equation and Its Applications,'', Springer-Verlag, (1988). Google Scholar

[8]

C. Cercignani, "Mathematical Methods in Kinetic Theory,'', Plenum Press, (1990). Google Scholar

[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. Google Scholar

[10]

C. Cercignani and F. Sernagiotto, Cylindrical Poiseuille flow of a rarefied gas,, Phys. Fluids, 9 (1966), 40. doi: 10.1063/1.1761530. Google Scholar

[11]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Nonuniform Gases,'', Cambridge University Press, (1970). Google Scholar

[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. Google Scholar

[13]

J. R. Dorfman and H. van Beijeren, The kinetic theory of gases,, in, (1977), 65. Google Scholar

[14]

A. I. Erofeev and O. G. Friedlander, Macroscopic models for non-equilibrium flows of monatomic gas and normal solutions,, in, (2007), 117. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

V. Garzó and A. Santos, "Kinetic Theory of Gases in Shear Flows. Nonlinear Transport,'', Kluwer Academic Publishers, (2003). Google Scholar

[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. Google Scholar

[22]

L. P. Kadanoff, G. R. McNamara and G. Zanetti, A Poiseuille viscometer for lattice gas automata,, Complex Syst., 1 (1987), 791. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

V. I. Nosik, Heat transfer between parallel plates in a mixture of gases of Maxwellian molecules,, Sov. Phys. Dokl., 25 (1981), 495. Google Scholar

[34]

V. I. Nosik, Degeneration of the Chapman-Enskog expansion in one-dimensional motions of Maxwellian molecule gases,, in, 13 (1983), 237. Google Scholar

[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. Google Scholar

[36]

M. C. Potter, Stability of plane Couette-Poiseuille flow,, J. Fluid Mech., 24 (1966), 609. doi: 10.1017/S0022112066000855. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[44]

A. Santos and M. Tij, Gravity-driven Poiseuille flow in dilute gases. Elastic and inelastic collisions,, in, (2006), 53. Google Scholar

[45]

Y. Sone, Asymptotic theory of a steady flow of a rarefied gas past bodies for small Knudsen numbers,, in, (1991), 19. Google Scholar

[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. Google Scholar

[47]

Y. Sone, "Kinetic Theory and Fluid Dynamics,'', Birkhäuser, (2002). Google Scholar

[48]

H. Struchtrup, "Macroscopic Transport Equations for Rarefied Gas Flows. Approximation Methods in Kinetic Theory,'', Springer, (2005). Google Scholar

[49]

H. Struchtrup and M Torrilhon, Higher-order effects in rarefied channel flows,, Phys. Rev. E, 78 (2008). doi: 10.1103/PhysRevE.78.046301. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[55]

M. Tij, M. Sabbane and A. Santos, Nonlinear Poiseuille flow in a gas,, Phys. Fluids, 10 (1998), 1021. doi: 10.1063/1.869621. Google Scholar

[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. Google Scholar

[57]

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