American Institute of Mathematical Sciences

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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

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 and 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), 175-219. doi: 10.1017/S0022112010000200.

[2]

M. Alaoui and A. Santos, Poiseuille flow driven by an external force, Phys. Fluids A, 4 (1992), 1273-1282. 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), 026315. 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), 892-894.

[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), 511-525. 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), 2842-2849. doi: 10.1103/PhysRevA.36.2842.

[7]

C. Cercignani, "The Boltzmann Equation and Its Applications,'' Springer-Verlag, New York, 1988.

[8]

C. Cercignani, "Mathematical Methods in Kinetic Theory,'' Plenum Press, New York, 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), 087102. doi: 10.1063/1.2335847.

[10]

C. Cercignani and F. Sernagiotto, Cylindrical Poiseuille flow of a rarefied gas, Phys. Fluids, 9 (1966), 40-44. doi: 10.1063/1.1761530.

[11]

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

[12]

V. Chikkadi and M. Alam, Slip velocity and stresses in granular Poiseuille flow via event-driven simulation, Phys. Rev. E, 80 (2009), 021303. doi: 10.1103/PhysRevE.80.021303.

[13]

J. R. Dorfman and H. van Beijeren, The kinetic theory of gases, in "Statistical Mechanics, Part B'' (ed. B. J. Berne), Plenum Press, New York, (1977), 65-179.

[14]

A. I. Erofeev and O. G. Friedlander, Macroscopic models for non-equilibrium flows of monatomic gas and normal solutions, in "Rarefied Gas Dynamics: 25th International Symposium on Rarefied Gas Dynamics,'' (eds. M. S. Ivanov and A. K. Rebrov), Publishing House of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, (2007), 117-124.

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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), 017104.

[17]

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L. S. García-Colín, R. M. Velasco and F. J. Uribe, Beyond the Navier-Stokes equations: Burnett hydrodynamics, Phys. Rep., 465 (2008), 149-189. doi: 10.1016/j.physrep.2008.04.010.

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V. Garzó and M. López de Haro, Nonlinear transport for a dilute gas in steady Couette flow, Phys. Fluids, 9 (1997), 776-787. doi: 10.1063/1.869232.

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

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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), 481-496. doi: 10.1016/S0378-4371(99)00254-X.

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L. P. Kadanoff, G. R. McNamara and G. Zanetti, A Poiseuille viscometer for lattice gas automata, Complex Syst., 1 (1987), 791-803.

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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), 4527-4541. 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), 328-338. 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), 7165-7174. doi: 10.1103/PhysRevA.40.7165.

[26]

G. M. Kremer, "An Introduction to the Boltzmann Equation and Transport Processes in Gases,'' Springer, Berlin, 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), 255-267. 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), 589-591.

[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), 367-375. 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), 1836-1842. 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), 3060-3073. 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), 389-399. 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), 495-497.

[34]

V. I. Nosik, Degeneration of the Chapman-Enskog expansion in one-dimensional motions of Maxwellian molecule gases, in "Rarefied Gas Dynamics,'' vol. 13 (eds. O. M. Belotserkovskii, M. N. Kogan, S. S. Kutateladze, and A. K. Rebrov), Plenum Press, New York, (1983), 237-244.

[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), 2042-2049. doi: 10.1063/1.857478.

[36]

M. C. Potter, Stability of plane Couette-Poiseuille flow, J. Fluid Mech., 24 (1966), 609-619. doi: 10.1017/S0022112066000855.

[37]

D. Risso and P. Cordero, Dilute gas Couette flow: Theory and molecular dynamics simulation, Phys. Rev. E, 56 (1997), 489-498. 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), 546-553. 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), 264-290. 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), 361-387. 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), 5047-5050. 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), 320-327. 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), 8018-8020. doi: 10.1103/PhysRevA.46.8018.

[44]

A. Santos and M. Tij, Gravity-driven Poiseuille flow in dilute gases. Elastic and inelastic collisions, in "Modelling and Numerics of Kinetic Dissipative Systems'' (eds. L. Pareschi, G. Russo and G. Toscani), Nova Science Publishers, New York, (2006), 53-67.

[45]

Y. Sone, Asymptotic theory of a steady flow of a rarefied gas past bodies for small Knudsen numbers, in "Advances in Kinetic Theory and Continuum Mechanics'' (eds. R. Gatignol and Soubbaramayer), Springer-Verlag, Berlin, (1991), 19-31.

[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), 779-811. doi: 10.1146/annurev.fluid.32.1.779.

[47]

Y. Sone, "Kinetic Theory and Fluid Dynamics,'' Birkhäuser, Boston, 2002.

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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), 028001. doi: 10.1103/PhysRevLett.104.028001.

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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), 175-219. doi: 10.1017/S0022112010000200.

[2]

M. Alaoui and A. Santos, Poiseuille flow driven by an external force, Phys. Fluids A, 4 (1992), 1273-1282. 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), 026315. 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), 892-894.

[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), 511-525. 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), 2842-2849. doi: 10.1103/PhysRevA.36.2842.

[7]

C. Cercignani, "The Boltzmann Equation and Its Applications,'' Springer-Verlag, New York, 1988.

[8]

C. Cercignani, "Mathematical Methods in Kinetic Theory,'' Plenum Press, New York, 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), 087102. doi: 10.1063/1.2335847.

[10]

C. Cercignani and F. Sernagiotto, Cylindrical Poiseuille flow of a rarefied gas, Phys. Fluids, 9 (1966), 40-44. doi: 10.1063/1.1761530.

[11]

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

[12]

V. Chikkadi and M. Alam, Slip velocity and stresses in granular Poiseuille flow via event-driven simulation, Phys. Rev. E, 80 (2009), 021303. doi: 10.1103/PhysRevE.80.021303.

[13]

J. R. Dorfman and H. van Beijeren, The kinetic theory of gases, in "Statistical Mechanics, Part B'' (ed. B. J. Berne), Plenum Press, New York, (1977), 65-179.

[14]

A. I. Erofeev and O. G. Friedlander, Macroscopic models for non-equilibrium flows of monatomic gas and normal solutions, in "Rarefied Gas Dynamics: 25th International Symposium on Rarefied Gas Dynamics,'' (eds. M. S. Ivanov and A. K. Rebrov), Publishing House of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, (2007), 117-124.

[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), 49-80. 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), 017104.

[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 "Rarefied Gas Dynamics: 25th International Symposium on Rarefied Gas Dynamics,'' (eds. M. S. Ivanov and A. K. Rebrov), Publishing House of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, (2007), 251-256.

[18]

L. S. García-Colín, R. M. Velasco and F. J. Uribe, Beyond the Navier-Stokes equations: Burnett hydrodynamics, Phys. Rep., 465 (2008), 149-189. 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), 776-787. doi: 10.1063/1.869232.

[20]

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

[21]

S. Hess and M. Malek Mansour, Temperature profile of a dilute gas undergoing a plane Poiseuille flow, Physica A, 272 (1999), 481-496. 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), 791-803.

[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), 4527-4541. 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), 328-338. 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), 7165-7174. doi: 10.1103/PhysRevA.40.7165.

[26]

G. M. Kremer, "An Introduction to the Boltzmann Equation and Transport Processes in Gases,'' Springer, Berlin, 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), 255-267. 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), 589-591.

[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), 367-375. 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), 1836-1842. 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), 3060-3073. 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), 389-399. 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), 495-497.

[34]

V. I. Nosik, Degeneration of the Chapman-Enskog expansion in one-dimensional motions of Maxwellian molecule gases, in "Rarefied Gas Dynamics,'' vol. 13 (eds. O. M. Belotserkovskii, M. N. Kogan, S. S. Kutateladze, and A. K. Rebrov), Plenum Press, New York, (1983), 237-244.

[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), 2042-2049. doi: 10.1063/1.857478.

[36]

M. C. Potter, Stability of plane Couette-Poiseuille flow, J. Fluid Mech., 24 (1966), 609-619. doi: 10.1017/S0022112066000855.

[37]

D. Risso and P. Cordero, Dilute gas Couette flow: Theory and molecular dynamics simulation, Phys. Rev. E, 56 (1997), 489-498. 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), 546-553. 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), 264-290. 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), 361-387. 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), 5047-5050. 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), 320-327. 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), 8018-8020. doi: 10.1103/PhysRevA.46.8018.

[44]

A. Santos and M. Tij, Gravity-driven Poiseuille flow in dilute gases. Elastic and inelastic collisions, in "Modelling and Numerics of Kinetic Dissipative Systems'' (eds. L. Pareschi, G. Russo and G. Toscani), Nova Science Publishers, New York, (2006), 53-67.

[45]

Y. Sone, Asymptotic theory of a steady flow of a rarefied gas past bodies for small Knudsen numbers, in "Advances in Kinetic Theory and Continuum Mechanics'' (eds. R. Gatignol and Soubbaramayer), Springer-Verlag, Berlin, (1991), 19-31.

[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), 779-811. doi: 10.1146/annurev.fluid.32.1.779.

[47]

Y. Sone, "Kinetic Theory and Fluid Dynamics,'' Birkhäuser, Boston, 2002.

[48]

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