Influence of a drag force on linear transport in low-density gases. stability analysis. (English) Zbl 1395.82209

Summary: The transport coefficients of a dilute classical gas in the presence of a drag force proportional to the velocity of the particle are determined from the Boltzmann equation. The viscous drag force could model the friction of solid particles with a surrounding fluid (interstitial gas phase). First, when the drag force is the only external action on the state of the system, the Boltzmann equation admits a Maxwellian solution \(f_0(\mathbf{v}, t)\) with a time-dependent temperature. Then, the Boltzmann equation is solved by means of the Chapman-Enskog expansion around the local version of the distribution \(f_0\) to obtain the relevant transport coefficients of the system: the shear viscosity \(\eta\), the thermal conductivity \(\kappa\), and a new transport coefficient \(\mu\) (which is also present in granular gases) relating the heat flux with the density gradient. The results indicate that while \(\eta\) is not affected by the drag force, the impact of this force on the transport coefficients \(\kappa\) and \(\mu\) may be significant. Finally, a stability analysis of the linear hydrodynamic equations with respect to the time-dependent equilibrium state is performed, showing that the onset of instability is associated with the transversal shear mode that could be unstable for wave numbers smaller than a certain critical wave number.


82C40 Kinetic theory of gases in time-dependent statistical mechanics
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[1] Evans, D. J.; Morriss, G. P., Comput. Phys. Rep., 1, 299, (1984)
[2] Evans, D. J.; Hoover, W. G., Annu. Rev. Fluid Mech., 18, 243, (1986)
[3] Evans, D. J.; Morriss, G. P., Statistical mechanics of nonequilibrium liquids, (1990), Academic Press London · Zbl 1145.82301
[4] Goldhirsch, I., Annu. Rev. Fluid Mech., 35, 267, (2003)
[5] Brilliantov, N. V.; Pöschel, T., Kinetic theory of granular gases, (2004), Oxford University Press Oxford · Zbl 1155.76386
[6] Puglisi, A.; Loreto, V.; Marconi, U. M.B.; Petri, A.; Vulpiani, A.; Puglisi, A.; Loreto, V.; Marini Bettolo Marconi, U.; Vulpiani, A.; Cafiero, R.; Luding, S.; Herrmann, H. J.; Prevost, A.; Egolf, D. A.; Urbach, J. S.; Puglisi, A.; Baldassarri, A.; Loreto, V.; Visco, P.; Puglisi, A.; Barrat, A.; Trizac, E.; van Wijland, F.; Fiege, A.; Aspelmeier, T.; Zippelius, A.; Sarracino, A.; Villamaina, D.; Gradenigo, G.; Puglisi, A.; Kranz, W. T.; Sperl, M.; Zippelius, A.; Puglisi, A.; Gnoli, A.; Gradenigo, G.; Sarracino, A.; Villamaina, D., Phys. Rev. Lett., Phys. Rev. E, Phys. Rev. Lett., Phys. Rev. Lett., Phys. Rev. E, J. Stat. Phys., Phys. Rev. Lett., EPL, Phys. Rev. Lett., J. Chem. Phys., 136, 014704, (2012), In the case of granular fluids, see for instance
[7] Dufty, J. W.; Santos, A.; Brey, J. J.; Rodríguez, R. F., Phys. Rev. A, 33, 459, (1986)
[8] Garzó, V.; Santos, A., Kinetic theory of gases in shear flows. nonlinear transport, (2003), Kluwer Academic Dordrecht · Zbl 1140.82030
[9] Koch, D. L., Phys. Fluids A, 2, 1711, (1990)
[10] Koch, D. L.; Hill, R. J., Annu. Rev. Fluid Mech., 33, 619, (2001)
[11] Garzó, V.; Tenneti, S.; Subramaniam, S.; Hrenya, C. M., J. Fluid Mech., 712, 129, (2012)
[12] Jackson, R., The dynamics of fluidized particles, (2000), Cambridge University Press Cambridge · Zbl 0956.76004
[13] Boyer, F.; Guazzelli, E.; Pouliquen, O., Phys. Rev. Lett., 107, 188301, (2011)
[14] Garzó, V.; Santos, A.; Brey, J. J., Physica A, 163, 651, (1990)
[15] Chapman, S.; Cowling, T. G., The mathematical theory of nonuniform gases, (1970), Cambridge University Press Cambridge · Zbl 0049.26102
[16] Brey, J. J.; Dufty, J. W.; Kim, C. S.; Santos, A., Phys. Rev. E, 58, 4638, (1998)
[17] Garzó, V.; Dufty, J. W., Phys. Rev. E, 59, 5895, (1999)
[18] Candela, D.; Walsworth, R. L., Amer. J. Phys., 75, 754, (2007)
[19] Goldhirsch, I.; Zanetti, G., Phys. Rev. Lett., 70, 1619, (1993)
[20] McNamara, S.; McNamara, S.; Young, W. R.; McNamara, S.; Young, W. R., Phys. Fluids A, Phys. Rev. E, Phys. Rev. E, 53, 5089, (1996)
[21] Cercignani, C., Mathematical methods in kinetic theory, (1990), Plenum Press New Yok · Zbl 0726.76083
[22] Montanero, J. M.; Garzó, V., Physica A, 313, 336, (2002)
[23] Tsao, H-K. W.; Koch, D. L., J. Fluid Mech., 296, 211, (1995)
[24] Lutsko, J. J., Phys. Rev. E, 63, 061211, (2001)
[25] Garzó, V.; Chamorro, M. G.; Vega Reyes, F., Phys. Rev. E, Phys. Rev. E, 87, 059906, (2013), Erratum
[26] Khalil, N.; Garzó, V., Phys. Rev. E, 88, 052201, (2013)
[27] Truesdell, C.; Muncaster, R. G., Fundamentals of maxwell’s kinetic theory of a simple monatonic gas, (1980), Academic Press New York
[28] Résibois, P.; de Leener, M., Classical kinetic theory of fluids, (1977), Wiley New York · Zbl 0152.46503
[29] Garzó, V., Phys. Rev. E, 72, 021106, (2005)
[30] Brey, J. J.; Ruiz-Montero, M. J.; Cubero, D., Phys. Rev. E, 60, 3150, (1993)
[31] Gradenigo, G.; Sarracino, A.; Villamaina, D.; Puglis, A., J. Stat. Mech., P08017, (2011)
[32] Astillero, A.; Santos, A., Phys. Rev. E, 72, 031309, (2005)
[33] Santos, A.; Garzó, V.; Santos, A.; Garzó, V.; Garzó, V.; Garzó, V.; Trizac, E.; Garzó, V.; Trizac, E.; Garzó, V.; Trizac, E., Physica A, J. Stat. Phys., J. Stat. Mech., J. Phys. A: Math. Theor., J. Non-Newtonian Fluid Mech., EPL, Phys. Rev. E, 85, 011302, (2012), See for instance · Zbl 0793.76077
[34] Karlin, I.; Dukek, G.; Nonnenmacher, T. F.; Karlin, I.; Dukek, G.; Nonnenmacher, T. F., Phys. Rev. E, Phys. Rev. E, 57, 3674, (1998)
[35] Santos, A., Phys. Rev. E, 62, 6597, (2000)
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