## 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.

### MSC:

 82C40 Kinetic theory of gases in time-dependent statistical mechanics
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### References:

 [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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