August  2019, 12(4): 765-789. doi: 10.3934/krm.2019030

Effect of abrupt change of the wall temperature in the kinetic theory

Department of Mathematics, National Cheng Kung University, Taiwan

Received  May 2018 Published  May 2019

Fund Project: The author is supported by MOST Grant 106-2115-M-006-009-MY2

We consider a semi-infinite expanse of a rarefied gas bounded by an infinite plane wall. The temperature of the wall is $ T_0 $, and the gas is initially in equilibrium with density $ \rho_0 $ and temperature $ T_0 $. The temperature of the wall is suddenly changed to $ T_w $ at time $ t = 0 $ and is kept at $ T_w $ afterward. We study the quantitative short time behavior of the gas in response to the abrupt change of the wall temperature on the basis of the linearized Boltzmann equation. Our approach is based on a straightforward calculation of the exact formulas derived by Duhamel's integral. Our method allows us to establish the pointwise estimates of the microscopic distribution and the macroscopic variables in short time. We show that the short-time solution consists of the free molecular flow and its perturbation, which exhibits logarithmic singularities along the characteristic line and on the boundary.

Citation: Hung-Wen Kuo. Effect of abrupt change of the wall temperature in the kinetic theory. Kinetic & Related Models, 2019, 12 (4) : 765-789. doi: 10.3934/krm.2019030
References:
[1]

K. Aoki and F. Golse, On the speed of approach to equilibrium for a collisionless gas, Kinet. Relat. Models, 4 (2011), 87-107. doi: 10.3934/krm.2011.4.87. Google Scholar

[2]

L. ArkerydR. EspositoR. Marra and A. Nouri, Ghost effect by curvature in planar couette flow, Kinet. Relat. Models, 4 (2011), 109-138. doi: 10.3934/krm.2011.4.109. Google Scholar

[3]

L. ArkerydR. EspositoR. Marra and A. Nouri, Stability for Rayleigh enard convective solutions of the Boltzmann equation, Arch. Rational Mech. Anal, 198 (2010), 125-187. doi: 10.1007/s00205-010-0292-z. Google Scholar

[4]

L. Arkeryd and A. Nouri, The stationary nonlinear Boltzmann equation in a Couette setting with multiple, isolated Lq -solutions and hydrodynamic limits, J. Stat. Phys, 118 (2005), 849-881. doi: 10.1007/s10955-004-2708-3. Google Scholar

[5]

C. Cercignani, The Boltzmann Equation and Its Applications, Springer, New York, 1988. doi: 10.1007/978-1-4612-1039-9. Google Scholar

[6]

C.-C. ChenI.-K. ChenT.-P. Liu and Y. Sone, Thermal transpiration for the linearized Boltzmann equation, Commun. Pure Appl. Math, 60 (2007), 147-163. doi: 10.1002/cpa.20167. Google Scholar

[7]

R. EspositoY. GuoC. Kim and R. Marra, Non-isothermal boundary in the Boltzmann theory and Fourier law, Commun. Math. Phys, 323 (2013), 177-239. doi: 10.1007/s00220-013-1766-2. Google Scholar

[8]

R. Esposito, Y. Guo, C. Kim and R. Marra, Stationary solutions to the Boltzmann equation in the hydrodynamic limit, Ann. PDE, 4 (2018), Art. 1,119 pp, https://doi.org/10.1007/s40818-017-0037-5. doi: 10.1007/s40818-017-0037-5. Google Scholar

[9]

R. EspositoJ. L. Lebowitz and R. Marra, Hydrodynamic limit of the stationary Boltzmann equation in a slab, Commun. Math. Phys, 160 (1994), 49-80. doi: 10.1007/BF02099789. Google Scholar

[10]

H. Grad, Asymptotic theory of the Boltzmann equation, Ⅱ, Int. Symp. on Rarefied Gas Dynamics, Third Symp., 1 (1962), 26-59. Google Scholar

[11]

H.-W. KuoT.-P. Liu and S. E. Noh, Mixture Lemma, Bulletin, Inst. Math., Academia Sinica (N.S.), 5 (2010), 1-10. Google Scholar

[12]

H.-W. Kuo, The initial layer for Rayleigh problem, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 137-170. doi: 10.3934/dcdsb.2011.15.137. Google Scholar

[13]

H.-W. KuoT.-P. Liu and L.-C. Tsai, Free molecular flow with boundary effect, Comm. Math. Phys., 318 (2013), 375-409. doi: 10.1007/s00220-013-1662-9. Google Scholar

[14]

H.-W. KuoT.-P. Liu and L.-C. Tsai, Equilibrating effects of boundary and collision in rarefied gases, Comm. Math. Phys., 328 (2014), 421-480. doi: 10.1007/s00220-014-2042-9. Google Scholar

[15]

H.-W. Kuo, Equilibrating effect of Maxwell-type boundary condition in highly rarefied gas, J. Stat. Phys., 161 (2015), 743-800. doi: 10.1007/s10955-015-1355-1. Google Scholar

[16]

H.-W. Kuo, Asymptotic behavior for Rayleigh problem based on kinetic theory, J. Stat. Phys., 166 (2017), 1247-1275. doi: 10.1007/s10955-017-1717-y. Google Scholar

[17]

T.-P. Liu and S.-H. Yu, Invariant manifolds for steady Boltzmann flows and applications, Arch. Rational Mech. Anal., 209 (2013), 869-997. doi: 10.1007/s00205-013-0640-x. Google Scholar

[18]

Y. Sone, Kinetic theory analysis of the linearized Rayleigh problem, Phys. Fluids, 7 (1964), 470-471. doi: 10.1063/1.1711221. Google Scholar

[19]

Y. Sone, Effect of sudden change of wall temperature in rarefied gas, J. Phys. Soc. Japan, 20 (1965), 222-229. doi: 10.1143/JPSJ.20.222. Google Scholar

[20]

Y. Sone, Molecular gas dynamics. Theory, techniques, and applications. Modeling and Simulation in Science, Engineering and Technology, Birkh user Boston, Boston, MA, 2007. doi: 10.1007/978-0-8176-4573-1. Google Scholar

[21]

T. TsujiK. Aoki and F. Golse, Relaxation of a free-molecular gas to equilibrium caused by interaction with vessel wall, J. Stat. Phys., 140 (2010), 518-543. doi: 10.1007/s10955-010-9997-5. Google Scholar

[22]

S. Ukai and K. Asano, Steady solutions of the Boltzmann equation for a gas flow past an obstacle, I. Existence, Arch. Rational Mech. Anal., 84 (1983), 249-291. doi: 10.1007/BF00281521. Google Scholar

[23]

S.-H. Yu, Stochastic formulation for the initial-boundary value problems of the Boltzmann equation, Arch. Ration. Mech. Anal., 192 (2009), 217-274. doi: 10.1007/s00205-008-0139-z. Google Scholar

show all references

References:
[1]

K. Aoki and F. Golse, On the speed of approach to equilibrium for a collisionless gas, Kinet. Relat. Models, 4 (2011), 87-107. doi: 10.3934/krm.2011.4.87. Google Scholar

[2]

L. ArkerydR. EspositoR. Marra and A. Nouri, Ghost effect by curvature in planar couette flow, Kinet. Relat. Models, 4 (2011), 109-138. doi: 10.3934/krm.2011.4.109. Google Scholar

[3]

L. ArkerydR. EspositoR. Marra and A. Nouri, Stability for Rayleigh enard convective solutions of the Boltzmann equation, Arch. Rational Mech. Anal, 198 (2010), 125-187. doi: 10.1007/s00205-010-0292-z. Google Scholar

[4]

L. Arkeryd and A. Nouri, The stationary nonlinear Boltzmann equation in a Couette setting with multiple, isolated Lq -solutions and hydrodynamic limits, J. Stat. Phys, 118 (2005), 849-881. doi: 10.1007/s10955-004-2708-3. Google Scholar

[5]

C. Cercignani, The Boltzmann Equation and Its Applications, Springer, New York, 1988. doi: 10.1007/978-1-4612-1039-9. Google Scholar

[6]

C.-C. ChenI.-K. ChenT.-P. Liu and Y. Sone, Thermal transpiration for the linearized Boltzmann equation, Commun. Pure Appl. Math, 60 (2007), 147-163. doi: 10.1002/cpa.20167. Google Scholar

[7]

R. EspositoY. GuoC. Kim and R. Marra, Non-isothermal boundary in the Boltzmann theory and Fourier law, Commun. Math. Phys, 323 (2013), 177-239. doi: 10.1007/s00220-013-1766-2. Google Scholar

[8]

R. Esposito, Y. Guo, C. Kim and R. Marra, Stationary solutions to the Boltzmann equation in the hydrodynamic limit, Ann. PDE, 4 (2018), Art. 1,119 pp, https://doi.org/10.1007/s40818-017-0037-5. doi: 10.1007/s40818-017-0037-5. Google Scholar

[9]

R. EspositoJ. L. Lebowitz and R. Marra, Hydrodynamic limit of the stationary Boltzmann equation in a slab, Commun. Math. Phys, 160 (1994), 49-80. doi: 10.1007/BF02099789. Google Scholar

[10]

H. Grad, Asymptotic theory of the Boltzmann equation, Ⅱ, Int. Symp. on Rarefied Gas Dynamics, Third Symp., 1 (1962), 26-59. Google Scholar

[11]

H.-W. KuoT.-P. Liu and S. E. Noh, Mixture Lemma, Bulletin, Inst. Math., Academia Sinica (N.S.), 5 (2010), 1-10. Google Scholar

[12]

H.-W. Kuo, The initial layer for Rayleigh problem, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 137-170. doi: 10.3934/dcdsb.2011.15.137. Google Scholar

[13]

H.-W. KuoT.-P. Liu and L.-C. Tsai, Free molecular flow with boundary effect, Comm. Math. Phys., 318 (2013), 375-409. doi: 10.1007/s00220-013-1662-9. Google Scholar

[14]

H.-W. KuoT.-P. Liu and L.-C. Tsai, Equilibrating effects of boundary and collision in rarefied gases, Comm. Math. Phys., 328 (2014), 421-480. doi: 10.1007/s00220-014-2042-9. Google Scholar

[15]

H.-W. Kuo, Equilibrating effect of Maxwell-type boundary condition in highly rarefied gas, J. Stat. Phys., 161 (2015), 743-800. doi: 10.1007/s10955-015-1355-1. Google Scholar

[16]

H.-W. Kuo, Asymptotic behavior for Rayleigh problem based on kinetic theory, J. Stat. Phys., 166 (2017), 1247-1275. doi: 10.1007/s10955-017-1717-y. Google Scholar

[17]

T.-P. Liu and S.-H. Yu, Invariant manifolds for steady Boltzmann flows and applications, Arch. Rational Mech. Anal., 209 (2013), 869-997. doi: 10.1007/s00205-013-0640-x. Google Scholar

[18]

Y. Sone, Kinetic theory analysis of the linearized Rayleigh problem, Phys. Fluids, 7 (1964), 470-471. doi: 10.1063/1.1711221. Google Scholar

[19]

Y. Sone, Effect of sudden change of wall temperature in rarefied gas, J. Phys. Soc. Japan, 20 (1965), 222-229. doi: 10.1143/JPSJ.20.222. Google Scholar

[20]

Y. Sone, Molecular gas dynamics. Theory, techniques, and applications. Modeling and Simulation in Science, Engineering and Technology, Birkh user Boston, Boston, MA, 2007. doi: 10.1007/978-0-8176-4573-1. Google Scholar

[21]

T. TsujiK. Aoki and F. Golse, Relaxation of a free-molecular gas to equilibrium caused by interaction with vessel wall, J. Stat. Phys., 140 (2010), 518-543. doi: 10.1007/s10955-010-9997-5. Google Scholar

[22]

S. Ukai and K. Asano, Steady solutions of the Boltzmann equation for a gas flow past an obstacle, I. Existence, Arch. Rational Mech. Anal., 84 (1983), 249-291. doi: 10.1007/BF00281521. Google Scholar

[23]

S.-H. Yu, Stochastic formulation for the initial-boundary value problems of the Boltzmann equation, Arch. Ration. Mech. Anal., 192 (2009), 217-274. doi: 10.1007/s00205-008-0139-z. Google Scholar

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