
-
Previous Article
Incompressible Limit of isentropic Navier-Stokes equations with Navier-slip boundary
- KRM Home
- This Issue
- Next Article
Wall effect on the motion of a rigid body immersed in a free molecular flow
School of Fundamental Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan |
Motion of a rigid body immersed in a semi-infinite expanse of free molecular gas in a $ d$-dimensional region bounded by an infinite plane wall is studied. The free molecular flow is described by the free Vlasov equation with the specular boundary condition. We show that the velocity $ V(t)$ of the body approaches its terminal velocity $ V_{∞}$ according to a power law $ V_{∞}-V(t)≈ t^{-(d-1)}$ by carefully analyzing the pre-collisions due to the presence of the wall. The exponent $ d-1$ is smaller than $ d+2$ for the case without the wall found in the classical work by Caprino, Marchioro and Pulvirenti [Comm. Math. Phys., 264 (2006), 167-189] and thus slower convergence results from the presence of the wall.
References:
[1] |
K. Aoki, G. Cavallaro, C. Marchioro and M. Pulvirenti,
On the motion of a body in thermal equilibrium immersed in a perfect gas, M2AN Math. Model. Numer. Anal., 42 (2008), 263-275.
doi: 10.1051/m2an:2008007. |
[2] |
A. Belmonte, J. Jacobsen and A. Jayaraman,
Monotone solutions of a nonautonomous differential equation for a sedimenting sphere, Electron. J. Differ. Eq., 2001 (2001), 1-17.
|
[3] |
P. Buttà, G. Cavallaro and C. Marchioro,
Mathematical Models of Viscous Friction, Lecture Notes in Mathematics, 2135 Springer, Cham, 2015.
doi: 10.1007/978-3-319-14759-8. |
[4] |
S. Caprino, G. Cavallaro and C. Marchioro,
On a microscopic model of viscous friction, Math. Models Methods Appl. Sci., 17 (2007), 1369-1403.
doi: 10.1142/S0218202507002315. |
[5] |
S. Caprino, C. Marchioro and M. Pulvirenti,
Approach to equilibrium in a microscopic model of friction, Comm. Math. Phys., 264 (2006), 167-189.
doi: 10.1007/s00220-006-1542-7. |
[6] |
G. Cavallaro,
On the motion of a convex body interacting with a perfect gas in the mean-field approximation, Rend. Mat. Appl., 27 (2007), 123-145.
|
[7] |
G. Cavallaro and C. Marchioro,
On the approach to equilibrium for a pendulum immersed in a Stokes fluid, Math. Models Methods Appl. Sci., 20 (2010), 1999-2019.
doi: 10.1142/S0218202510004854. |
[8] |
G. Cavallaro, C. Marchioro and T. Tsuji,
Approach to equilibrium of a rotating sphere in a Stokes flow, Ann. Univ. Ferrara Sez. Ⅶ Sci. Mat., 57 (2011), 211-228.
doi: 10.1007/s11565-011-0127-3. |
[9] |
X. Chen and W. Strauss,
Approach to equilibrium of a body colliding specularly and diffusely with a sea of particles, Arch. Ration. Mech. Anal., 211 (2014), 879-910.
doi: 10.1007/s00205-013-0675-z. |
[10] |
X. Chen and W. Strauss,
Velocity reversal criterion of a body immersed in a sea of particles, Comm. Math. Phys., 338 (2015), 139-168.
doi: 10.1007/s00220-015-2368-y. |
[11] |
C. Fanelli, F. Sisti and G. V. Stagno, Time dependent friction in a free gas, J. Math. Phys., 57 (2016), 033501, 12 pp.
doi: 10.1063/1.4943013. |
[12] |
C. Ricciuti and F. Sisti,
Effects of concavity on the motion of a body immersed in a Vlasov gas, SIAM J. Math. Anal., 46 (2014), 3579-3611.
doi: 10.1137/140954003. |
show all references
I thank Tatsuo Iguchi for reading the paper very carefully
References:
[1] |
K. Aoki, G. Cavallaro, C. Marchioro and M. Pulvirenti,
On the motion of a body in thermal equilibrium immersed in a perfect gas, M2AN Math. Model. Numer. Anal., 42 (2008), 263-275.
doi: 10.1051/m2an:2008007. |
[2] |
A. Belmonte, J. Jacobsen and A. Jayaraman,
Monotone solutions of a nonautonomous differential equation for a sedimenting sphere, Electron. J. Differ. Eq., 2001 (2001), 1-17.
|
[3] |
P. Buttà, G. Cavallaro and C. Marchioro,
Mathematical Models of Viscous Friction, Lecture Notes in Mathematics, 2135 Springer, Cham, 2015.
doi: 10.1007/978-3-319-14759-8. |
[4] |
S. Caprino, G. Cavallaro and C. Marchioro,
On a microscopic model of viscous friction, Math. Models Methods Appl. Sci., 17 (2007), 1369-1403.
doi: 10.1142/S0218202507002315. |
[5] |
S. Caprino, C. Marchioro and M. Pulvirenti,
Approach to equilibrium in a microscopic model of friction, Comm. Math. Phys., 264 (2006), 167-189.
doi: 10.1007/s00220-006-1542-7. |
[6] |
G. Cavallaro,
On the motion of a convex body interacting with a perfect gas in the mean-field approximation, Rend. Mat. Appl., 27 (2007), 123-145.
|
[7] |
G. Cavallaro and C. Marchioro,
On the approach to equilibrium for a pendulum immersed in a Stokes fluid, Math. Models Methods Appl. Sci., 20 (2010), 1999-2019.
doi: 10.1142/S0218202510004854. |
[8] |
G. Cavallaro, C. Marchioro and T. Tsuji,
Approach to equilibrium of a rotating sphere in a Stokes flow, Ann. Univ. Ferrara Sez. Ⅶ Sci. Mat., 57 (2011), 211-228.
doi: 10.1007/s11565-011-0127-3. |
[9] |
X. Chen and W. Strauss,
Approach to equilibrium of a body colliding specularly and diffusely with a sea of particles, Arch. Ration. Mech. Anal., 211 (2014), 879-910.
doi: 10.1007/s00205-013-0675-z. |
[10] |
X. Chen and W. Strauss,
Velocity reversal criterion of a body immersed in a sea of particles, Comm. Math. Phys., 338 (2015), 139-168.
doi: 10.1007/s00220-015-2368-y. |
[11] |
C. Fanelli, F. Sisti and G. V. Stagno, Time dependent friction in a free gas, J. Math. Phys., 57 (2016), 033501, 12 pp.
doi: 10.1063/1.4943013. |
[12] |
C. Ricciuti and F. Sisti,
Effects of concavity on the motion of a body immersed in a Vlasov gas, SIAM J. Math. Anal., 46 (2014), 3579-3611.
doi: 10.1137/140954003. |



[1] |
Renjun Duan, Shuangqian Liu. Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force. Kinetic and Related Models, 2013, 6 (4) : 687-700. doi: 10.3934/krm.2013.6.687 |
[2] |
Hiroshi Matsuzawa. A free boundary problem for the Fisher-KPP equation with a given moving boundary. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1821-1852. doi: 10.3934/cpaa.2018087 |
[3] |
Yuki Kaneko, Hiroshi Matsuzawa, Yoshio Yamada. A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions I : Classification of asymptotic behavior. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2719-2745. doi: 10.3934/dcds.2021209 |
[4] |
Chueh-Hsin Chang, Chiun-Chuan Chen, Chih-Chiang Huang. Traveling wave solutions of a free boundary problem with latent heat effect. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 1797-1809. doi: 10.3934/dcdsb.2021028 |
[5] |
José A. Carrillo, Young-Pil Choi, Yingping Peng. Large friction-high force fields limit for the nonlinear Vlasov–Poisson–Fokker–Planck system. Kinetic and Related Models, 2022, 15 (3) : 355-384. doi: 10.3934/krm.2021052 |
[6] |
Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003 |
[7] |
Chengxia Lei, Yihong Du. Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 895-911. doi: 10.3934/dcdsb.2017045 |
[8] |
Hua Chen, Shaohua Wu. The moving boundary problem in a chemotaxis model. Communications on Pure and Applied Analysis, 2012, 11 (2) : 735-746. doi: 10.3934/cpaa.2012.11.735 |
[9] |
Fujun Zhou, Junde Wu, Shangbin Cui. Existence and asymptotic behavior of solutions to a moving boundary problem modeling the growth of multi-layer tumors. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1669-1688. doi: 10.3934/cpaa.2009.8.1669 |
[10] |
Jie Wang, Xiaoqiang Wang. New asymptotic analysis method for phase field models in moving boundary problem with surface tension. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3185-3213. doi: 10.3934/dcdsb.2015.20.3185 |
[11] |
Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033 |
[12] |
Maria Rosaria Lancia, Paola Vernole. The Stokes problem in fractal domains: Asymptotic behaviour of the solutions. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1553-1565. doi: 10.3934/dcdss.2020088 |
[13] |
Giovambattista Amendola, Sandra Carillo, John Murrough Golden, Adele Manes. Viscoelastic fluids: Free energies, differential problems and asymptotic behaviour. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 1815-1835. doi: 10.3934/dcdsb.2014.19.1815 |
[14] |
Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393 |
[15] |
Khaled El Dika. Smoothing effect of the generalized BBM equation for localized solutions moving to the right. Discrete and Continuous Dynamical Systems, 2005, 12 (5) : 973-982. doi: 10.3934/dcds.2005.12.973 |
[16] |
Junde Wu, Shangbin Cui. Asymptotic behavior of solutions of a free boundary problem modelling the growth of tumors with Stokes equations. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 625-651. doi: 10.3934/dcds.2009.24.625 |
[17] |
Junde Wu, Shangbin Cui. Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 737-765. doi: 10.3934/dcds.2010.26.737 |
[18] |
Yuan Wu, Jin Liang, Bei Hu. A free boundary problem for defaultable corporate bond with credit rating migration risk and its asymptotic behavior. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1043-1058. doi: 10.3934/dcdsb.2019207 |
[19] |
Hung-Wen Kuo. Effect of abrupt change of the wall temperature in the kinetic theory. Kinetic and Related Models, 2019, 12 (4) : 765-789. doi: 10.3934/krm.2019030 |
[20] |
Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10 |
2021 Impact Factor: 1.398
Tools
Metrics
Other articles
by authors
[Back to Top]