# American Institute of Mathematical Sciences

June  2018, 11(3): 441-467. doi: 10.3934/krm.2018020

## 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

I thank Tatsuo Iguchi for reading the paper very carefully

Received  March 2017 Revised  June 2017 Published  March 2018

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.

Citation: Kai Koike. Wall effect on the motion of a rigid body immersed in a free molecular flow. Kinetic & Related Models, 2018, 11 (3) : 441-467. doi: 10.3934/krm.2018020
##### References:

show all references

##### References:
A two dimensional picture of a cylinder immersed in a semi-infinite expanse of gas in a region bounded by an infinite plane wall is shown. The radius of the cylinder is $R$ and the height is $h$. The distance between the cylinder and the wall is denoted by $X(t)$ and the velocity by $V(t) = dX(t)/dt$. A constant force $E$ is applied in the direction of the axis of the cylinder and a drag force $D_V(t)$ is exerted to the cylinder from the surrounding gas.
A two dimensional picture of a pre-collision at $C_{W}^{+}(\tilde{\tau}_1)$ is shown. The horizontal distance traversed by the cylinder and the characteristic curve $x(s)$ from $\tilde{\tau}_1$ to $t$ coincide.
Two dimensional picture of a pre-collision at $C_{W}^{-}(\tau_2)$ via pre-collision at the plane wall. The sum of the horizontal distance traversed by the cylinder and the characteristic curve $x(s)$ from $\tau_2$ to $t$ equals $2X(t)$.
 [1] Renjun Duan, Shuangqian Liu. Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force. Kinetic & 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 & Applied Analysis, 2018, 17 (5) : 1821-1852. doi: 10.3934/cpaa.2018087 [3] Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003 [4] Chengxia Lei, Yihong Du. Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 895-911. doi: 10.3934/dcdsb.2017045 [5] Hua Chen, Shaohua Wu. The moving boundary problem in a chemotaxis model. Communications on Pure & Applied Analysis, 2012, 11 (2) : 735-746. doi: 10.3934/cpaa.2012.11.735 [6] 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 & Applied Analysis, 2009, 8 (5) : 1669-1688. doi: 10.3934/cpaa.2009.8.1669 [7] Jie Wang, Xiaoqiang Wang. New asymptotic analysis method for phase field models in moving boundary problem with surface tension. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3185-3213. doi: 10.3934/dcdsb.2015.20.3185 [8] Khaled El Dika. Smoothing effect of the generalized BBM equation for localized solutions moving to the right. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 973-982. doi: 10.3934/dcds.2005.12.973 [9] Maria Rosaria Lancia, Paola Vernole. The Stokes problem in fractal domains: Asymptotic behaviour of the solutions. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-13. doi: 10.3934/dcdss.2020088 [10] Giovambattista Amendola, Sandra Carillo, John Murrough Golden, Adele Manes. Viscoelastic fluids: Free energies, differential problems and asymptotic behaviour. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1815-1835. doi: 10.3934/dcdsb.2014.19.1815 [11] Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393 [12] Junde Wu, Shangbin Cui. Asymptotic behavior of solutions of a free boundary problem modelling the growth of tumors with Stokes equations. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 625-651. doi: 10.3934/dcds.2009.24.625 [13] 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 & Continuous Dynamical Systems - A, 2010, 26 (2) : 737-765. doi: 10.3934/dcds.2010.26.737 [14] Yuan Wu, Jin Liang, Bei Hu. A free boundary problem for defaultable corporate bond with credit rating migration risk and its asymptotic behavior. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019207 [15] Renjun Duan, Tong Yang, Changjiang Zhu. Boltzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 253-277. doi: 10.3934/dcds.2006.16.253 [16] Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10 [17] 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 [18] Jian Yang. Asymptotic behavior of solutions for competitive models with a free boundary. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3253-3276. doi: 10.3934/dcds.2015.35.3253 [19] Khalid Latrach, Hatem Megdiche. Time asymptotic behaviour for Rotenberg's model with Maxwell boundary conditions. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 305-321. doi: 10.3934/dcds.2011.29.305 [20] Hayk Mikayelyan, Henrik Shahgholian. Convexity of the free boundary for an exterior free boundary problem involving the perimeter. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1431-1443. doi: 10.3934/cpaa.2013.12.1431

2018 Impact Factor: 1.38