# 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 and Related Models, 2018, 11 (3) : 441-467. doi: 10.3934/krm.2018020
##### 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.

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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.
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 $2X(t)$.
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