American Institute of Mathematical Sciences

Advanced
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

Kai Koike

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.

Keywords: Free Vlasov equation, moving boundary problem, drag/friction force, asymptotic behaviour, wall effect.
Mathematics Subject Classification: Primary: 35Q83, 35R37; Secondary: 70F40.
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:
[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  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.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  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.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  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 $2X(t)$.
Figure Options

Download full-size image

Download as PowerPoint slide

[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]

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 & Continuous Dynamical Systems, 2022  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 & 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 & Related Models, , () : -. doi: 10.3934/krm.2021052
[6]

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
[7]

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
[8]

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
[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 & 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 & 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 & 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 & 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 & Continuous Dynamical Systems - B, 2014, 19 (7) : 1815-1835. doi: 10.3934/dcdsb.2014.19.1815
[14]

Khaled El Dika. Smoothing effect of the generalized BBM equation for localized solutions moving to the right. Discrete & Continuous Dynamical Systems, 2005, 12 (5) : 973-982. doi: 10.3934/dcds.2005.12.973
[15]

Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393
[16]

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, 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 & 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 & 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 & Related Models, 2019, 12 (4) : 765-789. doi: 10.3934/krm.2019030
[20]

Yang Zhang. A free boundary problem of the cancer invasion. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021092

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (165)
  • HTML views (216)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy