American Institute of Mathematical Sciences

Advanced
September  2009, 25(3): 1061-1079. doi: 10.3934/dcds.2009.25.1061

Estimating thermal insulating ability of anisotropic coatings via Robin eigenvalues and eigenfunctions

Guojing Zhang 1, , Steve Rosencrans 2, , Xuefeng Wang 2, and  Kaijun Zhang 3,

1. 

Laboratory of Nonlinear Analysis, Department of Mathematics, Central China Normal University, 430079, China
2. 

Mathematics Department, Tulane University, New Orleans, LA 70118, United States
3. 

School of Mathematics and Statistics, Northeast Normal University, 130024, China

Received  October 2008 Revised  February 2009 Published  August 2009

The problem considered in this paper is the protection from overheating of a thermal conductor $\Omega_1$ by a thin anisotropic coating $\Omega_2$ (e.g. a space shuttle painted with a nano-insulator). We assume Newton's Cooling Law, so the temperature satisfies the Robin boundary condition on the outer boundary of the coating. Since the temperature function on $\Omega=\overline{\Omega}_1\cup\Omega_2$ can be expanded in terms of the eigenvalues and eigenfunctions of the elliptic operator $u\mapsto -\nabla (A \nabla u)$ with the Robin boundary condition on $\partial\Omega$, where $A$ is the thermal tensor of $\Omega$, we propose the following means to ensure the insulating ability of $\Omega_2$: (A) as many eigenvalues as possible should be small, in particular, the first eigenvalue should be small, (B) the first normalized eigenfunction should take large values on the body $\Omega_1$; we also argue that it is helpful for the understanding of the dynamics if (C) higher normalized eigenfunctions take small absolute values on $\Omega_1$. We assume that the thermal conductivity of $\Omega_2$ is small either in all directions or at least in the direction normal to $\partial\Omega_1$ (the case of "optimally aligned coating"). We study the asymptotic behavior of Robin eigenpairs as outcome of the interplay of the thermal tensor $A$, the thickness of $\Omega_2$ and the thermal transport coefficient in the Robin boundary condition, in the singular limit when either the thermal conductivity of $\Omega_2$, or the thickness of $\Omega_2$, or the thermal transport coefficient approaches $0$. By doing so, we identify the parameter ranges in which some or all of (A)-(C) occur.
Keywords: thin insulator, anisotropy, Overheating, Robin eigenvalue problem, asymptotics..
Mathematics Subject Classification: Primary: 35J05, 35J20; Secondary: 80A20, 80M3.
Citation: Guojing Zhang, Steve Rosencrans, Xuefeng Wang, Kaijun Zhang. Estimating thermal insulating ability of anisotropic coatings via Robin eigenvalues and eigenfunctions. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 1061-1079. doi: 10.3934/dcds.2009.25.1061
[1]

Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Dušan D. Repovš. Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2589-2618. doi: 10.3934/dcds.2017111
[2]

VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with $ p(x) $-growth and generalized Robin boundary value condition. Communications on Pure & Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003
[3]

Kazuaki Taira. The hypoelliptic Robin problem for quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1601-1618. doi: 10.3934/dcdss.2020091
[4]

Qianqian Hou, Tai-Chia Lin, Zhi-An Wang. On a singularly perturbed semi-linear problem with Robin boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 401-414. doi: 10.3934/dcdsb.2020083
[5]

Mihai Mihăilescu. An eigenvalue problem possessing a continuous family of eigenvalues plus an isolated eigenvalue. Communications on Pure & Applied Analysis, 2011, 10 (2) : 701-708. doi: 10.3934/cpaa.2011.10.701
[6]

Al-hassem Nayam. Asymptotics of an optimal compliance-network problem. Networks & Heterogeneous Media, 2013, 8 (2) : 573-589. doi: 10.3934/nhm.2013.8.573
[7]

Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure & Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012
[8]

Giacomo Bocerani, Dimitri Mugnai. A fractional eigenvalue problem in $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 619-629. doi: 10.3934/dcdss.2016016
[9]

David Colton, Yuk-J. Leung. On a transmission eigenvalue problem for a spherically stratified coated dielectric. Inverse Problems & Imaging, 2016, 10 (2) : 369-378. doi: 10.3934/ipi.2016004
[10]

Huan Gao, Zhibao Li, Haibin Zhang. A fast continuous method for the extreme eigenvalue problem. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1587-1599. doi: 10.3934/jimo.2017008
[11]

Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075
[12]

Isabeau Birindelli, Stefania Patrizi. A Neumann eigenvalue problem for fully nonlinear operators. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 845-863. doi: 10.3934/dcds.2010.28.845
[13]

Jean-Michel Rakotoson. Generalized eigenvalue problem for totally discontinuous operators. Discrete & Continuous Dynamical Systems, 2010, 28 (1) : 343-373. doi: 10.3934/dcds.2010.28.343
[14]

Francis Michael Russell, J. C. Eilbeck. Persistent mobile lattice excitations in a crystalline insulator. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1267-1285. doi: 10.3934/dcdss.2011.4.1267
[15]

Marcone C. Pereira, Ricardo P. Silva. Error estimates for a Neumann problem in highly oscillating thin domains. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 803-817. doi: 10.3934/dcds.2013.33.803
[16]

Fioralba Cakoni, Houssem Haddar, Isaac Harris. Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem. Inverse Problems & Imaging, 2015, 9 (4) : 1025-1049. doi: 10.3934/ipi.2015.9.1025
[17]

Maciek Korzec, Andreas Münch, Endre Süli, Barbara Wagner. Anisotropy in wavelet-based phase field models. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1167-1187. doi: 10.3934/dcdsb.2016.21.1167
[18]

Jiaoxiu Ling, Zhan Zhou. Positive solutions of the discrete Robin problem with $ \phi $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3183-3196. doi: 10.3934/dcdss.2020338
[19]

Nicolas Augier, Ugo Boscain, Mario Sigalotti. Semi-conical eigenvalue intersections and the ensemble controllability problem for quantum systems. Mathematical Control & Related Fields, 2020, 10 (4) : 877-911. doi: 10.3934/mcrf.2020023
[20]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (76)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy