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


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


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


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.
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

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


Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, 2021, 15 (3) : 445-474. doi: 10.3934/ipi.2020075


Kuan-Hsiang Wang. An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021030


V. V. Zhikov, S. E. Pastukhova. Korn inequalities on thin periodic structures. Networks & Heterogeneous Media, 2009, 4 (1) : 153-175. doi: 10.3934/nhm.2009.4.153


Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021028


Paul Deuring. Spatial asymptotics of mild solutions to the time-dependent Oseen system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021044


Jihoon Lee, Nguyen Thanh Nguyen. Gromov-Hausdorff stability of reaction diffusion equations with Robin boundary conditions under perturbations of the domain and equation. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1263-1296. doi: 10.3934/cpaa.2021020


Fuzhi Li, Dongmei Xu. Regular dynamics for stochastic Fitzhugh-Nagumo systems with additive noise on thin domains. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3517-3542. doi: 10.3934/dcdsb.2020244


Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3021-3029. doi: 10.3934/dcds.2020395


Chaoqian Li, Yajun Liu, Yaotang Li. Note on $ Z $-eigenvalue inclusion theorems for tensors. Journal of Industrial & Management Optimization, 2021, 17 (2) : 687-693. doi: 10.3934/jimo.2019129


Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397


Jun He, Guangjun Xu, Yanmin Liu. New Z-eigenvalue localization sets for tensors with applications. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021058


Hong-Yi Miao, Li Wang. Preconditioned inexact Newton-like method for large nonsymmetric eigenvalue problems. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021012


Caili Sang, Zhen Chen. Optimal $ Z $-eigenvalue inclusion intervals of tensors and their applications. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021075


Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022


Chonghu Guan, Xun Li, Rui Zhou, Wenxin Zhou. Free boundary problem for an optimal investment problem with a borrowing constraint. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021049


Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021


Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271


Giulio Ciraolo, Antonio Greco. An overdetermined problem associated to the Finsler Laplacian. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1025-1038. doi: 10.3934/cpaa.2021004


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

2019 Impact Factor: 1.338


  • PDF downloads (59)
  • HTML views (0)
  • Cited by (4)

[Back to Top]