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Instability thresholds in the microwave heating model with exponential non-linearity

Published online by Cambridge University Press:  08 February 2011

ADAM ALCOLADO ,
THEODORE KOLOKOLNIKOV  and
DAVID IRON
ADAM ALCOLADO
Affiliation:
Department of Mathematics and Statistics, Dalhousie University, Nova Scotia, Canada email: adam.alcolado@dal.ca, tkolokol@gmail.com, iron@mathstat.dal.ca
THEODORE KOLOKOLNIKOV
Affiliation:
Department of Mathematics and Statistics, Dalhousie University, Nova Scotia, Canada email: adam.alcolado@dal.ca, tkolokol@gmail.com, iron@mathstat.dal.ca
DAVID IRON
Affiliation:
Department of Mathematics and Statistics, Dalhousie University, Nova Scotia, Canada email: adam.alcolado@dal.ca, tkolokol@gmail.com, iron@mathstat.dal.ca
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Abstract

When ceramics are heated inside a microwave cavity, a well-known phenomenon is the occurrence of hot spots – localised regions of high temperature. This phenomenon was modelled by Kriegsmann ((1997), IMA J. Appl. Math. 59(2), pp. 123–146; (2001), IMA J. Appl. Math. 66(1), pp. 1–32) using a non-local evolution PDE. We investigate profile and the stability of hot spots in one and two dimensions by using Kriegsmann's model with exponential non-linearity. The linearised problem associated with hot-spot-type solutions possesses two classes of eigenvalues. The first type is the large eigenvalues associated with the stability of the hot-spot profile and in this particular model there cannot be instability associated with these eigenvalues. The second type is the small eigenvalues associated with translation invariance. We show that the hot spots can become unstable due to the presence of small eigenvalues, and we characterise the instability thresholds. In particular, we show that for the material with low heat conductivity (such as ceramics), and in the presence of a variable electric field, the hot spots are typically stable inside a plate (in two dimensions) but can become unstable for a slab (in one dimension) provided that the microwave power is sufficiently large. On the other hand, for materials with high heat conductivity, the interior hot spots are unstable and move to the boundary of the domain in either one or two dimensions. For materials with moderate heat conductivity, the stability of hot spots is determined by both the geometry and the electric field inside the microwave cavity.

Keywords

Microwave heatingBratu problemSpike solutions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 22 , Issue 3 , June 2011 , pp. 187 - 216
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Booty, M. R. & Kriegsmann, G. A. (1994) Microwave heating and joining of ceramic cylinders: A mathematical model. Methods Appl. Anal. 1 (4), 403–414.CrossRefGoogle Scholar
[2]Casten, R. & Holland, C. (1978) Instabifity results for reaction-diffusion equations with Neumann boundary conditions. J. Differ. Equ. 27, 266–273.CrossRefGoogle Scholar
[3]FlexPDE software. Accessed 27 January 2011, see URL: www.pdesolutions.com.Google Scholar
[4]Freitas, P. (1994) Bifurcation and stability of stationary solutions of nonlocal scalar reaction-diffusion equations. J. Dyn. Differ. Equ. 6 (4), 613–629.CrossRefGoogle Scholar
[5]Hill, J. M. & Marchant, T. R. (1996) Modelling microwave heating. Appl. Math. Modelling, 20, 3–15.CrossRefGoogle Scholar
[6]Iron, D. & Ward, M. (2000) A metastable spike solution for a nonlocal reaction-diffusion model. SIAM J. Appl. Math. 60 (3), 778–802.CrossRefGoogle Scholar
[7]Iron, D. & Ward, M. (2001) Spike pinning for the Gierer-Meinhardt model. J. Math. Comput. Simul. 55, 419–431.CrossRefGoogle Scholar
[8]Iron, D. & Ward, M. J. (2004) The stability and dynamics of hot-spot solutions for a one-dimensional microwave heating model. Anal. Appl. 2 (1), 21–70.CrossRefGoogle Scholar
[9]Iron, D., Ward, M. J. & Wei, J. (2001) The stability of spike solutions to the one-dimensional Gierer-Meinhardt model. Physica D 150, 25–62.Google Scholar
[10]Joseph, D. D. & Lundgren, T. S. (1973) Quasilinear Dirichlet problems driven by positive sources. Arch. Ration. Mech. Anal. 49, 241–269.CrossRefGoogle Scholar
[11]Kolokolnikov, T., Winter, M. & Wei, J. (2009) Existence and stability analysis of spiky solutions for the Gierer-Meinhardt system with large reaction rates. Physica D: Nonlinear Phenom. Vol. 238 (16), 1695–1710.CrossRefGoogle Scholar
[12]Kriegsmann, G. A. (1997) Hot spot formation in microwave heated ceramic fibres. IMA J. Appl. Math. 59 (2), 123–148.CrossRefGoogle Scholar
[13]Kriegsmann, G. A. (2001) Pattern formation in microwave heated ceramics: Cylinders and slabs. IMA J. Appl. Math. 66 (1), 1–32.CrossRefGoogle Scholar
[14]Kriegsmann, G. A. (2005) Microwave Heating of Materials: A Mathematical and Physical Overview, distinguished colloquium given at Institute of Applied Mathematics, University of British Columbia.Google Scholar
[15]Ma, L. & Wei, J. (2001) Convergence for a Liouville equation. Comment. Math. Helv. 76, 506–514.CrossRefGoogle Scholar
[16]Roussy, G., Bennani, A. & Thiebiant, J. (1987) Temperature runaway of microwave irradiated materials. J. Appl. Phys. 62, 1167–1170.CrossRefGoogle Scholar
[17]Strauss, W. A. (1992) Partial Differential Equations, an Introduction, John Wiley & Sons, New York.Google Scholar
[18]Wei, J. (1999) On single interior spike solutions of the Gierer-Meinhardt system: Uniqueness and spectrum estimate. Eur. J. Appl. Math. 10, 353–378.CrossRefGoogle Scholar