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The analysis of the stability of the Cauchy problem in the cylindrical double-layer area

Ciałkowski Michał, Joachimiak Magda, Mierzwiczak Magdalena, Frąckowiak Andrzej, Olejnik Aleksander, Kozakiewicz Adam

Archives of Thermodynamics

2023

Vol. 44, no 4

563--579

Ceramic protective coats, for instance, on turbine blades, create a double-layer area with various thermophysical properties and they require metal temperature control. In this paper, it is implemented by formulating a Cauchy problem for the equation of thermal conductivity in the metal cylindrical area with a ceramic layer. Due to the ill posed problem, a regularization method was applied consisting in the notation of thermal balance for the ceramic layer. A spectral radius for the equation matrix was taken as the stability measure of the Cauchy problem. Numerical calculations were performed for a varied thickness of the ceramic layer, with consideration of the non-linear thermophysical properties of steel and a ceramic layer (zirconium dioxide). A polynomial was determined which approximates temperature distribution in time for the protective layer. The stability of solutions was compared for undisturbed and disturbed temperature values, and thermophysical parameters with various ceramic layer thickness. The obtained calculation results confirmed the effectiveness of the proposed regularization method in obtaining stable solutions at random data disturbance.