Open AccessFORMULATION OF THE HEAT CONDUCTION EQUATION FOR HETEROGENEOUS MEDIA WITH MULTIPLE SPATIAL SCALES USING REITERATED HOMOGENIZATION
Author(s) -
Ernesto IglesiasRodríguez,
Manuel Cruz,
Julián BravoCastillero,
Raúl Guinovart-Dı́az,
Reinaldo Rodrı́guez-Ramos,
Leslie D. Pérez-Fernández
Publication year - 2016
Publication title -
engenharia térmica
Language(s) - English
Resource type - Journals
ISSN - 1676-1790
DOI - 10.5380/reterm.v15i1.62165
Subject(s) - homogenization (climate) , thermal conduction , heat equation , fourier number , scaling , fourier transform , statistical physics , scale (ratio) , length scale , homogeneous , thermal , materials science , mathematical analysis , physics , mathematics , mechanics , thermodynamics , heat transfer , geometry , heat flux , biodiversity , ecology , quantum mechanics , biology
Heterogeneous media with multiple spatial scales are finding increased importance in engineering. An example might be a large scale, otherwise homogeneous medium filled with dispersed small-scale particles that form aggregate structures at an intermediate scale. The objective in this paper is to formulate the strong-form Fourier heat conduction equation for such media using the method of reiterated homogenization. The phases are assumed to have a perfect thermal contact at the interface. The ratio of two successive length scales of the medium is a constant small parameter ε. The method is an up-scaling procedure that writes the temperature field as an asymptotic multiple-scale expansion in powers of the small parameter ε . The technique leads to two pairs of local and homogenized equations, linked by effective coefficients. In this manner the medium behavior at the smallest scales is seen to affect the macroscale behavior, which is the main interest in engineering. To facilitate the physical understanding of the formulation, an analytical solution is obtained for the heat conduction equation in a functionally graded material (FGM). The approach presented here may serve as a basis for future efforts to numerically compute effective properties of heterogeneous media with multiple spatial scales.