A micromechanically motivated higher‐order continuum formulation of linear thermal conduction

Abstract

AbstractA higher‐order continuum theory of linearized thermal conduction is developed where a rigid heat conductor is modeled as a material of grade‐2 with a view towards capturing the thermal effects in BGK‐Burnett type formulations for microfluidic flows. The construction is based on a second‐order thermal homogenization framework which leads to the identification of a thermal dissipation potential that is a function of the higher‐order gradients of the temperature. The dissipation potential delivers the expressions for the higher‐order fluxes and forms the basis of a discussion regarding the satisfaction of the second law of thermodynamics. Further restrictions on the constitutive expressions arise from material symmetry considerations. Strong and weak formulations of the associated boundary value problem are derived based on an appropriate identification of the boundary conditions. The methodologies employed and the implications of the theory are reminiscent of similar approaches in linearized elasticity formulations and are compared with micromorphic and Cahn‐Hilliard type formulations.