Transient linear and nonlinear heat conduction in weakly modulated domains

Heat conduction in two-dimensional domains with spatially periodic boundary is addressed in this study. The periodic modulation is assumed to be weak, but is of arbitrary shape. A regular perturbation approach is implemented to determine the temperature and heat flux throughout the domain. It is observed that the validity of the perturbation approach extends to include geometries of practical importance. Transient linear as well as steady nonlinear heat conduction problems are examined. The periodic domain is mapped onto the rectangular domain. For both steady and transient linear heat conduction, a fully analytical spectral solution becomes possible. The nonlinear problem is shown to reduce to a set of ordinary differential equations of the two-point-boundary-value type, which is solved using a variable-step-size finite-difference scheme. The perturbation approach is validated upon comparison with conventional methods; excellent agreement is obtained against the boundary- and finite-element methods.