Non-uniform perturbation temperature of thermoelectric material due to a smooth inhomogeneity

The two-dimensional thermoelectric coupling conduction problem of an inhomogeneity, which is characterized by a Laurent polynomial and embedded in a thermoelectric material subjected to uniform electric current density or uniform energy flux at infinity, is studied under the conditions of the electrical insulation and thermal conduction continuity. While the complex potential denoting the electric field has a compact form, the complex potential indicating the temperature field can be treated as a boundary value problem of analytic function. Then, an iterative strategy is developed to solve the series solution of the temperature fields inside and outside the inhomogeneity, expressed by Faber polynomials and their associated polynomials. Finally, the non-uniform temperature fields for the inhomogeneities shaped elliptic and polygonal shapes are carried out in a series form. After the convergence is guaranteed, the results are analyzed to show that the inhomogeneities with different shape characteristics exhibit different effects on the temperature distribution, and the temperature perturbation increase on the boundary is primarily determined by the relative thermal conductivity of the matrix to the inhomogeneity. The maximum curvature can be used to determine the severity of the maximum temperature perturbation on the boundary of inhomogeneities with the same area.