Analysis of the non-linearity of the heat transfer equation in case of a time-dependent heat source: application to the \(3\omega \) method

The \(3\omega \) method may be used to estimate the thermal conductivity of an electrically conducting wire. In this method, an alternating voltage with an angular frequency \(\omega \) is applied to the wire. The resulting low electrical tension \(U_{3\omega }\) of angular frequency \(3\omega \) that appears in the total electrical tension is extracted by a lock-in amplifier. The amplitude of \(U_{3\omega }\) is directly linked to the thermal conductivity of the wire and enables its estimation. All authors using the \(3\omega \) method for the determination of the thermal conductivity of an electric conducting wire considered that the heat flux produced by Joule effect in the wire is constant. This hypothesis leads to a linear form of the heat transfer equation. In this work, an analytical model taking into account the dependence of the heat flux on the temperature is developed, it leads to a non-linear form of the heat transfer equation. The importance of the non-linearity in certain cases is demonstrated and the analytical solution is used to define a unique criterion that must be verified to ensure the validity of the linear solution.