Linear and non-linear heat transfer in nanotechnology

The abstract addresses the fundamental principles of heat transfer, focusing on conduction, convection, and thermal radiation. It begins by applying the energy conservation law for closed and adjacent systems, resulting in a one-dimensional equation for temperature distribution based on Fourier’s law. The concept of thermal diffusivity as a measure of heat dispersion rate in a material is introduced. Subsequently, the process of thermal convection, both forced and natural, and the significance of the heat transfer coefficient h are discussed. A simplified example of body cooling through convection is presented. Lastly, thermal radiation is addressed, emphasizing the importance of the black body concept and the application of Stefan-Boltzmann’s law to determine radiant energy flux. This abstract provides a concise introduction to the critical principles of heat transfer. The porous medium equation (PME) represents a nonlinear form of the classical heat equation. The boundary conditions can be described as moving surfaces that expand over time. The theory of PME raises various questions concerning the existence, continuity, and uniqueness of solutions, as well as the regularity and the presence of Harnack-type inequalities. This thesis explores the dynamics of energy transport in 1D systems, focusing on the interplay between Hamiltonian and stochastic components. The transport properties of the system are investigated in the context of ballistic transport, where the quadratic potential and conserved quantities play a crucial role. To introduce stochasticity and explore anomalous transport, the Hamiltonian dynamics are augmented with a stochastic component while preserving the system’s conserved quantities. The paper compares two approaches: the traditional Hamiltonian method and the augmented stochastic method. For the open system configuration, the system is connected to heat reservoirs at different temperatures, resulting in non-linear temperature profiles. The temperature profiles are influenced by the interplay between the temperature-dependent thermal conductivity and the temperature difference between the reservoirs. The findings highlight the complex nature of energy transport in 1D systems and demonstrate the significance of incorporating stochastic components to capture anomalous transport phenomena. The fourth chapter presents an analysis of a Matlab code that simulates a non-linear trend in Fourier’s law. The goal is to observe convergence to UR across different intervals using various mathematical finite methods such as 1-D and 2-D diffusion, Burgers equations, and non-linear diffusion. The simulation results and temperature profiles are analyzed, providing insights into the behaviour of the system. This analysis contributes to understanding the complex dynamics of non-linear thermal transport and the implications for heat transfer processes.