Unsteady heat transfer in the harmonic heating of a dilute suspension of small particles

The governing equation for the temperature response of small particles subjected to diffusive and radiative heat transfer in a homogeneous medium is derived. The method used to derive this integro-differential equation is based on an extension of Duhamel’s superposition theorem and it is simpler than the Laplace-transform method traditionally used. This approach is also used to discuss the origin of the history term, which is shown to be a Riemann–Liouville–Weyl half-derivative of the temperature potential between the free-stream and the particle surface. This observation is used to derive the scaling of the unsteady and the quasi-steady contributions for harmonic perturbations of the background temperature field. We identify a scaling number SR, which is a normalized dimensionless frequency, that measures the importance of the history term as compared to the quasi-steady diffusion and radiation contributions. The relevance of the scaling analysis for turbulent flows is discussed.