Transient Temperatures in the Vicinity of an Asperity Contact

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Abstract
The two dimensional, transient temperature distribution in the vicinity of a small, stationary, circular heat source is derived. The source is assumed to be acting on the surface of a relatively much larger body which can be treated as semi-infinite. Formulations both with and without surface convection are considered. Successive integral transforms are used to obtain a direct solution of the governing differential equation. The transient of local temperature rise is found to be very short and very localized in the immediate vicinity of the source. The auxiliary problem of pure convective cooling, i.e., no heat input, is also presented. The initial temperature distribution is taken to be the steady distribution already derived with the heat input. The transient of local temperature in cooling-off is found to be even shorter. In both the main and the auxiliary problems, the governing parameters are source radius, heat flux, thermal conductivity and thermal diffusivity. The convective coefficient does not have a significant effect. This study is intended to represent the thermal behavior of a single asperity in an apparent area of contact. The auxiliary problem represents the cooling-off of the asperity as it moves out from the apparent contact; or it may also be the cooling-off between two consecutive asperity collisions within an apparent contact. The analysis can also be applied to multiple sources acting simultaneously, provided that they are located sufficiently far away from each other, and thermal interaction is negligible. Because the heat generated at the asperity interface must be partitioned, a partition coefficient is derived in the appendix.

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