Thermal contact resistance between mating surfaces occurs due to surface roughness and the resulting imperfect contact between the two, and plays a dominant rate-limiting role in several thermal systems. Thermal spreading/constriction around gaps due to surface roughness has been studied extensively in the past. However, most analytical solutions available in the literature only offer a limited range of validity in terms of the shape of the gap and contact fraction. Further, these papers mostly address only flat or circular arc shaped gaps. This work presents an exact theoretical analysis of thermal conduction around two-dimensional interfacial gaps of irregular shape, leading to an expression for the resulting thermal contact resistance. The gap region is treated as a material of zero thermal conductivity to enforce adiabatic gap boundaries. A unified expression for the thermal conductivity distribution in the entire domain is written, based on which, a series solution for the temperature field is derived. The resulting thermal contact resistance derived here is shown to agree well with previously reported results in their specific regimes of validity. Further, the versatility of the results derived here is illustrated by solving several problems comprising gaps of non-circular shapes. A problem comprising two highly curved circular gaps of different radii is also solved. Another problem that calculates thermal contact resistance using discrete surface roughness data from a profilometer measurement is also discussed. The theoretical analysis presented in this work makes it possible to treat gaps of irregular shape. While presented here in the context of two-dimensional gaps, the theoretical technique developed in this work is extendable to three-dimensional gaps of arbitrary shape.
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