Asymptotic solutions for forced convection in a shrouded longitudinal-fin heat sink

We consider laminar forced convection in a fully-shrouded longitudinal-fin heat sink (LFHS), as described by the pioneering study of Sparrow, Baliga and Patankar [1978, J. Heat Trans, 100(4)]. The base of the LFHS is isothermal and the fins are assumed thin such that the fins are not isothermal but the temperature distribution from their base to their tip is considered, and coupled to the fluid flow. Whereas Sparrow et al. solved the fully-developed flow and thermal problems numerically for a range of geometries (they also considered tip clearance), we consider here the physically realistic asymptotic limit where the fin spacing is small in comparison to their height. Using asymptotic analysis, we find approximate reduced-order solutions for the flow field, temperature field (in both the fluid and along the fins), and hence the local and average Nusselt numbers. We compare the solutions to full numerical results for a range of fin spacings, and assess the practical use of our reduced model.