Analytical formulae for flow resistance of a periodic heat sink

Analytical formulae are developed for the friction factor times Reynolds number (fRe) for fully-developed flow through a 5-sided duct. The value fRe is relevant to the flow resistance of longitudinal-fin heat sinks with or without clearance between the tips of the fins and a shroud as per the classic problem numerically resolved by Sparrow et al. [ASME J. Heat Transfer, 100, 1978]. Simple formulae for fRe in the limit of vanishing fin thickness are derived via matched asymptotic expansions in two settings: (i) when the fin spacing to fin height is small, and (ii) when the fin clearance to fin height is small. Numerical calculations show that the asymptotic formulae remain good approximants for fRe even outside the asymptotic parameter range for which they were derived.