Longitudinal-fin heat sinks (LFHSs) are ubiquitous in the thermal management of electronics. They consist of a bank of straight, but not isothermal, fins separated by gaps through which coolant fluid flows and they’re often not fully shrouded, i.e., a gap exists between the tips of the fins and a shroud over them. Analysis of them is heretofore numerical because it’s complicated by several facts. First, even assuming the fins are vanishingly fin insofar as the fluid portion of the domain, the boundary conditions along the fin and gap above them up to the shroud are, hydrodynamically, mixed, i.e., no-slip adjacent to shear free. Secondly, the conjugate thermal problem needs to be considered, i.e., conduction along the fins must be considered with temperature and heat flux continuity imposed at the solid-fluid interface. Moreover, above the fins are adiabatic lines of symmetry; therefore, the boundary conditions are far more complex than in the hydrodynamic problem. We consider this problem in the context of hydrodynamically and thermally fully-developed flow and define a small parameter (epsilon) equal to the ratio of the spacing between the fins to their height. It is shown that the Poiseuille and (mean) conjugate Nusselt numbers can then, via matched asymptotics and complex analysis, be computed to fourth and third orders in epsilon, respectively. This provides a simple means to optimize fin spacing and thickness to enhance the cooling of electronics. To conclude, we discuss a numerical solution to the more general problem of simultaneously-developing laminar flow through LFHSs. This work is in collaboration with colleagues on the Red Lotus Project, a collaboration between Mechanical Engineers at Tufts University and mathematicians at Imperial College London.