A model for uniform parallel flow over the surface of a rectangular source of heat on a conducting plate is used to demonstrate the use of analytic Greens functions to formulate the conjugate problem. The Greens functions are solutions to the temperature field that arise from a point source of heat on the surface. They provide a relationship between the local heat flux and surface temperature on the plate, effectively serving the same role as the heat transfer coefficient. By coupling the pointwise Greens function to a finite element discretization of the thin plate, the surface temperature and convective heat flux distributions on the heat source and its substrate are found by a non-iterative procedure. A parametric study showed that at high Peclet numbers, the heat transfer from the source approached the behavior of an infinite 2D source of heat. The average Nusselt numbers for rectangular sources of different aspect ratios were found to be insensitive to source aspect ratio at high Peclet numbers. Board conduction reduced the average Nusselt numbers over the source when it was defined in terms of the freestream temperature. New correlations for the source Nusselt number as a function of flow Peclet number and board conductivity are presented.
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