  In this article, we present the governing equations for the temperature field
upstream and downstream of the periodically developed flow region in channels
with arrays of periodic solid structures. From the ansatz that the temperature
field in this region is determined by exponentially decaying modes, just like
the quasi-developed flow field, we arrive at an eigenvalue problem which
governs the allowed temperature modes. This eigenvalue problem is virtually
identical to the one for periodically developed heat transfer in isothermal
solids, except that the conjugate heat transfer in the solid is effectively
taken into account.
