  We consider laminar, fully-developed, Poiseuille flows of liquid in the
Cassie state through diabatic, parallel-plate microchannels symmetrically
textured with isoflux ridges. Through the use of matched asymptotic expansions
we analytically develop expressions for (apparent hydrodynamic) slip lengths
and variously-defined Nusselt numbers. Our small parameter ($\epsilon$) is the
pitch of the ridges divided by the height of the microchannel. When the ridges
are oriented parallel to the flow, we quantify the error in the Nusselt number
expressions in the literature and provide a new closed-form result. The latter
is accurate to $O\left(\epsilon^2\right)$ and valid for any solid (ridge)
fraction, whereas those in the current literature are accurate to
$O\left(\epsilon^1\right)$ and breakdown in the important limit when solid
fraction approaches zero. When the ridges are oriented transverse to the
(periodically fully-developed) flow, the error associated with neglecting
inertial effects in the slip length is shown to be
$O\left(\epsilon^3\mathrm{Re}\right)$, where $\mathrm{Re}$ is the channel-scale
Reynolds number based on its hydraulic diameter. The corresponding Nusselt
number expressions are new and their accuracy is shown to be dependent on
Reynolds number, Peclet number and Prandtl number in addition to $\epsilon$.
Manipulating the solution to the inner temperature problem encountered in the
vicinity of the ridges shows that classic results for thermal spreading
resistance are better expressed in terms of polylogarithm functions.
