The present work is an extension of a novel methodology recently proposed by the authors for the analytical solution of conjugated heat transfer problems in channel flow, here taking into account the axial diffusion effects which are often of relevance in micro-channels. This methodology is based on a single domain formulation, which is proposed for modeling the heat transfer phenomena at both the fluid stream and the channel walls regions. By making use of coefficients represented as space variable functions, with abrupt transitions occurring at the fluid-wall interface, the mathematical model is fed with the information concerning the transition of the two domains, unifying the model into a single domain formulation with space variable coefficients. The Generalized Integral Transform Technique (GITT) is then employed in the hybrid numerical-analytical solution of the resulting convection-diffusion problem with variable coefficients. When the axial conduction term is included into the formulation, a non-classical eigenvalue problem must be employed in the solution procedure, which is itself handled with the GITT. In order to covalidate the results obtained by means of this solution path, we have also proposed an alternative solution, including a pseudo-transient term, with the aid of a classical Sturm-Liouville eigenvalue problem. The remarkable results demonstrate the feasibility of this single domain approach in handling conjugated heat transfer problems in micro-channels, as well as when fluid axial conduction cannot be neglected.

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