We investigate the entropy transport dynamics of a Newtonian fluid in a parallel-plate channel considering the effect of conjugate heat transfer into the analysis. We consider that the flow to be actuated by the combined influences of externally applied pressure gradient and movement of the upper plate of the channel. Using an analytical scheme associated with the thermal boundary conditions of third kind at the outer boundaries of the channel, we bring out the alteration in the underlying thermal transport characteristics of heat as attributable to the nonlinear interactions between flow dynamics and viscous heating. Our analysis demonstrates the influential role of different performance index parameters viz., the geometric parameters, material properties, flow conditions on the irreversibility generation rate of the system. In particular, we unveil an optimum value of wall thickness and wall to fluid conductivity ratio that leads to minimization of the entropy generation rate in the system owing to intricate interplay between the heat flow dynamics and viscous heating. Our results may have far ranging consequences in the design of various cooling and heat removal devices and systems, which are often used in different engineering applications. (C) 2015 Elsevier Ltd. All rights reserved.