This study investigates the inherent irreversibility in the flow of a variable (temperature-dependent) viscosity fluid through a channel with parallel plates. The channel is narrow so that the lubrication approximation may be applied, and the temperature-dependent nature of viscosity is assumed to follow an exponential model. The system is assumed to exchange heat with the ambient surroundings following Newton’s cooling law. Using a perturbation method coupled with a special type of Hermite–Padé approximation technique, the simplified governing nonlinear equations are solved and the important properties of overall flow structure, including velocity field, temperature field, and thermal criticality conditions are derived, which essentially expedite obtaining expressions for volumetric entropy generation numbers, irreversibility distribution ratio, and the Bejan number in the flow field. PACS Nos.: 44.10.+a, 47.11.–j, 47.15.gm