The present study attempts to investigate the effects the viscous dissipation on the unsteady temperature distribution in the conduction limit for both hydrodynamically and thermally fully developed, laminar flow of Newtonian fluid between two asymmetrically heated infinitely long parallel plates. Utilizing the assumptions routinely employed in the literature, we devise here a semi-analytical formalism to investigate the temperature distribution for two different flow configurations, i.e., the poiseuille flow and the simple shear driven flow. In the analysis, we give focus to the viscous dissipative effect arises because of the two individual aspects in case of shear-driven flow: the shear heating produced by the movable upper plate along with fluid friction, while only due to the internal fluid friction in case of Poiseuille flow. Finally, we show the variation of velocity and the temperature distribution in the flow field for several nondimensional parameters as emerge in the present study, and highlight their individual role in delineating the temperature distribution in the flow field, which essentially alters transient thermal transport characteristics of heat in different cases of flow dynamics.