This paper studies the two-dimensional unsteady incompressible Ag-water and CuO-water nanofluid flow in a semi-porous expanding-contracting channel in the presence of thermal radiation effect. The continuity equation, Navier-Stokes equation, and energy equation governing the model are transformed into a set of non-dimensional ordinary differential equations using appropriate transformations. These dimensionless governing equations are solved using power series with the aid of the Hermite-Padé approximation method. The influences of physical parameters such as Reynolds number, expansion ratio, solid volume fraction, Prandtl number, Magnetic parameter, and shape factor are depicted in velocity and temperature profiles. Moreover, the average Nusselt number and skin friction coefficient are also investigated with the effect of Reynolds number, solid volume fraction, and expansion ratio. It is observed that the heat transfer rate decreases significantly as the shape factor increases.