Solar energy is a plentiful and renewable resource; therefore, its use through clean energy technologies is necessary to reduce the consumption of fossil fuels and their consequent impact on the environment. Solar thermal energy can be harnessed to collect thermal energy through solar collectors for domestic or industrial use. In this work, a Y-shaped duct network was developed for a solar collector with a disc-shaped body. The shape and size of the network were defined using the constructal design method to harvest greater thermal energy from exposure to irradiance. For this purpose, the working fluid was a nanofluid composed of nanoparticles and a base fluid with non-Newtonian behavior, as described by the power-law model. The properties of the nanofluid were described using theoretical models that depend on the volume fraction of the nanoparticles in the base fluid. The aspect ratio of each element in the network was defined under the condition of minimum thermal resistance. For the shear-thinning nanofluid with a volume fraction of 0.04, the thermal resistance was reduced by 5.5% with an aspect ratio 5.8% lower than that of the pure base fluid. The minimum thermal resistance of the Y-shaped network arrangement occurs for shear-thinning fluids, whereas thermal resistance increases for shear-thickening fluids. This effect can be observed at different levels of branching and in pumping power. Results show that increasing the volume fraction of the nanofluid and base fluid with shear-thinning behavior aids in achieving minimal thermal resistance more easily with a smaller network. The above offers new design options for devices optimized to improve solar harvesting and thermal management, through shape, structure and rheological characteristics.