Abstract Copper nanoparticles are widely used in many sectors and research endeavors owing to their unique properties, including a large surface area, catalytic capabilities, and high thermal and electrical conductivity. The selection of the base fluid for copper nanoparticles should be contingent upon the anticipated application requirements since various fluids exhibit distinct characteristics that could potentially impact the mobility of the nanoparticles. The present investigation analyzes heat transfer phenomena occurring across a radially stretched surface. The research explores the effects of different states of Cu nanoparticles when combined with base fluids, such as water and silicone oil, on the heat transfer process. The momentum and energy equations are transformed into nonlinear ordinary differential equations by applying the similarity transformation. The boundary value problem-fourth-order (BVP4C) method numerically solves the governing ordinary differential equation for the modeled problem. In addition, the influence of various factors such as the slip parameter, solid volume fraction, Eckert number, Prandtl number, and unsteadiness parameter are examined. It has been discovered that blade-shaped nanoparticles transfer heat as quickly as possible via silicone oil and water. However, for platelet-shaped nanoparticles, a minimum heat transfer rate has been noted. The maximum skin friction coefficient is observed in platelet-shaped nanoparticles, while blade-shaped nanoparticles have the lowest skin friction coefficient.