The situation in which the fluid parameters are variable is one that has not garnered a great deal of attention up to this point. The vast majority of researchers make the assumption that the viscosity of the fluid is always the same, despite the fact that this assumption is inappropriate. The current study aims to determine the impact of joule heating and variable viscosity on an unsteady radially stretching sheet. Plate and needle-shaped \(A{l}_{2}{O}_{3}\) and \(Cu\) nanomaterials are infused into the \({H}_{2}O\) base fluid. The presentation regarding the physical model makes use of the fundamental principles of mass, momentum, and energy conservation. The effects of viscous dissipation, convective boundary conditions, and magnetohydrodynamics \(\left(MHD\right)\) are also considered for a realistic and thorough investigation. The traditional cylindrical coordinate system is chosen for the mathematical modeling of the physical problem. The governing nonlinear system of PDEs is transformed into a coupled system of ODEs by utilizing similarity analysis and then solved numerically with an effective technique, BVP4C in MATLAB. Additionally, by determining residual error, the numerical method is verified. The influence of emerging physical parameters on velocity, temperature, skin friction, and Nusselt numbers is discussed in detail and depicted through graphs and bar charts. The present investigation reveals that the variable viscosity was crucial in slowing down the fluid flow and lowering the fluid temperature, whereas the needle-shaped \(Cu-{H}_{2}O\) nanofluid has the lowest skin friction, highest velocity, and fastest rate of heat transfer.
