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The basic objective of this article is to highlight the impact of Joule heating, thermal radiation, and viscous dissipation on heat transfer rate and friction drag in the flow of viscous nanofluid subject to the curved stretching sheet. Moreover, the effects of magnetic field and two-heating process termed as prescribed surface temperature and prescribed heat flux is considered on the surface. The nanofluid is addressed as a mixture of alumina ( $ Al_2O_3 $ A l 2 O 3 ) nanoparticle and ethylene glycol ( $ C_2H_6O_2 $ C 2 H 6 O 2 ) base fluid. Initially, certain dimensionless variables have been addressed to modify the governing partial differential equation (PDEs) and the associated boundary condition into non-dimensional ordinary differential equations (ODEs) coupled with boundary conditions. Subsequently, the resulting equations are solved numerically by implementing bvp4c package in MATLAB. Multiple graphical results for heat transfer rate, friction drag, velocity, and temperature are obtained under the influence of several flow parameters like radiation parameter, Eckert number, curvature parameter, stretching parameter, Hartmann number, and volume fraction of nanoparticles. Moreover, several interesting results for the flat surface ( $ K \to \infty $ K → ∞), with prescribed heat flux and surface temperature have been determined. The current radiative and dissipative flow coupled with the Joule heating process have countless applications in various industrial and engineering techniques. [ABSTRACT FROM AUTHOR]