A numerical investigation on the transient-free convection flow of the multiphase nanofluid past a vertical cylinder, which has a power-law variation surface temperature along with the height, is presented. The problem has typical engineering applications involving cooling of vertical cylindrical rods in mechanical/manufacturing systems, cooling of nuclear reactors, and the design of other advanced cooling technologies. Buongiorno's model is applied in this research, which incorporates thermophoresis and Brownian diffusion effects of nanoparticles. The zero-volume flux condition is implemented for nanoparticle concentration at the boundary to obtain realistic results. A robust second-order accurate finite-difference scheme of Crank-Nicolson type is applied to tackle the system of coupled non-linear partial differential equations numerically. The impacts of time, variable surface temperature power-law exponent, Brownian and thermophoresis parameters are investigated on nanofluid flow and heat transfer aspects. The decisive finding suggests that the effect of the power-law exponent of the variable wall temperature is to reduce the nanoparticle relocation, velocity, and temperature in the nanofluid boundary layer causing the heat transfer enhancement. The skin-friction decreased significantly with the rise of the power-law exponent of the wall temperature. The present numerical scheme is corroborated by comparing the average skin-friction results with the available literature for clear fluid. Copyright © 2019 by ASME