This work presents numerical simulation of two-dimensional thermogravitational energy transport in a chamber flled with copper–water (Cu-H2O) nanoliquid under the uniform magnetic impact. In this study, our aim is to analyze the characteristic role of nanoliquid thermal conductivity under the infuence of various thermal boundary conditions with constant magnetic efect. The mathematical model of the fow physics consists of the Navier–Stokes (N–S) equations written using streamfunction–vorticity variables including the energy transport equation. The governing equations are solved by using a higher-order compact scheme based on fnite diference method. The impact of key characteristics including nanoliquid volume fraction (0 ≤ 휙 ≤ 0.04), Rayleigh number (104 ≤ Ra ≤ 106), Hartmann number (0 ≤ Ha ≤ 60) and amplitude of heating (0 ≤ I ≤ 1) is analyzed in detail. It is found that the energy transport augmentation occurs with nonuniform heating over uniform heating and the rate of thermal transmission rises with a growth of Ra and 휙 but it decreases with the growth of Ha number. In addition, the transient structures are very useful for understanding the thermo- and magnetohydrodynamic problems.