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Radiative–convective flow studies find wide range of applications in furnace design, solar fans, photochemical reactors, turbid water bodies, etc. The present article focusses on unsteady radiative–convective hydromagnetic couple stress fluid (CSF) flow from a vertical cylinder using the thermodynamic concept. The obtained governing equations of the present model are resolved by a well-organised numerical scheme. The unsteady nature of friction, entropy, coefficients of heat and mass transfer (HMT) along with the time-independent state pattern of flow-field profiles, are shown graphically for distinct values of governing radiation parameter, magnetic parameter, concentration parameter, and constant parameter to display important aspects of the solution. To analyse the HMT process in a 2D domain, Bejans flow visualisation is considered along with isotherms, streamlines, and isoconcentration lines. The Bejans HMT flow visualisation shows that the heat and mass function contours are denser in the foremost verge of the hot surface of the cylinder compared to other contours. The result indicates that the entropy generation (EG) parameter increases with decreasing values of radiation and magnetic parameters. Also, the entropy parameter increases for increasing values of concentration parameter or constant parameter. [ABSTRACT FROM AUTHOR]