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The purpose of this note is to provide general solutions for radiative magnetohydrodynamic natural convection flow.

To obtain exact solutions for such motions of Newtonian fluids, as seen in the existing literature, the Laplace transform technique is used.

General solutions are obtained for temperature, velocity and Nusselt number in the presence of heat source and shear stress on the boundary. They can generate exact solutions for any motion with technical relevance of this type. Fluid velocity is presented as the sum of mechanical and thermal components. Influence of physical parameters on temperature and velocity is graphically underlined for ramp-type heating plate that applies a constantly accelerating shear stress to the fluid. Thermal and mechanical effects are significant and must be taken into consideration.

For illustration, as well as for a check of results, three special cases with applications in engineering are considered and some known results are recovered.

Obtained solutions are presented in the simplest forms. In addition, the solutions corresponding to cosine oscillatory heating and oscillating shear are presented so that they can be immediately reduced to those corresponding to constant heating and uniform shear if the oscillations’ frequency becomes zero. Heat transfer characteristics with thermal radiation are graphically illustrated using one parameter only for such motions.