The analysis of hydromagnetic flow has been a concern of consideration for research scientists and engineers. In this treatise, the unsteady hydromagnetic free convective flow of an incompressible, viscous, and electrically conducting fluid and heat transfer in a vertical cylinder with Hall effects and transverse magnetic field, in the presence of heat source and sink, is studied. Exact solutions of dimensionless unsteady governing equations are obtained by using the Laplace transform technique. Transient fluid velocity and temperature field are depicted by graphs, whereas heat transfer rate, mass flux, and skinfriction are computed in a tabular form for different physical parametric values, such as the source/sink parameter, Hall current parameter, Prandtl number, and Hartmann number. The valuable result from the investigation is that the Hall parameter slightly accelerates the fluid velocity and it provides resistance to the skin friction at the cylindrical surface. Moreover, it is noticeable that temperature as well as velocity enhances and finally gets to its steady state with time. To be noted, the Hall effects have been widely applied in numerous experimental engineering applications, such as constrictions of turbines, Hall accelerators, Hall sensors.

The fully developed laminar magnetohydrodynamic free convection between two concentric vertical cylinders with Hall currents and heat source/sink, in the presence of the radial magnetic field, are studied. The governing thermal energy and momentum equations are changed into ordinary differential equations whose solutions are determined in closed-form expressions of the Bessel and modified Bessel functions of order zero. A parametric investigation illustrating the impacts of the Hall current, magnetic field, heat source, and radii ratio has been accomplished graphically to examine the changes in temperature as well as velocity while the Nusselt number, mass flux, and skin friction values have been presented in tabular forms. The results ensure that the Hall current has a strong and direct impact on the flow character, such that the influence of the Hall parameter enhances the velocity fields in the appearance of heat source and sink. The velocity remains almost constant as the Hall parameter value is greater than four.Moreover, the velocity and temperature fields have an increasing tendency due to the heat source and inversely for the heat sink. K E Y W O R D S Hall current, heat source/sink, magnetohydrodynamic, natural convection, Nusselt number, skin friction F I G U R E 1 Physical model of system

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The intention of the present article is to examine the transient effect on the free convection of the second‐grade fluid flowing in a vertical cylinder. We have found the analytical solution of the derived governing partial differential equations in nondimensional form corresponding to the velocity as well as the temperature fields by applying the Laplace transformation method in terms of the Bessel function of the first kind. To obtain the impacts of the appropriate parameters related to the given equations, such as the second‐grade fluid parameter and the Prandtl number, the numerical values of the velocity are discussed and displayed in graphs. Furthermore, the mass flow rate and the skin friction at the inner surface of the cylinder are given in a tabular form. The influence of the second‐grade fluid parameter is to decelerate the velocity as well as the mass flow rate and to enhance the steady‐state time.