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The main objective of this study is to examine an unsteady swirling flow of a non-Newtonian Eyring–Powell fluid between two coaxial disks. The lower and upper disks are considered to rotate with different angular velocities. The three-dimensional axisymmetric flow phenomenon and heat transfer mechanism are observed with the consequences of the magnetic field and variable thermal conductivity of the fluid. The variable thermal conductivity is taken to be dependent on the fluid temperature. The implementation of the Von Karman similarity transformations on the constituting equations of the flow phenomenon yields the dimensionless system of the non-linear equations. An optimal homotopy analysis technique is adopted to obtain analytical solutions for highly non-linear equations. In view of the same and opposite directions of disks rotation, the various aspects of the flow system corresponding to the pertinent parameters are discussed with physical significance. The obtained results indicate that both radial and axial fields are the escalating functions of the Eyring–Powell fluid parameter. Moreover, the heat transfer rate enhances with the improving variable thermal conductivity parameter.