In this research paper, an attempt has been made to investigate the torque transmission characteristics in a radial magnetorheological (MR) clutch disc with different groove profiles. To estimate the transmitted torque, a numerical procedure is initiated by the implementation of the Bingham constitutive model into a magnetic field analysis followed by computational fluid dynamic (CFD) analysis. CFD results are presented for the analysis of transmitted torque between a plane driving disc and three different driven discs, i.e., plane, radial groove, and circular groove disc, under the conditions of different working radii, magnetic fields, and rotational speeds. The MR fluid (MRF) domain is modeled using a technique which can be applied for any wall texture of driving and driven disc. To verify the numerical findings, an MR clutch with three different discs and a MR transmission test bed have been built, and the influence of groove profile on the transmission torque of MRFs has been investigated on the test bed. The results are presented to obtain the relationships among torque performance, disc radius, rotational speed, and magnetic field. Numerical results show good agreement of torque transmission for different groove profiles with experiments. Finally, temperature distribution patterns in different three MR clutches and an optimization of the radial groove profile have been analyzed numerically.