Rotational magnetohydrodynamic couple stress fluid flow between two eccentric spheres

Abstract This study presents a comprehensive analysis of the axisymmetric magnetohydrodynamic (MHD) flow behavior of a couple-stress fluid confined between two eccentric spheres. Under creeping flow conditions characterized by a low Reynolds number regime, the governing nonlinear momentum equations are formulated and solved using a semi-analytical computational framework based on the boundary collocation method. Particular emphasis is placed on elucidating the effects of the Hartmann number, rotational angular velocities, geometric size ratio, and couple-stress parameters on the hydrodynamic torque exerted on the inner sphere. The numerical findings reveal that the total hydrodynamic torque exhibits a pronounced enhancement with increasing values of the couple-stress parameter, Hartmann number, and eccentric separation distance, whereas an inverse dependence is observed with respect to the size ratio. Moreover, the flow characteristics demonstrate substantial sensitivity to the geometric configuration of the system. Specifically, for the case $a_1/a_2 = 0.5$, the hydrodynamic torque increases by approximately $9890\%$ as the separation parameter $\delta$ varies from a wide clearance of $\delta = 0.9$ to a tight confinement of $\delta = 0.01$. In contrast, at fixed values of $\eta = 0.05$ and $\delta = 0.01$, increasing the magnetic parameter $M$ results in a torque reduction of nearly $72.47\%$. To establish the reliability and accuracy of the present computational methodology, the obtained results are systematically validated against previously published benchmark solutions available in the literature, demonstrating excellent agreement and confirming the robustness of the proposed semi-analytical approach.