Magnetohydrodynamic (MHD) Unsteady Axial Flow in an Annulus: A semi-analytical Approach

This study presents a semi-analytical investigation of unsteady magnetohydrodynamic axial flow of an incompressible, viscous, electrically conducting Newtonian fluid through an annulus bounded by two stationary concentric cylinders. The flow is driven by a constant axial pressure gradient and subjected to a uniform transverse magnetic field. Under the low-magnetic-Reynolds-number approximation, the induced magnetic field is neglected and the Lorentz-force term appears as a linear resistance to the axial motion. The governing momentum equation is non-dimensionalized in terms of the radius ratio, dimensionless time, and Hartmann number. The resulting initial-boundary-value problem is solved using the Laplace transform method, while the Riemann-sum approximation is used to recover the transient velocity and skin-friction fields. Closed-form steady-state expressions are also obtained for the velocity and wall shear stress. The results show that the velocity increases with time and approaches the steady-state profile, while increasing the Hartmann number suppresses the axial velocity and reduces the wall shear magnitude. The transient skin friction on both annular walls approaches the corresponding steady-state value as time increases. The formulation provides a useful semi-analytical description of hydromagnetic start-up flow in annular geometries.