Unsteady magnetohydrodynamic Taylor–Couette flow with coupled static and dynamic wall slip

The start-up azimuthal motion of an incompressible, electrically conducting Newtonian fluid between concentric cylinders is analysed under an axial magnetic field and a coupled wall law containing Navier (static) slip and first-order interfacial (dynamic) slip. The inductionless, axisymmetric base-flow equations are transformed in time, solved in the Laplace domain with modified Bessel functions and inverted by a controlled Riemann-sum approximation (RSA). Exact steady solutions, the classical non-magnetic no-slip limit and an independent method-of-lines computation are used for validation. At the representative point r = 1.4 , δ = 1.6 and η = 0.2 , raising M from 1 to 2 reduces velocity by 14.8%, increases the magnitude of the inner-wall shear by 7.8% and reduces the outer-wall shear magnitude by 17.7%. Doubling static slip from β = 0.4 to 0.8 reduces the two shear magnitudes by 28.2% and 35.1%, respectively. Doubling dynamic slip from Δ = 0.4 to 0.8 delays the transient velocity by 39.9% but leaves the steady solution unchanged. With N = 800 , RSA and method-of-lines values agree within 0.03% for velocity and both wall shears. The results distinguish magnetic torque demand at the driven cylinder from torque transmission across the annulus and provide a quantitative basis for transient drag control in liquid-metal rotary devices.