Unsteady magnetohydrodynamic Dean flow with dynamic as well as static slips

This study investigates the unsteady magnetohydrodynamic Dean flow in an annulus subject to both dynamic and static slip boundary conditions. The governing equations for an incompressible, electrically conducting viscous fluid were formulated under axisymmetric assumptions and solved using Laplace transformation combined with the Riemann-sum approximation. The analysis provides detailed insight into the coupled effects of dynamic and static slips, magnetic damping, dimensionless time and dimensionless radial distance on the velocity distribution and wall shear stresses. The results demonstrate that static and dynamic slip conditions relax the no-slip constraint, leading to finite wall velocities, smoother velocity gradients, and significant reductions in boundary shear stresses. The Hartmann number introduces strong Lorentz damping, reducing velocity amplitudes and moderating shear stresses at both walls. Time evolution drives the system toward an asymptotic state, and the steady-state profiles for velocity and skin friction at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>r</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>r</mml:mi> <mml:mo>=</mml:mo> <mml:mi>δ</mml:mi> </mml:math> are obtained at large time. For instance, at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>η</mml:mi> <mml:mspace width="0.25em"/> <mml:mo>=</mml:mo> <mml:mn>0.2</mml:mn> </mml:math> , the peak velocity decreases by ∼24% as the Hartmann number increases from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>M</mml:mi> <mml:mspace width="0.25em"/> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>M</mml:mi> <mml:mspace width="0.25em"/> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:math> , while increasing static slip from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>β</mml:mi> <mml:mspace width="0.25em"/> <mml:mo>=</mml:mo> <mml:mn>0.4</mml:mn> </mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>β</mml:mi> <mml:mspace width="0.25em"/> <mml:mo>=</mml:mo> <mml:mn>0.6</mml:mn> </mml:math> leads to higher wall velocities and reduced shear stresses. The close agreement between Riemann-sum approximation, exact, and steady-state solutions confirms the reliability of the method. These findings demonstrate that the combined influence of magnetic damping and dynamic and static slip effects provides an effective mechanism for controlling velocity distributions and wall stresses in curved geometries, with implications for microfluidic devices, plasma systems, and biofluid transport. The Riemann-sum approximation results are rigorously validated by comparison with exact Laplace-domain solutions and steady-state analytical results, showing excellent agreement throughout the transient and asymptotic regimes.