Numerical study and fast method for unsteady magnetohydrodynamic flow, heat and mass transfer of the generalized Maxwell fluid in a vertical rectangular duct

Coupled magnetohydrodynamic (MHD) heat and mass transfer in electrically conducting viscoelastic fluids is relevant to magnetically controlled transport of polymeric liquids and electrolytes. Previous fractional MHD studies, however, have generally not combined two-dimensional duct confinement with independent memory descriptions for stress, heat flux, and mass flux. Here, we develop a transient generalized Maxwell model for a rectangular duct that assigns independent Caputo orders to the constitutive, heat-flux, and mass-flux relations. The model incorporates thermal and solutal buoyancy, Soret and Dufour cross-diffusion, Joule heating, viscous dissipation, and a first-order homogeneous chemical reaction. The coupled system is discretized using the L1 formula in time and Legendre–Gauss–Lobatto spectral collocation in space, with sum-of-exponentials approximations to accelerate the evaluation of the three fractional-history terms. Manufactured-solution tests show first-order temporal convergence and spectral spatial accuracy. At τ = 1 / 1280 , Richardson-extrapolated L ∞ errors are at most 2.6 × 1 0 − 5 across all three fields. For fractional-history evaluation, the acceleration reduces computational complexity from O ( N T 2 ) to O ( N T N e ) and the corresponding history storage from O ( N T ) to O ( N e ) . The fast and direct methods produce solutions that differ by less than 1 0 − 10 in the tested cases. The simulations show that stress memory redistributes velocity, heat-flux memory delays thermal penetration, and mass-flux memory changes the magnitude and position of concentration overshoots. Raising H a from 1 to 7 reduces the peak velocity at z = 1 by 46.5%, whereas raising τ q from 0 to 5 lowers the temperature at ( y , z ) = ( 0.5,1.8 ) by 75.6%. Raising S r from 1.66 to 6.89 increases the maximum concentration at z = 1.8 by 63.8%. The framework supports efficient analysis of coupled fractional MHD transport in confined ducts.