The objective of the study is to investigate and solve the model equations for an unsteady magnetohydrodynamic Carreau fluid flow using the overlapping grid spectral simple iteration method (O-SSIM). The study centres on the behavior of Carreau fluid flow over surfaces subjected to certain external forces and environmental factors. Numerical solutions are conducted to analyze the effects of inclined stretching/shrinking surfaces, and an inclined magnetic field on flow characteristics. The O-SSIM algorithm was implemented in MATLAB and tested for its convergence and accuracy. This research highlights the importance of the O-SSIM as a robust and accurate numerical technique for solving nonlinear differential equations. The R programming language was utilized to generate insightful plots and visualizations of the results to identify patterns and relationships between the local Nusselt number and skin friction coefficient. The results indicate several important trends and patterns, including a positive correlation between the small surface angle and magnetic field with the skin friction coefficient. Additionally, there is a positive relationship between increasing the thermal buoyancy parameter and a negative relationship between a higher Eckert number and the local Nusselt number. The paper provides valuable insights into the flow behavior of Carreau fluid and highlights the impact of an inclined magnetic field on the thermal and mechanical properties of the fluid.

The impacts of vertical throughflow, rotation, cross-diffusion, and vertical heterogeneous permeability on the double-diffusive convection in a finite rotating vertical porous cylinder have been studied. The fluid in the cylinder is warmed and salted from beneath, and its top and lower walls are taken to be isothermal, isosolutal and permeable. In the model formulation, the Brinkman model was adopted, coupled with the Boussinesq approximation. The normal mode technique is used to perform linear stability analysis and single term Galerkin technique is employed to solve the eigenvalue problem. Further, the influence of vertical heterogeneity, vertical throughflow, thermal and solute Rayleigh, Taylor, and the Soret and Dufour numbers on the fluid system instability has been investigated. We found, among other results, that vertical heterogeneity may either stabilize or destabilize the fluid system. The Dufour number delays both the stationary and oscillatory convection onsets. The positive Soret number is found to have a stabilizing effect on the stationary convection case, with a destabilizing effect on the oscillatory convection case.