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To provide a deeper insight of the transport phenomena inherent to the manufacturing of magnetic nano-polymer materials, in the present work a mathematical model is developed for time-dependent hydromagnetic rheological nano-polymer boundary layer flow and heat transfer over a stretching sheet in the presence of a transverse static magnetic field. Joule heating (Ohmic dissipation) and viscous heating effects are included since these phenomena arise frequently in magnetic materials processing. Stokes’ couple stress model is deployed to simulate non-Newtonian microstructural characteristics. The Tiwari–Das nanoscale model is adopted which permits different nanoparticles to be simulated (in this article, both copper–water and aluminium oxide–water nanofluids are considered). Similarity transformations are utilized to convert the governing partial differential conservation equations into a system of coupled, non-linear ordinary differential equations with appropriate wall and free stream boundary conditions. The shooting technique is used to solve the reduced non-linear coupled ordinary differential boundary value problem via MATLAB symbolic software. Validation with published results from the literature is included for the special cases of non-dissipative and Newtonian nanofluid flows. Fluid velocity and temperature profiles for both copper and aluminium oxide (Al2O3) nanofluids are observed to be enhanced with greater non-Newtonian couple stress parameter and magnetic parameter, whereas the opposite trend is computed with greater values of unsteadiness parameter. The boundary layer flow is accelerated with increasing buoyancy parameter, elastic sheet stretching parameter and convection parameter. Temperatures are generally increased with greater couple stress rheological parameter and are consistently higher for the aluminium oxide nanoparticle case. Temperatures are also boosted with magnetic parameter and exhibit an overshoot near the wall when magnetic parameter exceeds unity (magnetic force exceeds viscous force). A decrease in temperatures is induced with increasing sheet stretching parameter. Increasing Eckert number elevates temperatures considerably. With greater nanoparticle volume fraction, both skin friction and Nusselt number are elevated, and copper nanoparticles achieve higher magnitudes than aluminium oxide.

In the present study, the transient, free convective, boundary layer flow of a couple stress fluid flowing over a vertical cylinder is investigated, and the heat and mass functions for the final steady-state of the present flow are developed. The solution of the time dependent nonlinear and coupled governing equations is obtained with the aid of an unconditionally stable Crank–Nicolson type of numerical scheme. Numerical results for the time histories of the skin-friction coefficient, Nusselt number, and Sherwood number as well as the steady-state velocity, temperature, and concentration are presented graphically and discussed. Also, it is observed that time required for the flow variables to reach the steady-state increases with the increasing values of Schmidt and Prandtl numbers, while the opposite trend is observed with respect to the buoyancy ratio parameter. To analyze the flow variables in the steady-state, the heatlines and masslines are used in addition to streamlines, isotherms, and isoconcentration lines. When the heat and mass functions are properly made dimensionless, its dimensionless values are related to the local and overall Nusselt and Sherwood numbers. Boundary layer flow visualization indicates that the heatlines and masslines are dense in the vicinity of the hot wall, especially near the leading edge.

Motivated by magnetic polymer manufacturing applications, the present research article examines theoretically the hydromagnetic boundary layer flow of an electrically conducting non-Newtonian couple stress fluid due to a transient shrinking (contracting) porous sheet. The conservation partial differential equations for mass and momentum are rendered into a fifthorder nonlinear ordinary differential equation via similarity transformations with associated boundary conditions. A semi-analytical/numerical scheme employing Lagrangian multipliers and known as the variational iteration method (VIM) is implemented to solve the ordinary differential boundary value problem. Validation of the solutions is conducted by benchmarking against earlier Newtonian studies and very good agreement is achieved. A detailed assessment of the impact of couple stress (rheological), unsteadiness, magnetic body force parameter, and wall transpiration (suction/injection) parameter on flow characteristics is conducted with the aid of graphs. A significant deceleration in the flow is computed with increasing injection (acceleration is caused with greater suction) and acceleration is induced with higher unsteadiness parameter values. Increasing magnetic field (higher magnetic number) generates flow acceleration, rather