Dual Solutions of Magnetohydrodynamics Al2O3+Cu Hybrid Nanofluid Over a Vertical Exponentially Shrinking Sheet by Presences of Joule Heating and Thermal Slip Condition

Hybrid nanofluid has an extensive range of real-world applications. is a new and advanced modification extensively used to increase thermal efficiency in fluid flow systems. The main objective this research study magnetohydrodynamics hybrid numerically two dimensional over vertical exponentially shrinking sheet, considering the effects Joule heating slip condition. Furthermore, using Tiwari-Das model, influence suction parameter on variations reduced skin friction heat transfer also explored. Al2O3+Cu/water nanofluid, which water base fluid, types solid nanoparticles, namely Alumina (Al2O3) copper (Cu), are combined together. governing partial differential (PDEs) equations transformed into ordinary (ODEs) exponential similarity variables. resulting solved three-stage Labatto III-A technique "MATLAB software's" bvp4c solver. nanofluids have greater than fluid. Dual solutions obtained specified ranges suitable parameters. temperature profile rises both as Eckert value enhances. Besides, In first second solutions, boundary layer thickness decreased gradually increased. Finally, conclusions presented that solution duality exists when , while no possible .