Stability and heat transfer of mhd eyring powell tri hybrid nanofluid flow over a permeable shrinking cylinder.

Boundary layer flow over a moving and curved surface is very important in modern industrial processes like steel extrusion, polymer lamination, and advanced cooling technologies, where heat moves affects how efficient and high-quality the products are. This study investigates the dual solutions and temporal stability of Eyring-Powell tri-hybrid nanofluid flow and heat transfer over a permeable shrinking cylinder under an applied magnetic field and Joule heating. The tri-hybrid nanofluid consists of copper, aluminum oxide, and titanium dioxide nanoparticles suspended in water. The governing continuity, momentum, and energy equations are formulated under the boundary-layer approximation and transformed into a nonlinear system of ordinary differential equations using similarity transformations. The resulting boundary-value problem is solved numerically using MATLAB's bvp4c solver, while the temporal stability of the dual solutions is determined through an eigenvalue analysis. The numerical formulation is validated against published studies, with good agreement obtained for the limiting cases. The solution exhibits duality over a finite range of shrinking parameters, with the solutions merging at the critical value [Formula: see text]. The upper branch possesses positive smallest eigenvalues and is therefore temporally stable, whereas the lower branch is unstable. Increasing the Eckert number reduces the heat-transfer rate, while increasing the curvature parameter slightly extends the range of dual solutions. Suction enhances flow stability and heat transfer. The results demonstrate that the combined effects of non-Newtonian rheology, tri-hybrid nanoparticles, magnetic forces, Joule heating, and cylindrical curvature provide useful control of boundary-layer transport and stability in thermal-fluid systems.