MHD flow and heat transfer of nanotriple (Cu–Al[formula omitted]O[formula omitted]–Ag): Exact solutions

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This work presents an in-depth analytical study of the flow and heat transfer characteristics of a nanotriple fluid system—comprising copper, alumina, and silver nanoparticles—over a permeable, elastic, and deformable surface, subject to magnetohydrodynamics (MHD) and velocity slip conditions. Unlike many emerging numerical treatments of water-based nanotriple fluids, the primary objective is to derive exact, closed-form solutions, providing a substantial contribution to the analytical understanding of such complex systems. A unique aspect of this investigation is the identification of multiple algebraic-type solutions for the stretching/shrinking sheet problem, yielding dual solutions under injection and a single solution under suction conditions. In addition, critical numbers are identified as thresholds delineating the boundaries for the existence or absence of solutions. It is found that the number of solutions increases as the magnetic force strength decreases. Dual solutions are observed for both skin friction and thermal gradient in the exponential and algebraic cases. These analytical findings are further reinforced by extensive numerical computations, which offer robust validation of the exact solutions derived. Additionally, stability analysis is carried out in order to determine the stability of solutions, where the first branch demonstrates stability, and the second branch is unstable, highlighting the distinct behaviors within the solution branches.