The understanding of the mechanisms governing heat transfer in contexts characterized by slip flow related to either a stretching or shrinking sheet holds considerable practical significance across a diverse range of fields, such as materials processing, manufacturing, and thermal management systems. This comprehension facilitates the progress of innovative methodologies and systems to enhance heat transfer efficiency. Consequently, this study emphasizes the stability and multiplicity of solutions relevant to slip flow while also examining the dynamic behaviour and thermal properties of ternary hybrid nanofluids. It is proposed that these ternary hybrid nanofluids are situated on a stretching or contracting sheet. The trihybrid nanofluid has been synthesized through the dispersion of titanium dioxide ( \({\text{TiO}}_2\) ), aluminum oxide ( \({\text{Al}}_2\) O \(_3\) ), and copper oxide (CuO) suspended in water ( \({\text{H}}_2{\text{O}}\) ). The influences of non-linear thermal radiation, alongside velocity and thermal slip conditions, represent critical elements in the novelty of the proposed model. Two distinct sets of initial approximations are consecutively provided to the bvp4c algorithm to attain dual solutions. The stability criterion is determined by the lowest eigenvalue ( \(\gamma _1\) ). Although the flow exhibits diminished stability as the shrinking parameter becomes increasingly hostile, the alternative solution continues to demonstrate instability due to negative eigenvalues. The findings highlight that increasing the volume fraction of the third nanoparticle improves heat transfer efficiency, as demonstrated by a \(2.32\%\) increase in the Nusselt number. Furthermore, the analysis shows that stronger magnetic fields and velocity slip conditions significantly influence flow stability and energy transport. The study’s insights contribute to optimizing thermal management in engineering applications such as heat exchangers, cooling systems, and porous media flows, demonstrating the potential of THNFs for enhancing energy efficiency in industrial processes.
