Computational study of magneto-hydrodynamic hybrid nanofluid flow and heat transfer over a stretchable surface with temperature-dependent thermal conductivity under motile microbes and slip effect

Variable thermal conductivity plays important role in precisely modeling of hybrid nanofluid (HNF) flow and heat transfer. In practical applications, thermal conductivity frequently varies with temperature, nanoparticle concentration, and fluid composition, affecting the rate of energy transfer within the system. The present investigation offers an inclusive investigation of heat transfer in magnetized flow of a HNF consisting of zinc (Zn) and silicon dioxide (SiO2) nanoparticles dispersed in water (H2O), towards a rotating stretchable surface. The management of thermal energy in rotating, high-speed environments can be significantly enhanced through the use of magnetized nanofluids with optimized nanoparticle shapes. This study incorporates the effects velocity slip, and convective boundary conditions. The principal objective is to explore the transport phenomena of heat and mass transfer, as well as the bioconvective behavior induced by motile microorganisms within the hybrid nanofluid. By applying similarity transformation, the governing flow model of PDE’s for momentum, energy, concentration, and motile microorganism distribution are condensed to a set of coupled nonlinear ODE’s. These equations are numerically solved with MATLAB software, which employs a vigorous shooting method. A thoroughly parametric investigation is performed to assess the influence of several physical parameters, including the magnetic field strength ( $$M$$ ), rotational parameter ( $$\lambda $$ ), velocity slip coefficient ( $$\beta $$ ), thermal radiation parameter ( $$Rd$$ ), variable thermal conductivity ( $$\epsilon$$ ), and the volume fractions of the nanoparticles ( $${\phi}_{1}={\phi}_{2}$$ ), along with the nanoparticle shape factor. Furthermore, the effects of the Lewis number ( $$Le$$ ), Peclet number ( $$Pe$$ ) and bioconvective Lewis number ( $$Lb$$ ) are examined with respect to species concentration and microorganism. The numerical fallouts are presented both graphically and in tabular format, demonstrating that enhancements in magnetic field strength and radiation parameter notably affect the thermal and flow characteristics. The legitimacy of the suggested model is confirmed through excellent agreement with benchmark outcomes from existing literature across various Prandtl number ( $$Pr$$ ) values, thereby signifying the reliability and practical relevance of the current approach in advanced hybrid nanofluid-based thermal management systems.