The Hamilton-crosser thermal conductivity model in the quaternary nanofluid flow with second-order slip condition

Nanofluids, formed by dispersing nanoparticles into conventional base fluids, have gained significant attention as advanced heat transfer media in various sectors, including automotive, aerospace, nuclear reactors, biomedical systems, etc. Their enhanced thermophysical properties and particularly improved thermal conductivity, which make them superior to traditional fluids for high-efficiency thermal management. To accurately estimate the thermal conductivity of nanofluids, several theoretical models have been proposed, among which the Hamilton-Crosser (HC) model is widely adopted. This model extends Maxwell's classical theory by incorporating particle shape factors, providing more accurate predictions for fluids containing non-spherical or clustered nanoparticles. In the field of fluid mechanics, boundary layer flows influenced by magnetohydrodynamics (MHD) and thermal radiation are of increasing relevance in systems subjected to magnetic fields and high heat loads, such as in magnetic cooling, plasma processing and solar energy applications. According to this problem statement, this paper provides a numerical solution for the mathematical modelling of the MHD radiative water-based nanofluid flow. This nanofluid contains a combination of four nanoparticles: alumina, copper, graphene and silicon dioxide, which represents a quaternary hybrid nanofluid system. The mathematical modelling was initiated with the partial differential equations (PDEs) and finalised by ordinary differential equations (ODEs). The transformations from PDEs to ODEs simplify the mathematical model for numerical analysis, which can be done by using similarity transformations. The resulting ODEs are solved using MATLAB's built-in bvp4c solver. Key physical quantities for the profiles (velocity and temperature) and the quantities measured for the flow and thermal characteristics (skin friction coefficient and local Nusselt number) are graphically illustrated and described. The results show that the quaternary case produces the highest heat transfer rate compared to the nanofluid case, which has the lowest number of nanoparticle types (nanofluid case). In addition, the cylindrical shape is proven to have the highest local Nusselt number value for the stretching boundary case, while the spherical shape produces the highest local Nusselt number value for the shrinking boundary. Meanwhile, the magnetic field, thermal radiation, slip condition and projected angle at the sheet influence the distribution of the profile and physical parameters.