Quadratic convection and quadratic solutal convection on the nanofluid flow over a vertical frustum of a cone under Soret impact: An implementation of a spectral computational approach

Abstract The implication of buoyancy reveals complex, non-linear behavior in numerous mass and thermal transfer processes. Quadratic thermal and solutal buoyancy effects become significant in high-temperature and high-concentration systems where density variations are strongly non-linear. Additionally, cross thermo-solutal buoyancy interactions play a crucial role in alloy solidification, geophysical flows, and oceanography, where the stability and density of the flow are influenced by the combined action of thermal and solutal gradients. This study aims to investigate quadratic thermal and solutal buoyancy effects in nanofluid convection through a vertical frustum of a cone, emphasizing the Soret phenomenon. The partial differential equations derived from the mathematical formulation using non-similarity variables were solved numerically. The governing non-similar partial differential equations are solved numerically using the bivariate pseudo-spectral local linearization method, which combines quasi-linearization and spectral collocation for enhanced accuracy. The results are compared to asymptotic series solutions in order to evaluate the reliability of the proposed methodology. Graphical results illustrate how quadratic and mixed buoyancy parameters influence momentum, heat, and mass transfer characteristics. The findings provide new insights into the physics of non-linear buoyancy-driven nanofluid convection in conical geometries relevant to thermal, geophysical, and chemical systems.