Nonlinear stability analysis for the rheology of Oldroyd-B nanofluids embedded by Darcy–Brinkman porous media using a two-phase model

An alternative to the Darcy model called the Darcy–Brinkman is employed to investigate a linear and nonlinear analysis of thermal convection in porous media saturated by Oldroyd-B nanofluids. The two-phase model is utilized for a nanofluid which is based on the theory of nanoparticles–fluid relative velocity. The analytical expressions for oscillatory and stationary thermal Rayleigh numbers have been derived using the Galerkin scheme and normal mode technique to study the impacts of viscoelastic and other non-dimensional parameters governing the stability of the system. Nonlinear analysis is made using minimal double Fourier series with a weekly period. Both time-dependent and independent variations of the Nusselt number and Sherwood number for emerging parameters are represented to study heat and mass transport. The novelty of the paper lies in the fact that due to the viscoelasticity of the fluid oscillatory mode of convection came into existence for top-heavy nanoparticle arrangement which was otherwise not possible for nanofluids with Newtonian base fluids. The results of the mathematical model revealed that heat and mass transfer is enhanced with the higher values of viscoelastic parameters, namely stress relaxation and strain retardation numbers. The increasing Darcy number is observed to have substantial control over the stability as it inhibits heat and mass transfer rate. Graphical interpretations reveal that viscoelastic non-Newtonian nanofluids have higher heat and mass transfer rates than Newtonian nanofluids for the system under consideration.