Multiplicity and stability of solutions in transport phenomena of non-Newtonian Carreau nanofluid with nonlinear Rosseland thermal approximations and neural prediction

Predicting multiple solutions and conducting stability analyses for heat transfer in non-Newtonian nanofluids are of significant interest in nonlinear mathematics and engineering applications. The impact of non-linear boundaries and Rosseland thermal radiation (nonlinear) approximations on the Carreau nanofluid flow induced by stretching and shrinking permeable sheets in porous media is investigated. To capture the occurrence of multiple solutions not only in the case of a shrinking sheet but also in the case of a stretching sheet, an appropriate scaling is applied using the stretching/shrinking parameter (χ) followed by the implementation of Lie group theory. In the case of a shrinking surface, multiple solutions are identified for the Weissenberg number (0<We<Wec=1.0149), the non-Newtonian parameter (1<n<nc=2.0384), the power-law index (−0.5<m<mc=1.2085), the shrinking parameter (χc=−1.0032<χ<0) and the suction parameter (0>ζ>ζc=2.4943), while keeping other parameters fixed. Subsequently, perturbation theory is employed to formulate an eigenvalue problem to assess the stability of the solutions. To analyse the relationship between the three Nusselt numbers and various controlling parameters, artificial neural network (ANN) training algorithms are utilized and compared, incorporating five input variables. The optimal number of neurons in a single hidden layer is determined and compared using metrics R2 and RMSE for each algorithm. Bayesian regularization and Levenberg–Marquardt optimization models consistently provide better predictions in all cases. The study aims to apply the findings in dynamic systems involving heat transformation, such as manufacturing, solar distillation, CPU cooling, and other applications.