This struggle is part of a responsible method for the development of nanoscience and nanotechnology inspect closely of nanofluid. The conformist melting phenomena for steady Falkner–Skan flow of Cross nanofluid is considered. The Buongiorno model is used to discuss the thermal efficacies of the fluid flows in the presence of nanoparticles. MATLAB’s scheme of bvp4c is adopted to solve these non-linear ODEs and graphical results are presented in the form of graphs and tables. The main findings of the study are velocity boost up for melting heat and velocity ratio parameter. Concentration goes down for the Brownian motion of molecules and arises for thermophoresis diffusion.

Mathematically, complex chemical reactions can be simplified by “model reduction,” that is, the rigorous way of approximating and representing a complex model in simplified form. Furthermore, to reduce the dimension of the reaction mechanism there are different available model reduction techniques (MRT). Two MRT Spectral Quasi Equilibrium Manifold (SQEM) and Intrinsic Low Dimensional Manifold (ILDM) are applied here. Both techniques are good approximation techniques but not completely analytical. While for a case study, the complex behavior of the two-route reaction mechanism allied graphically. The characteristic properties of nodes and trees of reaction mechanism are addressed with the graph theory. The invariant solution curves and their comparison for two-route reaction mechanism have never been reported so far. Therefore, the aim of this article is to study such facts. The possible approaches to their solution and key problems are based on the Horiuti’s rules and their slow invariants. Both model-reduction techniques are applied to solve the high dimensional complex problem and comparison between invariant curves of both routes are presented graphically. The efficiency of ILDM with respect to time is compared with the SQEM graphically and in tabulated form by using MATLAB.