This study explores thermal criticality and dissipation involving a two-step reaction in a hyperbolic tangential fluid flow and quadratic Boussinesq approximation to model the complex internal heat transfer mechanisms during combustion. Subject to suitable convective boundary conditions, the transformed energy and momentum equations are numerically solved using Galerkin approximation integration coupled with a weighted residual scheme. The outcomes are disseminated using a variety of graphs to illustrate for parametric sensitivities of the thermal and velocity profiles. Based on the results, it is discovered that increases in the Frank-Kamenetskii parameter, Brinkman number, Weissenberg number, activation energy, activation ratio term, and second step term all aid in the complete combustion of hydrocarbons. Monitoring all terms that stimulate internal heat generation is essential to avoid system blow-ups.