A post-transitional, spatially-developing, exothermically reacting shear-layer is simulated using the transport-element method. The scheme, which solves the unaveraged, time-dependent coupled scalar-transport and Navier-Stokes equations, in their gradient and vorticity transport forms, respectively, is Lagrangian, grid-free and adaptive. The Shvab- Zeldovich formulation is exploited to provide solutions for both moderately fast and infinitely fast reactions. In the case of a finite reaction rate, Arrhenius kinetics are assumed. Numerical results are used to evaluate the effectiveness- with respect to predicting the dynamic effects of combustion exothermicity on the flow- of the infinite reaction rate model in approximating finite reaction rate combustion. Results indicate that as long as the reaction is last compared to the flow and abrupt non-equilibrium transient phenomena, such as quenching, are avoided, the computationally more efficient infinite reaction rate model offers a reasonable substitute of its finite reaction rate counterpart. For both models the externally-forced flow growth rate is reduced in the presence of combustion exothermicity through the alteration of the mechanism of interaction of the vortical structures from paring to tearing and the alignment of the eddy larger dimension with the streamwise direction. By comparing cases with different enthalpies of reaction we show that the performance of the infinite reaction rate model improves as the Damkohler number is raised.