A powerful approach is developed to predict and optimize flow pattern and to improve heat and mass transfer in vortex burners. New analytical solutions of the Navier-Stokes, heat, and diffusion equations are obtained, and the technique of matching asymptotic expansions is developed. This allows us to describe complex swirling flows with recirculation zones and three-dimensional fields of temperature and concentration. This approach helps in deducing the appropriate flow pattern, shape and position of a flame front, heat transfer, and geometry of the vortex burner. Optimal parameters are found for flame stabilization and flame-surface expansion, prolonging the residence time of the reactants favorable for complete combustion. This approach greatly facilitates finding the optimum because it is much less laborious than computational fluid dynamics methods and allows a wider parametric search.