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Asymptotic techniques are used to investigate two different phenomena, namely, acoustically driven counterflows and the flow field surrounding fire whirls. In Part I of the dissertation, the interaction of non-premixed flamelets with acoustic waves of large characteristic wavelength, central to the development of acoustic instabilities in liquid-propellant rocket engines, is investigated using as model the counterflow diffusion flame subject to harmonic pressure and strain variations, with the presentation given in this work proceeding with increasing levels of complexity, outlined below. In order to relate to typical experimental realizations of counterflow diffusion flames, the presentation begins with the investigation of the high-Reynolds and low-Mach number collision of two chemically frozen gaseous streams of different density. The self-similarity of the stagnation-point region is analyzed, with the strain-rate and stagnation point location, amongst other properties relevant for counterflow-flame studies, given as functions of the macroscopic properties of the experimental setup, including the nozzle-separation to semi-width ratio, for irrotational and rotational flows, with explicit formulas given for the former. A general formulation is then provided for reacting mixing layers in counterflows subject to both time-varying strain and pressure using an inverse-thermal-conductivity-weighted coordinate which is shown to have benefits when compared to the classic Howarth-Dorodnitzyn variable. The formulation is applied to the interaction of acoustic waves with non-premixed flamelets by consideration of small amplitude harmonic oscillations of the pressure or strain-rate, with the amplitude serving as small parameter in the perturbative analysis for model one-step Arrhenius chemistry. First, the limit of infinitely fast reaction for non-unity Lewis numbers is considered. It is shown that differential-diffusion effects promote fluctuations of the flame location and reactant consumption rates. In connection with acoustic instabilities characterized by the Rayleigh index, the analysis predicts acoustic amplification for all frequencies in the pressure response, whereas a critical crossover frequency is identified in the strain response demarcating a transition between amplification and attenuation. Next, finite-rate effects are considered for systems with large activation energies. The results indicate the response for typical propellant combinations leads to acoustic amplification, not attenuation, the amplification being larger at higher strain rates. These results, drawn on the basis of one-step model chemistry, are supplemented with numerical computations of hydrogen-air systems employing realistic chemical-kinetic mechanisms. The Rayleigh index is employed as a vehicle for quantifying inaccuracies of predictions caused by the introduction of reduced chemistry to decrease computation times. The computations indicate that inaccuracies of a systematically reduced 2-step mechanism, derived from a detailed 12-step mechanism for hydrogen-air systems, are small at low strain rates but become appreciable as extinction is approached.Part II of the dissertation describes the steady axisymmetric structure of the cold boundary-layer flow surrounding fire whirls developing over localized fuel sources lying on a horizontal surface. The structure is shown to consist of three separate regions, including an outer inviscid swirling region, a near-wall boundary layer and a near-axis non-slender collision region, each described sequentially. Particular attention is given to the terminal shape of the boundary-layer velocity near the axis, displaying a three-layered structure described by matched asymptotic expansions. The resulting composite expansion, dependent on the level of ambient swirl, is useful in mathematical formulations of localized fire-whirl flows, providing consistent boundary conditions for further numerical investigations.