The present dissertation tackles two different problems in fluid mechanics. In the first problem, we study the stability of free or natural convection of a compressible fluid over a heated plate underlying a semi-infinite vertical slot and underlying a closed vessel. For both settings, we show linear and non-linear stability of the reference solution by constructing respective Lyapunov functions and using LaSalle's Invariance Principle. The construction of the Lyapunov function is motivated by the assumption that the fluid particles evolve according to a Markov chain. In the second problem, we study the flow dynamics of a viscous stably stratified fluid in a channel with time-periodic temperature variations applied at the sidewalls. Seeking solutions in the form of simple harmonic oscillations, we obtain analytical temperature and velocity profiles for the one-dimensional time-dependent flow. We then use these solutions to study the possibility of resonance of the fluid flow with the periodic oscillations of the externally supplied temperatures at the walls. Results indicate the existence of resonance depicted as prominent peaks of physical quantities at certain frequencies of the external temperature oscillations.