Mathematical Issues Arising from the Onset of Fluid Instabilities

Instabilities occurring in fluid flows are sometimes characterized as either oscillatory or stationary. This is true in diverse situations with the effects of temperature, rotation, seepage and gravity. Several unifying properties have been identified leading to some simple criteria. The theory of positive operators plays an important role in the analysis. Applications are given for B\'{e}nard convection with heat sources and variable gravity, flows in porous media, Taylor-Couette flow and Langmuir circulations.