On the Onset of Convection in a Hele-Shaw Geometry under the Influence of Mechanical Vibrations

The onset of natural convection in a vertical Hele-Shaw configuration under the action of mechanical vibration is studied.Vibration is considered to be a harmonic one and its direction is considered parallel to the temperature gradient.Linear stability analysis for small Pr number leads us to the Mathieu equation.The so-called time averaged method is applied to this equation which gives us a closed form relation for the critical Ra T number for the harmonic (synchronous) solution.The results are in a perfect agreement with published numerical results.