Thermal Stability Analysis of the Darcy-Benard Problem under Mechanical Vibration
Yazdan Pedram Razi1*, K Maliwan2, A Mojtabi3 and MC Charrier Mojtabi3
1Aerospace Department, San Jose State University, CA, USA
2Prince of Songkla University, Thailand
3IMFT, UMR CNRS/INP/UPS N°5502, UFR MIG, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse Cedex, France
*Corresponding Author: Yazdan Pedram Razi, Aerospace Department, San Jose State University, CA, USA.
Published: July 08, 2025
DOI: 10.55162/MCET.08.285
Abstract
This paper focuses on different standpoints regarding the thermal stability of a horizontal porous layer under the action of vertical vibration. Different time scales are distinguished (i.e. vibrational, viscous, conductive and buoyancy). For the case in which the vibrational time-scale is much smaller than other time scales and the amplitude of vibration is smaller than the height of the layer, the so-called time-averaged method is adopted. The linear stability analysis reveals that the response of the system in this case is harmonic and vibration has a stabilizing effect. Weakly non- linear stability analysis reveals that bifurcation is of the supercritical pitchfork type and an expression for the Nusselt number has been obtained. Although the time-averaged method has the advantage of simplifying the governing equations and providing closed form expression for the optimum choice of control parameters, it fails to describe sub-harmonic modes. In the follow up, it is shown that the stability analysis under arbitrary time-scale relations, called the direct method, leads to the study of a second order parametric oscillator (Mathieu equation). The harmonic and sub-harmonic responses are distinguished. We have put forward a set of conditions from which the stability results obtained by the direct method may be compared with those of the time-averaged method. The validity of the time- averaged method for thermos-vibrational problem has been proved for the first time in porous media. The criterion for the onset of convection has been obtained and this result has been generalized. The behavior of sub-harmonic solution is emphasized. It is shown that at very high frequency, the onset of convection corresponding to this mode does not depend on frequency and is independent of gravity. The response of the solution will be in sub-harmonic mode.
.