Onset of filtration convection in a vibrating medium: The Brinkman model

Abstract

The effect of vibrations on the onset of convection in a horizontal fluid saturated porous layer heated either from below or from above is investigated. Vertical vibrations are considered with arbitrary amplitude and frequency. The Brinkman model of flow through porous media is used to describe the non-Darcian effect and the Oberbeck–Boussinesq approximation is invoked. Both continued fraction and Hill’s infinite determinant methods are used to determine the convective threshold for the stress free and rigid boundaries with the aid of Floquet theory. The marginal curves exhibiting synchronous and subharmonic resonant regions of dynamic instability and their critical boundaries are determined. The results obtained for Da=10−1 corresponding to the Brinkman regime is compared with Da=10−4 corresponding to the Darcy regime. It is demonstrated that vibrations can produce a stabilizing or a destabilizing effect depending on their amplitude and frequency for a porous layer heated from below. However in the case of a porous layer heated from above vibrations of increasing amplitude always result in a destabilizing effect irrespective their frequency. It is also shown that an increase in Da/φ, the ratio of Darcy number to porosity restricts competition between the synchronous and subharmonic modes to a lower range of vibrational frequencies. The present study reproduces already established results in the literature as particular cases.

Onset of Vibrational Convection in a Binary Fluid Saturated Non-Darcy Porous Layer Heated from Above

“…We now explain this transition in terms of marginal curves. The solution through the Floquet analysis revealed marginal curves in the form of a group of loop shaped branches enclosing the region of instability [9]. The bottommost loop was S for very small ω and the minimum of this loop determined the onset condition.…”

“…Also we fixed Pr = 1, κ = 1 and c = Da = 0.1 corresponding to Brinkman porous medium throughout the study. The expression (16) contains, in particular, the corresponding result of Strong [20] for Da = 0, binary fluid case for the Darcy model and Saravanan and Sivakumar [9] for Ra S = 0, pure fluid case for the Brinkman model.…”

“…Recently, Strong [8] reported the results of this problem for arbitrary range of modulation parameters by the use of continued fraction method. There are also few recent works available in the literature dealing with thermovia e-mail: thirumalaisiva@gmail.com brational convection in a more general Brinkman porous medium (see [9][10][11][12]). In our recent study [9], the stability of convection in a Brinkman porous layer in the presence of vibrations of arbitrary amplitude and frequency was considered.…”

“…There are also few recent works available in the literature dealing with thermovia e-mail: thirumalaisiva@gmail.com brational convection in a more general Brinkman porous medium (see [9][10][11][12]). In our recent study [9], the stability of convection in a Brinkman porous layer in the presence of vibrations of arbitrary amplitude and frequency was considered. It was demonstrated that the vibration amplitude favors as well as suppresses the setting up of convection depending on the vibration frequency for a layer heated from below whereas it always favors convection irrespective of the vibration frequency for a layer heated from above.…”

Effect of double thermal modulation on heat transfer in a square cavity heated from bellow

“…However, for high frequencies, the flow and interface deformation converges toward their free state value. Saravanan and Sivakumar ( 2010) recently studied the effect of vibration with arbitrary amplitude and frequency in a porous horizontal saturated fluid heated from below. They demonstrated that these vibrations can produce stabilization or destabilization in function of the chosen amplitude and frequency.…”