Finite Darcy–Prandtl Number and Maximum Density Effects on Gill's Stability Problem

Abstract

The simultaneous effect of a time-dependent velocity term in the momentum equation and a maximum density property on the stability of natural convection in a vertical layer of Darcy porous medium is investigated. The density is assumed to vary quadratically with temperature and as a result, the basic velocity distribution becomes asymmetric. The problem has been analyzed separately with (case 1) and without (case 2) time-dependent velocity term. It is established that Gill's proof of linear stability effective for case 2 but found to be ineffective for case 1. Due to the lack of Gill's proof for case1, the stability eigenvalue problem is solved numerically and observed that the instability sets in always via traveling-wave mode when the Darcy–Prandtl number is not larger than 7.08. The neutral stability curves and isolines are presented for different governing parameters. The critical values of Darcy–Rayleigh number corresponding to quadratic density variation with respect to temperature, critical wave number, and the critical wave speed are computed for different values of governing parameters. It is found that the system becomes more stable with increasing Darcy–Rayleigh number corresponding to linear density variation with respect to temperature and the Darcy–Prandtl number.