Stability of double-diffusive natural convection in a vertical fluid layer

Abstract

The stability of basic buoyant flow in a vertical fluid layer induced by temperature and solute concentration differences between the vertical boundaries is investigated. The linear dynamics of the perturbed flow is formulated as an eigenvalue problem and solved numerically by employing the Chebyshev collocation method. The validity of Squire's theorem is proved, and therefore, two-dimensional motions are considered. The neutral stability curves defining the threshold of linear instability and the critical values of the thermal Grashof number and wave number at the onset of instability are determined for various values of the Prandtl number Pr, the solute Grashof number GS, and the Lewis number Le. The magnitude of the Prandtl number at which the transition from stationary to travelling-wave mode occurs can be either increased or decreased by tuning the values of GS and Le. For certain combinations of the parameters, there exist one or two closed disconnected travelling-wave neutral curves emphasizing the necessity of multiple thermal Grashof numbers to embark upon the stability of fluid flow, a result of contrast to that of the single-diffusive fluid layer. The mechanism of modal instability is deciphered by using the method of energy budget and four different modes of instability are identified, one of which is new and due entirely to the presence of solutal buoyancy.