Linear and nonlinear stability analysis of double diffusive convection in a fluid layer

Abstract

Instability in Rayleigh Bénard problem occurs because of density difference that is the cause of temperature difference across the fluid; however, if the salt is included in the fluid, then instability occurs because of density difference that is the cause of temperature as well as concentration differences and phenomena of convection arises, in this case, is called double-diffusive convection. In this thesis, an effort is made to investigate the double-diffusive convection problem in a fluid layer. We consider Navier - Stokes equation for governing the fluid motion. The law of conservation of energy regulates the temperature distribution within the system. Since the effect of mass transfer is also taken into consideration thus, the equation for the concentration of solute will govern the movement of salt concentration. The linear analysis is utilized with the help of the normal mode technique. Stern, (1960) developed the linear theory for a system having linear opposing gradients between the two horizontal boundaries. Later, Veronis, (1965), (1968) worked on the double-diffusive convection problem and attained fascinating results that directly apply to oceanography. The linear theory provides a bound above which the solution remains unstable. However, the nonlinear stability analysis provides a bound below which the solution remains overall stable [see, Straughan, (1993)]. The study on double-diffusive convection has gained the attention of engineers and scientists due to its global appearance in engineering and sciences. The gravitational instability in the ocean occurs because of evaporation, and the unstable region in stars is due to the higher inner temperature gradient of stars. Moreover, the study is essential in geological reservoirs, soil salinization, crude oil extraction, transfer of energy in a wet cooling tower, and electrical power from the moving conducting fluid. newline