Couple stresses effect on instability and nonlinear stability in a double diffusive convection

In this study, we have addressed the problem of double-diffusive convection in a reacting fluid with the effect of couple stresses. In this system, there are two competing effects which are the temperature gradation that leads to instability and a salt gradient which increases the stability of the system. The density is assumed to have quadratic dependence on the temperature and a linear dependence on the concentration. Linear instability and nonlinear stability analyses were performed. The standard energy method does not give an unconditional stability so a weighted energy analysis is used to achieve global results. Moreover, in addition to the weighted energy analysis, a global nonlinear stability analysis (unconditional) was proposed. The eigenvalue systems, which result from the linear and nonlinear theories, have been solved using the Chebyshev collocation method. Then, the accuracy of this method has been tested using the analytical solution of eigenvalue system for linear instability theory. The results show that the Chebyshev collocation method was very accurate even with high order derivatives which are produced by the couple stresses term. Finally, numerical results for the linear instability, weighted energy and global nonlinear thresholds were computed and discussed in detail.