Research Projects

In this project, we consider the flow of a fluid with 2 solutes present in it. A linear stability analysis along with a nonlinear analysis using the energy method will be conducted. We shall deal with a nonlinear stability analysis as it varies significantly from the classical linear stability analysis. Linear stability analysis only tells us about linear instability but nothing about the stability. It is also well-known that there are many equations where the nonlinear solutions become unstable much before the linear solutions predict the instability of it. Hence it is important that we also conduct a nonlinear analysis of the flow. We focus on three types of porous medium flows: anisotropic, bidisperse and local thermal nonequilibrium. All the flows will be of the multicomponent diffusive convection type where the flows are in a horizontal layer heated from below. We model each flow considering the Darcy – Brinkman equation coupled with the temperature equation and two solute concentration equations considering the Soret and Dufour effects. We consider the no-slip condition for the velocity with fixed temperature and concentration values at the boundary. With these considerations, we obtain three problems, which need to be solved. The governing equations will be nondimensionalized along with the respective boundary conditions. A linear stability analysis will be conducted using the normal modes. The obtained set of linearized equations will be solved using the Galerkin method using a trigonometric approximation as the trial function. We thus obtain the dispersion relation and use it to study the interplay between the important flow parameters such as temperature and concentration Rayleigh numbers, wave number, Prandtl number, Lewis number, Soret parameter, Dufour parameters. We will also be able to identify the critical Rayleigh numbers and critical wave number. These critical numbers tell us the point when instability sets in the flow. An analysis of the stationary and oscillatory convections will also be done. The nondimensionalized equations will then be used for the nonlinear analysis using the energy method. In this part, we find out if the energy of the flow dissipates or grows after a large amount of time. The Galerkin method will once again be used to obtain the nonlinear energy bound of the flow. The results and output of the project is expected to give a better insight into the flow stability of multicomponent diffusive convection. Obtaining the critical Rayleigh numbers will tell us when instability sets in the flow. Also, the energy method tells us directly if the energy in the flow dissipates or grows with time. To fully understand a variety of industrial or natural processes, this analysis is fundamental. The design of aircraft or pipes for the transportation of gas and oil are examples of engineering applications, in addition to atmospheric circulation and oceanic currents.