| In this thesis we analyze a model for incompressible chemically reacting flows where reactants enter the domain, react and then leave the domain. In the interior of the domain we have the Navier-Stokes equations for the fluid flow coupled with reaction diffusion equations for the chemistry and temperature. On the boundary we use mixed Neumann and inhomogeneous Robin boundary conditions for the chemistry and temperature equations which fix the amount of chemicals and heat flowing into the system.;For three dimensional domains we show the existence of weak solutions which are physically reasonable, i.e., which satisfy certain maximum principle properties. These solutions are sufficient to provide the basis for a dynamical systems study of these reacting flows. We do not show that the solutions are unique, however we are able to show the existence of global attractors for the system.;We then consider the existence and properties of strong solutions for the model. We show results which exactly parallel those of the Navier-Stokes equations, i.e., in two dimensions strong solutions exist for all time, and in three dimensions we show existence only for small times. In two dimensions, we also show the existence of global attractors which are compact in $Lsp2(Omega)$. In this analysis we present our arguments so that they apply both to the mixed Robin-Neumann boundary conditions and also other boundary conditions appearing in the literature.;Finally, we consider the relationship between the chemically reacting flow model and continuous flow stirred tank reactors (CSTR). We show that in the limit as the chemical and thermal diffusivities go to infinity, the solutions of the reacting flow PDE approach the solutions to the CSTR ODE. We further show that the global attractors for the reacting flow come arbitrarily close to the CSTR global attractor. In this analysis the mixed Robin-Neumann boundary conditions play an important role, since they mimic the CSTR inflow and outflow terms. The key in our analysis is an examination of how the Laplacian with Robin-Neumann boundary conditions converges to the Laplacian with Neumann boundary conditions as the diffusivity goes to infinity. |
