(583d) Observations and Analysis of Index Infinity Dae Systems Describing Reactors and Reacting Flows

Mathematical models that describe the steady-state behavior of chemical reactors and reacting flows in which volumetric and/or wall reactions occur can be generally classified as elliptic or parabolic. The behavior of elliptic models is well studied in the literature and they are known to display a variety of solutions and bifurcations (e.g. multiple homogeneous solutions and various types of patterned/asymmetric solutions). In the literature, the so called “boundary layer models (in which axial diffusion/conduction is neglected) used to describe these systems are treated as parabolic equations and having a unique solution. Such a solution is usually obtained by discretization of the transverse coordinate and integration in the axial direction. However, it was shown in our earlier work that these are index infinity differential-algebraic equation (DAE) systems, and could have an infinite number of solutions. Further, the integration of such models along the flow direction (forward) and against the flow direction (backward) could lead to different solutions. In this work, we present further observations and analytical as well as numerical methods for determining the various solutions of such index infinity DAE systems.  We consider three specific systems to illustrate the main ideas and techniques : (i) a DAE system consisting of one differential equation (in axial coordinate) and one algebraic equation, having infinite number of solutions (such a system is obtained by transverse averaging of the partial differential equation models) (ii) a parabolic equation in two variables (axial and transverse) that has nonlinearity in the boundary conditions (iii) a parabolic equation in two variables with linear boundary conditions but a nonlinear source term. For these cases, we present bifurcation analysis as well as numerical methods for determining all the solutions with particular emphasis on how to eliminate the Gibbs’ phenomenon (that arises due to discontinuous solutions) and numerical diffusion (that can lead to spurious solutions).