Erich Miersemann

Microimaging of Transient Concentration Profiles of Reactant and Product Molecules during Catalytic Conversion in Nanoporous Materials

Microimaging by IR microscopy is applied to the recording of the evolution of the concentration profiles of reactant and product molecules during catalytic reaction, notably during the hydrogenation of benzene to cyclohexane by nickel dispersed within a nanoporous glass. Being defined as the ratio between the reaction rate in the presence of and without diffusion limitation, the effectiveness factors of catalytic reactions were previously determined by deliberately varying the extent of transport limitation by changing a suitably chosen system parameter, such as the particle size and by comparison of the respective reaction rates. With the novel options of microimaging, effectiveness factors become accessible in a single measurement by simply monitoring the distribution of the reactant molecules over the catalyst particles.

Stability and continuation of solutions to obstacle problems

Abstract In this paper we will give a summary of some of our results which we have obtained recently. We mainly consider the question whether solutions to variational inequalities with an eigenvalue parameter are stable in the sense defined in Section 1. More precisely, we ask whether a solution to the variational inequality yields a strict local minimum of an associated energy functional defined on a closed convex subset of a real Hilbert space. This nonlinearity of the space of admissible vectors implies a new and interesting stability behavior of the solutions which is not present in the case of equations. Moreover, it is noteworthy that optimal regularity properties of the solutions to the variational inequality are needed for the stability criterion which we will describe in Section 2. Applications to the beam and plate are considered in Sections 4 and 5. In the case of a plate, numerical computations are crucial because it is impossible to find an analytical expression for a branch of solutions to the variational inequality which is not also a solution to the free problem. Closely connected to the question of stability of a given solution to a variational inequality is the question of the continuation of this solution, which we will discuss in Section 3. In Section 6 a survey will be given on the methods used for the computation of stability bounds. This includes in particular a short introduction to continuation algorithms for both equations and variational inequalities. Frequent references will be made to the literature of direct relevance to the material presented. A few additional related research papers or monographs have been included in the bibliography (Courant and Hilbert (1962/1968), Fichera (1972), Funk (1962), Glowinski et al. (1981), Kikuchi and Oden (1988), Landau and Lifschitz (1970), Lions (1971) and Lions and Stampacchia (1967)).

Stability in obstacle problems for the von Karman plate

The buckling beyond the critical load of a plate governed by the von Karman equations is studied. A variational inequality formulation of the problem is derived. The deflection of the plate is subject to an obstacle and the question of the stability of the state with a nontrivial contact set is considered. A stability criterion characterizing the bound through a Rayleigh quotient is proved in the general case. It is specialized to simply connected plates for which also a stress function is introduced. For a square plate numerical continuation along the variational inequality branch yields solutions whose stability is then checked through evaluation of the stability criterion. Stability bounds for both clamped and simply supported plates are obtained.