General Principles

Abstract

In the general principle, we give an overview of the recently used methods and schemes to solve multicomponent and multiscale systems. While multicomponent systems are evolution equations based on each single species, which are coupled with the other species, e.g. with reaction-, diffusion-processes, multiscale systems are evolution equations based on different scales for each species, e.g. macroscopicor microscopic scale. We give the general criteria for practically performing the different splitting and multiscale methods, such that a modification to practical applications of the splitting schemes to a real-life problem can be done. Multicomponent SystemsIn the following, we deal with multicomponent systems. Multicomponent systems concentrate on disparate components in the underlying multiscale models, while they can be coupled by linear or nonlinear functions or differential systems, e.g. reactions or transport phenomena. Often, also different scales are important to resolve to understand the interactions of the different components. Here, we have to apply multicomponent schemes that are also related to disparate spatial and timescales to resolve such complexities, see algorithmic ideas of multicomponent problems in [1].In the following sections, we concentrate on modelling or algorithmical aspects of multicomponent systems for the following applications:• Multicomponent flows, see [2,3].• Multicomponent transport, see [1,4].We simulate the flow and transport systems based on their interactions with the different components. Hence, we allow to study weakly or strongly coupled components in the underlying modelling equation systems. Multicomponent FlowsThe class of multicomponent flows can be defined as a mixture of different chemical species on a molecular level, which are flown with the same velocity and temperature, The modelling of such behaviours is important in engineering, e.g. reactor design (Chemical vapour deposition reactors, see [6]) in chemical engineering. Such processes are very complex, while different physical processes occur, e.g. injection, heating, mixturing, homogeneous and heterogeneous chemistry and further, see [7].In the following, we present some typical problems in multicomponent flow problems:• Ionized Species, e.g. plasma problems, see [4].• Combustion of oil, coal or natural gas, see [8].• Chemical reaction processes in chemical engineering, see [6,9,10].• Atmospheric pollution, see [11].Remark 1.1 In the different applications, the term multiphase flow is often used. Here, we define multiphase flows where the phases are immiscible and not chemically related, see [2,12]. So each phase has a separately defined volume fraction and velocity field. Therefore, also the conservation equations for the flow of each species and their interchange between the phases are different from the multicomponent flow. Here, one is taken into account to define a common pressure field, while each phase is related to the gradient of this field and its volume fraction, see [13]. The applications a...