# Heterogeneous Models

### Heterogeneous Models

The research unit features three different physical scenarios of heterogeneous physical models which share many modeling aspects and numerical requirements and serve as role models for a wider range of applications. We will briefly introduce these scenarios below.

##### Magnetized Plasma

This scenario is part of the wider field of models of magnetohydrodynamics Goedbloed, Keppens, Poedts (2019), in which an electrically conducting fluid interacts with magnetic fields. It is one of the most fundamental examples of a coupling between fluid flow and electrodynamics. The fluid experiences the electromagnetic fields, while the motion of the fluid influences the evolution of the fields. Typically, the fluid follows the equations of continuum mechanics, which include a mass-, momentum- and energy-balance, and possibly viscous and heat-conducting effects. The electrodynamic fields follow from the Maxwell equations. Between both these systems momentum- and energy-exchange takes place, such that conservation of these quantities only makes sense for the system as a whole, see Project B1, where in addition the evolution of entropy is considered.

The situation gets more involved through the fact that both the fluid and electrodynamics can be modeled by different descriptions. The charged particles may be considered as single quasi-neutral fluid or separated into independent species with different charges, see Project A2. The Maxwell equations could be used in eddy current formulation or full form including all electromagnetic waves. In certain regimes, the fluid particles must be described by kinetic theory using, for instance, the Vlasov equation Nicholson (1983), see Project B2, or by a moment hierarchy Zhdanov (2002), see Project B4. In all these formulations, conservation properties need to be clarified.

##### Complex Fluids

This scenario shows in its coupling patterns a striking similarity to magnetized plasmas. Typically, complex fluids consider a suspension of mesoscopic particles, like macromolecules or nano-particles, in a liquid. Here, we will consider a model of particles with rod-like microscopic structure that is described in a configuration space of orientation or angle Doi, Edwards (1986). The particles move with the liquid and change their orientation through the flow. In return, the particles impose forces onto the liquid which are modeled as internal friction of the fluid. This process is described by a kinetic equation for the distribution of orientation coupled to the liquid velocity. Again, there is momentum and energy exchange and the conservation properties must be formulated for the entire system.

Compared with the Vlasov-Maxwell system, this setting is slightly simpler (Vlasov’s velocity space is all $$R^3$$, while orientation is $$S^2$$) and will also serve as a test compound for numerical strategies. Additionally, the kinetic description of the particles can be replaced by a moment hierarchy, see Project B3. Different flow regimes may require locally different moment descriptions. This extends the discussion to structural properties like conservation or entropy laws.

##### Electrochemical Systems

This scenario requires more development in terms of modeling before numerical methods can be applied, and also reaches out to diffusive rather than convection-dominated processes. We consider Maxwell’s equations of electrodynamics coupled to a multicomponent, non-neutral, possibly polarizable and magnetizable material De Groot, Mazur (1962), Dreyer, Guhlke, Müller (2013), Griffith, Peskin (2013), Mirabella (2010). In this case, thDe Groot, Mazur (1962)e diffusion contains intriguing cross effects in which electric fields generate diffusion fluxes and vice versa. The consistent modeling poses the question how the constitutive relations of polarization and magnetization of a multi-component system interact with momentum and energy conservation and an entropy law. The focus of research will be on the appropriate formulation of the model equations, and to identify the relevant structural elements that a numerical coupling procedure will be based on. This includes the proper momentum and energy exchange mechanisms and an H-theorem, see Project A1, but also multi-scale aspects and suitable model hierarchies, see Project A3.