Multi-field formulation

Keywords Ionic polymer gels Modelling Numerical simulation Chemical stimulation Electrical stimulation Multi-field formulation Finite elements Discrete elements... 

To gain a more precise insight into the phenomena occurring in gels, the Theory of Porous Media (TPM), or the coupled chemo-electro-mechanical formulation is a good choice. Normally, the TPM is applied for the gel only, while the coupled multi-field formulation incorporates the gel and the surrounding solution. 

The coupled chemo-electro-mechanical multi-field formulation is a model on the micro-mesoscopic scale. It has been developed to gain a more precise insight into... 

In this chapter numerical simulations by applying the coupled chemo-electro-mechanical multi-field formulation on a deforming mesh have been performed. It has been shown that this formulation is an excellent method for describing the coupled behaviour of polymer gels in a solution. This method is capable to explain the mechanisms of the behaviour of polyelectrolyte gels under chemical and electrical stimulation, the ion distributions as well as the swelling and bending behaviour. 

Ionic polymer gels Polyelectrolyte gels Modeling Numerical simulation Statistical theory Transport model Continuum model Coupled multi-field formulation... 

In the previous sections, the Born-Oppenheimer approximation was assumed, i.e. nuclear motion on a single PES was considered. However, in many simations, the dynamics of the nuclei need to be treated on several PESs corresponding to coupled electronic state. The coupling between the electronic states can be due to the presence of an external electric field or to internal vibronic interactions. There exists two different ways of treating several coupled electronic states with the MCTDH method [60], the single-set formulation and the multi-set formulation. 

In this section the embedded interface (El) method (frequently referred to as a front tracking method) developed for direct numerical simulations of viscous multi-fluid flows is outlined and discussed. The unsteady model is based on the whole field formulation in which a sharp interface separates immiscible fluids, thus the different phases are treated as one fluid with variable material properties. Therefore, Eqs. (3.14) and (3.15) account for both the differences in the material properties of the different phases as well as surface tension effects at the phase boundary. The bulk fluids are... 

Werner H-J 1987 Matrix-formulated direct multiconfigurational self-consistent field and multi reference configuration interaction methods Adv. Chem. Phys. 69 1... 

Note that A, and , will, in general, depend on multi-point information from the random fields U and 0. For example, they will depend on the velocity/scalar gradients and the velocity/scalar Laplacians. Since these quantities are not contained in the one-point formulation for U(x, t) and 0(x, f), we will lump them all into an unknown random vector Z(x, f).16 Denoting the one-point joint PDF of U, 0, and Z by /u,,z(V, ip, z x, t), we can express it in terms of an unknown conditional joint PDF and the known joint velocity, composition PDF ... 

This chapter is concerned with the design and improvement of chemically-active ship bottom paints known as antifouling paints. The aims have been to illustrate the challenges involved in working with such multi-component, functional products and to show which scientific and engineering tools are available. The research in this field includes both purely empirical formulation and test methods and advanced tools including mathematical modelling of paint behaviour. 

Bioassay wt = wind tunnel, f = field traps. Other issues chem = pheromone chemistry, mult = multi-species lures, cr = cross attraction, seas = seasonal patterns, tp = trapping parameters such as height, type, pheromone dose, pheromone formulation/aging, or host-related synergist, wh = warehouse environment. 

Topical formulations, by their very nature, are multi-component and it is not surprising that neural networks have been applied to deal with this complexity. The first work in this field was performed on hydrogel formulations containing ketoprofen, at Hoshi University in Japan followed up by further studies in 999[65] 2001.In all cases, the objective—that... 

Unfortunately, the present models are still on a level aiming at reasonable solutions with several model parameters tuned to known flow fields. For predictive purposes, these models are hardly able to predict unknown flow fields with reasonable degree of accuracy. It appears that the CFD evaluations of bubble columns by use of multi-dimensional multi-fluid models still have very limited inherent capabilities to fully replace the empirical based analysis (i.e., in the framework of axial dispersion models) in use today [63]. After two decades performing fluid dynamic modeling of bubble columns, it has been realized that there is a limit for how accurate one will be able to formulate closure laws adopting the Eulerian framework. In the subsequent sections a survay of the present status on bubble column modeling is given. 

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