Mathematical model main forces

Common physical features may evidently lead to a uniform theoretical representation of the phenomena. Many authors, particularly those presented in this book, introduced a distributed mass force in the momentum equations (or a distributed source term for the diffusion problem) to represent the influence of obstructions on the flow. The question was whether these mathematical models can cover the main flow features. Below, the overview of several such models will be given. We will start from simple one-dimensional models allowing an analytical investigation, generalize the results to two-dimensional problems, and apply them to some practical problems. 

Albeit the problem structure for branching pipehnes is generally similar to a pipehne structure with multiple depots located along one main pipeline, it comphcates the mathematical formulation by forcing to track batches along the pipeline branches which equals the modelling of additional pipelines. In the follow-up paper (MirHas-... 

Broadening the scope, we may briefly consider a nonexhaustive panorama of various types and features of supramolecular polymers depending on their constitution, characterized by three main parameters the nature of the core/framework of the monomers, the type of noncovalent interaction(s), and the eventual incorporation of functional subunits. The interactions may involve complementary arrays of hydrogen-bonding sites, electrostatic forces, electronic donor-acceptor interactions, metalion coordination, etc. The polyassociated structure itself may be of main-chain, side-chain, or branched, dendritic type, depending on the number and disposition of the interaction subunits. The central question is that of the size and the polydispersity of the polymeric supramolecular species formed. Of course their size is expected to increase with concentration and the polydispersity depends on the stability constants for successive associations. The dependence of the molecular weight distribution on these parameters may be simulated by a mathematical model [19]. These features are detailed in Chapters 2, 3, and 6 for various growth mechanisms. 

Heat being generated in the fuel plates is mainly transfered by heat conduction across the fuel plates and removed by the forced convection of the coolant. The physical and mathematical model of the heat transfer of HEATHYD includes the equation for thermal conduction and Newton s law of cooling. The equation of heat conduction for slab geometry of the fuel plates is expressed in the following form and numericaly solved using iteration method. 

We now wish to extend the considerations above dealing with several different structures extending simultaneously within the same space. Thus, for example, a macromolecule attached to carbon black at one point along its length may be cross-linked to other molecules at one or more points on either or both sides of the black particle, while another may pass fairly close to the black without contacting it and be equally cross-linked at the distant points. In order to be able to develop the mathematical relationships required for the development of this theory, it is necessary to represent the state of affairs by means of a highly idealized model. However, the model proposed here is believed to contain all the elements of a uniformly filled and cross-linked elastomeric compound. The possible existence of crystallization and inter-and intra-molecular forces other than main-chain carbon to carbon links, cross links and bonds between rubber and carbon black as hypothesized earlier, is specifically neglected. 

Force field methods use mathematical functions—interatomic potentials— to describe the attraction of different atom types to each other and the strain exerted on the molecular configuration by the presence of other atoms. They have become more elaborate over time, progressing from mainly non-bonded van der Waals interactions to more complicated 4-body terms and shell models of atomic distortion. CmciaUy, however, force field methods do not allow for drastic changes in the electronic configuration of a system—i.e. bond making or breaking—and only describe the system well when the configuration is near equilibrium. 

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