Kinetic theory hydrodynamic model

By contrast, when both the reactive solute molecules are of a size similar to or smaller than the solvent molecules, reaction cannot be described satisfactorily by Langevin, Fokker—Planck or diffusion equation analysis. Recently, theories of chemical reaction in solution have been developed by several groups. Those of Kapral and co-workers [37, 285, 286] use the kinetic theory of liquids to treat solute and solvent molecules as hard spheres, but on an equal basis (see Chap. 12). While this approach in its simplest approximation leads to an identical result to that of Smoluchowski, it is relatively straightforward to include more details of molecular motion. Furthermore, re-encounter events can be discussed very much more satisfactorily because the motion of both reactants and also the surrounding solvent is followed. An unreactive collision between reactant molecules necessarily leads to a correlation in the motion of both reactants. Even after collision with solvent molecules, some correlation of motion between reactants remains. Subsequent encounters between reactants are more or less probable than predicted by a random walk model (loss of correlation on each jump) and so reaction rates may be expected to depart from those predicted by the Smoluchowski analysis. Furthermore, such analysis based on the kinetic theory of liquids leads to both an easy incorporation of competitive effects (see Sect. 2.3 and Chap. 9, Sect. 5) and back reaction (see Sect. 3.3). Cukier et al. have found that to include hydrodynamic repulsion in a kinetic theory analysis is a much more difficult task [454]. 

Table. Hydrodynamic kinetic theory model for SBCR...

Hydrodynamics. Our historical interlude has made reference to effective theories of both gases (kinetic theory) and solids (elasticity), and now we take up yet a third example of enormous importance to modeling the natural world in general, and which serves as an example of the type of multiscale efforts of interest here, namely, the study of fluids. 

In 1944 Kramers [1] published a phase-space kinetic theory for the steady-state potential flow of monodisperse dilute polymer systems in which the polymer molecule is modeled as a freely jointed bead-rod chain. Subsequent scholars developed kinetic theories for shearing flows of monodisperse dilute polymer solutions Kirkwood [2] for freely rotating bead-rod chains with equilibnum-averaged hydrodynamic interaction. Rouse [3] and Zimm [4] for freely jointed bead-spring chains, and others. These theories were all formulated m the configuration space of a single polymer chain. 

The cut-off radius rc t is defined arbitrarily and reveals the range of interaction between the fluid particles. DPD model with longer cut-off radius reproduces better dynamical properties of realistic fluids expressed in terms of velocity correlation function [80]. Simultaneously, for a shorter cut-off radius, the efficiency of DPD codes increases as 0(1 /t ut). which allows for more precise computation of thermodynamic properties of the particle system from statistical mechanics point of view. A strong background drawn from statistical mechanics has been provided to DPD [43,80,81] from which explicit formulas for transport coefficients in terms of the particle interactions can be derived. The kinetic theory for standard hydrodynamic behavior in the DPD model was developed by Marsh et al. [81] for the low-friction (small value of yin Equation (26.25)), low-density case and vanishing conservative interactions Fc. In this weak scattering theory, the interactions between the dissipative particles produce only small deflections. 

The kinetic theory of gases is a classical model of relationship between the kinetic energy of the molecnles composing a gas and its hydrodynamical energy (volume energy). This model is based on the interaction between a wall and molecules colliding elastically with it (without dissipation). During a collision, a molecule exerts a force on the wall and this force is translated into a pressure within the container of the gas. 

The severe implications of these facts have been partially uncovered in reference [5] as a result of formulating a kinetic theory for granular flow without interaction with the ambient medium. These implications, as well as additional difficulties due to the necessity to calculate the energy supply to the particle fluctuations, make somewhat problematic, at the present state of the art, the formulation of a reliable and sufficiently simple hydrodynamic model even for coarse dispersions. We have succeeded in this respect only at the expense of making certain supplementary assumptions. These assumptions cne ... 

Now the whole set of SFM hydrodynamic equations with EMMS mesoscale modeling is estabhshed, where the drag and stress closures are based on the bimodal distribution. However, the mass exchange (or phase change) between the dilute and dense phases is still a big challenge to the kinetic theory derivation. More elaborate efforts are needed on this topic. 

In order to take account of the fact that the solvent is made up of discrete molecules, one must abandon the simple hydrodynamically-based model and treat the solvent as a many-body system. The simplest theoretical approach is to focus on the encounters of a specific pair of molecules. Their interactions may be handled by calculating the radial distribution function, whose variations with time and distance describe the behaviour of a pair of molecules which are initially separated but eventually collide. Such a treatment leads (as has long been known) to the same limiting equations for the rate constant as the hydro-dynamically based treatments, including the term fco through which an activation requirement can be expressed, and the time-dependent term in (Equation (2.13)) [17]. The procedure can be developed, but the mathematics is somewhat complex. Non-equilibrium statistical thermodynamics provides an alternative approach [16]. The kinetic theory of liquids provides another model that readily permits the inclusion of a variety of interactions the mathematics is again fairly complex [37,a]. In the computer age, however, mathematical complexity is no bar to progress. Refinement of the model is considered further below (Section (2.6)). 

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