Continuum equations

Finally, before leaving our exploration of the dusty gas model, we must compare the large pore (or high pressure) limiting form of its flux relations with the corresponding results derived in Chapter 4 by detailed solution of the continuum equations in a long capillary. The relevant equations are (4,23) and (4,25), to be compared with the corresponding scalar forms of equations (5.23) and (5.24). Equations (4.25) and (5.24).are seen to be identical, while (4,23) and (5.23) differ only in the pressure diffusion term, which takes the form... 

In this section, we discuss the role of numerical simulations in studying the response of materials and structures to large deformation or shock loading. The methods we consider here are based on solving discrete approximations to the continuum equations of mass, momentum, and energy balance. Such computational techniques have found widespread use for research and engineering applications in government, industry, and academia. 

In the complete Eulerian description of multiphase flows, the dispersed phase may well be conceived as a second continuous phase that interpenetrates the real continuous phase, the carrier phase this approach is often referred to as two-fluid formulation. The resulting simultaneous presence of two continua is taken into account by their respective volume fractions. All other variables such as velocities need to be averaged, in some way, in proportion to their presence various techniques have been proposed to that purpose leading, however, to different formulations of the continuum equations. The method of ensemble averaging (based on a statistical average of individual realizations) is now generally accepted as most appropriate. 

Figure 3. Typical images of the surfaces obtained by the integration of the continuum equation. The rescaled time is shown below the images.

In order to quantitatively compare the simulation with the continuum equation, we introduce a generalized free energy of the surface, and write the equation of motion in variational form. 

At this point we are able to compare the coarsening process of the Monte-Carlo simulation with that of the continuum equation, (5). We rescale the surfaces obtained in the simulation to the dimensionless variables Fl(X,T), and compare the time evolution of the free energy associated with the rescaled surface of the simulation (with different parameter values) with the free energy of the continuum equation. As we expect, the free energy (Fig. 4) decreases in time. But it turns out that the... 

Figure 4. The (a) nonequilibrium and the (b) equilibrium part of the free energy. (The free energy itself is the sum of these two.) The dashed line corresponds to the continuum equation, the solid lines are the rescaled curves of the simulation for different parameter values F, S).

It can be seen from the comparison that the non-equilibrium part (Fig. 4a), which in most cases dominates the free energy, is consistent with that of the continuum equation. But on the other hand, although the equilibrium part (Fig. 4b) more-or-less coincides with the result of the continuum equation for some parameter values of the simulation, for an another domain of the parameter space it does not. This could mean (and later we will argue that it does) that the term of Eq. (4) is important in those cases. We will give an explanation for this later in this paper. 

Figure 5. The value of the coarsening exponent n of the Monte-Carlo surfaces and two experimental surfaces as a function of the growth parameters. The points (o) where n = 1/6 coincide with those simulations where the equilibrium part of the free energy did not match that of the continuum equation. The error bars show the parameter range/uncertanity of an Fe/Fe( 100) experiment of Ref. 10 (o measured n = 0.16 0.04) and Ref 12 ( measured n = 0.23 0.02). The estimate of the Ehrlich-Schwoebel barrier is taken from Ref 13 (thin line) and Ref. 14 (thick line).

We have also compared the surface profiles from the simulation with results from a numerical integration of Eq. (31). Figure 10 shows a sequence of profiles obtained by taking = W- xi WI6). The dashed line in Fig. 10 is a reproduction of the thick line in Fig. 7. The agreement between the two profiles is nearly perfect The procedure also fixes the time unit v in the continuum equation. Profiles at earlier times also agree reasonably well. There is however one major difference the simulation took a week to complete on a Sun SPARC-20 workstation, while the numerical integration finished in the twinkling of an eye. 

We now modify the 2D continuum equations of step motion, Eqs. (7) and (8), in order to study some aspects of the dynamics of faceting. We assume the system is in the nucleation regime where the critical width Wc is much larger than the average step spacing In the simplest approximation discussed here, we incorporate the physics of the two state critical width model into the definition of the effective interaction term V(w) in Eq. (2), which in turn modifies the step chemical potential terms in Eqs. (7) and (8). Again we set V(w) = w/ l/w) as in Eq. (4) but now we use the /from Eq. (10) that takes account of reconstruction if a terrace is sufficiently wide. Note that this use of the two state model to describe an individual terrace with width w is more accurate than is the use of Eq. (10) to describe the properties of a macroscopic surface with average slope s = Mw. 

Macroscopic treatments of diffusion result in continuum equations for the fluxes of particles and the evolution of their concentration fields. The continuum models involve the diffusivity, D, which is a kinetic factor related to the diffusive motion of the particles. In this chapter, the microscopic physics of this motion is treated and atomistic models are developed. The displacement of a particular particle can be modeled as the result of a series of thermally activated discrete movements (or jumps) between neighboring positions of local minimum energy. The rate at which each jump occurs depends on the vibration rate of the particle in its minimum-energy position and the excitation energy required for the jump. The average of such displacements over many particles over a period of time is related to the macroscopic diffusivity. Analyses of random walks produce relationships between individual atomic displacements and macroscopic diffusivity. 

When motion of the fluid consists of only small fluctuations about a state of near-rest, Lhe continuum equations are linearized by neglecting nonlinear terms and they become lhe equalions of acoustics. A large variety of fluid motions are described as sound waves when the small-motion or acoustic description can be used, the principle of superposition is valid. This powerful principle allows addition of simple simultaneous motions to represent a more complex motion, such as the sound reaching lhe audience from the instruments of a symphony orchestra. The superposition principle does not apply to large-scale (nonacoustical) motions, and the subject of fluid dynamics (in distinction from acoustics) treats nonlinear flows. i.e.. those that cannot be described as superpositions of other flows. 

Euler-Euler models assume interpenetrating continua to derive averaged continuum equations for both phases. The probability that a phase exists at a certain position at a certain time is given by a phase indicator function, which, for steady-state processes, is equivalent to the volume of fraction of the correspondent phase (volume-of-fluid technique). The phase-averaging process introduces further unknowns into the basic conservation equations their description requires empirical and problem-dependent input (94). In principal, Euler-Euler models are applicable to all multiphase flows. Advantages and disadvantages of both methods are compared, e.g., in Refs. 95 and 96. 

Such a simplification is possible through the introduction of a continuum mathematical description of the gas-solid flow processes where this continuum description is based upon spatial averaging techniques. With this methodology, point variables, describing thermohydrodynamic processes on the scale of the particle size, are replaced by averaged variables which describe these processes on a scale large compared to the particle size but small compared to the size of the reactor. There is an extensive literature of such derivations of continuum equations for multiphase systems (17, 18, 19). In the present study, we have developed (17, ) a system of equations for... 

According to reference [1] four flow regimes for gases exist continuum flow (0iKn<0.001), slip flow (0.00l Kn<0.1), transition flow (0.l SKn<10), and free molecular flow (lOsKn). Continuum equations are valid for Kn- >0, while kinetic theory is applicable for Kn>8. Slip flow occurs when gases are at low pressure or in micro conduits. The gas slip at the surface, while in continuum flow at the surface it is immobilized. 

Within the FPM, we can extend further the capabilities of the discrete particle method to the mesoscopic regime and show that they are competitive to standard simulation techniques with continuum equations. These methods establish a foundation for cross-scale computations ranging from nanoscales to microns and can provide a framework for studying the interaction of microstructures and large-scale flow, which may be of value in blood flow and other applications in polymeric flows (Banfield et al. 2000 Schwertman et al 1999 Hiemstra and VanReimsdijk 1999). 

For phenol, PA (phenolate) = 350 kcalmol", AGt of the proton is about 260 kcalmol" and AGt((PhO ) — (PhOH)) may be roughly estimated assuming that it is mainly given by the Born free-energy of solvation of charged cavities immersed in a dielectric continuum (equation 16),... 

---