Phase diffusion equation linearized

Equation (3.3.5) represents a nonlinear phase diffusion equation. It is equivalent to the Burgers equation in the case of one space dimension (Chap. 6). It is known that the Burgers equation can be reduced to a linear diffusion equation through a transformation called the Hopf-Cole transformation (Burgers, 1974), and essentially the same is true for (3.3.5) in an arbitrary dimension. We shall take advantage of this fact in Chap. 6 when analytically discussing a certain form of chemical waves. 

This shows the anticipated fact that the instability of the uniform oscillation to long wavelength fluctuations corresponds precisely to the negative sign of the phase diffusion constant. The equality = —y may also be confirmed, where y is the quantity which appeared in (4.2.36) and is the abbreviation of —a> defined in (4.2.35). More generally, it is possible to prove that the dispersion curve of the phaselike branch has an exact correspondence to the linearized form of the phase diffusion equation (4.2.36), or one may possibly have... 

Linear stability analysis then yields a phase diffusion equation [2,28,53] for the slow phase modulations Y, T) that evolve on time and length scales larger than those of the modulus of A ... 

The linear phase diffusion equations have also been derived for patterns of hexagonal symmetry (M — 3) [58, 59] for a variational model (otherwise... 

The superscripts o and w denote the organic and aqueous phases, respectively. Equation (6) suggests a linear dependence off c ) on t for a diffusion-controlled process. 

In all considered above models, the equilibrium morphology is chosen from the set of possible candidates, which makes these approaches unsuitable for discovery of new unknown structures. However, the SCFT equation can be solved in the real space without any assumptions about the phase symmetry [130], The box under the periodic boundary conditions in considered. The initial quest for uy(r) is produced by a random number generator. Equations (42)-(44) are used to produce density distributions T(r) and pressure field ,(r). The diffusion equations are numerically integrated to obtain q and for 0 < s < 1. The right-hand size of Eq. (47) is evaluated to obtain new density profiles. The volume fractions at the next iterations are obtained by a linear mixing of new and old solutions. The iterations are performed repeatedly until the free-energy change... 

It is seen that the Golay equation produces a curve identical to the Van Deemter equation but with no contribution from a multipath term. It is also seen that, the value of (H) is solely dependent on the diffusivity of the solute in the mobile phase and the linear mobile phase velocity, It is clear that the capillary column can, therefore, provide a simple means of determining the diffusivity of a solute in any given liquid. 

Assuming a constant surface area, dissolution at a solution-solid interface (Case I) results in linear kinetics in which the rate of mass transfer is constant with time (equation 1). Analytical solutions to the diffusion equation result in parabolic rates of mass transfer (, 16) (equation 2). This result is obtained whether the boundary conditions are defined so diffusion occurs across a progressively thickening, leached layer within the silicate phase (Case II), or across a growing precipitate layer forming on the silicate surface (Case III). Another case of linear kinetics (equation 1) may occur when the rate of formation of a metastable product or leached layer at the fresh silicate surface becomes equal to the rate at which this layer is destroyed at the aqueous... 

Next, Fig. 5 shows a typical center of mass trajectory for the case of a full chaotic internal phase space. The eyecatching new feature is that the motion is no more restricted to some bounded volume of phase space. The trajectory of the CM motion of the hydrogen atom in the plane perpendicular to the magnetic field now closely resembles the random motion of a Brownian particle. In fact, the underlying equation of motion at Eq. (35) for the CM motion is a Langevin-type equation without friction. The corresponding stochastic Lan-gevin force is replaced by our intrinsic chaotic force — e B x r). A main characteristic of random Brownian motion is the diffusion law, i.e. the linear dependence of the travelled mean-square distance on time. We have plotted in Fig. 6 for our case of a chaotic force for 500 CM trajectories the mean-square distance as a function of time. Within statistical accuracy the plot shows a linear dependence. The mean square distance

of the CM after time t, therefore, obeys the diffusion equation... 

If the flows are unsteady, the terms containing apo can be added on both sides of Eq. (7.10) (refer to Section 6.4). It must be noted that for multiphase flows, the inflow and outflow terms require considerations of interpolations of phase volume fractions in addition to the usual interpolations of velocity and the coefficient of diffusive transport. The source term linearization practices discussed in the previous chapter are also applicable to multiphase flows. It is useful to recognize that special sources for multiphase flows, for example, an interphase mass transfer, is often constituted of terms having similar significance to the a and b terms. Such discretized equations can be formulated for each variable at each computational cell. The issues related to the phase continuity equation, momentum equations and the pressure correction equation are discussed below. 

The dimensionless mean retention time, Hi/to is independent of the carrier gas velocity and is only a function of the thermodynamic properties of the polymer-solute system. The dimensionless variance, i2 /tc2. is a function of the thermodynamic and transport properties of the system. The first term of Equation 30 represents the contribution of the slow stationary phase diffusion to peak dispersion. The second term represents the contribution of axial molecular diffusion in the gas phase. At high carrier gas velocities, the dimensionless second moment is a linear function of velocity with the slope inversely proportional to the diffusion coefficient. 

The Kinetics of Spinodal Decomposition. Cahn s kinetic theory of spinodal decomposition (2) was based on the diffuse interface theory of Cahn and Hilliard (13). By considering the local free energy a function of both composition and composition gradients, Cahn arrived at the following modified linearized diffusion equation (Equation 3) to describe the early stages of phase separation within the unstable region. In this equation, 2 is an Onsager-type... 

Because of the diffusion resistance, the solvent concentration at the interface is less then in the bulk, k < ko. Writing the equation of phase equilibrium in linear approximation with respect to Ak = ko - k, from [7.2.55] one can receive ... 

We wish to see what the overall conversion of a continuous mixture will be, but first we have to ask which parameters will depend on x, the index variable of the continuous mixture. Clearly k, the rate constant in the Damkdhler number, will be a function of x, and if monotonic, it can be put equal to Da.x. The parameter P is clearly hydrodynamic, and so, for the most part, are the terms in the Davidson number. The only term in Equation 6.21 of Davidson and Harrison that might depend on x is the gas phase diffusivity, and this appears under a square root sign in the second of two terms. Tr was found to be virtually constant with a value close to three in a series of experiments by Orcutt which Davidson and Harrison analyze. We will, therefore, assume that only the Damkdhler number varies with x, and that this variation is linear. Then for the well-mixed model... 

For kinetic separations by PSA, a simple parameter could be defined as the ratio of the amounts of uptake for the two competing components during the adsorption step. Assuming a step change in the gas phase concentration, clean beds initially, and linear isotherms, the amounts can be expressed by the short-time solution of the diffusion equation (Carslaw and Jaeger, 1959) ... 

As mentioned in Sect. 5.5, in the classical diffusion theory for a porous medium, adsorption is described by a distribution coefficient Kd resulting from the transfer of the species from the fluid phase to the solid phase through the linearized equation of equilibrium adsorption isotherm (5.113). 

As a consequence of the Maxwell equations, linear field gradients are accompanied by additional spatially dependent field components. Then, an asymmetric coil generates concomitant field terms of zeroth and first order in space. The formalism was used experimentally to compensate for artifacts observed in three different imaging methods an image shift in standard echo planar imaging (EPI), an echo shift in diffusion-weighted EPI, and a phase shift in a flow quantification technique based on phase contrast images. 

This method operates fundamentally in non-stationary regime. Assuming that the organic phase was totally at rest diffusion equations could be solved analytically and the flux of matter could be expressed as a function of time. The reaction at the interface was considered as irreversible and characterized by a first-order aqueous-to-organic extraction rate constant k It was found that the amount of matter extracted at time t after phase contact was a linear function of with an intercept proportional to D/k, with D the diffusion coefficient of the species in the aqueous phase. 

Even though this correlation shows no dependences on the channel diameter and liquid-phase diffusivity, it demonstrates a good agreement with experimental data. Figure 12.9a presents the influence of the linear velocity on the kj a for the experimental data and modeling results according to Equation 12.29 under different conditions. In Figure 12.9b, one can see that the Berac-Pintar correlation describes the relationship of the process parameters and kj a within the experimental error and shows better results than an earlier model of Irandoust et al. [54]. 

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