Dispersion linearized equation

Some problems with diffusion or dispersion give rise to the second order linear equation with constant coefficients,... 

Collective Modes and Time-Correlation Functions. Our linear equations (6.15) and (6.16) describe two characteristic kinds of collective modes in gels the longitudinal part of u obeys Tanaka s equation (4.16) and g = V u is governed by the diffusion equation (4.18), while the transverse parts of u and are coupled to form a slow transverse sound at small wave numbers. By assuming the space-time dependence as exp(i[Pg.99]

Figure 4-24 For Tapioca Starch Dispersions Two Linear Equations or a Single Exponential Equation (not shown) Provided a Better Fit of the Data between cQ and (A / o (A is the diameter of the heated starch granule and A is that of the unheated granule) (Rao and Tattiyakul, 1998).

The different components of the C6 dispersion coefficients in the LaTbM scheme for (i) two different linear molecules, and (ii) an atom and a linear molecule, are given in Table 11.2 of Magnasco and Ottonelli (1999) in terms of the symmetry-adapted combinations of the elementary dispersion constants (Equations (4.27). For identical molecules, C = B in (4.27), and the (020) and (200) coefficients are equal. 

Minimum mass transport occurs even in the absence of water filtration. We will review uni dimensional (linear) mass transport. If V = 0 and q, = 0, advective-dispersive equation acquires the format of a linear equation Fick s second law (equation 3.12). 

The zero viscosity of the liquid means that the amplitude can neither decrease nor increase, but the wave can be dispersive, i.e. co can depend on k. It is shown in [2] that perturbations with small amplitude as compared to the wavelength (a 2) are described by the linearized equation... 

At very low polymer volume fractions, the entry follows Smoluchowski kinetics Sm = 1), whereas for polymer volume fractions greater than 0.1% Smis increased (Figure 25.4). The BD simulations reveal an almost linear dependence of the radical capture rate coefficient on the polymer volume fraction in the dispersion, cf. Equation 25.26 ... 

To deseribe the transport and reaetion of these eompounds in the subsurfaee, one-dimensional adveetion, three-dimensional dispersion, linear adsorption, and sequential first order biodegradation are assumed as shown in the equations below. All equations, but the first, are eoupled to another equation through the reaetion term. 

The coated material used on the pipe is usually a viscoelastic layer adhered on the pipe. The internal losses in the coated material are modeled according to the theory of linear viscoelasticity, which is also the model implemented in the software DISPERSE [17] used for the wave structure analysis. The shear velocity and shear attenuation of bitumen are obtained from the result of Simonetti measurement [16] for the software used to predict the attenuation of guided wave. The material properties of the other two coated materials are found from the data bank in the DISPERSE software. The theory of linear viscoelasticity for isotropic and homogenous media is modeled in the frequency domain, which leads to linear equation of motion [18]. Thus... 

Here, the simple dispersion (oiq) v g was used for the acoustic phonons (if whole molecules vibrate with respect to each other, like the change of the stacking distance in a stack one speaks of acoustic phonons, because they have a long wavelength comparable to those of acoustic waves) and the sound velocity v, is determined by the relation o, = Ci/p), where c, is the longitudinal elastic constant and p is the mass density. One should remark that this very simple linear dispersion relation (o q) = v,g is not necessarily correct. With the help of the FG method described in Section 9.1 one can obtain more accurate dispersion curves. Equation (9.48) can now be used to calculate the charge carrier mobilities and free paths, defined in this case hy p= e xlm ) and A = (t ), respectively, where... 

First we investigate the dispersion relation for the linearized version of the field equations (5.35). For simplicity, the linear equations are for a mixture initially at rest... 

This is an approximation to the complete dispersion equation [131]. The amplitude of a train of waves originating from an infinitely long linear source decays exponentially with the distance x from the source... 

All nonlinear (electric field) spectroscopies are to be found in all temis of equation (B 1.3.1) except for the first. The latter exclusively accounts for the standard linear spectroscopies—one-photon absorption and emission (Class I) and linear dispersion (Class II). For example, the temi at third order contains by far the majority of the modem Raman spectroscopies (table B 1.3.1 and tableBl.3.2). 

Axial Dispersion Effects In adsorption bed calculations, axial dispersion effects are typically accounted for by the axial diffusionhke term in the bed conservation equations [Eqs. (16-51) and (16-52)]. For nearly linear isotherms (0.5 < R < 1.5), the combined effects of axial dispersion and mass-transfer resistances on the adsorption behavior of packed beds can be expressed approximately in terms of an apparent rate coefficient for use with a fluid-phase driving force (column 1, Table 16-12) ... 

Dispersion Model An impulse input to a stream flowing through a vessel may spread axially because of a combination of molecular diffusion and eddy currents that together are called dispersion. Mathematically, the process can be represented by Fick s equation with a dispersion coefficient replacing the diffusion coefficient. The dispersion coefficient is associated with a linear dimension L and a linear velocity in the Peclet number, Pe = uL/D. In plug flow, = 0 and Pe oq and in a CSTR, oa and Pe = 0. 

The original Rate Theory which describes dispersion in packed beds evolved over a number of years, probably starting with the work of Lapidus and Amundson [6] in 1952, extended by that of Glueckauf [7] and Tunitski [8] in 1954. The final form of the equation that described dispersion in packed beds as a function of the linear... 

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