Solution of the Dispersion Equation

Equation (11.61) has three roots three real, or one real and two complex, depending on the value of determinant D = q - -Pl (Korn and Korn 1968). Since our aim is to determine the complex frequency D, we will consider the complex solution of Eq. (11.61)only. 

---

The form of the solution of the dispersion equation (11.61) depends on the sign of the determinant D = q + Pl, i.e., on the values of the characteristic parameters g and P. The latter are determined by the physical properties of the liquid and its vapor, as well as the values of the Peclet number. This allows us to use g and P as some general characteristics of the problem considered here. 

The solution of the dispersion equation along a stream line is given in problem P5.08.08. That Equation (10) can be written in the form... 

These qualitative statements are confirmed by the numerical solution of the dispersion equation for the wave in a two-wall nanotube. It can be seen from Fig. 1 that the wave retardation exceed 100 in that case (c = 3x 10 ° cm/s). 

The slow exchange limit can be investigated using a perturbation solution of the dispersion equation... 

Solution of the dispersion equation for the impulse response, using a slightly different set of boundary conditions for equation (5-20), gives... 

In Sec. 3 our presentation is focused on the most important results obtained by different authors in the framework of the rephca Ornstein-Zernike (ROZ) integral equations and by simulations of simple fluids in microporous matrices. For illustrative purposes, we discuss some original results obtained recently in our laboratory. Those allow us to show the application of the ROZ equations to the structure and thermodynamics of fluids adsorbed in disordered porous media. In particular, we present a solution of the ROZ equations for a hard sphere mixture that is highly asymmetric by size, adsorbed in a matrix of hard spheres. This example is relevant in describing the structure of colloidal dispersions in a disordered microporous medium. On the other hand, we present some of the results for the adsorption of a hard sphere fluid in a disordered medium of spherical permeable membranes. The theory developed for the description of this model agrees well with computer simulation data. Finally, in this section we demonstrate the applications of the ROZ theory and present simulation data for adsorption of a hard sphere fluid in a matrix of short chain molecules. This example serves to show the relevance of the theory of Wertheim to chemical association for a set of problems focused on adsorption of fluids and mixtures in disordered microporous matrices prepared by polymerization of species. 

There are several closed form approximate solutions to both the general and first-order forms of the dispersion equations (11.2.9 and 11.2.10). For example, Levenspiel and Bischoff... 

Though this new algorithm still requires some time step refinement for computations with highly inelastic particles, it turns out that most computations can be carried out with acceptable time steps of 10 5 s or larger. An alternative numerical method that is also based on the compressibility of the dispersed particulate phase is presented by Laux (1998). In this so-called compressible disperse-phase method the shear stresses in the momentum equations are implicitly taken into account, which further enhances the stability of the code in the quasi-static state near minimum fluidization, especially when frictional shear is taken into account. In theory, the stability of the numerical solution method can be further enhanced by fully implicit discretization and simultaneous solution of all governing equations. This latter is however not expected to result in faster solution of the TFM equations since the numerical efforts per time step increase. 

Two numerical methods have been used for the solution of the spray equation. In the first method, i.e., the full spray equation method 543 544 the full distribution function / is found approximately by subdividing the domain of coordinates accessible to the droplets, including their physical positions, velocities, sizes, and temperatures, into computational cells and keeping a value of / in each cell. The computational cells are fixed in time as in an Eulerian fluid dynamics calculation, and derivatives off are approximated by taking finite differences of the cell values. This approach suffersfrom two principal drawbacks (a) large numerical diffusion and dispersion... 

The root time method of data analysis for diffusion coefficient determination was developed by Mohamed and Yong [142] and Mohamed et al. [153]. The procedure used for computing the diffusion coefficient utilizes the analytical solution of the differential equation of solute transport in soil-solids (i.e., the diffusion-dispersion equation) ... 

Any point on either branch of the dispersion surface is an equally good solution of the Maxwell equations. However, the only points that will be selected are... 

The coefficients a, (3, y and5 are obtained by imposing the Bloch conditions with periodicity A,p, the continuity conditions of the wave function and its derivative at L/2, and finally by normalization in the surface unit. The solution of the eigenvalue equation for E gives the electronic energy dispersion for the n-th subband with energy... 

For nonlinear systems the solution of the governing equations must generally be obtained numerically, but such solutions can be obtained without undue difficulty for any desired rate expression with or without axial dispersion. The case of a Langmuir system with linear driving force rate expression and negligible axial dispersion is a special case that is amenable to analytical solution by an elegant nonlinear transformation. 

D. L. Hovhannisyan, Analytic solution of the wave equation describing dispersion-free propagation of a femtosecond laser pulse in a medium with cubic and fifth-order nonlinearity, Optics Commun. 196, 103 (2001)... 

Fig. 4 General solution for the dispersion equation on water at 25 °C. The damping coefficient a vs. the real capillary wave frequency o> , for isopleths of constant dynamic dilation elasticity ed (solid radial curves), and dilational viscosity k (dashed circular curves). The plot was generated for a reference subphase at k = 32431 m 1, ad = 71.97 mN m-1, /i = 0mNsm 1, p = 997.0kgm 3, jj = 0.894mPas and g = 9.80ms 2. The limits correspond to I = Pure Liquid Limit, II = Maximum Velocity Limit for a Purely Elastic Surface Film, III = Maximum Damping Coefficient for the same, IV = Minimum Velocity Limit, V = Surface Film with an Infinite Lateral Modulus and VI = Maximum Damping Coefficient for a Perfectly Viscous Surface Film...

Continuous Stirred Tank Reactors. Biesenberger (8) solved for the MWD with condensation polymerization in a CSTR, analogous to the treatment Denbigh (14) provided for the other two mechanisms. In this case, the variable residence time distribution leads to an extremely broad MWD with even the maximum weight fraction at the lowest molecular weight (monomer). The dispersion index approaches infinity as the condensation is driven to completion in a stirred tank reactor. A sequential analytical solution of the algebraic equations was obtained with a numerical evaluation of the consecutive equations. 

Fig. 26 Numeric dispersion and oscillation effects for the numeric solution of the transport equation (after Kovarik, 2000)...

Vergnes107 has outlined a simple method for the determination of the Peclet number when the tracer injection is sharp enough to be considered as a true Dirac 6 impulse and the response equation is described by the solution of the dispersion model for a system open at both ends. The method does not require knowledge of the tracer quantity added, the sensitivity of the recorder, or the chart speed. 

Relation (3.108) gives the analytical solution of the axial dispersion model which contains relations (3.97), (3.99) and (3.100). Here the proper values of are the solutions of the transcendent equation (3.109) ... 

First order hyperbolic differential equations transmit discontinuities without dispersion or dissipation. Unfortunately, as Carver (10) and Carver and Hinds (11) point out, the use of spatial finite difference formulas introduces unwanted dispersion and spurious oscillation problems into the numerical solution of the differential equations. They suggest the use of upwind difference formulas as a way to diminish the oscillation problem. This follows directly from the concept of domain of influence. For hyperbolic systems, the domain of influence of a given variable is downstream from the point of reference, and therefore, a natural consequence is to use upstream difference formulas to estimate downstream conditions. When necessary, the unwanted dispersion problem can be reduced by using low order upwind difference formulas. 

---