Dispersivity variation with scale

In the special kind of reactive system discussed above, a filler is generated in the course of the reaction. Assuming that t, we can obtain an expression for the kinetics of the visct ity variation in gelation processes quite similar to a scaling formula in Ref. [9], However, the given approach has a fundamental difference which consists in the fact that the viscosity of the dispersion medium is not a constant but varies in the course of the process [94]. Then, the viscosity variation with time can be written as ... 

The scaled surface area and its variation with d> are of crucial importance in the definition and evaluation of the osmotic pressure , H, of a foam or emulsion. We introduced the concept in Ref 37, where it was referred to as the compressive pressure , P. It has turned out to be an extremely finitful concept (22,27,38). The term osmotic was chosen, with some hesitation, because of the operational similarity with the more familiar usage in solutions. In foams and emulsions, the role of the solute molecules is played by the drops or bubbles that of the solvent by the continuous phase, although it must be remembered that the nature of the interaetions is entirely different. Thus, the osmotic pressure is denned as the pressure that needs to be applied to a semipermeable, freely movable membrane, separating a fluid/fluid dispersion from its continuous phase, to prevent the latter from entering the former and to reduce thereby the augmented surface free energy (Fig. 4). The membrane is permeable to all the components of the continuous phase but not to the drops or bubbles. As we wish to postpone diseussion of compressibility effects in foams until latter, we assume that the total volume (and therefore the volume of the dispersed phase) is held constant. 

Thus, the dispersion relation for Eqn. (1.4.3), is the statement of governing equation in the spectral plane and tells us that the scale of space variation and the scale of time variation are not independent and they are related. For many other problems, the dispersion relation will be consequence of boundary conditions, as is often derived for water waves developing for an equilibrium solution given by the Laplace s equation. Equation (1.4.5) implies that each frequency component will travel in space with the... 

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