Initially inhomogeneous distribution

In gases the track structures usually contain a small number of active particles and are separated from each other by considerable distance. Owing to efficient diffusion, the initial inhomogeneity in the distribution of active particles rapidly smoothes out, and by the time the chemical reactions begin, the intermediate chemically active particles are distributed practically homogeneously in the irradiated volume. For this reason the influence of tracks on radiation-chemical processes in gaseous media... 

The choice of initial inhomogeneity, such as flie dye concentration distribution or the relative positioning of chemical streams, can have a significant influence on the progression of mixing in a system. Consider, for example, mixing with the linear shear flow... 

It is important to note that while the rate of reaction depends on the concentrations of the reactants, the rate constant is independent of these and is the parameter commonly referred to in discussion of reaction kinetics. While for normal reaction systems the rate constant is naturally independent of time, for systems featuring an initially inhomogeneous distribution of reactants—as, for example, along the track of an a-particle or laser pulse immediately after discharge—the rate constant varies with time until homogeneity is achieved. 

Illustration Short-time behavior in well mixed systems. Consider the initial evolution of the size distribution of an aggregation process for small deviations from monodisperse initial conditions. Assume, as well, that the system is well-mixed so that spatial inhomogeneities may be ignored. Of particular interest is the growth rate of the average cluster size and how the polydispersity scales with the average cluster size. 

In the previous section we discussed the ultimate strength of a polymer fibre o0. This value corresponds to the stress at which all secondary bonds in the fibre are broken. Due to the presence of the chain orientation distribution alone, it follows that even the strength of a polymer fibre without any flaws will never attain this value. Yet, fracture in a real fibre may not always initiate in the most disoriented domains. If there are inhomogeneities that lead to stress concentrations, fracture can also occur in domains at a smaller angle to the fibre axis. 

The results from Fig. 14.1 show the developing turbulent flame zone. The nonsymmetries of the reaction rate field are due to inhomogeneity of the polydis-persed mixture, i.e., nonsymmetrical distribution of model particles and their velocities. The reaction front is under formation oxygen and partially the volatiles in the center are burnt out, but the reaction front is not sphere-shaped yet. The nonuniformity of the model particles distribution was induced initially due to the stochastic modeling of the particulate phase. 

The preliminary results obtained show that the initiation limits for polydispersed mixtures and stability of flame propagation strongly depend on inhomogeneity of particles (droplets) concentration distribution typical for the majority of practical cases wherein the ignition and combustion of polydispersed mixtures take place. Thus to ensure stable ignition and combustion characteristics... 

The active membrane separates two compartments and it is possible to get this pH value throughout the system, in presence of the two substrates, by the transient use of a buffer. The pH values outside are controlled and H+ fluxes measured by pH-stat systems. After small asymmetrical perturbations of the pH values at the boundaries (0.05), an inhomogeneous pH distribution arises spontaneously inside the membrane. The initial perturbations are amplified and the pH values in the compartments tend to evolve in opposite directions. The H+ fluxes entering and leaving the membrane can be determined by pH-stat measurements. If the boundary pH values are not maintained constant by a pH stat, the system evolves to a new stationary state characterized by a pH gradient of two pH units across the membrane. 

We can think of the reactant concentration and some initial spatial distribution of the intermediate concentration and temperature profiles specifying a point on Fig. 10.9. If we choose a point above the neutral stability curve, then the first response of the system will be for spatial inhomogeneity to disappear. If the value of /r lies outside the range given by (10.79), then the system adjusts to a stable spatially uniform stationary state. If ji lies between H and n, we may find uniform oscillations. 

To solve the diffusion equation and obtain the appropriate rate coefficient with these initial distributions is less easy than with the random distribution. As already remarked, the random distribution is a solution of the diffusion equation, while the other distributions are not. The substitution of Z for r(p(r,s) — p(r, 0)/s) is not possible because an inhomogeneous equation results. This requires either the variation of parameters or Green s function methods to be used (they are equivalent). Appendix A discusses these points. The Green s function g(r, t r0) is called the fundamental solution of the diffusion equation and is the solution to the... 

For unhydrolyzed HEC, we obtained the following values M = 1,261,000 MW = 1,360,000 Mz = 1,467,000. Since MJMn equals 1.08, the initial molecular weight distribution is narrow. Comparing this result with the Mw found in our light-scattering experiments, we found only a difference of 5%. This difference could be caused by inhomogeneities in the bulk HEC powder. 

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