Variables similarity

The power law developed above uses the ratio of the two different shear rates as the variable in terms of which changes in 17 are expressed. Suppose that instead of some reference shear rate, values of 7 were expressed relative to some other rate, something characteristic of the flow process itself. In that case Eq. (2.14) or its equivalent would take on a more fundamental significance. In the model we shall examine, the rate of flow is compared to the rate of a chemical reaction. The latter is characterized by a specific rate constant we shall see that such a constant can also be visualized for the flow process. Accordingly, we anticipate that the molecular theory we develop will replace the variable 7/7. by a similar variable 7/kj, where kj is the rate constant for the flow process. 

Similarity Variables The physical meaning of the term similarity relates to internal similitude, or self-similitude. Thus, similar solutions in boundaiy-layer flow over a horizontal flat plate are those for which the horizontal component of velocity u has the property that two velocity profiles located at different coordinates x differ only by a scale factor. The mathematical interpretation of the term similarity is a transformation of variables carried out so that a reduction in the number of independent variables is achieved. There are essentially two methods for finding similarity variables, separation of variables (not the classical concept) and the use of continuous transformation groups. The basic theoiy is available in Ames (see the references). 

Combining Eqs. (76) and (77) results in a total integrodifferential equation in a single similarity variable p... 

The situation becomes bleaker still when we recognize that, for a given substance, there may be many different conditions A under which its adverse effects and their dose-response characteristics have been investigated Results may be available from several different epidemiological studies, in different groups exposed under different circumstances, and with results that are not entirely consistent with each other. Some of the conditions may involve experimental data, similarly variable in outcome and in how they will be interpreted by different scientists. So, when we are faced with toxic hazard and dose-response data from studies involving conditions A1 through A12, which, if any, are most useful and relevant for extrapolation to condition B ... 

Filaments, or fibrils, are similarly variable in size. When the terms refer to portions of a fiber, their dimensions are also usually unspecified but are relative to the fiber itself. Fibrils of glass or inorganic polymers that have been measured are often less than 10 nanometers in diameter and form bundles of aligned fibrils 0.2 to 10 micrometers in diameter and up to 100 micrometers in length (Ray, 1978). Occasionally, minerals with diameters of a few millimeters and lengths of hundreds of centimeters have been described as fibrous. Most objects called fibers, however, are microscopic, with maximum dimensions of less than a millimeter. 

Three 3-alkyl-2-methoxypyrazines (24b, 24c, and 24d) are detected as odor components in the monarch butterfly, Danaus plexippus (Table III). The wide variability in pyrazine content observed with this insect is correlated with similar variability in the larval food plants, Asclepias sp. It seems possible that the pyrazines may be one of the factors implicated in the food choice mechanism (69). 

In the third portion of the study, the results using five different sampler and analytical method combinations were compared. When obvious outliers were excluded from the data, the normalized percentage differences compared to the mean value for sulfur varied from -21 to +23%. Pairwise comparisons for other elements showed similar variability. The agreement overall for X-ray fluorescence compared to PIXE was good, although there was scatter in the individual measurements, perhaps due to differences in sampling (Nejedly et al., 1998). 

Since the diffusion coefficient is small, the depth of penetration by diffusion is also small. Consequently, it is likely that the depth of penetration by diffusion is smaller than the thickness of the fluid element flowing along the wall for all values of x < x . Therefore, the distribution of concentration can be approximated by that valid in a semiinfinite fluid. The similarity variable... 

Selected Applications of Equation 18. Equation 18 does not provide useful information for similarity variables that are constants during flow, as is the case, for example, for the composition variables in mixtures containing only a single species A2 and its dissociation product A. Thus, for the reaction... 

Before we turn to this issue, we would like to substantiate the above discussion of basic features of nonlinear diffusion with some examples based upon the well-known similarity solutions of the Cauchy problems for the relevant diffusion equations. Similarity solutions are particularly instructive because they express the intrinsic symmetry features of the equation [6], [28], [29], Recall that those are the shape-preserving solutions in the sense that they are composed of some function of time only, multiplied by another function of a product of some powers of the time and space coordinates, termed the similarity variable. This latter can usually be constructed from dimensional arguments. Accordingly, a similarity solution may only be available when the Cauchy problem under consideration lacks an explicit length scale. Thus, the two types of initial conditions compatible with the similarity requirement are those corresponding to an instantaneous point source and to a piecewise constant initial profile, respectively, of the form... 

Let us reiterate that whenever a similarity solution to (3.2.5), (3.2.4c) exists, the physical space coordinate of any given value of concentration between 0 and Cq propagates as const s/t. The shape of the solution (the concentration profile) evolves accordingly in terms of the physical space variable x, whereas it is preserved unchanged (either after a rescaling with some function of time only as in (3.2.9) or without it as in (3.2.13b)) in terms of the similarity variable. 

An interesting class of exact self-similar solutions (H2) can be deduced for the case where the newly formed phase density is a function of temperature only. The method involves a transformation to Lagrangian coordinates, based upon the principle of conservation of mass within the new phase. A similarity variable akin to that employed by Zener (Z2) is then introduced which immobilizes the moving boundary in the transformed space. A particular case which has been studied in detail is that of a column of liquid, initially at the saturation temperature T , in contact with a flat, horizontal plate whose temperature is suddenly increased to a large value, Tw T . Suppose that the density of nucleation sites is so great that individual bubbles coalesce immediately upon formation into a continuous vapor film of uniform thickness, which increases with time. Eventually the liquid-vapor interface becomes severely distorted, in part due to Taylor instability but the vapor film growth, before such effects become important, can be treated as a one-dimensional problem. This problem is closely related to reactor safety problems associated with fast power transients. The assumptions made are ... 

It is not surprising that the approximate bounds give the correct time dependence for the free boundary motion, since X is identical, except for constant factors, with the Boltzmann similarity variable. For a block of ice whose surface temperature is subjected to a step increase of 5°C. the upper and lower bounds are within 3% of each other, and the approximate growth constant calculated from Eq. (238) is about 0.5% from the exact value. 

Along with the methods of similarity theory, Ya.B. extensively used and enriched the important concept of self-similarity. Ya.B. discovered the property of self-similarity in many problems which he studied, beginning with his hydrodynamic papers in 1937 and his first papers on nitrogen oxidation (25, 26). Let us mention his joint work with A. S. Kompaneets [7] on selfsimilar solutions of nonlinear thermal conduction problems. A remarkable property of strong thermal waves before whose front the thermal conduction is zero was discovered here for the first time their finite propagation velocity. Independently, but somewhat later, similar results were obtained by G. I. Barenblatt in another physical problem, the filtration of gas and underground water. But these were classical self-similarities the exponents in the self-similar variables were obtained in these problems from dimensional analysis and the conservation laws. 

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