Kolmogoroff

The final result is very well known in experimental studies and is usually used to evaluate a32 from known and s values. Other studies in this field include the work by Shinnar and Church (S7) who used the Kolmogoroff theory of local isotropy to predict particle size in agitated dispersions, and an analysis by Levich (L3) on the breakup of bubbles. Levich derived an expression for the critical bubble radius (the radius at which breakup begins) ... 

This case can also be approached using Kolmogoroff s (K9, H15) theory of local isotropic turbulence to predict the velocity of suspended particles relative to a homogeneous and isotropic turbulent flow. By examining this situation for spherical particles moving with a constant relative velocity, varying randomly in direction, Levich, (L3) has demonstrated that... 

Kolmogoroff A (1958) Collected works on the statistical theory of turbulence. Akademic Verlag, Berlin... 

Stresses acting on micro-organisms in (a) to (c) are derived on the premise that the flow forces originate from the turbulent motion of the carrier medium. In almost all cases, turbulence is assumed to be locally isotropic and homogeneous which greatly simplifies the analysis and allows the application of the Kolmogoroff s theory of turbulence to the problem [81]. The Kolomogoroff micro-scale of turbulence,... 

The Kolmogoroff microscale, tik, has the physically meaning of the size of the region within which energy is dissipated by viscous forces. It can be estimated by ... 

The existence of a normal distribution can only be confirmed by a goodness-of-fit test (e.g., yf, according to Kolmogoroff [1941] and Smirnoff [1948]). 

Kolmogoroff AN (1941) Confidence limits for an unknown distribution function. Ann Math Statist 12 461... 

KOH plants, environmental awareness in, 20 634. See also Potassium hydroxide KOH solution, 12 215 meta-Koksowy coal grade (Poland), 6 713t Kolbe-Schmidt reaction, 2 208 of salicylic acid synthesis, 22 7-8 Kolmogoroff microscale, 16 697 Konica Dry Color System, 19 348 Konjac glucomannan, 4 7241... 

When the fluid is in turbulent flow, or where turbulence is generated by some external agent such as an agitator, the drag coefficient may be substantially increased. Brucato et al.(24> have shown that the increase in drag coefficient may be expressed in terms of the Kolmogoroff scale of the eddies (kE) given by ... 

Kolmogoroff, A. N. Sammelband zur Theorie der statistischen Turbulenz Akademieverlag Berlin, 1958. 

Reynolds (Rl, R2) was one of the earlier investigators to appreciate the random nature of turbulence. The dimensionless parameter bearing his name is widely used as a measure of the physical characteristics of steady, uniform flow. Such a measure is essentially macroscopic and does not describe the local or transient behavior at a point in the stream. In recent years much effort has been devoted to understanding the basic mechanism of momentum transport by turbulence. The early work of Prandtl (P6), Taylor (Tl), Karmdn (Kl), and Howarth (K4) laid a basis for the statistical theory of turbulence which is apparently in reasonable agreement with experiment. More recently Onsager (03), Corrsin (C6), and Kolmogoroff (K10) extended the statistical theory of turbulence to describe the available experimental data in terms of kinetic-energy... 

Little detailed experimental information is available on the value of eddy transport properties under conditions of simultaneous thermal and material transport. If it is assumed that the Reynolds analogy is applicable, it follows that the eddy diffusivity and eddy conductivity are equal and independent of cross linking. Such an assumption is probably not true since it is to be expected that a substantial part of the eddy transport is associated with molecular transport particularly as the eddies become small in accordance with Kolmogoroff s (K10) principle. For this reason it is to be expected that temperature gradients in turbulent streams will influence to some extent the material transport in the same... 

A. Kolmogoroff, I. Petrovsky, and N. Piscounoff, Etude de I equation de la diffusion avec croissance de la quantite de matier et son application a un probleme biologique, Moscow Univ. Bull. Math., 1 (1937), pp. 1-25. 

It is interesting to notice that a simple formula for the addition of transition probabilities exists only in extremely simple models of stochastic processes. An example is the Chapman-Smoluchowski-Kolmogoroff equation.30... 

We have in this way obtained a generalization of Einstein s theory of the interaction between matter and radiation including multiple photon processes and involving transition probabilities. But there is a basic difference. The operator definite positive. We no longer have a simple addition of transition probabilities. This corresponds exactly to the interference of probabilities discussed in Section IV. The process is not of the simple Chapman-Smoluchowski-Kolmogoroff type (Eq. (11)) the operator transition probability. As the result, the second of the two sequences discussed above may decrease the effect of the first one. It is very interesting that even in the limit of classical mechanics (which may be performed easily in the case of anharmonic oscillators) this interference of probabilities persists. This is in agreement with our conclusion in Section IV. 

A. Kolmogoroff, op. cit. in LI A. Friedman, Stochastic Differential Equations and Applications I (Academic Press, New York 1975) ch. I. See, however, the footnote on p. 296 of A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York 1965). 

Several workers (Kolbel et al. [40, 41], Deckwer et al. [17], Michael and Reicheit [42]) have investigated the heat transfer in BSCR versus solid concentration and particle diameters. Deckwer et al. [17] applied Kolmogoroff s theory of isotropic turbulence in combination with the surface renewal theory of Higbie [43] and suggested the following expression for the heat transfer coefficient in the Fischer-Tropsch synthesis in BSCR ... 

Accepting Kolmogoroff s theory of local isotropy, they find for the critical drop size the relation... 

From Kolmogoroff s theory of local isotropic turbulence [see Batchelor (Bl)] it follows that u cc e1/3rf1/3 where e is the turbulent energy dissipation per unit time per unit volume. Therefore,... 

An equation of the type of eq. 7 for crystallization was first proposed by Kolmogoroff, A. N. On the crystallization process in metals. Isvest. Akad. Nauk SSSR Ser. Math. 1, 335 (1937) and than independently derived by Avrami and others... 

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