A stream tube model for miscible flow

A simple theoretical model is described for deriving a 1-dimensional equation for the spreading of a tracer in a steady flow at the field scale. The originality of the model is to use a stochastic appoach not in the 3-dimensional space but in the 1-D space of the stream tubes. The simplicity of calculation comes from the local relationship between permeability and velocity in a 1-D flow. The spreading of a tracer front is due to local variations in the cross-sectional area of the stream tubes, which induces randomness in travel time. The derived transport equation is averaged in the main flow direction. It differs from the standard dispersion equation. The roles of time and space variables are exchanged. This result can be explained by using the statistical theory of Continuous Time Random Walk instead of a standard Random Walk. However, the two equations are very close, since their solutions have the same first and second moments. Dispersivity is found to be equal to the product of the correlation length by the variance of the logarithm of permeability, a result similar to Gelhar's macrodispersion.Nomenclature A total cross-section area of the sample - C (resident) concentration of tracer - D,D ^* dispersion coefficient - F flux of tracer - G probability distribution function for permeability in the stream-tube segments - I tracer intensity (mass crossing a surface per unit time) - K permeability - L length of the medium - M number of stream tubes in the medium - N number of segments along a stream tube - P pressure - Q total flow rate in the sample - a length of an elementary stream-tube segment - g probability distribution function for permeability in the space - i, j indices, tube numbers - q flow rate in each stream tube - s variable cross-section area of a stream tube - t, t time - u front velocity - x space variable in the flow direction - small local variation in time - , _t longitudinal, transverse dispersivity - porosity of the porous medium - correlation length in the permeability field - viscosity of the fluid - time for filling an elementary stream tube segment - standard deviation of a stochastic variable - probability distribution of arrival times (Gaussian)     

ContinuousK-theory

We study arithmetical properties of homotopy groups of thel-adic completion of Quillen'sK-theory space of number field, with a view on the Dwyer-Friedlander comparison map into étaleK-theory. The relation of these groups toK-theory is a complete analogy to the relation of continuous étale cohomology to étale cohomology. We identify the torsion subgroup of the resulting term with the subgroup of divisible elements inK _2n (F). We prove that this term is sent isomorphically into étaleK-theory, giving some further evidence for the Lichtenbaum-Quillen conjectures.     

Some remarks on the modulus of continuity of a conformal mapping of the disk onto a Jordan domain

Letd(;z, t) be the smallest diameter of the arcs of a Jordan curve with endsz andt. Consider the rapidity of decreasing ofd(;)=sup{d(;z, t):z, t , ¦z–t¦} (as 0,0) as a measure of nicety of . Letg(x) (x0) be a continuous and nondecreasing function such thatg(x)x,g(0)=0. Put¯g(x)=g(x)+x, h(x)=(¯g(x))². LetH(x) be an arbitrary primitive of 1/h ^–1(x). Note that the functionH ^–1 x is positive and increasing on (–, +),H ^–1 0 asx– andH ^–1+ asx +. The following statement is proved in the paper.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 176–184, August, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00236 and by the International Science Foundation under grant No. NCF000.     

The uniform modulus of continuity of iterated Brownian motion

LetX be a Brownian motion defined on the line (withX(0)=0) and letY be an independent Brownian motion defined on the nonnegative real numbers. For allt0, we define theiterated Brownian motion (IBM),Z, by setting . In this paper we determine the exact uniform modulus of continuity of the process Z.Research supported by NSF grant DMS-9122242.     

Chains of strongly non-reflexive dual groups of integer-valued continuous functions

Answering a question of Eklof-Mekler (Almost free modules, set-theoretic methods, North-Holland, Amsterdam, 1990), we prove: (1) If there exists a non-reflecting stationary set of consisting of ordinals of cofinality for each , then there exist abelian groups such that and for each . (2) There exist abelian groups such that for each and for each . The groups are the groups of -valued continuous functions on a topological space and their dual groups.

     

Improved oxygen delivery in a continuous-roller-bottle reactor

Variations to the original aeration system in a continuous roller bottle reactor of novel design have been tested and compared for optimal oxygen (O) delivery. Reactor operating parameters that affect O transfer are rotation rate, liquid-volume level, fresh-feed rate, and supplementary-aeration rate. Design modifications to enhance gas-liquid O transfer include the addition of wall baffles and center baffles. The number and location of each of these baffles are compared for their effect on k_La values in the reaction chamber. The liquid feed into the system has been modified to improve the axial liquid mixing and O transfer.     

Singularity spectrum of fractal signals from wavelet analysis: Exact results

The multifractal formalism for singular measures is revisited using the wavelet transform. For Bernoulli invariant measures of some expanding Markov maps, the generalized fractal dimensions are proved to be transition points for the scaling exponents of some partition functions defined from the wavelet transform modulus maxima. The generalization of this formalism to fractal signals is established for the class of distribution functions of these singular invariant measures. It is demonstrated that the Hausdorff dimensionD(h) of the set of singularities of Hölder exponenth can be directly determined from the wavelet transform modulus maxima. The singularity spectrum so obtained is shown to be not disturbed by the presence, in the signal, of a superimposed polynomial behavior of ordern, provided one uses an analyzing wavelet that possesses at leastN>n vanishing moments. However, it is shown that aC behavior generally induces a phase transition in theD(h) singularity spectrum that somewhat masks the weakest singularities. This phase transition actually depends on the numberN of vanishing moments of the analyzing wavelet; its observation is emphasized as a reliable experimental test for the existence of nonsingular behavior in the considered signal. These theoretical results are illustrated with numerical examples. They are likely to be valid for a large class of fractal functions as suggested by recent applications to fractional Brownian motions and turbulent velocity signals.     

Continuous chromatographic separation of glucose-fructose mixtures using anion-exchange resins

Summary A semi-continuous, counter-current chromatographic refiner (SCCR) was packed with an anion resin in the bisulphite form and used to separate mixtures of sugars from different feedstocks. Products with purities of more than 99% were achieved and solid concentrations of the fructose products of 12% w/v were possible. The deterioration of the anion resin was controlled by simple precautions and thus its useful life was substantially prolonged.     

Effect of molecular orientation distribution and crystallinity on the measurement of the crystal lattice modulus of nylon 6 by x-ray diffraction

The crystal lattice modulus of nylon 6 (-type) was measured by x-ray diffraction using nylon 6 films drawn up to five times. The measured crystal lattice modulus was 173–175 GPa for all specimens whose crystallinity and the Young's modulus were beyond 46% and 3.75 GPa, respectively. These results indicate that a state of homogenous stress can be achieved. In contrast, the values were scattered for the speciments whose crystallinity and Young's modulus are less than the above values. To study the origin, a numerical calculation of the crystal lattice modulus, as measured by x-ray diffraction, was carried out by considering effects on the orientation factors of molecular chains and crystallinity. In this calculation, a previously introduced model was employed, in which oriented crystalline layers are surrounded by oriented amorphous phases so that the strains of the two phases at the boundary are identical. The theoretical results calculated by the introduced model indicated that the crystal lattice modulus by x-ray diffraction is almost equal to the intrinsic crystal modulus if the morphology of the test specimen can be represented as a series model. In contrast, if a parallel model is more appropriate, the difference between the measured modulus and the intrinsic value can be pronounced. Such morphological dependence was found to be less pronounced with increasing high degree of molecular orientation and crystallinity.     

Mechanical properties and molecular mobility in oriented polyolefines

The static modulus of elasticity (E) and the correlation time of rotation ( _c ) of 2,-2,6,6-tetramethylpiperidine-1-oxyl are studied as a function of the temperature (210<T <350°K) for oriented films of isotactic polypropylene and polyethylene of high and low density.E and _c change both upon heating and polymer orientation; this result indicates that sample properties are influenced by the microstructure of the amorphous phase where probes are localized.