The influence of the hall effect on the current structure in a plasma flow with a spatially periodic magnetic field

The structure of the electric fields and current is studied for stationary plasma flow in an axially symmetric, spatially periodic magnetic field. The problem is solved in the magnetohydrodynamic approximation with allowance for the Hall term in the generalized Ohm's law equation. It is assumed that the magnetic Reynolds number and the interaction parameter are small.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 11–16, September–October, 1972.The author thanks N. A. Khizhnyak and A. A. Kalmykov for useful discussions.     

Density distribution in rarefaction wave produced by rupture of diaphragm in a shock tube

The effect of the finite time of opening of the diaphragm in a shock tube on the formation of the rarefaction wave was investigated experimentally. The density distribution in the rarefaction wave was measured in relation to the coordinate and time and was compared with the known self-similar solution.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 130–131, March–April, 1972.     

Generalization of the Krook kinetic relaxation equation

One of the most significant achievements in rarefied gas theory in the last 20 years is the Krook model for the Boltzmann equation [1]. The Krook model relaxation equation retains all the features of the Boltzmann equation which are associated with free molecular motion and describes approximately, in a mean-statistical fashion, the molecular collisions. The structure of the collisional term in the Krook formula is the simplest of all possible structures which reflect the nature of the phenomenon. Careful and thorough study of the model relaxation equation [2–4], and also solution of several problems for this equation, have aided in providing a deeper understanding of the processes in a rarefied gas. However, the quantitative results obtained from the Krook model equation, with the exception of certain rare cases, differ from the corresponding results based on the exact solution of the Boltzmann equation. At least one of the sources of error is obvious. It is that, in going over to a continuum, the relaxation equation yields a Prandtl number equal to unity, while the exact value for a monatomic gas is 2/3.In a comparatively recent study [5] Holway proposed the use of the maximal probability principle to obtain a model kinetic equation which would yield in going over to a continuum the expressions for the stress tensor and the thermal flux vector with the proper viscosity and thermal conductivity.In the following we propose a technique for constructing a sequence of model equations which provide the correct Prandtl number. The technique is based on an approximation of the Boltzmann equation for pseudo-Maxwellian molecules using the method suggested by the author previously in [6], For arbitrary molecules each approximating equation may be considered a model equation. A comparison is made of our results with those of [5].     

Two-layer quasi-one-dimensional model for calculating gas-particle mixture flow in nozzles

The one-dimensional approximation is widely used at the present time to calculate gas-particle (solid or liquid) mixture flows in nozzles within the framework of the two-velocity (or multi-velocity) continuum model. Other studies have been made [1–6] in which the calculations of the two-phase flow in the supersonic part of the nozzle was made by the method of characteristics, and, within the limits of the model adopted, these results may be considered exact. Comparison of the exact and approximate results [6] has shown that even for nozzles of quite simple form (nearly conical) the accuracy of the one-dimensional approximation in the case of mixture flow is considerably lower than for the pure gas, and the computation error increases with increase in the relative particle flow rate. This deterioration of the accuracy is to a considerable degree caused by flow stratification, which arises because of particle lag and leads to the formation of a wall region of pure gas. For high particle content, the wall layer, in which the gas is not subjected to thermal and dynamic input from the particles, has the nature of a low-entropy, low-temperature, high-velocity layer with parameters which differ significantly from the gas parameters in the region occupied by the particles.Therefore, in the present study a modification was made in the one-dimensional theory, based on separate averaging of the flow in the wall layer and in the core, where the gas flows together with the foreign particles. Comparison of the exact results with those obtained with the aid of conventional one-dimensional theory and the proposed two-layer model showed that this modification of one-dimensional theory led to a considerable reduction in the errors of calculation for the flow parameters.In conclusion, the authors wish to thank S. Yu. Krasheninnikov for suggesting this study and also N. S. Galyun, A. M. Konkin, and L. P. Frolov for assistance in the investigation.     

Reservoir fluid flow through vertical fracture of thick stratum with underlying fluid interface

An exact solution is obtained for the problem of steady-state filtration of a heavy dense incompressible fluid in a thin, infinitely deep, inclined reservoir having a crack of given depth along the reservoir rise. The region of filtration of the lighter liquid (oil) has an impermeable upper boundary in the form of a horizontal fault line. Below the filtration region there is a free boundary, below which lies the region of stationary fluid (bottom water). The interface of the fluids, the fissure profile, and the reservoir fluid flow rate are determined from the solution of the problem on the basis of the given parameters (permeability of the reservoir and of the material filling the fissure, viscosity of the filtering fluid, specific weight of the upper and lower fluids, depth of the fissure, pressure differential between a point at the fissure and a point at the interface of the fluids). In the case when the thin reservoir is a vertical filtering layer, the considered flow is interpreted as the motion of the reservoir fluid through a vertical fissure of a thick reservoir (half-space) in the presence of an underlying fluid interface. The problem is solved in finite form with the aid of known analytic functions using integrals of the Cauchy type. The fundamental solution is first found of the special problem of flow with a point singularity. The fundamental solution is also of independent importance as an extension of the solution of certain known problems [1–4].     

Calibration and use of the Epprecht Rheomat

Summary This paper deals with the calibration and use of an Epprecht Rheomat with particular reference to non Newtonian liquids. A method of calculating accurate rates of shear for non-Newtonian flow is derived using theEuler-Maclaurin sum formula. Some examples are given of the calculation of shear of poly vinyl chlorideplasticiser suspensions.

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Zusammenfassung Die vorliegende Abhandlung befaßt sich mit der Eichung und dem Gebrauch eines Epprecht-Rheomats mit besonderer Hinsicht auf nicht-Newtonische Flüssigkeiten.Ein Verfahren zur exakten Berechnung von Scherwerten für nicht-Newtonische Flüssigkeiten läßt sich aus derEuler-Maclaurinschen Summenformel ableiten.Einige Beispiele zur Berechnung von Scherwerten für Weichmacher von Polyvinylchlorid sind beigefügt.

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Paper presented at the Conference on Experimental Rheology, University of Bradford, April 17–19, 1968.     

Internal-strain measurements of longitudinal pulses in conical bars

Longitudinal stress waves in a truncated 20-deg solid cone were investigated using embedded semiconductor strain gages. The cone, composed of an aluminum-filled epoxy, was struck normally at its small end with a 1/2-in.-diam steel ball traveling at a velocity of 170 jps. The results show the magnitude of the resulting stress wave to be nonuniform over a plane cross section perpendicular to the cone axis, the strain being greater at the center of the cone than near the surface, and the nonuniformity to increase with distance of travel from the impact end. The surface-strain measurements were compared with the one-dimensional theory of longitudinal waves in cones developed by Landon and Quinney as solved by Kenner and Goldsmith for a onehalf cycle sine-squared input pulse, and found to be in qualitative agreement with this theory, but to vary significantly in strain magnitude due to the strain nonuniformity over plane cross sections. The nonuniformity was compared with the Pochhammer-Chree theory for stress waves in cylindrical bars when that theory was evaluated for a cross section equivalent to the cone cross section. The trends of the deviations were similar, but the variations measured in the cone were consistently greater than that predicted by the theory.     

Stability of non isothermal flow of a film of viscous liquid on an inclined plane

A study is made of the stability of nonisothermal flow of a film of viscous liquid down an inclined plane under the influence of gravity with allowance for dissipation of energy in the flow. It is assumed that the liquid is incompressible, and that its physical properties do not depend on the temperature. On the free surface of the film, allowance is made for evaporation and condensation effects. The treatment is in the long-wavelength approximation of the method proposed by Yih Chia-shun [1]. The expression obtained for the critical Reynolds number at which the flow becomes unstable indicates that viscous dissipation plays a destabilizing part in a nonisothermal flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 145–148, July–August, 1979.     

Flows of dilute polymer solutions through packed porous chromatographic columns

Summary We have found that diffusion of polystyrene into the micropores of packed porous chromatographic columns of a type used for size separations of macromolecules is strongly influenced by the flow rate of the polymer solution through the column and also by the macromolecular size and internal pore size of the column packing. Dilute polymer solutions (0.05 wt.% polystyrene in THF) were pumped continously through the columns (Dupont SEC). Step changes in flow rate were made and the polymer concentration in the column effluent was monitored continuously. When the flow rate was increased, the partition coefficient for polymer between mobile and stationary phase shifted toward more polymer retention in the column, as evidenced by the drop in outlet concentration for several minutes following the step change. Molecular weight, pore size and solvent effects have all been thoroughly investigated. The conclusion is that polymer molecules diffuse into the internal pores of the column packing at a flow ratedependent rate and moreover, they do so to a greater extent the larger the polymer molecule, provided they are not sterically excluded from the pores. This dynamic diffusional partitioning is clearly important in its effect on resolution characteristics of an SEC column and also sheds some light on qualitatively similar observations in polymer flooding for enhanced oil recovery.

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Zusammenfassung Es wurde gefunden, daß die Diffusion von Polystyrol in die Mikroporen von porösen Packungen chromatographischer Säulen, wie sie für die Trennung von Makromolekeln verschiedener Teilchengrößen Verwendung finden, sowohl von der Fließgeschwindigkeit der Polymerlösung durch die Säule als auch von der Größe der Makromolekeln und der inneren Poren erheblich beeinflußt werden. Verdünnte Polymerlösungen (0,05 Gew.-% Polystyrol in THF) wurden kontinuierlich durch die Säulen (Dupont SEC) gepumpt. Dabei wurde die Fließgeschwindigkeit sprunghaft geändert und die Polymerkonzentration im Auslauf kontinuierlich registriert. Wenn die Durch-flußgeschwindigkeit erhöht wurde, verschob sich der Verteilungskoeffizient der Polymeren zwischen bewegter und stationärer Phase zu höheren Polymeranteilen in der Säule hin, was durch einen Konzentrationsabfall in der ausfließenden Lösung während einiger Minuten nach dem Geschwindigkeitssprung angezeigt wurde. Molekulargewicht, Porenweite und Lösungsmitteleinfluß wurden eingehend untersucht. Dabei ergab sich, daß die Polymermolekeln in die inneren Poren der Packung mit einer fließgeschwindigkeitsabhängigen Geschwindigkeit hineindiffundieren und dies in einem um so höheren Maße, je größer die Molekeln sind, vorausgesetzt, daß sie nicht aus sterischen Gründen zurückgehalten werden. Diese dynamische diffusive Verteilung ist natürlich von Bedeutung für die Trenncharakteristik einer GPC-Säule und wirft ebenso einiges Licht auf verschiedene qualitativ ähnliche Beobachtungen beim Fluten mit Polymeren zur Ausbeutesteigerung bei der Ölförderung.

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With 8 figures and 3 tables     

Die Berechnung des Blasenwachstums beim Verdampfen von Flüssigkeiten an Heizflächen als numerische Lösung der Erhaltungsgleichungen

A numerical procedure on the basis of the Marker and Cell-method [1] was developed in order to solve the conservation equations for mass, momentum and energy for the case of bubble growth on a heating surface. This procedure was used to calculate steam bubble growth on a horizontal stainless steel heating surface under saturated pool boiling conditions at a system pressure of 1 bar and different superheatings. The essential results obtained are:

-Good agreement was found between calculations and experiments concerning bubble growth rates, bubble shape and temperature field in the liquid surrounding the bubble.

-During its growth the bubble penetrates the temperature boundary layer formed in the liquid on the heating surface, simultaneously liquid is displaced aside.

-The microlayer evaporation fraction of the total bubble growth increases with growth time from 20 % to 50%.