Statistical hydromechanics of disperse systems. Part 2. Solution of the kinetic equation for suspended particles

To solve the kinetic equation for particles of a monodisperse two-phase mixture the method of successive approximations is developed; this resembles in its main features the well-known Chapman-Enskog method in the kinetic theory of gases. This method is applicable for a mixture whose state differs slightly from the equilibrium, i.e., when time and space derivatives of the dynamic variables describing the mean flow of both phases of the mixture are sufficiently small. Accordingly, the solution obtained is valid when the time and space scales of the mean flow exceed considerably those for random pseudo-turbulent motion of particles and a fluid. The conservation equations for determination of all the dynamic variables are formulated in approximations which have the same meaning as those of Euler and Navier & Stokes in hydromechanics of one-phase media.

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Direct Measurement of the Mean Flow Velocity Vector

Journal of Fluids Engineering ◽

10.1115/1.3448471 ◽

1976 ◽

Vol 98 (4) ◽

pp. 736-739 ◽

Cited By ~ 1

Author(s):

R. H. Kirchhoff ◽

R. M. Struziak

Keyword(s):

Flow Velocity ◽

Fundamental Frequency ◽

Velocity Vector ◽

Mean Flow ◽

Two Phase ◽

Response Equation ◽

The Mean ◽

Lock In ◽

Mean Flow Velocity

The response of a single inclined rotating hot wire anemometer was analyzed. The mean flow anemometer response equation was expanded in a Fourier Series about the fundamental frequency of rotation. Utilizing the d-c level and the first two harmonics of the response it is possible to construct the mean flow velocity vector within a solid angle determined by the mounting angle of the wire. The rotating anemometer response was measured using the technique of two phase lock-in detection to determine the first two harmonics and their phases relative to the fundamental frequency of rotation. Determination of the mean flow velocity vector using this technique was found to be feasible.

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Towards a mathematical model of limestone pavement vegetation. III. Estimation of the determinants of species frequencies

Canadian Journal of Botany ◽

10.1139/b70-210 ◽

1970 ◽

Vol 48 (7) ◽

pp. 1387-1404 ◽

Cited By ~ 13

Author(s):

G. A. Yarranton

Keyword(s):

Mathematical Model ◽

Multiple Regression ◽

Two Phase ◽

Surface Ph ◽

Expected Frequency ◽

Species Frequency ◽

Independent Variables ◽

The Mean ◽

Species Frequencies

Models of the determination of the frequencies of 31 species of bryophytes, lichens, and algae in limestone pavement grikes were calculated by multiple regression. Independent variables were parameters of microclimate and surface pH. Percentages of variance accounted for varied from 14 to 72. In most cases the balance is attributable to replicate errors in the estimates of species frequencies and the abundance of zeros in the data. The models were used to predict species frequencies in other grikes, and in most species the mean difference between predicted and observed frequencies was less than 10%. Determinants of species frequency were found to be similar for species growing near the bottoms of the grikes but differed for those occurring near the tops.It is suggested that a two-phase analysis would be more informative than straightforward regression. In phase (a) the conditions under which the expected frequency is nonzero would be delimited; in phase (b) multiple regression would be performed on the observations falling within this region. This proposal is shown to be related to Hutchinson's concepts of fundamental and realized niches. It would also eliminate the problem of the multiple zero.

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Determination of the Buckling Load of Struts with Successive Approximations

Journal of the Royal Aeronautical Society ◽

10.1017/s0368393100099442 ◽

1943 ◽

Vol 47 (387) ◽

pp. 103-105

Author(s):

J. Ratzersdorfer

Keyword(s):

Differential Equation ◽

Compressive Force ◽

Buckling Load ◽

Moment Of Inertia ◽

Successive Approximations ◽

Method Of Successive Approximations ◽

Exact Determination ◽

The Moment ◽

Image Position

In cases of tapered struts with hinged or built-in ends where the exact determination of the buckling load is complicated it may be useful to apply a method of successive approximations.Let us first consider a bar of the length l with hinged ends under the action of the compressive force P. The differential equation of the bending line becomeswhere v is the deflection at the section u, v with the moment of inertia I (u) and E is Young's modulus. At the ends of the bar the deflection v is equal to zero (Fig. I).

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Nozzle Admittance and Damping Analysis Using the LEE Method

International Journal of Turbo and Jet Engines ◽

10.1515/tjj-2015-0050 ◽

2017 ◽

Vol 34 (1) ◽

Cited By ~ 1

Author(s):

Wang Mu-xin ◽

Liu Pei-jin ◽

Yang Wen-jing ◽

Wei Xiang-geng

Keyword(s):

Theoretical Analysis ◽

Acoustic Energy ◽

Mean Flow ◽

Flow Properties ◽

Linearized Euler Equations ◽

Oscillation Modes ◽

Nozzle Size ◽

Half Angle ◽

The Mean

AbstractThe nozzle admittance is very important in the theoretical analysis of nozzle damping in combustion instability. The linearized Euler equations (LEE) are used to determine the nozzle admittance with consideration of the mean flow properties. The acoustic energy flux through the nozzle is calculated to evaluate the nozzle damping upon longitudinal oscillation modes. Then the parametric study, involving the nozzle convergent geometry, convergent half angle and nozzle size, is carried out. It is shown that the imaginary part of the nozzle admittance plays a non-negligible role in the determination of the nozzle damping. Under the conditions considered in this work (

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Resolution and Analysis of Enantiomers of Amphetamines by Liquid Chromatography on a Chiral Stationary Phase: Collaborative Study

Journal of AOAC INTERNATIONAL ◽

10.1093/jaoac/71.3.530 ◽

1988 ◽

Vol 71 (3) ◽

pp. 530-533

Author(s):

Marc C Alembik ◽

Irving W Wainer

Keyword(s):

Stationary Phase ◽

Chiral Stationary Phase ◽

Collaborative Study ◽

Accurate Method ◽

Enantiomeric Composition ◽

Amide Derivatives ◽

The Mean ◽

Bulk Drug ◽

Derivatives Of

Abstract A rapid, accurate method for separating and determining the enantiomeric composition of amphetamine bulk drug and commercial preparations was developed and subjected to collaborative study. Amide derivatives of the amphetamine enantiomers are formed by using achiral 2-naphthoyl chloride. The resulting enantiomeric amides are then chromatographed on a commercially available chiral stationary phase with hexane-isopropyl alcohol-acetonitrile (97 + 3 + 0.5) mobile phase, with detection at 254 nm. Seven collaborators received bulk drug and commercial samples of amphetamine. The collaborators and authors determined the mean percent /- and rf-amphetamine from 2 injections of each sample. The method can detect the presence of as little as 0.5% of the /-enantiomer in ^-amphetamine, with reproducibility between laboratories of ±71.3%. The method has been adopted official first action for determination of the enantiomeric composition of amphetamine bulk drug and preparations

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A Series-Solution Method for Cometary Orbits

Symposium - International Astronomical Union ◽

10.1017/s0074180900006240 ◽

1972 ◽

Vol 45 ◽

pp. 43-51

Author(s):

P. E. Nacozy

Keyword(s):

Series Solution ◽

Solution Method ◽

Successive Approximations ◽

Method Of Successive Approximations ◽

Reference Solution ◽

Successive Interval ◽

First Order ◽

Order Solution ◽

The Mean ◽

The Masses

A series-solution method for highly-eccentric perturbed orbits using a modified form of Hansen's method of partial anomalies is presented. Series in Chebyshev polynomials in the eccentric anomaly of a comet and the mean anomaly at an epoch of a planet provide a theory valid to first order with respect to the masses. The first-order solution becomes a reference solution about which higher-order perturbations are obtained by the method of successive approximations. The first-order solutions are valid approximations for long durations of time, whereas the higher orders are valid only over the interval of time that is selected for the Chebyshev expansions. The method is somewhat similar to Encke's method of special perturbations except that for each successive interval of time perturbations about a first-order solution are calculated instead of perturbations about a conic solution.

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Peristaltic Transport of a Particle-Fluid Suspension

Journal of Biomechanical Engineering ◽

10.1115/1.3168358 ◽

1989 ◽

Vol 111 (2) ◽

pp. 157-165 ◽

Cited By ~ 64

Author(s):

L. M. Srivastava ◽

V. P. Srivastava

Keyword(s):

Particle Concentration ◽

Amplitude Ratio ◽

Flow Reversal ◽

Spherical Particles ◽

Free Fluid ◽

Mean Flow ◽

Two Phase ◽

Flow Process ◽

Fluid Mixture ◽

The Mean

Peristaltic pumping by a sinusoidal traveling wave in the walls of a two-dimensional channel filled with a viscous incompressible fluid in which are distributed identical rigid spherical particles, is investigated theoretically. A perturbation solution is obtained which satisfies the momentum equations for the case in which amplitude ratio (wave amplitude/channel half width) is small. The results show that the fluid phase mean axial velocity decreases with increase in the particle concentration. The phenomenon of reflux (the mean flow reversal) is discussed. A reversal of velocity in the neighborhood of the centerline occurs when the pressure gradient is greater than that of the critical reflux condition. It is found that the critical reflux pressure is lower for the particle-fluid suspension than for the particle-free fluid. It is further observed that the mean flow reversal is strongly dependent on the particle concentration and the presence of particles in the fluid favors the reversal flow. A motivation of the present analysis has been the hope that such a theory of two-phase flow process is very useful in understanding the role of peristaltic muscular contraction in transporting bio-fluid behaving like a particle-fluid mixture. Also the theory is important to the engineering applications of pumping solid-fluid mixtures by peristalsis.

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CALCULATION OF POLYNOMIAL RATIO APPROXIMATED SURFACES OF BIFOCAL LENS

Issues of radio electronics ◽

10.21778/2218-5453-2018-4-29-33 ◽

2018 ◽

pp. 29-33

Author(s):

Yu. S. Nikulina ◽

A. V. Nikulin ◽

M. A. Stepanov

Keyword(s):

Polynomial Coefficient ◽

Successive Approximations ◽

Analytical Determination ◽

Method Of Successive Approximations ◽

Synthesis Methods ◽

Methods Of Synthesis ◽

Equality Condition ◽

Lens Surface ◽

Lens Center

General imperfection of existent synthesis methods of bifocal lens surface is use complexity emerging over lack of analytical solution. Method of successive approximations is relatively simple, its main problem is determination of power polynomial coefficient approximating illuminated and shadow surface of being synthesized lens. To solve this problem it is suggest analytical determination method of polynomial coefficient approximating illuminated and shadow surface of bifocal lens. This method based on equality condition of electric path length of edge beam and beam passing through lens center. It is obtained analytical equation which permit to determine interdependence polynomials coefficients approximating illuminated and shadow surface of bifocal lens. Moreover, been finded equation allow to calculate bifocal lens thickness, that impossible using known methods of synthesis methods of bifocal lens surface. Cross-section of bifocal lens was be calculated with being suggested analytical method.

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Dynamics of zonal flows: failure of wave-kinetic theory, and new geometrical optics approximations

Journal of Plasma Physics ◽

10.1017/s0022377816001021 ◽

2016 ◽

Vol 82 (6) ◽

Cited By ~ 16

Author(s):

Jeffrey B. Parker

Keyword(s):

Kinetic Equation ◽

Large Scale ◽

Zonal Flow ◽

Geometrical Optics ◽

Ultraviolet Divergence ◽

New Wave ◽

Mean Flow ◽

Zonal Flows ◽

The Mean ◽

Third Derivative

The self-organisation of turbulence into regular zonal flows can be fruitfully investigated with quasi-linear methods and statistical descriptions. A wave-kinetic equation that assumes asymptotically large-scale zonal flows leads to ultraviolet divergence. From an exact description of quasi-linear dynamics emerges two better geometrical optics approximations. These involve not only the mean flow shear but also the second and third derivative of the mean flow. One approximation takes the form of a new wave-kinetic equation, but is only valid when the zonal flow is quasi-static and wave action is conserved.

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