The stability of viscous fluid flow between rotating cylinders

1937 ◽  
Vol 33 (1) ◽  
pp. 41-61 ◽  
Author(s):  
S. Goldstein
Keyword(s):  
Angular Velocity ◽  
Wave Length ◽  
Steady Motion ◽  
Circular Cylinders ◽  
Critical Flow ◽  
Average Value ◽  
Time Period ◽  
Component Velocity ◽  
The Stability ◽  
Image Position

The stability of the motion of viscous incompressible fluid, of density ρ and kinematic viscosity ν, between two infinitely long coaxial circular cylinders, of radiiaanda+d, whered/ais small, is investigated mathematically by the method of small oscillations. The inner cylinder is rotating with angular velocity ω and the outer one with angular velocity αω, and there is a constant pressure gradient parallel to the axis. The fluid therefore has a component velocityWparallel to the axis, in addition to the velocity round the axis. A disturbance is assumed which is symmetrical about the axis and periodic along it. The critical disturbance, which neither increases nor decreases with the time, is periodic with respect to the time (except whenW= 0, when the critical disturbance is a steady motion). As Reynolds number of the flow we take ||d/ν, whereis the average value ofWacross the annulus, and we denote bylthe wave-length of the disturbance along the axis, by σ/2π the time period of the critical flow, bycthe wavelength of the critical flow, byωcthe critical value of ω, and we putapproximately, if α is not nearly equal to 1.

On the disturbance of the steady flow of an inviscid liquid between parallel planes

1934 ◽  
Vol 234 (732) ◽  
pp. 43-78 ◽  
Keyword(s):  
Mathematical Proof ◽  
Fluid Motion ◽  
Steady Motion ◽  
Circular Cylinders ◽  
General Validity ◽  
Inviscid Liquid ◽  
Characteristic Value ◽  
Characteristic Values ◽  
The Stability ◽  
Liquid Flows

1—In a number of papers dealing with the stability of fluid motion, RAYLEIGH employed a certain method, which we may refer to as the “characteristic-value” method. For some problems this method gives results in agreement with observation. For example, it establishes that a heterogeneous inviscid liquid at rest under gravity is stable if the density decreases steadily as we pass upward; it establishes that an inviscid liquid rotating between concentric circular cylinders is stable if, and only if, the square of the circulation increases steadily as we pass outward. This result was stated by RAYLEIGH, and its validity appears to be confirmed by the experiments of TAYLOR, but a simple < d 2 u 0 /dy 2 retains the same sign throughout the liquid, u 0 being the velocity in the steady motion and y the distance from one of the planes. This result is deduced from the fact that mathematical proof by the characteristic value method was not given. I have recently supplied such a proof, extending the problem to include a heterogeneous liquid. But when the method is applied to some other problems, the situation is not so satisfactory. Among the results to which Rayleigh was led is the following. If an inviscid liquid flows between parallel planes, the motion is stable if the characteristic values of a parameter in a certain differential equation cannot be complex, the implication being that they are therefore real. Rayleigh further claimed that the method established the stability of a uniform shearing motion, for which d 2 u 0 /dy 2 =0. KELVIN and LOVE criticized the method, and a review of the situation in 1907 was given by ORR. In spite of the fact that its general validity remains obscure, the characteristic-value method has been widely employed. It is not the purpose of the present paper to attempt to justify or to discredit the characteristic-value method in general. The paper deals only with the simplest of all stability problems, that of an inviscid liquid flowing between fixed parallel planes. In §2 the method is discussed in some detail and in §3 an argument is developed to show that Rayleigh’s criterion for stability, mentioned above, cannot be legitimately deduced by his method. He proved that complex characteristic values are impossible, and I now prove that real characteristic values are also impossible. The conclusion to be drawn is that the characteristic-value method is not applicable to this case.

The flow due to a rotating disc

1934 ◽  
Vol 30 (3) ◽  
pp. 365-375 ◽  
Author(s):  
W. G. Cochran
Keyword(s):  
Angular Velocity ◽  
Viscous Fluid ◽  
Equations Of Motion ◽  
Steady Motion ◽  
Constant Angular Velocity ◽  
Incompressible Viscous Fluid ◽  
Polar Coordinates ◽  
Rotating Disc ◽  
Cylindrical Polar Coordinates ◽  
Image Position

1. The steady motion of an incompressible viscous fluid, due to an infinite rotating plane lamina, has been considered by Kármán. If r, θ, z are cylindrical polar coordinates, the plane lamina is taken to be z = 0; it is rotating with constant angular velocity ω about the axis r = 0. We consider the motion of the fluid on the side of the plane for which z is positive; the fluid is infinite in extent and z = 0 is the only boundary. If u, v, w are the components of the velocity of the fluid in the directions of r, θ and z increasing, respectively, and p is the pressure, then Kármán shows that the equations of motion and continuity are satisfied by taking

Is there a “universal” MAC equation?

1990 ◽  
Vol 48 (2) ◽  
pp. 228-229
Author(s):  
Robert E. Ogilvie
Keyword(s):  
Absorption Coefficient ◽  
Atomic Absorption ◽  
Atomic Number ◽  
X Ray ◽  
Time Period ◽  
Image Position

The search for an empirical absorption equation begins with the work of Siegbahn (1) in 1914. At that time Siegbahn showed that the value of (μ/ρ) for a given element could be expressed as a function of the wavelength (λ) of the x-ray photon by the following equationwhere C is a constant for a given material, which will have sudden jumps in value at critial absorption limits. Siegbahn found that n varied from 2.66 to 2.71 for various solids, and from 2.66 to 2.94 for various gases.Bragg and Pierce (2) , at this same time period, showed that their results on materials ranging from Al(13) to Au(79) could be represented by the followingwhere μa is the atomic absorption coefficient, Z the atomic number. Today equation (2) is known as the “Bragg-Pierce” Law. The exponent of 5/2(n) was questioned by many investigators, and that n should be closer to 3. The work of Wingardh (3) showed that the exponent of Z should be much lower, p = 2.95, however, this is much lower than that found by most investigators.

Utilization of remote sensing for estimation of scale of cutting and growing of forest zones

2017 ◽  
Vol 920 (2) ◽  
pp. 57-60
Author(s):  
F.E. Guliyeva
Keyword(s):  
Remote Sensing ◽  
Fixed Time ◽  
Time Interval ◽  
Time Points ◽  
Average Value ◽  
Time Period ◽  
Remote Estimation ◽  
The Given ◽  
Forest Cutting

The study of results of relevant works on remote sensing of forests has shown that the known methods of remote estimation of forest cuts and growth don’t allow to calculate the objective average value of forests cut volume during the fixed time period. The existing mathematical estimates are not monotonous and make it possible to estimate primitively the scale of cutting by computing the ratio of data in two fixed time points. In the article the extreme properties of the considered estimates for deforestation and reforestation models are researched. The extreme features of integrated averaged values of given estimates upon limitations applied on variables, characterizing the deforestation and reforestation processes are studied. The integrated parameter, making it possible to calculate the averaged value of estimates of forest cutting, computed for all fixed time period with a fixed step is suggested. It is shown mathematically that the given estimate has a monotonous feature in regard of value of given time interval and make it possible to evaluate objectively the scales of forest cutting.

Stability of Ring-Type MEMS Gyroscopes Subjected to Stochastic Angular Speed Fluctuation

2017 ◽  
Vol 139 (4) ◽  
Author(s):  
Samuel F. Asokanthan ◽  
Soroush Arghavan ◽  
Mohamed Bognash
Keyword(s):  
Angular Velocity ◽  
Degrees Of Freedom ◽  
Microelectromechanical Systems ◽  
Damping Ratio ◽  
Operating Conditions ◽  
System Stability ◽  
System Response ◽  
Ring Type ◽  
Mems Gyroscopes ◽  
The Stability

Effect of stochastic fluctuations in angular velocity on the stability of two degrees-of-freedom ring-type microelectromechanical systems (MEMS) gyroscopes is investigated. The governing stochastic differential equations (SDEs) are discretized using the higher-order Milstein scheme in order to numerically predict the system response assuming the fluctuations to be white noise. Simulations via Euler scheme as well as a measure of largest Lyapunov exponents (LLEs) are employed for validation purposes due to lack of similar analytical or experimental data. The response of the gyroscope under different noise fluctuation magnitudes has been computed to ascertain the stability behavior of the system. External noise that affect the gyroscope dynamic behavior typically results from environment factors and the nature of the system operation can be exerted on the system at any frequency range depending on the source. Hence, a parametric study is performed to assess the noise intensity stability threshold for a number of damping ratio values. The stability investigation predicts the form of threshold fluctuation intensity dependence on damping ratio. Under typical gyroscope operating conditions, nominal input angular velocity magnitude and mass mismatch appear to have minimal influence on system stability.

Development and Effects of Super-Critical Taylor-Vortex Flow in a Lightly Loaded Journal Bearing

1974 ◽  
Vol 96 (1) ◽  
pp. 28-35 ◽  
Author(s):  
R. C. DiPrima ◽  
J. T. Stuart
Keyword(s):  
Vortex Flow ◽  
Axial Direction ◽  
Basic Flow ◽  
Circular Cylinders ◽  
Taylor Vortex ◽  
Taylor Vortices ◽  
Taylor Vortex Flow ◽  
Secondary Motion ◽  
The Stability ◽  
Unified Description

At sufficiently high operating speeds in lightly loaded journal bearings the basic laminar flow will be unstable. The instability leads to a new steady secondary motion of ring vortices around the cylinders with a regular periodicity in the axial direction and a strength that depends on the azimuthial position (Taylor vortices). Very recently published work on the basic flow and the stability of the basic flow between eccentric circular cylinders with the inner cylinder rotating is summarized so as to provide a unified description. A procedure for calculating the Taylor-vortex flow is developed, a comparison with observed properties of the flow field is made, and formulas for the load and torque are given.

Stability of a Rigid Body With an Oscillating Particle: An Application of MACSYMA

1985 ◽  
Vol 52 (3) ◽  
pp. 686-692 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Angular Velocity ◽  
Rigid Body ◽  
Classical Problem ◽  
Moment Of Inertia ◽  
Model Parameters ◽  
Stability Chart ◽  
The Stability ◽  
Transition Curves ◽  
Minimum Moment ◽  
Small Angular Velocity

This problem is a generalization of the classical problem of the stability of a spinning rigid body. We obtain the stability chart by using: (i) the computer algebra system MACSYMA in conjunction with a perturbation method, and (ii) numerical integration based on Floquet theory. We show that the form of the stability chart is different for each of the three cases in which the spin axis is the minimum, maximum, or middle principal moment of inertia axis. In particular, a rotation with arbitrarily small angular velocity about the maximum moment of inertia axis can be made unstable by appropriately choosing the model parameters. In contrast, a rotation about the minimum moment of inertia axis is always stable for a sufficiently small angular velocity. The MACSYMA program, which we used to obtain the transition curves, is included in the Appendix.

scholarly journals Characteristics and Kinetics of Rosin Pentaerythritol Ester via Oxidation Process under Ultraviolet Irradiation

Molecules ◽  
2018 ◽  
Vol 23 (11) ◽  
pp. 2816 ◽  
Author(s):  
Yuanlin Li ◽  
Xiongmin Liu ◽  
Qiang Zhang ◽  
Bo Wang ◽  
Chang Yu ◽  
...  
Keyword(s):  
Light Intensity ◽  
Oxidation Kinetics ◽  
Pseudo First Order Reaction ◽  
Initial Oxidation ◽  
Photo Oxidation ◽  
Average Value ◽  
Pentaerythritol Ester ◽  
Fundamental Information ◽  
The Stability ◽  
Kinetics Of

A self-designed reaction device was used as a promising equipment to investigate the oxidation characteristics and kinetics of rosin pentaerythritol ester (RPE) under UV irradiation. Photo-oxidation kinetics and the initial quantum yield (Φ) of RPE were calculated. The initial oxidation product of the photo-oxidation reaction—peroxide was analyzed by iodimetry. The peroxide concentration is related to the light intensity (I) and the temperature (T), and the increasing T and I would destabilize the RPE by accelerating peroxide forming. Photo-oxidation of RPE follows the pseudo first-order reaction kinetics. The relationship between activation energy and logarithm of light intensity (ln I) is linear, and it is expressed as Ea = −4.937ln I + 45.565. Φ was calculated by the photo-oxidation kinetics, and the average value of Φ was 7.19% in the light intensity range of 200–800 μW cm−2. This research can provide fundamental information for application of RPE, and help obtain a better understanding of the stability of rosin esters.

scholarly journals Tensile Strength of NaCl Ice

1962 ◽  
Vol 4 (31) ◽  
pp. 25-52 ◽  
Author(s):  
W. F. Weeks
Keyword(s):  
Tensile Strength ◽  
Sea Ice ◽  
Fresh Water ◽  
Eutectic Point ◽  
Water Ice ◽  
Average Value ◽  
Freshwater Ice ◽  
The One ◽  
Image Position ◽  
Bodies Of Water

AbstractTo resolve some of the factors causing strength variation in natural sea ice, fresh water and five different NaCl–H2O solutions were frozen in a tank designed to simulate the one-dimensional cooling of natural bodies of water. The resulting ice was structurally similar to lake and sea ice. The salinity of the salt ice varied from 1‰ to 22‰. Tables of brine volumes and densities were computed for these salinities in the temperature range 0° to −35° C. The ring-tensile strength σ of fresh-water ice was found to be essentially temperature independent from −10° to −30°C., with an average value of 29.6±8.5 kg./cm.2at −10° C. The strength of salt ice at temperatures above the eutectic point (–21.2° C.) significantly decreases with brine volumev;. The σ–axis intercept of this line is comparable to the a values determined for fresh ice indicating that there is little, if any, difference in stress concentration between sea and lake ice as a result of the presence of brine pockets. The strength of ice containing NaCl.2H2O is slightly less than the strength of freshwater ice and is independent of the volume of solid salt and the ice temperature. No evidence was found for the existence of either phase or geometric hysteresis in NaCl ice. The strength of ice at sub-eutectic temperatures, however, is decreased appreciably if the ice has been subjected to temperatures above the eutectic point; this is the result of the redistribution of brine during the warm-temperature period. Short-term cooling produces an appreciable (20 per cent) decrease in strength, in fresh-water and NaCl.2H2O ice. The present results are compared with tests on natural sea ice and it is suggested that the strength of freshwater ice is a limit which is approached but not exceeded by cold sea ice and that the reinforcement of brine pockets by Na2SO4.10H2O is either lacking or much less than previously assumed.

2. On the Rotation of the Plane of Polarization by Quartz, and its relation to Wave-Length

1882 ◽  
Vol 11 ◽  
pp. 815-818 ◽  
Author(s):  
W. Peddie
Keyword(s):  
Wave Length ◽  
Angular Rotation ◽  
Light Rays ◽  
Image Position ◽  
Polarization Of Light

The angular rotation of the plane of polarization of light-rays in their passage through quartz is a function of the wave-length, and is roughly represented by the formulawhere A is a constant depending on the quartz. This formula is only approximate, however, and one object of the experiments described below was to ascertain how closely the rotation might be represented by three terms of the equation

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