The Differentiation Method in Rheology: II. Characteristic Derivatives of Ideal Models in Poiseuille Flow

1962 ◽  
Vol 2 (04) ◽  
pp. 309-316 ◽  
Author(s):  
J.G. Savins ◽  
G.C. Wallick ◽  
W.R. Foster
Keyword(s):  
Poiseuille Flow ◽  
Integration Method ◽  
Historical Review ◽  
Real Data ◽  
Background Information ◽  
Characteristic Functions ◽  
Integration Methods ◽  
Rheological Analysis ◽  
Response Characteristics ◽  
Differentiation Method

Abstract Inasmuch as the differentiation and integration methods represent different modes of rheological analyses, a dual scheme of analysis using both methods should lead to a generalized method of data analysis. A dual differentiation. integration method of analysis is applied here to the Poiseuille flow of a variety of ideal Generalized Newtonian and viscoplastic models. Using machine processing techniques, the result is a spectrum of response patterns which are expressed in terms of certain derivative functions. It is shown that these characteristic functions form the basis of a highly-sensitive analytic technique for optimizing the selection of the most appropriate functional relationship between shear rate and shearing stress. Introduction In the first paper of this series, Savins, Wallick and Foster presented an historical review of the salient features of the differentiation method of rheological analysis in Poiseuille flow, and also indicated how the method could be applied to problems involving plane Poiseuille flow. It was shown that the differentiation and integration methods, although basically not incompatible, do represent different modes of rheological analysis. This suggests that valuable background information regarding the probable response characteristics of real data obtained with the differentiation method can be obtained from an integration method-differentiation method analysis of the response of a variety of ideal rheological models. The present paper describes how this dual method of analysis has been applied to suites of idealized models representing a wide variety of Generalized Newtonian and viscoplastic behavior which have received attention at various places in the literature. THEORETICAL CONSIDERATIONS NEWTONIAN LIQUID For a liquid of constant viscosity .............................(1) By substituting Eq. 1 in Eq. 9 of Ref. 1 and integrating, it is easily shown that for Poiseuille flow .......................(2) and, hence, ....................(3) ...........................(4) GENERALIZED NEWTONIAN SYSTEMS Odd Power This model is of the form ......................(5a) Note that it represents a Maclaurin-type series expansion, based on the Newtonian model, which is restricted to the odd powers of the stress. SPEJ P. 309^

The Differentiation Method in Rheology: I. Poiseuille-Type Flow

1962 ◽  
Vol 2 (03) ◽  
pp. 211-215 ◽  
Author(s):  
J.G. Savins ◽  
G.C. Wallick ◽  
W.R. Foster
Keyword(s):  
Integral Equation ◽  
Data Analysis ◽  
Integration Method ◽  
Flow Behavior ◽  
Rheological Parameters ◽  
Flow Properties ◽  
Ideal Model ◽  
Rheological Analysis ◽  
Differentiation Method ◽  
General Method

Abstract A comprehensive review of the salient features of the differentiation method of rheological analysis in Poiseuille flow from its inception circa 1928 is presented. Here no initial assumptions regarding the nature of the function relating rheological parameters to observed kinematical and dynamical parameters are required in the data-analysis process. In contrast, the integration method involves interpreting flow properties in terms of a particular ideal model. It is shown that, although both methods represent modes of solution of the same integral equation, being relatively bias-free, the differentiation method offers a more discriminating procedure for rheological analysis. The application to problems involving plane Poiseuille flow is also described. Introduction In most instances, the approach to the problem of interpreting the rheological properties of various compositions as they ate affected by changes in chemical or physical environment, as saying the characteristics of a particular constituent of a suspension, analyzing flow behavior in terms of interactions between components in a system, to cite but a few examples, has been in terms of what Hersey terms the integration method. Briefly, it consists of interpreting flow properties in terms of a particular ideal model. The usual practice of the integration method is to choose a model with a minimum number of parameters because, other things being equal, it is desirable to use the simplest model which will describe the behavior of a real material and yet be mathematically tract able for the requirements of data analysis. This expression is then substituted into an equation which relates observed kinematical and dynamical quantities, such as volume flux Q and pressure gradient J, and angular velocity and torque T, in a capillary and concentric cylinder apparatus, respectively. The rheological parameters appear on integrating, in an expression relating the pairs of observable quantities such as those just given. In many instances a particular model provides a good representation of rheological behavior over a reasonable range of compositional and environmental changes. just as often, however, it is obvious that the interpretation of rheological changes by the integration method is not providing realistic information about changes in flow behavior. A more general method of interpreting rheological data for a given material is to make no initial assumptions regarding the nature of the function relating rheological parameters to observed kinematical and dynamical quantities, e.g., flow rate and pressure drop in capillary flow or angular velocity and torque in a rotational viscometer. This general method Hersey terms the differentiation method. Instead of integrating, one differentiates the integral equation with respect to one of the limits, i.e., one of the boundary conditions; the resulting expression contains the same observable quantities just given, their derivatives, and the rheological function evaluated at that boundary. By obtaining these derivatives from experimental ‘data, graphically or by a computer routine, they can be substituted into the differential equation and a graphical form of the function derived. THEORY OF THE DIFFERENTIATION METHOD FOR POISEUILLE-TYPE FLOWS In this introductory paper, two flow cases which are important in viscometry are considered (one for the first time) from the differentiation method of analysis, flow in a cylindrical tube and flow between fixed parallel surfaces of infinite extent, the basic integral equations being formulated in a manner analogous to the way they originally appeared in the literature. In addition, the following ideal conditions will be assumed:an absence of anomalous wall effects,isotropic behavior everywhere, andsteady laminar flow conditions. SPEJ P. 211^

scholarly journals Investigating the optimal integration of airborne, ship-borne, satellite and terrestrial gravity data for use in geoid determination

2021 ◽  
Author(s):  
◽  
Rachelle Winefield
Keyword(s):  
Least Squares ◽  
Gravity Field ◽  
Integration Method ◽  
Bouguer Anomaly ◽  
Least Squares Collocation ◽  
Integration Methods ◽  
Case Study Area ◽  
Gravity Observation ◽  
Gravity Map

<p>Each gravity observation technique has different parameters and contributes to different pieces of the gravity spectrum. This means that no one gravity dataset is able to model the Earth’s gravity field completely and the best gravity map is one derived from many sources. Therefore, one of the challenges in gravity field modelling is combining multiple types of heterogeneous gravity datasets.  The aim of this study is to determine the optimal method to produce a single gravity map of the Canterbury case study area, for the purposes of use in geoid modelling.  This objective is realised through the identification and application of a four-step integration process: purpose, data, combination and assessment. This includes the evaluation of three integration methods: natural neighbour, ordinary kriging and least squares collocation.  As geoid modelling requires the combination of gravity datasets collected at various altitudes, it is beneficial to be able to combine the dataset using an integration method which operates in a three-dimensional space. Of the three integration methods assessed, least squares collocation is the only integration method which is able to perform this type of reduction.  The resulting product is a Bouguer anomaly map of the Canterbury case study area, which combines satellite altimetry, terrestrial, ship-borne, airborne, and satellite gravimetry using least squares collocation.</p>

Analysis of the characteristics‐integration method for one‐dimensional wave inversion

Geophysics ◽  
1988 ◽  
Vol 53 (8) ◽  
pp. 1034-1044 ◽  
Author(s):  
Nei‐Mao Chen ◽  
Yu‐Hua Chu ◽  
John T. Kuo
Keyword(s):  
Wave Equation ◽  
Inverse Method ◽  
Integration Method ◽  
Low Frequency ◽  
Real Data ◽  
Frequency Component ◽  
One Dimensional ◽  
Reflection Data ◽  
Wave Inversion ◽  
Impedance Profile

Basing our work on the one‐dimensional (1-D) wave equation, we present an inverse method which we call the characteristics‐integration method. The method is derived from integration along characteristic families of straight lines of the wave equation in the time domain. With the source function known and reflection data recorded on the surface, the characteristics‐integration method can efficiently and economically recover the subsurface impedance profile, provided that the structure is inhomogeneous only in the depth direction. In general, when seismic data are contaminated by noise, the characteristics‐integration method, like any other 1-D inverse method, suffers from instability. We find that, for a smoothly varying impedance profile, the instability of inversions using characteristic methods depends heavily on the bandwidth of the source wavelet. We devised a resampling technique to stabilize the inverse scheme and to suppress the growth of errors. Numerical examples, including data contaminated by noise, data missing the low‐frequency component, and real data cases, show the feasibility of recovering impedance profiles using the characteristics‐integration method.

The `seed-skewness' integration method generalized for three-dimensional Bragg peaks

2003 ◽  
Vol 36 (6) ◽  
pp. 1475-1479 ◽  
Author(s):  
J. Peters
Keyword(s):  
Statistical Analysis ◽  
High Resolution ◽  
Integration Method ◽  
A Priori ◽  
Reciprocal Lattice ◽  
Three Dimensional ◽  
Real Data ◽  
Area Detector ◽  
Peak Integration ◽  
Electronic Detector

The integration of the three-dimensional profile of each node of the reciprocal lattice without ana priorimodelling of the shape of the reflections is a prerequisite in order to improve the capability of area detectors in diffraction studies. Bolotovskyet al.[J. Appl. Cryst.(1995),28, 86–95] published a new method of area-detector peak integration based on a statistical analysis of pixel intensities and suggested its generalization for processing of high-resolution three-dimensional electronic detector data. This has been done in the present work, respecting the special requirements of data collected from neutron diffraction. The results are compared with other integration methods. It is shown that the seed-skewness method is successful in giving very reliable results and simultaneously optimizes the standard deviations. The integration procedures are applied to real data, which are refined and compared with benchmark results.

scholarly journals DEVELOPMENT OF INTEGRATION AND ADJUSTMENT METHOD FOR SEQUENTIAL RANGE IMAGES

2015 ◽  
Vol XL-4/W5 ◽  
pp. 177-181
Author(s):  
K. Nagara ◽  
T. Fuse
Keyword(s):  
Point Cloud ◽  
Integration Method ◽  
Three Dimensional ◽  
Real Data ◽  
Bundle Adjustment ◽  
Single Shot ◽  
Range Image ◽  
Range Images ◽  
Error Adjustment ◽  
Iterative Closest Point Algorithm

With increasing widespread use of three-dimensional data, the demand for simplified data acquisition is also increasing. The range camera, which is a simplified sensor, can acquire a dense-range image in a single shot; however, its measuring coverage is narrow and its measuring accuracy is limited. The former drawback had be overcome by registering sequential range images. This method, however, assumes that the point cloud is error-free. In this paper, we develop an integration method for sequential range images with error adjustment of the point cloud. The proposed method consists of ICP (Iterative Closest Point) algorithm and self-calibration bundle adjustment. The ICP algorithm is considered an initial specification for the bundle adjustment. By applying the bundle adjustment, coordinates of the point cloud are modified and the camera poses are updated. Through experimentation on real data, the efficiency of the proposed method has been confirmed.

The Clinical Utility of Psychometric Tests

2021 ◽  
pp. 304-323
Author(s):  
Luis Anunciação ◽  
Marco A. Arruda ◽  
J. Landeira-Fernandez
Keyword(s):  
Clinical Utility ◽  
Historical Review ◽  
Real Data ◽  
Statistical Decision Theory ◽  
Likelihood Ratios ◽  
Statistical Decision ◽  
Predictive Values ◽  
Receiver Operating Characteristic Curves ◽  
Wide Range ◽  
Sensitivity Specificity

The clinical utility of a measure involves its ability to support a wide range of decisions that enhance its pragmatism and use. Although several statistics are part of this feature, one centerpiece of this concept is the ability of an instrument to provide cutoff scores that can accurately discriminate between groups that consist of patients and non-patients. This latter aspect leads to such concepts as sensitivity, specificity, positive and negative predictive values and likelihood ratios, accuracy, and receiver operating characteristic curves. This chapter addresses these topics from two perspectives. First, because these features of clinical utility are encompassed as a subfield of statistical decision theory, the authors provide a historical review that links null hypothesis significance testing (NHST), signal detection theory (SDT), and psychological testing. Second, a real-data approach is used to demonstrate these concepts. Additionally, a free software program was developed to present these concepts.

The Differentiation Method in Rheology: III. Couette Flow

1963 ◽  
Vol 3 (01) ◽  
pp. 14-18 ◽  
Author(s):  
J.G. Savins ◽  
G.C. Wallick ◽  
W.R. Foster
Keyword(s):  
Integral Equation ◽  
Difference Equation ◽  
Yield Stress ◽  
Couette Flow ◽  
Yield Point ◽  
Poiseuille Flow ◽  
Shearing Stress ◽  
Differentiation Method ◽  
Type Flow ◽  
Flow Experiment

Abstract The theory of the differentiation method for the Couette flow experiment is reviewed. Particular attention is given to the requirements on data analyses in the case of the class of non-Newtonian materials described as viscoplastics, i. e., materials characterized by a yield point or yield stress. Here, changes in boundary conditions arise when the shearing stress attains a critical value with the result that the form of the basic integral equation for Couette flow is determined by the flow conditions existing during the measurement. Introduction In the preceding papers in this series, the salient features of the differentiation method of rheological analysis in Poiseuille-type flow were discussed. It was shown that a dual differentiation- integration method analysis of the Poiseuille flow of idealized generalized Newtonian and visco-plastic models could be used to develop a spectrum of highly sensitive response patterns in terms of certain characteristic derivative functions. These functions were shown to optimize the selection of the most appropriate functional relationship between f(p) and p from the Poiseuille flow experiment. The present paper reviews the theory of the differentiation method as applied to the equally important Couette flow experiment. We will also discuss the range of variables over which the basic integral equation for Couette flow is applicable when the non-Newtonian material is of the viscoplastic type, i.e., characterized by a yield point or yield stress. THEORY Having described the application of the differentiation method to Poiseuille-type flow in the preceding papers, we now proceed to the case where the test liquid is confined to the annular space between coaxial cylinders of length L, one of which is in motion, i.e., Couette flow, formulating the basic integral equation after the method of Mooney. The observed kinematical and dynamical quantities are the angular velocityand the torque T. Here, the one nonvanishing component of the shear-rate tensor is ........................(1) and the corresponding component of the shearing-stress tensor at any point r is given by ..........................(2) The shearing stresses at the inner surface of radius R(1) and the outer surface of radius R(2) are related by .................(3) Combining Eqs. 1, 2 and 3, letting = 0 at p = p1 and = at p = p2 and integrating yield .........................(4) Note that the definite integral has a finite lower limit. Differentiating Eq. 4 with respect to p1, following the rule of Leibnitz (i.e., in Eq. 11 of Ref. 1), gives a difference equation in the desired function ..................(5) This result was initially obtained by Mooney who used it as a starting point for an approximate solution. Several other approximate solutions of the difference equation have been described, the principal results of which are described in the succeeding sections. The interested reader is referred to the original papers for the details. SPEJ P. 14^

The Differentiation Method in Rheology: IV. Characteristic Derivatives of Ideal Models in Couette Flow

1963 ◽  
Vol 3 (02) ◽  
pp. 177-184 ◽  
Author(s):  
J.G. Savins ◽  
G.C. Wallick ◽  
W.R. Foster
Keyword(s):  
Couette Flow ◽  
Poiseuille Flow ◽  
Integration Method ◽  
Flow Behavior ◽  
Preceding Paper ◽  
Applicable Range ◽  
Dual Differentiation ◽  
Viscoplastic Models ◽  
Basic Equations ◽  
Processing Techniques

Abstract The dual differentiation-integration method of rheological analysis is applied to Couette flow. Using machine processing techniques, a spectrum of characteristic derivative functions for a variety of ideal Generalized Newtonian and viscoplastic models has been developed: As for the case of Poiseuille flow the functions for Generalized Newtonian models as a class are strikingly different from the corresponding functions for the class of viscoplastic models. It is shown that this dual scheme of analysis is a highly sensitive analytic method for determining the applicable range of a rheological model. The unique characteristics of this function in the case of viscoplastics may lead to a more precise detection and evaluation of the yield point or yield stress in these materials. Introduction It has been shown that the dual differentiation- integration method of analysis is a discriminating and flexible method for interpreting flow behavior and determining the applicable range of a particular model from Poiseuille flow experiments. In the preceding paper the basic equations governing the Couette flow experiment were reviewed and the significance of changes in boundary conditions on data analysis in the case of the viscoplastic discussed. In this paper the dual method of analysis is applied to Couette flow using the suites of rheological models considered previously. THEORETICAL CONSIDERATIONS NEWTONIAN LIQUID f (p) = p......................(1) Substituting Eq. 1 in Eq. 4 of Ref. 3 and integrating yields Dv = p1........................(2) where Dv is the nominal shear rate at R1, viz:R2, and B is the radii ratio. Here R1 ............(3) ...................(4) ....................(5) GENERALIZED NEWTONIAN SYSTEMS Odd Power ..........(6) Integration after substitution of Eq. 6 in Eq. 4 of Ref. 3 yields ......(7) and hence, ...(8) ........(9) = .............(10) Series (General) ............(11) SPEJ P. 177^

scholarly journals Semi-numerical analysis of a two-stage series composite planetary transmission considering incremental harmonic balance and multi-scale perturbation methods

2021 ◽  
Vol 12 (2) ◽  
pp. 701-714
Author(s):  
Xigui Wang ◽  
Siyuan An ◽  
Yongmei Wang ◽  
Jiafu Ruan ◽  
Baixue Fu
Keyword(s):  
Frequency Response ◽  
Harmonic Balance ◽  
Integration Method ◽  
Numerical Integration Method ◽  
Superharmonic Resonance ◽  
Response Characteristics ◽  
Multi Scale ◽  
Incremental Harmonic Balance ◽  
Scale Perturbation ◽  
Planetary Transmission

Abstract. This study conducts an analytical investigation of the dynamic response characteristics of a two-stage series composite system (TsSCS) with a planetary transmission consisting of dual-power branches. An improved incremental harmonic balance (IHB) method, which solves the closed solution of incremental parameters passing through the singularity point of the analytical path, based on the arc length extension technique, is proposed. The results are compared with those of the numerical integration method to verify the feasibility and effectiveness of the improved method. Following that, the multi-scale perturbation (MsP) method is applied to the TsSCS proposed in this subject to analyze the parameter excitation and gap nonlinear equations and then to obtain the analytical frequency response functions including the fundamental, subharmonic, and superharmonic resonance responses. The frequency response equations of the primary resonance, subharmonic resonance, and superharmonic resonance are solved to generate the frequency response characteristic curves of the planetary gear system (PGS) in this method. A comparison between the results obtained by the MsP method and the numerical integration method proves that the former is ideal and credible in most regions. Considering the parameters of TsSCS excitation frequency and damping, the nonlinear response characteristics of steady-state motion are mutually converted. The effects of the time-varying parameters and the nonlinear deenthing caused by the gear teeth clearance on the amplitude–frequency characteristics of TsSCS components are studied in this special topic.

scholarly journals Integration method of non-elementary exponential functions using iterated Fubinni integrals

2021 ◽  
Vol 9 (9) ◽  
pp. 169-177
Author(s):  
Odirley Willians Miranda Saraiva ◽  
Gustavo Nogueira Dias ◽  
Fabricio da Silva Lobato ◽  
José Carlos Barros de Souza Júnior ◽  
Washington Luiz Pedrosa da Silva Junior ◽  
...  
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Exponential Function ◽  
Taylor Series ◽  
Integration Method ◽  
Exponential Functions ◽  
Integration Methods ◽  
Elliptic Differential Equations ◽  
Double Integrals ◽  
Statistical Functions

The present work presents a new method of integration of non-elementary exponential functions where Fubinni's iterated integrals were used. In this research, some approximations were used in order to generalize the results obtained through mathematical series, in addition to integration methods and double integrals. In addition to the integration methods, the Taylor series was used, where the value found and compatible with the values ​​of the power series that are used to calculate the value of the exponential function demonstrated in the work was verified. In addition to the methods described, a comparison of the values ​​obtained by the series and the values ​​described in the method was improvised, where it was noticed that the higher the value of the variable, the closer the results show a stability for the variable greater than the value 4, described in table 01. The conclusions point to a great improvement, mainly for solving elliptic differential equations and statistical functions.

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