XVI. On the motion of fluid, part of which is moving rotationally and part irrotationally

1884 ◽  
Vol 175 ◽  
pp. 363-409 ◽  
Keyword(s):  
Single Equation ◽  
Plane Motion ◽  
Vortex Lines ◽  
Rectangular Coordinates ◽  
Irrotational Motion ◽  
Arbitrary Velocity ◽  
Complicated Nature

Clebsch has shown that the components of the velocity of a fluid u , v , w , parallel to rectangular axes x , y , z , may always be expressed thus u = dχ / dx + λ dΨ / dx , v = dχ / dy + λ dΨ / dy , w = dχ / dz + λ dΨ / dz ; where λ, Ψ are systems of surfaces whose intersections determine the vortex lines; and the pressure satisfies an equation which is equivalent to the following p / ρ + V = – dχ / dt –½{( dχ / dx ) 2 + ( dχ / dy ) 2 +( dχ / dz ) 2 } + ½ λ 2 {( dΨ / dx ) 2 +( dΨ / dy ) 2 +( dΨ / dz ) 2 } where p is the pressure, ρ the density, and V the potential of the forces acting on the liquid. It is shown in this paper that an equation of a complicated nature in λ only can be obtained in the following cases (that is to say, as in cases of irrotational motion, the determination of the motion depends on the solution of a single equation only):— (1.) Plane motion, referred to rectangular coordinates x , y . The equation is somewhat simpler when the vortex surfaces are of invariable form, and move parallel to one of the axes of coordinates with arbitrary velocity.

scholarly journals I. On the motion of fluid, part of which is moving rotationally and part irrotationally

1883 ◽  
Vol 36 (228-231) ◽  
pp. 276-284 ◽  
Keyword(s):  
Single Equation ◽  
Plane Motion ◽  
Vortex Lines ◽  
Irrotational Motion

Clebsch has show that the components of the velocity of a fluid u, v, w , parallel to rectangular axes x, y, z , may always be expressed thus— u = dX / dx + λ dψ / dx . v = dX / dy + λ dψ / dy , w = dX / dz +λ dψ / dz ; Where λ, ψ are systems of surfaces whose intersections determine the vortex lines; and the pressure satisfies an equation which is equivalent to the following— p /ρ + V = - dX / dt -1/2{( dX / dx ) 2 + ( dX / dy ) 2 + ( dX / dz ) 2 } + 1/2 λ 2 {( dψ / dx ) 2 + ( dψ / dy ) 2 + ( dψ / dz ) 2 } where p is the pressure, ρ the density, and V the potential of the forces acting on the liquid. It is shown in this paper that an equation in λ only can be obtained in the following cases (that is to say, as in cases of irrotational motion, the determination of the motion depends on the solution of a single equation only):— (1.) Plane motion, referred to rectangular co-ordinates x , y .

scholarly journals Methods for Transformation of Rectangular Spatial Coordinates to Geodetic Coordinates

2018 ◽  
Vol 7 (4.38) ◽  
pp. 1179
Author(s):  
Pavel Aleksandrovich Medvedev ◽  
Leonid Vasilevich Bykov ◽  
Vasiliy Leonidovich Bykov ◽  
Marina Vladimirovna Novorodskaya ◽  
Svetlana Ivanovna Sherstneva
Keyword(s):  
Transcendental Equation ◽  
Curvilinear Coordinates ◽  
Quartic Equation ◽  
Iterative Processes ◽  
Geodetic Height ◽  
Rectangular Coordinates ◽  
Spatial Coordinates ◽  
Root Isolation ◽  
Chord Method

The article gives a brief analysis of methods and algorithms for the transformation of spatial rectangular coordinates to curvilinear coordinates - geodetic latitude, geodetic longitude, geodetic height. Two algorithms for solving the equation for determining longitude are considered. Three formulas used to calculate the height are analyzed, with an estimate of their errors due to the approximate latitude. The shortcomings of mathematical solutions to these problems are revealed. A study of different approaches and methods for solving the transcendental equation for determining the latitude, based on the theory of separation of the root of the equation, is performed. Using this technique, iterative processes were performed to calculate the reduced latitude , using trigonometric identities, by introducing an auxiliary angle and transforming it to an algebraic quartic equation, which Borkowski solves by the Ferrari's method. The determination of the root isolation interval allowed using the chord method (proportional parts) to determine the latitude. In all cases, estimates of the convergence of the iterative processes that facilitate the comparative analysis of the proposed solutions are obtained. By further decreasing the separation interval of the root, the accuracy of the non-iterative determination of the latitude is improved by the Newton method.  

scholarly journals About the decision regularized equations of perturbed motion body

2019 ◽  
Vol 110 ◽  
pp. 01031
Author(s):  
Vera Petelina
Keyword(s):  
Higher Order ◽  
The Body ◽  
Body Motion ◽  
Motion Velocity ◽  
Intermediate Point ◽  
Rectangular Coordinates ◽  
The Right ◽  
Approximating Polynomials ◽  
Disturbed Motion

The article deals with determination of the second- and higher-order perturbations in Cartesian coordinates and body motion velocity constituents. A special perturbed motion differential equations system is constructed. The right-hand sides of this system are finite polynomials relative to an independent regularizing variable. This allows constructing a single algorithm to determine the second and higher order perturbations in the form of finite polynomials relative to some regularizing variables that are chosen at each approximation step. Following the calculations results with the use of the developed method, the coefficients of approximating polynomials representing rectangular coordinates and components of the regularized body speed were obtained. Comparison with the results of numerical integration of the equations of disturbed motion shows close agreement of the results. The developed methods make it possible to calculate, by the approximating polynomials, any intermediate point of the motion trajectory of the body.

scholarly journals Models of unionism and unemployment

2011 ◽  
Vol 17 (2) ◽  
pp. 141-152 ◽  
Author(s):  
Daniele Checchi ◽  
Luca Nunziata
Keyword(s):  
Labour Market ◽  
Simultaneous Determination ◽  
Empirical Analysis ◽  
Union Density ◽  
Single Equation ◽  
Income Support ◽  
Market Institutions ◽  
The Unemployed ◽  
The Relationship

We investigate the problem of simultaneous determination of labour market institutions and outcomes in single equation multi-country estimations by presenting an empirical analysis of unemployment and union density in 20 OECD countries. When explicitly modelling potential endogeneity and heterogeneity, our results suggest that unions contribute to explaining unemployment in different ways than previously thought. In addition, the relationship between unemployment and union density is heterogeneous across countries, depending on the way in which income support for the unemployed is organized.

scholarly journals DETERMINATION OF GEODESIC LATITUDE BY SPATIAL RIGHT-MUGAL COORDINATES BY USING A DIFFERENTIAL AMENDMENT

2019 ◽  
Vol 1 (2) ◽  
pp. 3-8
Author(s):  
Konstantin Afonin ◽  
Yulia Trifonova
Keyword(s):  
Iterative Methods ◽  
Practical Application ◽  
Third Way ◽  
Differential Correction ◽  
Numerical Example ◽  
The Third ◽  
Rectangular Coordinates

GNSS technologies are currently essential for coordinate support of territories. However, theymakeit possible to obtain spatial rectangular coordinates of the points being determined. While most users need the flat rectangular coordinates of Gauss – Kruger. And these coordinates can be calculated only by geodetic latitudes and longitudes. The special literature describes more than a dozen methods for calculating the geodetic latitude in spatial rectangular coordinates. To solve this problem, usually use any iterative or non-iterative methods. Both those and others have their ad-vantages and disadvantages. In the work applied the third way to solve the problem. It is proposed to calculate and use the differential correction to the initial (approximate) value of the geodesic lati-tude. Received working formulas that implement this idea. A numerical example is given showing the possibility of practical application oftheproposedmethod.

Determination of the geodetic latitude and height by spatial geocentric coordinates

2021 ◽  
Vol 977 (11) ◽  
pp. 2-7
Author(s):  
P.D. Penev ◽  
E.P. Peneva
Keyword(s):  
Geodetic Height ◽  
Rectangular Coordinates ◽  
Geocentric Coordinates

The authors propose to derive the formulas given in [1, 2] for determining the height and latitude based on the Cartesian rectangular coordinates X, Y, Z, giving an accuracy for the geodetic height H of 1 mm for heights up to 50 km and for geodetic latitude B of 0,0001 arc seconds for H < 10 km. The formulas proposed in [1, 2] apply to all values of latitude and longitude (B and L). In [3], we propose two new formulas for H and B. In this paper, it is shown that the formulas proposed in [3] apply to points of ellipsoid surface and points with geodetic latitude of 0° and 90°. For the same formulas proposed in [3], the corrections are derived to ensure an accuracy of H of 1 mm at H ≤ 10 km, which apply to all values of B and L. Basing on the presented geometric conclusions, calculations and analyzes, a new solution for H and B respectively is proposed for given X, Y, Z, which provides an accuracy for H less than 1 mm for H ≤ 100 km and for B of 0,0001 arc seconds for H ≤ 50 km.

scholarly journals Characterization of Kinetic Binding Properties of Unlabeled Ligands via a Preincubation Endpoint Binding Approach

2016 ◽  
Vol 21 (7) ◽  
pp. 729-737 ◽  
Author(s):  
Yuji Shimizu ◽  
Kazumasa Ogawa ◽  
Masaharu Nakayama
Keyword(s):  
Dissociation Constants ◽  
Early Stage ◽  
Single Equation ◽  
Dissociation Rate ◽  
Response Curves ◽  
Kinetic Measurements ◽  
Simple Method ◽  
Endpoint Assay

The dissociation rates of unlabeled drugs have been well studied by kinetic binding analyses. Since kinetic assays are laborious, we developed a simple method to determine the kinetic binding parameters of unlabeled competitors by a preincubation endpoint assay. The probe binding after preincubation of a competitor can be described by a single equation as a function of time. Simulations using the equation revealed the degree of IC50 change induced by preincubation of a competitor depended on the dissociation rate koff of the competitor but not on the association rate kon. To validate the model, an in vitro binding assay was performed using a smoothened receptor (SMO) and [3H]TAK-441, a SMO antagonist. The equilibrium dissociation constants (KI) and koff of SMO antagonists determined by globally fitting the model to the concentration–response curves obtained with and without 24 h preincubation correlated well with those determined by other methods. This approach could be useful for early-stage optimization of drug candidates by enabling determination of binding kinetics in a high-throughput manner because it does not require kinetic measurements, an intermediate washout step during the reaction, or prior determination of competitors’ KI values.

scholarly journals Design procedure of first-order perturbations for collisions trajectories with a perturbing body

2019 ◽  
Vol 138 ◽  
pp. 01034
Author(s):  
Vera Petelina
Keyword(s):  
Design Procedure ◽  
Equation System ◽  
Body Motion ◽  
Differential Equation System ◽  
Motion Equations ◽  
Rectangular Coordinates ◽  
The Right ◽  
Approximating Polynomials ◽  
Disturbed Motion

The article is devoted to the determination of firstand secondorder perturbations in rectangular coordinates and velocity components of body motion. Special differential equation system of perturbed motion is constructed. The right-hand sides of this system are finitesimal polynomials in powers of an independent regularizing variate. This allows constructing a single algorithm to determine firstand second-order perturbations in the form of finitesimal polynomials in powers of regularizing variates that are chosen at each approximation step. Following the calculations results with the developed method use, the coefficients of approximating polynomials representing rectangular coordinates and components of the regularized body speed were obtained. Comparison with the numerical results of the disturbed motion equations shows their close agreement. The developed method make it possible to calculate any visa point of body motion by the approximating polynomials.

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