Drift of a Charged Particle in a Magnetic Field of Constant Gradient

1959 ◽  
Vol 12 (4) ◽  
pp. 309 ◽  
Author(s):  
PW Seymour
Keyword(s):  
Magnetic Field ◽  
Charged Particle ◽  
Small Perturbation ◽  
Drift Velocity ◽  
Order Theory ◽  
Simple Expression ◽  
Uniform Field ◽  
First Order ◽  
First Order Theory ◽  
Constant Gradient

A simple expression for the drift velocity of a charged particle moving in an inhomo, geneous magnetic field has been obtained by Alfven, who, in his first-order theory, considered the inhomogeneity as a small perturbation of a uniform field,

scholarly journals Drift of a Charged Particle in a Static Magnetic Field Having a Power Law Dependence

1975 ◽  
Vol 28 (3) ◽  
pp. 289 ◽  
Author(s):  
M Headland ◽  
PW Seymour
Keyword(s):  
Magnetic Field ◽  
Charged Particle ◽  
Exact Solutions ◽  
Drift Velocity ◽  
Static Magnetic Field ◽  
Special Solutions ◽  
Complete Elliptic Integrals ◽  
Bound Orbits ◽  
Constant Gradient ◽  
Velocity Region

As a generalization of Seymour's (1959) exact solution for the drift velocity of a charged particle in a static magnetic field of constant gradient, exact solutions are obtained for charged particle drift in a static magnetic field represented by B. = AX', where A and IX are constants. Four cases of bound orbits are analysed. Exact solutions in terms of hypergeometric, confluent hypergeometric and gamma functions are obtained for the displacement Ay per cycle, the periodic time T and the drift velocity Vd. The special solutions in terms of complete elliptic integrals obtained by Seymour (1959) are also recovered. Calculated exact drift velocity characteristics for representative conditions are presented, and the manner in which the exact curves merge into the Alfven approximate drift velocity region is indicated.

Supercavitating Flows–Small Perturbation Theory

1964 ◽  
Vol 8 (01) ◽  
pp. 16-37 ◽  
Author(s):  
Marshall P. Tulin
Keyword(s):  
Small Perturbation ◽  
Order Theory ◽  
Second Order ◽  
Extensive Literature ◽  
Flat Plates ◽  
First Order ◽  
First Order Theory ◽  
Analytical Results ◽  
Special Cases ◽  
Linearized Theory

This paper contains analytical results which are both old and new. It also contains an historical introduction briefly outlining events leading up to the formulation of linearized theory and indicating the scope of its subsequent development and engineering applications. A large number of references to the extensive literature are presented. The new analytical results include: The logical development of the linearized theory of steady supercavitating flows; the presentation of a general solution for the linearized problem; introduction of a new nonlinear model for finite cavity flows; the formulation of a simple second-order theory; and the application of this theory to the case of flat plates and symmetric wedges. The logical development of the linearized theory involves a small perturbation expansion of the nonlinear problem in which the dependent variable is the complex function In Ψ' and the independent variable is the complex potential Ψ; the problem is formulated for a nonlinear finite-cavity model which features cavity termination in spiral vortices followed by a "smooth" wake which is closed at infinity. The resulting boundary-value problems are of identical form for both the first and second-order theories. The first-order theory is identical to the usual linearized theory, which has previously been formulated by more intuitive means. The simple second-order theory leads in the special cases of inclined flat plates and symmetric wedges in supercavitating flow to results for lift and drag in simple (no quadratures necessary) terms of the first-order results. The old analytical results contained herein refer particularly to: The hydrofoil-airfoil equivalence and results related thereto, which has been so useful in practice for designing hydrofoils at zero cavitation number; and also to problems involving hydrofoils operating at high speeds beneath (hydrofoil boats) and over (supercavitating propeller sections) free surfaces.

scholarly journals A first-order theory of Ulm type

Computability ◽  
2019 ◽  
Vol 8 (3-4) ◽  
pp. 347-358
Author(s):  
Matthew Harrison-Trainor
Keyword(s):  
Order Theory ◽  
First Order ◽  
First Order Theory

scholarly journals Quasi-decidability of a Fragment of the First-Order Theory of Real Numbers

2015 ◽  
Vol 57 (2) ◽  
pp. 157-185 ◽  
Author(s):  
Peter Franek ◽  
Stefan Ratschan ◽  
Piotr Zgliczynski
Keyword(s):  
Order Theory ◽  
Real Numbers ◽  
First Order ◽  
First Order Theory
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