Convergence of the Keck‐Beyer Perturbation Solution for Plane Waves of Finite Amplitude in a Viscous Fluid

1966 ◽  
Vol 39 (2) ◽  
pp. 411-413 ◽  
Author(s):  
David T. Blackstock
Keyword(s):  
Viscous Fluid ◽  
Plane Waves ◽  
Finite Amplitude ◽  
Perturbation Solution

Further analysis of nonlinear, periodic, highly superluminous waves in a magnetized plasma

1982 ◽  
Vol 27 (1) ◽  
pp. 177-187 ◽  
Author(s):  
P. C. Clemmow
Keyword(s):  
Magnetic Field ◽  
Power Series ◽  
Periodic Solutions ◽  
Wave Speed ◽  
Nonlinear Differential Equations ◽  
Plane Waves ◽  
Electron Plasma ◽  
Finite Amplitude ◽  
Magnetized Plasma ◽  
Cold Electron

A perturbation method is applied to the pair of second-order, coupled, nonlinear differential equations that describe the propagation, through a cold electron plasma, of plane waves of fixed profile, with direction of propagation and electric vector perpendicular to the ambient magnetic field. The equations are expressed in terms of polar variables π, φ, and solutions are sought as power series in the small parameter n, where c/n is the wave speed. When n = 0 periodic solutions are represented in the (π,φ) plane by circles π = constant, and when n is small it is found that there are corresponding periodic solutions represented to order n2 by ellipses. It is noted that further investigation is required to relate these finite-amplitude solutions to the conventional solutions of linear theory, and to determine their behaviour in the vicinity of certain resonances that arise in the perturbation treatment.

Determination of the nonlinear parameter by propagating and modeling finite amplitude plane waves

2006 ◽  
Vol 119 (5) ◽  
pp. 2639-2644 ◽  
Author(s):  
F. Chavrier ◽  
C. Lafon ◽  
A. Birer ◽  
C. Barrière ◽  
X. Jacob ◽  
...  
Keyword(s):  
Plane Waves ◽  
Finite Amplitude ◽  
Nonlinear Parameter

Distortion and necking of a viscous inclusion in Stokes flow

1989 ◽  
Vol 422 (1862) ◽  
pp. 173-192 ◽  
Keyword(s):  
Viscous Fluid ◽  
Stokes Flow ◽  
Numerical Solutions ◽  
Finite Amplitude ◽  
Elliptic Solution ◽  
Small Perturbations ◽  
All Solutions ◽  
Elliptic Profile ◽  
The Stability ◽  
Small Disturbances

Spence & Wilmott (1988) considered the deformation of a slender inclusion of highly-viscous fluid in a Stokes flow of less viscous fluid, and derived a coupled pair of equations to describe its evolution. The equations possess self-preserving solutions for elliptical inclusions, previously known from the work of Bilby et al . (1975, 1976) and Bilby & Kolbuszewski (1977). In the present paper numerical solutions are presented for non-elliptic shapes. These have been obtained on a CRAY XMP-22 by use of an eigenfunction expansion suggested by the elliptic solution, leading to an initial value problem for the coefficients. The solutions show that large deformations can develop in finite time from small perturbations of the ellipse, particularly at large viscosity ratios. Special attention is directed to the phenomenon of boudinage, whereby alternate swelling and necking develops as the inclusion is stretched by the Stokes flow. This appears to characterize all solutions that depart significantly from pure elliptic shape. In the Appendix the stability of the elliptic profile to small disturbances is examined. It is found that all subharmonics of a given disturbance are excited to a finite amplitude that increases with the viscosity ratio, but that higher harmonics are not excited in linear theory, although nonlinear coupling leads to the eventual excitation of all modes.

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