PhyThematics 17 – The Zakharov-Kuznetsov equation and its applications

The Zakharov-Kuznetsov (ZK) equation is a partial differential equation that describes the evolution of weakly nonlinear dispersive waves in two spatial dimensions. The equation was introduced by V.E. Zakharov and E.A. Kuznetsov in 1972.

The Korteweg-de Vries (KdV) equation, which explains the development of one-dimensional weakly nonlinear dispersive waves, is generalized by the ZK equation. The ZK equation may be used to examine two-dimensional waves since it has an extra element that takes into consideration the coupling between waves moving in the x and y axes.

Some intriguing characteristics of the ZK equation include the existence of soliton solutions and the potential for wave breaking. It is often employed in the research of many different physical phenomena, such as fluid dynamics, nonlinear optics, and plasma physics.

NB-“Phythematics” is meant to be a short calligraphic text containing calculus or content related to science, aiming to maximize understanding of physics and natural processes and helping science benefit society. For the fun of it. Remember, for better learning:

• Write notes in books
• Add unique visual notes to digital documents
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Some codes in python or python notebooks will be available here on GitHub: https://github.com/mjgpinheiro/Physics_models

Phythematics 9-Breathing mode in Hall Thrusters

Breathing mode is an ionization instability which is observed ubiquitously in the operation of Hall thrusters. It is recognized as a relatively low frequency (10–30 kHz) longitudinal oscillation of the discharge current and the plasma parameters.

REF:

[1] Anode sheath in Hall thrusters, Appl. Phys. Lett. 83, 2551 (2003); https://doi.org/10.1063/1.1615307

L. Dorf and V. SemenovInstitute of Applied Physics of Russian Academy of Science, Nizhny Novgorod, Russia 603155

Y. RaitsesPrinceton Plasma Physics Laboratory, Princeton, New Jersey 08543

[2] O. Chapurin, A. I Smolyakov, G. Hagelaar, Y. Raitses. On the mechanism of ionization oscillations in
Hall thrusters. Journal of Applied Physics, American Institute of Physics, 2021, 129 (23), pp.233307.
￿10.1063/5.0049105

NB-“Phythematics” is meant to be a short calligraphic text containing calculus aiming to maximize understanding of physics and natural processes. For the fun of it. Remember, for better learnig:

• Write notes in books
• Add unique visual notes to digital documents
• Stick with paper and pen for creative pursuits.

Phythematics 4: Hypersonic cone and turbulence

«Laminar–turbulent transition in hypersonic boundary layers has a dramatic effect on heat transfer, skin friction, and separation. This effect is critical to reentry vehicles and airbreathing cruise vehicles, yet the physics of the transition process is not yet well enough understood to be used for predictive purposes. »

REF: Steven P. Schneider, Hypersonic laminar–turbulent transition on circular cones and scramjet forebodies, Progress in Aerospace Sciences, Volume 40, Issues 1–2, 2004, Pages 1-50.

NB-“Phythematics” is meant to be a short calligraphic text containing calculus aiming to maximize understanding of physics and natural processes. For the fun of it.