scholarly journals Long-wave equation for a confined ferrofluid interface: periodic interfacial waves as dissipative solitons

2021 ◽  
Vol 477 (2256) ◽  
Author(s):  
Zongxin Yu ◽  
Ivan C. Christov
Keyword(s):  
Magnetic Field ◽  
Wave Equation ◽  
Propagation Velocity ◽  
Linear Prediction ◽  
Travelling Waves ◽  
Type Equation ◽  
Linear Instability ◽  
Interfacial Waves ◽  
Long Wave ◽  
Dissipative Solitons

We study the dynamics of a ferrofluid thin film confined in a Hele-Shaw cell, and subjected to a tilted non-uniform magnetic field. It is shown that the interface between the ferrofluid and an inviscid outer fluid (air) supports travelling waves, governed by a novel modified Kuramoto–Sivashinsky-type equation derived under the long-wave approximation. The balance between energy production and dissipation in this long-wave equation allows for the existence of dissipative solitons. These permanent travelling waves’ propagation velocity and profile shape are shown to be tunable via the external magnetic field. A multiple-scale analysis is performed to obtain the correction to the linear prediction of the propagation velocity, and to reveal how the nonlinearity arrests the linear instability. The travelling periodic interfacial waves discovered are identified as fixed points in an energy phase plane. It is shown that transitions between states (wave profiles) occur. These transitions are explained via the spectral stability of the travelling waves. Interestingly, multi-periodic waves, which are a non-integrable analogue of the double cnoidal wave, are also found to propagate under the model long-wave equation. These multi-periodic solutions are investigated numerically, and they are found to be long-lived transients, but ultimately abruptly transition to one of the stable periodic states identified.

Stability of interfacial waves in aluminium reduction cells

1998 ◽  
Vol 362 ◽  
pp. 273-295 ◽  
Author(s):  
P. A. DAVIDSON ◽  
R. I. LINDSAY
Keyword(s):  
Magnetic Field ◽  
Wave Equation ◽  
Fourier Analysis ◽  
Lorentz Force ◽  
Travelling Waves ◽  
Fluid Motion ◽  
Standing Waves ◽  
Background Magnetic Field ◽  
The Stability ◽  
Aluminium Reduction

We investigate the stability of interfacial waves in conducting fluids under the influence of a vertical current density, paying particular attention to aluminium reduction cells in which such instabilities are commonly observed. We develop a wave equation for the interface in which the Lorentz force is expressed explicitly in terms of the fluid motion. Our wave equation differs from previous models, most notably that developed by Urata (1985), in that earlier formulations rested on a more complex, implicit coupling between the fluid motion and the Lorentz force. Our formulation furnishes a number of quite general stability results without the need to resort to Fourier analysis. (The need for Fourier analysis typifies previous studies.) Moreover, our equation supports both travelling and standing waves. We investigate each in turn.We obtain three new results. First, we show that travelling waves may become unstable in the presence of a uniform, vertical magnetic field. (Our travelling waves are quite different to those discovered by previous investigators (Sneyd 1985 and Moreau & Ziegler 1986) which require more complex magnetic fields to become unstable.) Second, in line with previous studies we confirm that standing waves may also become unstable. In this context we derive a simple energy criterion which shows which types of motion may extract energy from the background magnetic field. This indicates that a rotating, tilted interface is particularly prone to instability, and indeed such a motion is often seen in practice. Finally, we use Gershgorin's theorem to produce a sufficient condition for the stability of standing waves in a finite domain. This allows us to place a lower bound on the critical value of the background magnetic field at which an instability first appears, without solving the governing equations of motion.

scholarly journals The internal conversion of gamma-rays.—Part II

1928 ◽  
Vol 121 (787) ◽  
pp. 447-456
Keyword(s):  
Magnetic Field ◽  
Quantum Mechanics ◽  
Wave Equation ◽  
Gamma Rays ◽  
Internal Conversion ◽  
Scalar Potential ◽  
X Ray ◽  
Electro Magnetic Field ◽  
Γ Rays ◽  
Electro Magnetic

In a previous paper the absorption of γ-rays in the K-X-ray levels of the atom in which they are emitted was calculated according to the Quantum Mechanics, supposing the γ-rays to be emitted from a doublet of moment f ( t ) at the centre of the atom. The non-relativity wave equation derived from the relativity wave equation for an electron of charge — ε moving in an electro-magnetic field of vector potential K and scalar potential V is h 2 ∇ 2 ϕ + 2μ ( ih ∂/∂ t + εV + ih ε/μ c (K. grad)) ϕ = 0. (1) Suppose, however, that K involves the space co-ordinates. Then, (K. grad) ϕ ≠ (grad . K) ϕ , and the expression (K . grad) ϕ is not Hermitic. Equation (1) cannot therefore be the correct non-relativity wave equation for a single electron in an electron agnetic field, and we must substitute h 2 ∇ 2 ϕ + 2μ ( ih ∂/∂ t + εV) ϕ + ih ε/ c ((K. grad) ϕ + (grad. K) ϕ ) = 0. (2)

scholarly journals On Galilean Invariant and Energy Preserving BBM-Type Equations

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 878
Author(s):  
Alexei Cheviakov ◽  
Denys Dutykh ◽  
Aidar Assylbekuly
Keyword(s):  
Wave Equation ◽  
Conservation Laws ◽  
Numerical Simulations ◽  
Solitary Waves ◽  
Local Symmetry ◽  
Symmetry And Conservation Laws ◽  
Higher Order ◽  
Special Equation ◽  
Long Wave ◽  
Local Symmetries

We investigate a family of higher-order Benjamin–Bona–Mahony-type equations, which appeared in the course of study towards finding a Galilei-invariant, energy-preserving long wave equation. We perform local symmetry and conservation laws classification for this family of Partial Differential Equations (PDEs). The analysis reveals that this family includes a special equation which admits additional, higher-order local symmetries and conservation laws. We compute its solitary waves and simulate their collisions. The numerical simulations show that their collision is elastic, which is an indication of its S−integrability. This particular PDE turns out to be a rescaled version of the celebrated Camassa–Holm equation, which confirms its integrability.

Euler operators and conservation laws of the BBM equation

1979 ◽  
Vol 85 (1) ◽  
pp. 143-160 ◽  
Author(s):  
Peter J. Olver
Keyword(s):  
Wave Equation ◽  
Conservation Laws ◽  
Calculus Of Variations ◽  
Long Wave ◽  
Regularized Long Wave Equation ◽  
Formal Calculus ◽  
Euler Operators ◽  
Bbm Equation

AbstractThe BBM or Regularized Long Wave Equation is shown to possess only three non-trivial independent conservation laws. In order to prove this result, a new theory of Euler-type operators in the formal calculus of variations will be developed in detail.

On the existence of compressional MHD oscillations in an inhomogeneous magnetoplasma

1990 ◽  
Vol 44 (2) ◽  
pp. 361-375 ◽  
Author(s):  
Andrew N. Wright
Keyword(s):  
Magnetic Field ◽  
Wave Equation ◽  
Density Distribution ◽  
Cold Plasma ◽  
Wave Equations ◽  
Perturbation Field ◽  
Plasma Density Distribution ◽  
Background Magnetic Field ◽  
Transverse Fields ◽  
The Magnetic Field

In a cold plasma the wave equation for solely compressional magnetic field perturbations appears to decouple in any surface orthogonal to the background magnetic field. However, the compressional fields in any two of these surfaces are related to each other by the condition that the perturbation field b be divergence-free. Hence the wave equations in these surfaces are not truly decoupled from one another. If the two solutions happen to be ‘matched’ (i.e. V.b = 0) then the medium may execute a solely compressional oscillation. If the two solutions are unmatched then transverse fields must evolve. We consider two classes of compressional solutions and derive a set of criteria for when the medium will be able to support pure compressional field oscillations. These criteria relate to the geometry of the magnetic field and the plasma density distribution. We present the conditions in such a manner that it is easy to see if a given magnetoplasma is able to executive either of the compressional solutions we investigate.

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