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Yu Z, Christov IC. Long-wave equation for a confined ferrofluid interface: periodic interfacial waves as dissipative solitons. Proc Math Phys Eng Sci 2021. [DOI: 10.1098/rspa.2021.0550] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/18/2023] Open
Abstract
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.
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Affiliation(s)
- Zongxin Yu
- School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA
| | - Ivan C. Christov
- School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA
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Lyubimova TP, Lyubimov DV, Sadilov ES, Popov DM. Stability of the fluid interface in a Hele-Shaw cell subjected to horizontal vibrations. Phys Rev E 2018; 96:013108. [PMID: 29347178 DOI: 10.1103/physreve.96.013108] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/27/2016] [Indexed: 11/07/2022]
Abstract
The stability of the horizontal interface of two immiscible viscous fluids in a Hele-Shaw cell subject to gravity and horizontal vibrations is studied. The problem is reduced to the generalized Hill equation, which is solved analytically by the multiple scale method and numerically. The long-wave instability, the resonance (parametric resonance) excitation of waves at finite frequencies of vibrations (for the first three resonances), and the limit of high-frequency vibrations are studied analytically under the assumption of small amplitudes of vibrations and small viscosity. For finite amplitudes of vibrations, finite wave numbers, and finite viscosity, the study is performed numerically. The influence of the specific natural control parameters and physical parameters of the system on its instability threshold is discussed. The results provide extension to other results [J. Bouchgl, S. Aniss, and M. Souhar, Phys. Rev. E 88, 023027 (2013)10.1103/PhysRevE.88.023027], where the authors considered a similar problem but took into account viscosity in the basic state and did not consider it in the equations for perturbations.
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Affiliation(s)
- T P Lyubimova
- Institute of Continuous Media Mechanics, UB RAS, 1 Koroleva Street, 614013 Perm, Russia.,Perm State University, 15 Bukireva Street, 614990 Perm, Russia
| | - D V Lyubimov
- Perm State University, 15 Bukireva Street, 614990 Perm, Russia
| | - E S Sadilov
- Institute of Continuous Media Mechanics, UB RAS, 1 Koroleva Street, 614013 Perm, Russia
| | - D M Popov
- Institute of Continuous Media Mechanics, UB RAS, 1 Koroleva Street, 614013 Perm, Russia
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Lesur M, Itoh K, Ido T, Osakabe M, Ogawa K, Shimizu A, Sasaki M, Ida K, Inagaki S, Itoh SI. Nonlinear Excitation of Subcritical Instabilities in a Toroidal Plasma. PHYSICAL REVIEW LETTERS 2016; 116:015003. [PMID: 26799024 DOI: 10.1103/physrevlett.116.015003] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/11/2015] [Indexed: 06/05/2023]
Abstract
In a collisionless plasma, it is known that linearly stable modes can be destabilized (subcritically) by the presence of structures in phase space. However, nonlinear growth requires the presence of a seed structure with a relatively large threshold in amplitude. We demonstrate that, in the presence of another, linearly unstable (supercritical) mode, wave-wave coupling can provide a seed, which is significantly below the threshold, but can still grow by (and only by) the collaboration of fluid and kinetic nonlinearities. By modeling the subcritical mode kinetically, and the impact of the supercritical mode by simple wave-wave coupling equations, it is shown that this new kind of subcritical instability can be triggered, even when the frequency of the supercritical mode is rapidly sweeping. The model is applied to the bursty onset of geodesic acoustic modes in a LHD experiment. The model recovers several key features such as relative amplitude, time scales, and phase relations. It suggests that the strongest bursts are subcritical instabilities, driven by this mechanism of combined fluid and kinetic nonlinearities.
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Affiliation(s)
- M Lesur
- Research Institute for Applied Mechanics, Kyushu University, Kasuga 816-8580, Japan
| | - K Itoh
- National Institute for Fusion Science, Toki 509-5292, Japan
- Research Center for Plasma Turbulence, Kyushu University, Kasuga 816-8580, Japan
| | - T Ido
- National Institute for Fusion Science, Toki 509-5292, Japan
| | - M Osakabe
- National Institute for Fusion Science, Toki 509-5292, Japan
- SOKENDAI (The Graduate University for Advanced Studies), Toki 509-5292, Japan
| | - K Ogawa
- National Institute for Fusion Science, Toki 509-5292, Japan
- SOKENDAI (The Graduate University for Advanced Studies), Toki 509-5292, Japan
| | - A Shimizu
- National Institute for Fusion Science, Toki 509-5292, Japan
| | - M Sasaki
- Research Institute for Applied Mechanics, Kyushu University, Kasuga 816-8580, Japan
- Research Center for Plasma Turbulence, Kyushu University, Kasuga 816-8580, Japan
| | - K Ida
- National Institute for Fusion Science, Toki 509-5292, Japan
- SOKENDAI (The Graduate University for Advanced Studies), Toki 509-5292, Japan
| | - S Inagaki
- Research Institute for Applied Mechanics, Kyushu University, Kasuga 816-8580, Japan
- Research Center for Plasma Turbulence, Kyushu University, Kasuga 816-8580, Japan
| | - S-I Itoh
- Research Institute for Applied Mechanics, Kyushu University, Kasuga 816-8580, Japan
- Research Center for Plasma Turbulence, Kyushu University, Kasuga 816-8580, Japan
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Lira SA, Miranda JA. Nonlinear traveling waves in confined ferrofluids. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:056301. [PMID: 23214870 DOI: 10.1103/physreve.86.056301] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/02/2012] [Indexed: 06/01/2023]
Abstract
We study the development of nonlinear traveling waves on the interface separating two viscous fluids flowing in parallel in a vertical Hele-Shaw cell. One of the fluids is a ferrofluid and a uniform magnetic field is applied in the plane of the cell, making an angle with the initially undisturbed interface. We employ a mode-coupling theory that predicts the possibility of controlling the speed of the waves by purely magnetic means. The influence of the tilted magnetic field on the waves shape profile and the establishment of stationary traveling wave structures are investigated.
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Affiliation(s)
- Sérgio A Lira
- Departamento de Física, Universidade Federal de Pernambuco, Recife, PE 50670-901 Brazil
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Jung D, Lücke M. Traveling wave fronts and localized traveling wave convection in binary fluid mixtures. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:026307. [PMID: 16196711 DOI: 10.1103/physreve.72.026307] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/04/2005] [Indexed: 05/04/2023]
Abstract
Nonlinear fronts between spatially extended traveling wave (TW) convection and quiescent fluid and spatially localized traveling waves (LTWs) are investigated in quantitative detail in the bistable regime of binary fluid mixtures heated from below. A finite-difference method is used to solve the full hydrodynamic field equations in a vertical cross section of the layer perpendicular to the convection roll axes. Results are presented for ethanol-water parameters with several strongly negative separation ratios where TW solutions bifurcate subcritically. Fronts and LTWs are compared with each other and similarities and differences are elucidated. Phase propagation out of the quiescent fluid into the convective structure entails a unique selection of the latter while fronts and interfaces where the phase moves into the quiescent state behave differently. Interpretations of various experimental observations are suggested.
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Affiliation(s)
- D Jung
- Institut für Theoretische Physik, Universität des Saarlandes, Postfach 151150, D-66041 Saarbrücken, Germany
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