A re-examination of weakly-nonlinear acoustic traveling waves in thermoviscous fluids under Rubin–Rosenau–Gottlieb theory

AbstractInstantaneous acoustic heating of a fluid with thermodynamic relaxation is the subject of investigation. Among others, viscoelastic biological media described by the Maxwell model of the viscous stress tensor, belong to this type of fluid. The governing equation of acoustic heating is derived by means of the special linear combination of conservation equations in differential form, allowing the reduction of all acoustic terms in the linear part of the final equation, but preserving terms belonging to the thermal mode responsible for heating. The procedure of decomposition is valid for weakly nonlinear flows, resulting in the nonlinear terms responsible for the modes interaction. Nonlinear acoustic terms form a source of acoustic heating in the case of dominative sound, which reflects the thermoviscous and dispersive properties of a fluid. The method of deriving the governing equations does not need averaging over the sound period, and the final governing dynamic equation of the thermal mode is instantaneous. Some examples of acoustic heating are illustrated and discussed, conclusions about efficiency of heating caused by different sound impulses are made.

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Nonlinear Influence of Sound on the Vibrational Energy of Molecules in a Relaxing Gas

Archives of Acoustics ◽

10.2478/v10168-012-0012-9 ◽

2012 ◽

Vol 37 (1) ◽

pp. 89-96 ◽

Cited By ~ 1

Author(s):

Anna Perelomova

Keyword(s):

High Frequency ◽

Degrees Of Freedom ◽

Vibrational Energy ◽

Acoustic Energy ◽

Acoustic Source ◽

Weakly Nonlinear ◽

Dispersive Flow ◽

Nonlinear Character ◽

Nonlinear Acoustic ◽

Relaxing Gas

AbstractDynamics of a weakly nonlinear and weakly dispersive flow of a gas where molecular vibrational relaxation takes place is studied. Variations in the vibrational energy in the field of intense sound is considered. These variations are caused by a nonlinear transfer of the acoustic energy into energy of vibrational degrees of freedom in a relaxing gas. The final dynamic equation which describes this is instantaneous, it includes a quadratic nonlinear acoustic source reflecting the nonlinear character of interaction of high-frequency acoustic and non-acoustic motions in a gas. All types of sound, periodic or aperiodic, may serve as an acoustic source. Some conclusions about temporal behavior of the vibrational mode caused by periodic and aperiodic sounds are made.

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Acoustic heating produced in the thermoviscous flow of a Bingham plastic

Open Physics ◽

10.2478/s11534-010-0043-7 ◽

2011 ◽

Vol 9 (1) ◽

Author(s):

Anna Perelomova

Keyword(s):

Linear Part ◽

Acoustic Pulse ◽

Nonlinear Flow ◽

Thermal Mode ◽

Viscous Stress ◽

Viscous Stress Tensor ◽

Bingham Plastic ◽

Weakly Nonlinear ◽

Nonlinear Acoustic ◽

Final Equation

AbstractThis study is devoted to the instantaneous acoustic heating of a Bingham plastic. The model of the Bingham plastic’s viscous stress tensor includes the yield stress along with the shear viscosity, which differentiates a Bingham plastic from a viscous Newtonian fluid. A special linear combination of the conservation equations in differential form makes it possible to reduce all acoustic terms in the linear part of of the final equation governing acoustic heating, and to retain those belonging to the thermal mode. The nonlinear terms of the final equation are a result of interaction between sounds and the thermal mode. In the field of intense sound, the resulting nonlinear acoustic terms form a driving force for the heating. The final governing dynamic equation of the thermal mode is valid in a weakly nonlinear flow. It is instantaneous, and does not imply that sounds be periodic. The equations governing the dynamics of both sounds and the thermal mode depend on sign of the shear rate. An example of the propagation of a bipolar initially acoustic pulse and the evolution of the heating induced by it is illustrated and discussed.

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Synchronization of traveling waves in coupled dispersive systems

10.5194/egusphere-egu2020-7455 ◽

2020 ◽

Author(s):

Nikolay Makarenko ◽

Zakhar Makridin

Keyword(s):

Traveling Waves ◽

Harmonic Wave ◽

Existence Condition ◽

Weakly Nonlinear ◽

Rfbr Grant ◽

Wave Trains ◽

Basic Equations ◽

Wave Limit ◽

Kdv Type Equations ◽

Dispersive Models

<p>&#1057;oupled KdV-type equations arise in multimodal dispersive models such as the Gear &#8211; Grimshaw system which describes weakly nonlinear internal waves in neighboring pycnoclines. Coupling occurs when two or more phase speeds of different modes are close together. &#160;This phenomenon of kissing modes is known as the Eckart resonance providing energy transfer between pycnoclines in stratified fluid. Decoupled basic equations generate separated modes of traveling waves with different phase shifts. In this context, synchronization means the existence of coupled phase-shifted solutions which can be constructed from decoupled modes by appropriate perturbation procedure. &#160;In the present paper, we consider analytic conditions which provide the existence of periodic solutions describing synchronized cnoidal-type wave trains. Application of the Lyapunov &#8211; Schmidt method reduces this problem to the nonlinear system of implicit bifurcation equations for unknown phase shift and wave amplitude. Asymptotic analysis of these equations results sufficient condition of synchronization, which involves the Poincare &#8211; Pontryagin function depending on coupling nonlinear terms. In addition, we illustrate two different limit cases which lead to the same existence condition. &#160;First of them corresponds to a solitary-wave limit for cnoidal waves (i.e. a nonlinear long-wave limit), and the second one is adapted to a small-amplitude limit of coupled harmonic wave packets.</p><p>This paper was supported by RFBR (grant No 18-01-00648).</p>

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