Thermoviscous Medium Research Articles

Existence of Classical Solutions for a Class of the Khokhlov-Zabolotskaya-Kuznetsov Type Equations

In medical sciences, during medical exploration and diagnosis of tissues or in medical imaging, we often use mathematical models to answer questions related to these examinations. Among these models, the nonlinear partial differential equation of the Khokhlov--Zabolotskaya--Kuznetsov type (abbreviated as the KZK equation) is of proven interest in ultrasound acoustics problems. This mathematical model describes the nonlinear propagation of a sound pulse of finite amplitude in a thermo-viscous medium. The equation is obtained by combining the conservation of mass equation, the conservation of momentum equation and the equations of state. It should be noted that for this equation little mathematical analysis is reserved. This equation takes into account three combined effects: the diffraction of the wave, the absorption of energy and the nonlinearity of the medium in which the wave propagates. KZK-type equation introduced in this paper is a modified version of the KZK model known in acoustics. We study a class of~the~Khokhlov--Zabolotskaya--Kuznetsov type equations for the existence of global classical solutions. We~give conditions under which the considered equations have at least one and at least two classical solutions. To prove our main results, we propose a new approach based on recent theoretical results.

Dispersed media: scattering of sound waves, stratification in swirling flows and sedimentation processes

A brief review on three problems from the multiphase media mechanics area related to the study of the dynamic behavior of disperse systems under various conditions and methods of interaction with external fields or the environment is presented. The problems are an integral part of the scientific project “Hydrodynamic effects in multiphase and thermoviscous media under wave and thermal effects”. The review is carried out in the context of the vector of development of modern research in the relevant areas. As a part of solving the problem of acoustic wave scattering by a set of spherical bubbles or drops, taking into account their sound permeability, the fast multipole method was developed to expand the possibilities of its application in the considered cases. On the basis of addition theorems for spherical wave functions, a new formula for the total scattering cross section for a set of interacting sound-permeable spheres arbitrarily located in space is obtained. An important aspect of the research was the estimation of the region in the parameter space of the problem, in which the effects of multiple scattering are significant. The second problem is related to mathematical modeling of a swirling turbulent flow containing particles of a dispersed phase. For numerical studies of temperature stratification in a vortex tube, an algorithm and a computer code were created using an orthogonalized finite-volume mesh with separation of the near-wall layer. A number of parametric studies have been carried out, in particular, the dependence of the temperature of the outgoing air in the cold diaphragm channel depending on the diameter of the diaphragm has been considered. In order to increase the efficiency of geological exploration in solving the third problem for describing the process of magmatic ore formation, a system of equations based on the methods of mechanics of multiphase media and thermohydrodynamics is proposed. The mathematical model provides for the heat exchange of the magmatic melt flow with the surrounding host rocks, as well as the release of heavy and light fractions from basaltic magma during its cooling. The results obtained in the course of the computational experiment indicate the possibility of a periodically inhomogeneous nature of the distribution of ore-forming fractions.

Comparison between the Gol’dberg number and the Morfey-Howell Q/S for finite-amplitude plane waves

In this paper we compare the two most popular indicators of nonlinearity, namely, the Gol’dberg number [Sov. Phys. Acoust. 2, 346-350 (1956)] and the Morfey-Howell Q/S [AIAA J. 19, 986-992 (1981)] for the case of an initially-sinusoidal, finite-amplitude plane wave in a thermoviscous medium. The Gol’dberg number Γ, defined as the ratio of the absorption length at the fundamental frequency to the shock formation distance, is a parameter that single-handedly dictates the entire history of the wave [for example, weak (Γ < 1) and strong (Γ >> 1) waves]. However, it fails to quantify the strength of nonlinearity that each individual harmonic “sees” at any given point. In other words, not every harmonic sees the apparent strength of nonlinearity according to Γ. The Q/S, on the other hand, is defined for each harmonic as a function of propagation distance, and hence keeps track of the ever-changing strength of nonlinearity during the course of propagation. One interesting ramification of this is that using Q/S it is possible to pinpoint the distance at which the value of nonlinearity relative to absorption actually becomes Γ for the fundamental component.

Evolution of the average steepening factor for nonlinearly propagating waves.

Difficulties arise in attempting to discern the effects of nonlinearity in near-field jet-noise measurements due to the complicated source structure of high-velocity jets. This article describes a measure that may be used to help quantify the effects of nonlinearity on waveform propagation. This measure, called the average steepening factor (ASF), is the ratio of the average positive slope in a time waveform to the average negative slope. The ASF is the inverse of the wave steepening factor defined originally by Gallagher [AIAA Paper No. 82-0416 (1982)]. An analytical description of the ASF evolution is given for benchmark cases-initially sinusoidal plane waves propagating through lossless and thermoviscous media. The effects of finite sampling rates and measurement noise on ASF estimation from measured waveforms are discussed. The evolution of initially broadband Gaussian noise and signals propagating in media with realistic absorption are described using numerical and experimental methods. The ASF is found to be relatively sensitive to measurement noise but is a relatively robust measure for limited sampling rates. The ASF is found to increase more slowly for initially Gaussian noise signals than for initially sinusoidal signals of the same level, indicating the average distortion within noise waveforms occur more slowly.

Driving force of acoustic streaming caused by aperiodic sound beam in unbounded volumes

Instantaneous driving force of acoustic streaming in the thermoviscous medium is the subject of investigation. Dynamic equation of the Eulerian streaming velocity is a result of splitting the hydrodynamic equations into acoustic and non-acoustic parts. The acoustic force represents a sum of three parts, one is the classic one, which being averaged over the sound period coincides with the well-known expression. The second one is connected to the periodicity of the sound, it becomes exact zero after averaging for the strictly periodic sound but is not zero for other acoustic wave. The last term originates from the sound divergence. All terms are nonlinear and proportional to the overall attenuation. The consistent comparative analysis of both formula for quasi-periodic and modulated sound is proceeded.

On the derivation of the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and validation of the KZK-approximation for viscous and non-viscous thermo-elastic media

We consider the derivation of the Khokhlov-Zabolotskaya-Kuznetzov (KZK) equation from the nonlinear isentropic Navier-Stokes and Euler systems. The KZK equation is a mathematical model that describes the nonlinear propagation of a finite-amplitude sound pulse in a thermo-viscous medium. The derivation of the KZK equation has to date been based on the paraxial approximation of small perturbations around a given state of the Navier-Stokes system. However, this method does not guarantee that the solution of the initial Navier-Stokes system can be reconstructed from the solution of the KZK equation. We introduce a corrector function in the derivation ansatz that allows one to validate the KZK-approximation. We also give the analysis of other types of derivation. We prove the validation of the KZK-approximation for the non-viscous case, as well as for the viscous nonlinear and linear cases. The results are obtained in Sobolev spaces for functions periodic in one of the space variables and with a mean value of zero. The existence of a unique regular solution of the isentropic Navier-Stokes system in the half space with boundary conditions that are both periodic and mean value zero in time is also obtained.

A FOURTH-ORDER ACCURATE SCHEME FOR SOLVING ONE-DIMENSIONAL HIGHLY NONLINEAR STANDING WAVE EQUATION IN DIFFERENT THERMOVISCOUS FLUIDS

Combination of a fourth-order Padé compact finite difference discretization in space and a fourth-order Runge–Kutta time stepping scheme is shown to yield an effective method for solving highly nonlinear standing waves in a thermoviscous medium. This accurate and fast-solver numerical scheme can predict the pressure, particle velocity, and density along the standing wave resonator filled with a thermoviscous fluid from linear to strongly nonlinear levels of the excitation amplitude. The stability analysis is performed to determine the stability region of the scheme. Beside the fourth-order accuracy in both time and space, another advantage of the given numerical scheme is that no additional attenuation is required to get numerical stability. As it is well known, the results show that the pressure and particle velocity waveforms for highly nonlinear waves are significantly different from that of the linear waves, in both time and space. For highly nonlinear waves, the results also indicate the presence of a wavefront that travels along the resonator with very high pressure and velocity gradients. Two gases, air and CO 2, are considered. It is observed that the slopes of the traveling velocity and pressure gradients are higher for CO 2 than those for air. For highly nonlinear waves, the results also indicate the higher asymmetry in pressure for CO 2 than that for air.

QUALITATIVE ANALYSIS OF THE KHOKHLOV–ZABOLOTSKAYA–KUZNETSOV (KZK) EQUATION

The Khokhlov–Zabolotskaya–Kuznetzov (KZK) equation is considered as a model of nonlinear acoustic which describes the nonlinear propagation of a finite-amplitude focused sound beam which is essentially one-directional, in the thermo-viscous medium.1,7,8The aim of this paper is the study of the existence, uniqueness, stability, regularity, continuous dependence on the initial value and blow-up of solution of the KZK equation in Sobolev spaces Hsof periodic on x functions and with mean value zero. Existence of shock waves for the model with zero viscosity is proved using S. Alinhac's method.2Global existence in time of the beam's propagation in viscous media is established for small enough initial data. Existence result is proved by two methods: first by the fractional step method in the particular case ℝ3and s = 3 to justify the numerical results of Thierry Le Pollès25and second for the general case ℝnand s > [n/2] + 1 by the approach used in Refs. 12 and 13 for the Kadomtsev–Petviashvili (KP) equation.

Finite element simulation of nonlinear wave propagation in thermoviscous fluids including dissipation.

A recently developed finite element method (FEM) for the numerical simulation of nonlinear sound wave propagation in thermoviscous fluids is presented. Based on the nonlinear wave equation as derived by Kuznetsov, typical effects associated with nonlinear acoustics, such as generation of higher harmonics and dissipation resulting from the propagation of a finite amplitude wave through a thermoviscous medium, are covered. An efficient time-stepping algorithm based on a modification of the standard Newmark method is used for solving the non-linear semidiscrete equation system. The method is verified by comparison with the well-known Fubini and Fay solutions for plane wave problems, where good agreement is found. As a practical application, a high intensity focused ultrasound (HIFU) source is considered. Impedance simulations of the piezoelectric transducer and the complete HIFU source loaded with air and water are performed and compared with measured data. Measurements of radiated low and high amplitude pressure pulses are compared with corresponding simulation results. The obtained good agreement demonstrates validity and applicability of the nonlinear FEM.

Non-linear propagation of waves induced by general vibrations of plates

First-order uniform expansions for the non-linear propagation of waves induced by general motions of plates are obtained. The method of renormalization is used to determine the non-linear propagation of two waves and 2 N waves in a lossless medium. The strained co-ordinates contain terms that account for the effects of self-interaction and mutual interaction of the waves. The mutual interaction terms are negligible if the induced waves are pulses or periodic functions. The method of multiple scales is used to determine the propagation of two waves in a thermoviscous medium. The result consists of two uncoupled equations that can be reduced to Burgers' equations. They account for the effects of dissipation and self-interaction of the waves but not the effects of mutual interaction. A combination of the methods of renormalization and multiple scales can be used to account for dissipation and all interactions.

Asymptotic solutions of model equations in nonlinear acoustics

This paper uses the method of matched asymptotic expansions to derive asymptotic solutions to various problems in nonlinear acoustics. Model equations, generalizing the well known Burgers equation to include effects of cylindrical or spherical spreading and of non-equilibrium relaxation, are given and regarded as governing the propagation. Solutions are sought for initial or boundary conditions of N-wave or harmonic wave form. For a thermoviscous medium, the small parameter upon which the asymptotic expansions are predicated is an inverse acoustic Reynolds number; for a relaxing medium it is the product of wave frequency and relaxation time. The complete asymptotic solution for N-waves in a thermoviscous fluid is known, in the case of plane motion, from the Cole—Hopf solution of Burgers’s equation. Here a similarly complete solution is found for spherical N-waves, with the exception of one region of space—time in which an irreducible nonlinear problem remains unsolved. In this region the outer limiting behaviour is, nevertheless, determined, so that the solutions in all other regions are completely fixed. For cylindrical N-waves an irreducible problem again results, but the motion can be followed right through into its ‘old age’ phase aside from an undetermined purely numerical constant. Correct results are obtained here for the ‘correction due to diffusivity' to the weak-shock theory prediction of shock centre location for plane, cylindrical and spherical N-waves. These results indicate a non-uniformity in weak shock theory at large times, and also, in the case of spherical N-waves, reveal a large time non-uniformity in the Taylor shock solution. Harmonic waves, plane, cylindrical and spherical, in thermoviscous fluids and relaxing fluids are considered, and the asymptotic solutions are found to leading order in most of the many overlapping asymptotic regions of space-time. A single dimensionless function remains undetermined in the important case of spherical harmonic waves. We have also been unable to find scalings and differential equations describing precisely how a discontinuity is formed at the front of a partly dispersed shock in a relaxing gas, though the shock centre is located for both fully and partly dispersed shocks. The harmonic wave solutions unify and extend certain solutions (the Fay, Fubini and old-age solutions) which are well known in the nonlinear acoustics literature, and the amplitude saturation and scaling laws for the old age regime are in accord with experiments on high amplitude spherical waves in water.

Notes on the Exact Equations Governing the Propagation of Sound in Fluids

The assumptions underlying the exact equations of motion for a thermoviscous fluid are reviewed and the complete equations are given, for reference convenience, in both tensor and vector form. The first- and second-order acoustic equations are then exhibited and used to obtain the source terms that account for the generation of vorticity and streaming. In order to preserve a broad base from which to make the approximations appropriate under various circumstances, all terms are retained explicitly including those arising from any functional dependence of the viscosity and thermal coefficients on the state variables. The distinction between spatial and material coordinate systems is carefully drawn and conversion transforms are derived rigorously and their use illustrated. The general properties of finite-amplitude waves are demonstrated by including the second-order terms in a plane-wave solution of the exact wave equation in material coordinates, with special concern for the effects of large amplitude on speed of propagation and on wave-form distortion. Sound absorption and dispersion measures for a viscous conducting fluid are analyzed in terms of Truesdell's recent exact solution of the first-order secular equation. These differ characteristically from the corresponding measures predicted for pure relaxation in a two-fluid mixture. It is concluded that a complete and adequate theory of sound absorption and dispersion will need to take into account both relaxation and viscothermal phenomena as well as their interaction, and that until such a general theory is available, the exact theory of viscothermal effects—rather than the crude linear approximation commonly, but inappropriately, called “classical”—should be used in computing the “excess” absorption and dispersion to be accounted for by relaxation processes. The exact solutions of the secular equation permit a new evaluation, in series form, of the characteristic acoustic impedance for a thermoviscous medium. The notes conclude with a revised account of the spectral character of thermal noise in the acoustic medium based on the quantum hypothesis and a merger of the concepts of architectural acoustics and specific-heat theory.