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We study the stability of a steady Eckart streaming jet that is acoustically forced at one end of a closed cylindrical cavity and impinges the wall at the other end, … We study the stability of a steady Eckart streaming jet that is acoustically forced at one end of a closed cylindrical cavity and impinges the wall at the other end, where a recirculation forms. This configuration generically represents industrial processes where acoustic forcing offers a contactless means of stirring or controlling confined flows. Successfully doing so, however, requires sufficient insight into the topology of the acoustically forced flow. This raises the question of whether the base acoustic streaming jet is stable and, when not, of which alternative states emerge. Using Linear Stability Analysis (LSA) and three-dimensional nonlinear simulations, we identify the instability mechanisms and determine the nature of the bifurcations that ensue. We show that the ratio $C_R$ between the cavity and the maximum beam radii determines the dominant unstable mode. For $4 \leq C_R \leq 6$, a non-oscillatory perturbation rooted in the jet impingement triggers a supercritical bifurcation. For $C_R = 3$, the flow destabilises through a subcritical non-oscillatory bifurcation. Further reducing $C_R$ increases the shear within the flow, and gradually relocates the instability in the shear layer between impingement-induced vortices: for $C_R = 2$, an unstable travelling wave grows out of a subcritical bifurcation, which becomes supercritical for $C_R=1$. For each geometry, the nonlinear 3D simulations validate the LSA, identify the saturated nonlinear state and its stability. This study offers fundamental insight into the stability of acoustically-driven flows in general, but also opens possible pathways to either induce turbulence acoustically, or to avoid it in realistic configurations.
This work identifies the physical mechanisms at play in the different flow regions along an Eckart acoustic streaming jet by means of numerical simulation based on a novel modeling of … This work identifies the physical mechanisms at play in the different flow regions along an Eckart acoustic streaming jet by means of numerical simulation based on a novel modeling of the driving acoustic force including attenuation effects. The flow is forced by an axisymmetric beam of progressive sound waves attenuating over a significant part of a closed cylindrical vessel where the jet is confined. We focus on the steady, axisymmetric and laminar regime. The jet typically displays a strong acceleration close to the source before reaching a peak velocity. At further distances from the transducer, the on-axis jet velocity smoothly decays before reaching the opposite wall. For each of these flow regions along the jet, we derive scaling laws for the on-axis velocity with the magnitude of the acoustic force and the diffraction of the driving acoustic beam. These laws highlight the different flow regimes along the jet and establish a clear picture of its spatial structure, able to inform the design of experimental or industrial setups involving Eckart streaming jets.
We consider the unsteady regimes of an acoustically-driven jet that forces a recirculating flow through successive reflections on the walls of a square cavity. The specific question being addressed is … We consider the unsteady regimes of an acoustically-driven jet that forces a recirculating flow through successive reflections on the walls of a square cavity. The specific question being addressed is to know whether the system can sustain states of low-dimensional chaos when the acoustic intensity driving the jet is increased, and, if so, to characterise the pathway and underlying physical mechanisms. We adopt two complementary approaches, both based on data extracted from numerical simulations: (i) We first characterise successive bifurcations through the analysis of leading frequencies. Two successive phases in the evolution of the system are singled out in this way, both leading to potentially chaotic states. The two phases are separated by a drastic simplification of the dynamics that immediately follows the emergence of intermittency. The second phase also features a second intermediate state where the dynamics is simplified due to frequency-locking. (ii) Nonlinear time series analysis enables us to reconstruct the attractor of the underlying dynamical system, and to calculate its correlation dimension and leading Lyapunov exponent. Both these quantities bring confirmation that the state preceding the dynamic simplification that initiates the second phase is chaotic. Poincar\'e maps further reveal that this chaotic state in fact results from a dynamic instability of the system between two non-chaotic states respectively observed at slightly lower and slightly higher acoustic forcing.
This study is a linear stability analysis of the flows induced by ultrasound acoustic waves (Eckart streaming) within an infinite horizontal fluid layer heated from below. We first investigate the … This study is a linear stability analysis of the flows induced by ultrasound acoustic waves (Eckart streaming) within an infinite horizontal fluid layer heated from below. We first investigate the dependence of the instability threshold on the normalized acoustic beam width H(b) for an isothermal fluid layer. The critical curve, given by the critical values of the acoustic streaming parameter, A(c), has a minimum for a beam width H(b) ≈ 0.32. This curve, which corresponds to the onset of oscillatory instabilities, compares well with that obtained for a two-dimensional cavity of large aspect ratio [A(x) = (length/height) = 10]. For a fluid layer heated from below subject to acoustic waves (the Rayleigh-Bénard-Eckart problem), the influence of the acoustic streaming parameter A on the stability threshold is investigated for various values of the beam width H(b) and different Prandtl numbers Pr. It is shown that, for not too small values of the Prandtl number (Pr > Pr(l)), the acoustic streaming delays the appearance of the instabilities in some range of the acoustic streaming parameter A. The critical curves display two behaviors. For small or moderate values of A, the critical Rayleigh number Ra(c) increases with A up to a maximum. Then, when A is further increased, Ra(c) undergoes a decrease and eventually goes to 0 at A = A(c), i.e., at the critical value of the isothermal case. Large beam widths and large Prandtl numbers give a better stabilizing effect. In contrast, for Prandtl numbers below the limiting value Pr(l) (which depends on H(b)), stabilization cannot be obtained. The instabilities in the Rayleigh-Bénard-Eckart problem are oscillatory and correspond to right- or left-traveling waves, depending on the parameter values. Finally, energy analyses of the instabilities at threshold have indicated that the change of the thresholds can be connected to the modifications induced by the streaming flow on the critical perturbations.
We prove infinite-dimensional versions of the shadowing lenuna and Smale's theorem ( for a transverse homoclinic orbit) of a C1 map, not a diffeomorphism, using the notion of an exponential … We prove infinite-dimensional versions of the shadowing lenuna and Smale's theorem ( for a transverse homoclinic orbit) of a C1 map, not a diffeomorphism, using the notion of an exponential dichotomy.
We announce a result concerning the continuous differentiability of the unknown boundary curve defined by a weak solution of the one-dimensional two-phase Stefan problem.We deal with the following two-phase Stefan … We announce a result concerning the continuous differentiability of the unknown boundary curve defined by a weak solution of the one-dimensional two-phase Stefan problem.We deal with the following two-phase Stefan problem: to determine u(x, t) for 0<:t<:T, 0<:X<:1 and s(t) for 0^t<±Tsuch that (i) 0<s(t)< 1, s(0)=b; (ii) u^pjUn for 0<t^T 9 0<x<s(t) and u t ==p 2 u xx for 0<t^T, s(t)<x<\; (iii) w(0, 0=/i(0>0 and u(l,t)=f 2 (t)<0 for O^t^T; (iv) u(x, 0)=Y>(JC) for 0^x<l; (v) u(s(t) 9 1)=0 for 0<st<>T; and (vi) ai(0= -u a (s(t)-0 9 t)+u x (s(t)+0, t) for 0<t<^T.Here and throughout, ft and a are positive parameters, b e (0, 1), f { and xp are continuous functions with/i(0)=y>(0),/ 2 (0)=^(1), ¥>(ô)=0, y>(x)>0 for 0<^x<b, y>(x)<0 for b<x£l 9 and \ip(x)\£K\b-x\ for O^x^l.Cannon and Primicerio [3], following the work of Cannon, Douglas and Hill [2] showed that this problem has a unique classical solution (one for which the expressions appearing in (vi) are defined and continuous for 0</^r) on condition that the f t and y> are bounded by certain constants which depend on the parameters of the problem. A. Friedman [4], AMS (MOS) subject classifications (1970).

Commonly Cited References

AVNER FRIEDMAN^)Introduction.The Stefan problem is a free boundary problem for parabolic equations.The solution is required to satisfy the usual initial-boundary conditions, but a part of the boundary is free.Naturally, an … AVNER FRIEDMAN^)Introduction.The Stefan problem is a free boundary problem for parabolic equations.The solution is required to satisfy the usual initial-boundary conditions, but a part of the boundary is free.Naturally, an additional condition is imposed at the free boundary.A two-phase problem is such that on both sides of the free boundary there are given parabolic equations and initial-boundary conditions, and neither of the solutions is identically constant.In case the space-dimension is one, there are numerous results concerning existence, uniqueness, stability, and asymptotic behavior of the solution; we refer to [1] and the literature quoted there (see also [8]).In the case of several space variables the problem is much harder.The difficulty is not merely due to mathematical shortcomings but also to complications in the physical situation.Thus, even if the data are very smooth the solution need not be smooth, in general.For example, when a body of ice having the shape keeps growing, the interfaces AB and CD may eventually coincide.Then, in the next moment the whole joint boundary will disappear.Thus the free boundary varies in a discontinuous manner.This example motivates one to look for "weak" solutions.In [4] the concept of a weak solution is defined.Furthermore, existence and uniqueness theorems are proved.The existence proofs are based on a finite-difference approximation.In the present work we give a simpler derivation of the existence theorems of [4].Our method has also the advantages that (i) it yields better inequalities on the solution and on its first derivatives than in [4], and (ii) it enables us to find certain
We describe the implementation of methods of nonlinear time series analysis which are based on the paradigm of deterministic chaos. A variety of algorithms for data representation, prediction, noise reduction, … We describe the implementation of methods of nonlinear time series analysis which are based on the paradigm of deterministic chaos. A variety of algorithms for data representation, prediction, noise reduction, dimension and Lyapunov estimation, and nonlinearity testing are discussed with particular emphasis on issues of implementation and choice of parameters. Computer programs that implement the resulting strategies are publicly available as the TISEAN software package. The use of each algorithm will be illustrated with a typical application. As to the theoretical background, we will essentially give pointers to the literature. (c) 1999 American Institute of Physics.
b (1.2) 9 1h(A;)1J(a),) + 9J3(x,4)V(x,A)dx = 6(i), ~~~~~~~~a where the matrix 0J3(i)Q) is free from x and {ah} is a finite or infinite set of points on a fundamental … b (1.2) 9 1h(A;)1J(a),) + 9J3(x,4)V(x,A)dx = 6(i), ~~~~~~~~a where the matrix 0J3(i)Q) is free from x and {ah} is a finite or infinite set of points on a fundamental interval [a, b]. Tamarkin [5] has considered the general case, but many of his results were obtained by limiting the discussion to the situation where 03(x,2)-?) or where no boundary points exist in the interior of the fundamental interval. In particular, he did not define the adjoint system for the general case. Wilder [9] and Cole [1] have treated the case of a finite set of points and no integral term. Langer [2] has developed the theory associated with a finite set of boundary points in a complex domain. Whyburn has made substantial contributions to the problem, including a summary [6; 8] of known results. He has also shown [7] that the condition (1.2) is, in a certain sense, equivalent to
(1979). Generic properties of nonlinear boundary value problems. Communications in Partial Differential Equations: Vol. 4, No. 3, pp. 293-319. (1979). Generic properties of nonlinear boundary value problems. Communications in Partial Differential Equations: Vol. 4, No. 3, pp. 293-319.
We announce a result concerning the continuous differentiability of the unknown boundary curve defined by a weak solution of the one-dimensional two-phase Stefan problem.We deal with the following two-phase Stefan … We announce a result concerning the continuous differentiability of the unknown boundary curve defined by a weak solution of the one-dimensional two-phase Stefan problem.We deal with the following two-phase Stefan problem: to determine u(x, t) for 0<:t<:T, 0<:X<:1 and s(t) for 0^t<±Tsuch that (i) 0<s(t)< 1, s(0)=b; (ii) u^pjUn for 0<t^T 9 0<x<s(t) and u t ==p 2 u xx for 0<t^T, s(t)<x<\; (iii) w(0, 0=/i(0>0 and u(l,t)=f 2 (t)<0 for O^t^T; (iv) u(x, 0)=Y>(JC) for 0^x<l; (v) u(s(t) 9 1)=0 for 0<st<>T; and (vi) ai(0= -u a (s(t)-0 9 t)+u x (s(t)+0, t) for 0<t<^T.Here and throughout, ft and a are positive parameters, b e (0, 1), f { and xp are continuous functions with/i(0)=y>(0),/ 2 (0)=^(1), ¥>(ô)=0, y>(x)>0 for 0<^x<b, y>(x)<0 for b<x£l 9 and \ip(x)\£K\b-x\ for O^x^l.Cannon and Primicerio [3], following the work of Cannon, Douglas and Hill [2] showed that this problem has a unique classical solution (one for which the expressions appearing in (vi) are defined and continuous for 0</^r) on condition that the f t and y> are bounded by certain constants which depend on the parameters of the problem. A. Friedman [4], AMS (MOS) subject classifications (1970).
The prerequisite for the study of this book is a knowledge of matrices and the essentials of functions of a complex variable. It has been developed from courses given by … The prerequisite for the study of this book is a knowledge of matrices and the essentials of functions of a complex variable. It has been developed from courses given by the authors and probably contains more material than will ordinarily be covered in a one-year course. It is hoped that the book will be a useful text in the application of differential equations as well as for the pure mathematician.
Abstract In Section 1 we present a general principle for associating nonlinear equations evolutions with linear operators so that the eigenvalues of the linear operator integrals of the nonlinear equation. … Abstract In Section 1 we present a general principle for associating nonlinear equations evolutions with linear operators so that the eigenvalues of the linear operator integrals of the nonlinear equation. A striking instance of such a procedure discovery by Gardner, Miura and Kruskal that the eigenvalues of the Schrödinger operator are integrals of the Korteweg‐de Vries equation. In Section 2 we prove the simplest case of a conjecture of Kruskal and Zabusky concerning the existence of double wave solutions of the Korteweg‐de Vries equation, i.e., of solutions which for |I| large behave as the superposition of two solitary waves travelling at different speeds. The main tool used is the first of remarkable series of integrals discovered by Kruskal and Zabusky.
Improvements are made on the theory for the stability of solitary waves developed by T. B. Benjamin. The results apply equally to the Kortewegde Vries equation and to an alternative … Improvements are made on the theory for the stability of solitary waves developed by T. B. Benjamin. The results apply equally to the Kortewegde Vries equation and to an alternative model equation for the propagation of long waves in nonlinear dispersive media.
The general linear difference-differential equation takes the form where x is a real variable, ν( x ) and A μν ( x ) are known functions and The general linear difference-differential equation takes the form where x is a real variable, ν( x ) and A μν ( x ) are known functions and
The Korteweg-de Vries equation, which describes the unidirectional propagation of long waves in a wide class of nonlinear dispersive systems, is well known to have solutions representing solitary waves. The … The Korteweg-de Vries equation, which describes the unidirectional propagation of long waves in a wide class of nonlinear dispersive systems, is well known to have solutions representing solitary waves. The present analysis establishes that these solutions are stable, confirming a property that has for a long time been presumed. The demonstration of stability hinges on two nonlinear functionals which for solutions of the Korteweg-de Vries equation are invariant with time: these are introduced in § 2, where it is recalled that Boussinesq recognized their significance in relation to the stability of solitary waves. The principles upon which the stability theory is based are explained in § 3, being supported by a few elementary ideas from functional analysis. A proof that solitary wave solutions are stable is completed in § 4, the most exacting steps of which are accomplished by means of spectral theory. In appendix A a method deriving from the calculus of variations is presented, whereby results needed for the proof of stability may be obtained independently of spectral theory as used in § 4. In appendix B it is shown how the stability analysis may readily be adapted to solitary-wave solutions of the ‘regularized long-wave equation’ that has recently been advocated by Benjamin, Bona &amp; Mahony as an alternative to the Korteweg-de Vries equation. In appendix C a variational principle is demonstrated relating to the exact boundaryvalue problem for solitary waves in water: this is a counterpart to a principle used in the present work (introduced in §2) and offers some prospect of proving the stability of exact solitary waves.
We Consider the boundary value problem [d] Here λ is a non negative parameter; f is a given real valuede function defined and a class C2 [d] is an arbitrarily … We Consider the boundary value problem [d] Here λ is a non negative parameter; f is a given real valuede function defined and a class C2 [d] is an arbitrarily specified function of class C1 on [0, n] satisfying [d] = 0. Under suitable hypotheses concerning f, we investigate the existence and stability properties of stationary solutions for (*). Our approach is to interpret (*) as a dynamical system in an appropriately chosen Banach space, and then to apply to (*) certain known results in the theory of Liapunov stability for general dynamical systems
This study is a linear stability analysis of the flows induced by ultrasound acoustic waves (Eckart streaming) within an infinite horizontal fluid layer heated from below. We first investigate the … This study is a linear stability analysis of the flows induced by ultrasound acoustic waves (Eckart streaming) within an infinite horizontal fluid layer heated from below. We first investigate the dependence of the instability threshold on the normalized acoustic beam width H(b) for an isothermal fluid layer. The critical curve, given by the critical values of the acoustic streaming parameter, A(c), has a minimum for a beam width H(b) ≈ 0.32. This curve, which corresponds to the onset of oscillatory instabilities, compares well with that obtained for a two-dimensional cavity of large aspect ratio [A(x) = (length/height) = 10]. For a fluid layer heated from below subject to acoustic waves (the Rayleigh-Bénard-Eckart problem), the influence of the acoustic streaming parameter A on the stability threshold is investigated for various values of the beam width H(b) and different Prandtl numbers Pr. It is shown that, for not too small values of the Prandtl number (Pr > Pr(l)), the acoustic streaming delays the appearance of the instabilities in some range of the acoustic streaming parameter A. The critical curves display two behaviors. For small or moderate values of A, the critical Rayleigh number Ra(c) increases with A up to a maximum. Then, when A is further increased, Ra(c) undergoes a decrease and eventually goes to 0 at A = A(c), i.e., at the critical value of the isothermal case. Large beam widths and large Prandtl numbers give a better stabilizing effect. In contrast, for Prandtl numbers below the limiting value Pr(l) (which depends on H(b)), stabilization cannot be obtained. The instabilities in the Rayleigh-Bénard-Eckart problem are oscillatory and correspond to right- or left-traveling waves, depending on the parameter values. Finally, energy analyses of the instabilities at threshold have indicated that the change of the thresholds can be connected to the modifications induced by the streaming flow on the critical perturbations.
We report the experimental evidence of the existence of a random attractor in a fully developed turbulent swirling flow. By defining a global observable which tracks the asymmetry in the … We report the experimental evidence of the existence of a random attractor in a fully developed turbulent swirling flow. By defining a global observable which tracks the asymmetry in the flux of angular momentum imparted to the flow, we can first reconstruct the associated turbulent attractor and then follow its route towards chaos. We further show that the experimental attractor can be modeled by stochastic Duffing equations, that match the quantitative properties of the experimental flow, namely, the number of quasistationary states and transition rates among them, the effective dimensions, and the continuity of the first Lyapunov exponents. Such properties can be recovered neither using deterministic models nor using stochastic differential equations based on effective potentials obtained by inverting the probability distributions of the experimental global observables. Our findings open the way to low-dimensional modeling of systems featuring a large number of degrees of freedom and multiple quasistationary states.
The idea of co-limiting sets borrowed from the theory of dynamical systems has been employed in many authors' works including [1] and [5 ] to investigate the asymptotic properties of … The idea of co-limiting sets borrowed from the theory of dynamical systems has been employed in many authors' works including [1] and [5 ] to investigate the asymptotic properties of solutions of autonomous parabolic initial-boundary value problem s. In the case of parabolic systems, it is well-known that the w-limiting set of a solution often contains plural elements (in fact infinitely many elements). But, to the best of our knowledge, it has not yet been made clear in the case of single equations whether there exists such a solution as has plural co-limiting points. The present paper forms part of the answer to this question. That is, we show that the co-limiting set of any solution contains at most one element providing that the space dimension is one. This result leads to the conclusion that in the case of single onedimensional equations any solution that neither blows up in a finite time nor grows up as t—*oo should converge to some equilibrium solution as t tends to infinity.