On the stability of mixed convective motions in a vertical layer with wave-like boundaries

The stability of the stationary and oscillatory convective motions which develop in a vertical layer with periodically curved boundaries is studied for the case of longitudinal fluid injection. The amplitude of the boundary undulations and the flow of fluid along the layer are both assumed to be small, and methods of perturbation theory are used. The characteristic properties of the incremental spectrum of the spatially periodic motions are studied and the most dangerous types of perturbations as well as the forms of the stability regions are determined.

Theoretical investigations of the effect of spatial inhomogeneity of the boundary conditions on the stability of convection were sparse, and they deal mainly with horizontal layers of fluid /1–3/. Stationary, spatially periodic motions in a vertical layer with curved boundaries were investigated in /4/ for the case of free convection (when the flow was closed), and their stability was investigated in /5/. It was established that the presence of a small but finite flow of fluid along the layer leads to an increase in the number of different modes of flow, and to the appearance of non-stationary convective motions in the region near the threshold.     

Stability of viscous shock wave under periodic perturbation for compressible Navier–Stokes–Korteweg system

In this paper, we study the stability of a viscous shock wave for the isentropic Navier–Stokes–Korteweg (N-S-K) equations under space-periodic perturbation. It is shown that if the initial perturbation around the shock and the amplitude of the shock are small, then the solution of the N-S-K equations tends to the viscous shock.     

New estimates of the stability of one-dimensional plane-parallel flows of a viscous incompressible fluid

The stability of a number of one-dimensional plane-parallel steady flows of a viscous incompressible fluid is investigated analytically using the method of integral relations. The mathematical formulation is reduced to eigenvalue problems for the Orr–Sommerfeld equation. One of three versions is chosen as the boundary conditions: all the components of the velocity perturbation are equal to zero on both boundaries of the layer (in this case we have the classical Orr–Sommerfeld problem), all the components of the velocity perturbation on one of the boundaries are equal to zero and the perturbations of the shear component of the stress vector and of the normal component of the velocity are equal to zero on the other, and all the components of the velocity perturbation are equal to zero on one boundary and the other boundary should be free. The boundary conditions derived in the latter case, are characterized by the occurrence of a spectral parameter in them. For kinematic conditions the lower estimates of the critical Reynolds number – the Joseph–Yih estimates, are improved. In the remaining cases the technique of the integral-relations method is developed, leading to new estimates of the stability. Analogs of Squire's theorem are derived for the boundary conditions of all the types mentioned above. Upper estimates of the increment of the increase in perturbations in eigenvalue problems for the Rayleigh equation with two types of boundary conditions are given.     

Calculation of pressure in the linear problem of vibrator in a supersonic boundary layer : PMM vol. 43, no. 6, 1979, pp. 1014–1028

The supersonic flow over a body consisting of a triangular oscillating plate the vibrator— mounted between two flat plates is investigated. The body is assumed to be thermally insulated, and the vibrator dimensions and the oscillation frequencies to be such that the flow can be defined by equations of a boundary layer with self-induced pressure [1 — 5]. The oscillation amplitude is assumed small so that these equations can be linearized. The Fourier transform of the longitudinal coordinate is used for solution derivation. The inverse Fourier transform is obtained by numerical methods. It is shown that the perturbations of flow parameters induced by the vibrator are damped upstream and downstream in accordance with an exponential law.     

Effects of suction/blowing on the lower branch modes of the incompressible boundary layer flow due to a rotating-disk

In this study a theoretical approach is pursued to investigate the effects of suction and blowing on the structure of the lower branch neutral stability modes of three-dimensional small disturbances imposed on the incompressible von Karman’s boundary layer flow induced by a rotating-disk. Particular interest is placed upon the short-wavelength, non-linear and nonstationary crossflow vortex modes developing within the presence of suction/blowing at sufficiently high Reynolds numbers with reasonably small scaled frequencies. Following closely the asymptotic framework introduced in [1], the role of suction on the non-linear disturbances of the lower branch described first in [2] for the stationary modes only, is extended in order to obtain an understanding of the behavior of non-stationary perturbations. The analysis using the rational asymptotic technique based on the triple-deck theory enables us to derive initially an eigenrelation which describes the evolution of linear modes. The asymptotic linear modes calculated at high Reynolds number limit are found to be destabilizing as far as the non-parallelism accounted by the approach is concerned, and they compare fairly well with the numerical results generated directly by solving the linearized system with the usual parallel flow approximation. An amplitude equation is derived next to account for the effects of non-linearity. Even though the form of this equation is the same as that of found in [2] for no suction, it is under the strong influence of suction and blowing. This amplitude equation is shown to be adjusted by a balance between viscous and Coriolis forces, and it describes the evolution of not only the stationary but also the non-stationary modes for both suction and injection applied at the disk surface. A close investigation of the amplitude equation shows that the non-linearity is highly destabilizing for both positive and negative frequency waves, though finite amplitude growth of a disturbance having positive frequency close to the neutral location is more effective at destabilization of the flow under consideration. Finally, a smaller initial amplitude of a disturbance is found to be sufficient for the non-linear amplification of the modes in the case of suction, whereas a larger amplitude is required if injection is active on the surface of the disk.     

The non-linear Schrödinger equation with a periodic δ-interaction

We study the existence and stability of space-periodic standing waves for the space-periodic cubic nonlinear Schrödinger equation with a point defect determined by a space-periodic Dirac distributionat the origin. This equation admits a smooth curve of positive space-periodic solutions with a profile given by the Jacobi elliptic function of dnoidal type. Via a perturbationmethod and continuation argument, we prove that in the case of an attractive defect the standing wave solutions are stable in H _per ¹ ([?π, π]) with respect to perturbations which have the same space-periodic as the wave itself. In the case of a repulsive defect, the standing wave solutions are stable in the subspace of even functions of H _per ¹ ([?π, π]) and unstable in H _per ¹ ([?π, π]) with respect to perturbations which have the same space-periodic as the wave itself.     

On the Onset of Rayleigh-Bénard Convection in a Fluid Layer of Slowly Increasing Depth

A two-scaling approach is used to investigate the onset of convection in a fluid layer whose depth is a slowly increasing function of horizontal distance. It is shown that whatever the value of the imposed temperature difference between the boundaries (provided, of course, that the lower one is hotter) there are regions which are stable and regions which are unstable to small perturbations. As the depth increases the amplitude of steady solutions increases from exponentially small values to take on the familiar square-root behavior of weakly nonlinear solutions. The solution in this narrow transition region is described in terms of the second Painlevé transcendent. In the exceptional case when the perturbation takes the form of longitudinal rolls, this equation needs some modification in that the second derivative is replaced by the fourth. The flow in a horizontal layer when the temperature difference between the boundaries increases slowly may be treated in exactly the same way. The necessary modifications to theory and results are given in an Appendix.     

Stationary single waves in turbulent open-channel flow

Steady two-dimensional turbulent open-channel flow is considered. Stationary single-wave solutions are investigated. The fully-developed oncoming flow is slightly supercritical. The Reynolds number is very large. The analysis is kept free of turbulence modelling. As stationary solitary waves cannot exist in turbulent flow for a plane bottom with constant roughness [1], two particular perturbations of the conditions at the channel bottom are examined: 1) We revisit the case [1] where the friction coefficient locally differs slightly by a constant from the reference value upstream; 2) An unevenness of very small height in the channel bottom (bump, ramp) is admitted, with the bottom roughness taken constant. An analogy between these cases is presented. In both cases, three stationary solutions for the surface elevation are found: A stable and an unstable solitary wave, respectively, and a single wave of a second kind with smaller amplitude. For the latter, an analysis for weak dissipation yields a uniformly valid solution that is in good agreement with the numerical results for various parameters. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Zur Stabilität diskreter Teilspektren singulärer Differentialoperatoren zweiter Ordnung gegenüber Störungen der Koeffizienten und der Definitionsgebiete

The stability ofL ²-eigenvalues and associated eigenspaces of singular second order differential operators of Schrödinger-type is shown for asymptotic perturbations of the coefficients and the domain of definition. The perturbations involved are more general than those studied in [3] and [5], because we do not postulate the convergence of the coefficients “from above” or of the domains “from inside” or “from outside”. Moreover, the domain of definition is allowed to be perturbed in its interior. The underlying abstract perturbation theory was established in a previous paper [9].     

Stability of steady flows of perfect incompressible fluid with a free boundary : PMM vol. 42, no. 6, 1978, pp. 1049–1055

The criterion of stability of steady flow of a perfect incompressible fluid bounded by solid walls, indicated by Amol'd [1, 2], is extended to the case when a part of the flow region boundary is free and subjected to surface tension.     

The linear stability of inviscid shear flow of a vibrationally excited diatomic gas

The problem of the linear stability of plane-parallel shear flows of a vibrationally excited compressible diatomic gas is investigated using a two-temperature gas dynamics model. The necessary and sufficient conditions for stability of the flows considered are obtained using the energy integrals of the corresponding linearized system for the perturbations. It is proved that thermal relaxation produces an additional dissipation factor, which enhances the flow stability. A region of eigenvalues of unstable perturbations is distinguished in the upper complex half-plane. Numerical calculations of the eigenvalues and eigenfunctions of the unstable inviscid modes are carried out. The dependence on the Mach number of the carrier stream, the vibrational relaxation time τ and the degree of non-equilibrium of the vibrational mode is analysed. The most unstable modes with maximum growth rate are obtained. It is shown that in the limit there is a continuous transition to well-known results for an ideal fluid as the Mach number and τ approach zero and for an ideal gas when τ → 0.     

Vibration of a stamp partially adhering to an elastic medium : PMM, vol. 42, no. 6, 1978, pp. 1085–1092

There is examined the problem of vibration of a stamp of arbitrary planform occupying a space Ω and vibrating harmonically in an elastic medium with plane boundaries. It is assumed that the elastic medium is a packet of layers with parallel boundaries, at rest in the stiff or elastic half-space. Contact of three kinds is realized under the stamp: rigid adhesion in the domain Ω₁, friction-free contact in domain Ω₂, there are no tangential contact stresses, and “film” contact without normal force in domain Ω₃ (there are no normal contact stresses, only tangential stresses are present.). It is assumed that the boundaries of all the domains have twice continuously differentiable curvature and Ω = Ω₁ Ω₂ Ω₃.

The problem under consideration assumes the presence of a static load pressing the stamp to the layer and hindering the formation of a separation zone. Moreover, a dynamic load, harmonic in time, acts on the stamp causing dynamical stresses which are of the greatest interest since the solution of the static problem is obtained as a particular case of the dynamic problem for ω = 0 (ω is the frequency of vibration). The general solution is constructed in the form of a sum of static and dynamic solutions.

A uniqueness theorem is established for the integral equation of the problem mentioned and for the case of axisymmetric vibration of a circular stamp partially coupled rigidly to the layer, partially making friction-free contact, the problem is reduced to an effectively solvable system of integral equations of the second kind, which reduce easily to a Fredholm system.

These results are an extension of the method elucidated in [1], where by the approach in [1] must be altered qualitatively to obtain them.     

Nonmonotone functionals in the Lyapunov direct method for equations of the neutral type

We present new tests for the stability and asymptotic stability of trivial solutions of equations with deviating argument of the neutral type. Unlike well-known results, here we use nonmonotone indefinite Lyapunov functionals. Our class of functionals contains both Lyapunov-Krasovskii functionals and Lyapunov-Razumikhin functions as natural special cases. This class of functionals is broad enough that, in a number of stability tests, we have been able to omit the a priori requirement of stability of the corresponding difference operator. In addition, we present tests for the asymptotic stability of solutions of equations of the neutral type with unbounded right-hand side and new estimates for the magnitude of perturbations that do not violate the asymptotic stability if it holds for the unperturbed equation. The obtained estimates single out domains of the phase space in which perturbations should be small and domains in which essentially no constraints are imposed on the perturbation magnitude.     

A linear stability analysis of a mixed convection plume from a line source: higher order effects

Aiding mixed convection flow resulting from the vertical flow of a uniform stream past a horizontal line source of heat is of importance in many practical situations such as hot-wire anemometry, etc. In this paper, the stability of such a flow to small disturbances is analyzed in terms of the linear stability theory. The analysis treats the presence of the free stream as a perturbation of a natural convection plume generated by the line source of heat. The base flow as well as the disturbance field are determined by means of a systematic perturbation expansion.The results presented here extend the results of an earlier investigation [1], by considering second-order mixed convection effects. The results reveal that the free stream has a stabilizing effect. As expected, consideration of second-order mixed convection effects further enhances the stability of the flow. The reported results are valid at a large distance from the source where the flow field is dominated by buoyancy effects.

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Résumé Aidant la convection mixte d'un écoulement résultant d'un écoulement uniforme vertical, à l'arrière d'une source rectiligne horizontale de chaleur, est important en plusieurs situations pratiques, comme celui de l'anémomètre à fil chaud, etc. Dans cet article, la stabilité de ce genre d'écoulement aux pertubations est analysée par la théorie de la stabilité linéaire. L'analyse traits la présence de l'écoulement libre comme perturbation d'une convection naturelle du panache crée par la source rectiligne de chaleur. L'écoulement de base ainsi que le champ exposé aux perturbations sont déterminés au moyen d'une série de perturbations systématiques. L'analyse prend en considération la nature non-parallèle de l'écoulement de base.Les résultats présentés ici, élargissent les résultats d'une étude précédente [1], en considérant le second ordre des effets de la convection mixte. La consideration du second ordre des effets de la convection mixte augmente la stabilité de l'écoulement. De même, l'effet de la convection mixte sur la courbe neutre semble différent de celle-ci pour une paroi limitée des écoulements.

     

Deriving amplitude equations for weakly-nonlinear oscillators and their generalizations

Results by physicists on renormalization group techniques have recently sparked interest in the singular perturbations community of applied mathematicians. The survey paper, [Phys. Rev. E 54(1) (1996) 376–394], by Chen et al. demonstrated that many problems which applied mathematicians solve using disparate methods can be solved using a single approach. Analysis of that renormalization group method by Mudavanhu and O’Malley [Stud. Appl. Math. 107(1) (2001) 63–79; SIAM J. Appl. Math. 63(2) (2002) 373–397], among others, indicates that the technique can be streamlined. This paper carries that analysis several steps further to present an amplitude equation technique which is both well adapted for use with a computer algebra system and easy to relate to the classical methods of averaging and multiple scales.     

The Algebraic Perturbation Method for Generalized Inverses

Algebraic perturbation methods were first proposed for the solution of nonsingular linear systems by R. E. Lynch and T. J. Aird [2]. Since then, the algebraic perturbation methods for generalized inverses have been discussed by many scholars [3]-[6]. In [4], a singular square matrix was perturbed algebraically to obtain a nonsingular matrix, resulting in the algebraic perturbation method for the Moore-Penrose generalized inverse. In [5], some results on the relations between nonsingular perturbations and generalized inverses of $m\times n$ matrices were obtained, which generalized the results in [4]. For the Drazin generalized inverse, the author has derived an algebraic perturbation method in [6]. In this paper, we will discuss the algebraic perturbation method for generalized inverses with prescribed range and null space, which generalizes the results in [5] and [6]. We remark that the algebraic perturbation methods for generalized inverses are quite useful. The applications can be found in [5] and [8]. In this paper, we use the same terms and notations as in [1].     

Effect of interconvertibility of electromagnetic and gravitational waves in strong external electromagnetic fields and the propagation of waves in the field of a charged “black hole” : PMM vol. 38, n 6, 1974, pp. 1122–1129

It is shown that the behavior of an arbitrary wave propagating in the field of a nonrotating charged black hole is defined (with the use of quadratures) by four functions. Each of these functions obeys its second order equation of the wave kind. Short electromagnetic waves falling onto a black hole are reflected by its field in the form of gravitational and electromagnetic waves whose amplitude was explicitly determined. In the case of the wave carrying rays winding around the limit cycle the reflection and transmission coefficients were obtained in the form of analytic expressions.Various physical processes taking place inside, as well as outside a collapsing star, may induce perturbations of the gravitational, electromagnetic and other fields, and lead to the appearance in the surrounding space of waves of various kinds which propagate over a distorted background and are dissipated along its inhomogeneities.In the absence of rotation and charge in a star, the analysis of small perturbations of the gravitational fields is based on the system of Einstein equations linearized around the Schwarzschild solution. In [1, 2] this system of equations, after expansion of perturbations in spherical harmonics and Fourier transformation with respect to time, was reduced to two independent linear ordinary differential equations of second order of the form of the stationary Schrödinger equation for a particle in a potential force field. Each of these equations defines one of two possible independent perturbation kinds: “even” and “odd” (the different behavior of spherical tensor harmonics at coordinate inversion is the deciding factor in the determination of the kind of perturbation [1, 2]). Although these equations were derived with the superposition on the perturbations of the metric of specific coordinate conditions, they define, as shown in [4], the behavior of invariants of the perturbed gravitational field, which imparts to the potential barriers appearing in these equations an invariant meaning.The system of Maxwell equations on the background of Schwarzschild solution also reduces to similar equations, which differ from the above only by the form of potential barriers appearing in these [5].In the presence in the unperturbed solution of a strong electromagnetic field the gravitational and electromagnetic waves interact with each other, and transmutation takes place. The train of short periodic electromagnetic waves generates the accompanying train of gravitational waves. This phenomenon was first analyzed in [6] on and arbitrary background. It was shown in [7, 8] that dense stars surrounded by hot plasma may acquire a charge owing to splitting of charges by radiation pressure and the “sweeping out” of positrons nascent in vapors in strong electrostatic fields. The interaction of waves becomes particularly clearly evident in the neighborhood of black holes which may serve as “valves” by maintaining equilibrium between the relict electromagnetic and gravitational radiation in the Universe. Rotation of black holes intensifies this effect [6].If a nonrotating star possesses an electrostatic charge, the definition of perturbations of the electromagnetic and gravitational fields must be based on the complete system of Einstein-Maxwell equations linearized around the Nordström-Reissner solution. (Small perturbations of electromagnetic field outside a charged black hole were considered in [9, 10] on the basis of the system of Maxwell equations on a “rigid” background of the Nordström-Reissner solution, without taking into account the interconvertibility of gravitational and electromagnetic waves, which materially affects their behavior in the neighborhood of a charged black hole). Here this system of equations which define the interacting gravitational and electromagnetic perturbations are reduced to four independent second order differential equations, two for each kind of perturbations (an importsnt part is played here by the coordinate conditions imposed on the perturbations of the metric, proposed by the authors in [4]). Perturbation components of the metric and of the electromagnetic field are determined in quadratures by the solutions of these equations. If the charge of a star tends to vanish, two of the derived equations convert to equations for gravitational waves on the background of the Schwarzschild solution [1, 2], while the twoothers become equations which are equivalent to Maxwell solutions on the same background. The short-wave asymptotics of derived equations is determined throughout including the neighborhood of the limit cycle for the wave carrying rays. These solutions far away from the point of turn coincide with those obtained in [6] for any arbitrary background. Approximation of geometric optics does not provide correct asymptotics for impact parameters of rays which are close to critical for which the Isotropie and geodesic parameters wind around the limit cycle. This case is investigated below.A similar situation in the Schwarzschild field was analyzed in [11], where analytic expressions for the wave reflection and transmission coefficients were determined, and the integral radiation stream trapped by a black hole produced by another radiation component of the dual system was calculated.     

经过修正的层流流动的流动稳定性问题(Ⅲ)——经过修正的平行剪切流的非线性稳定性研究

本文根据文[1]给出的经过修正的层流流动的流动稳定性理论及平行剪切流中平均速度的一类修正剖面,研究了平行剪切流的非线性稳定性性质,并在本文的假设下,把背景湍流噪声的干扰引入了流动稳定性计算,对于平面Poiseuille流动和圆管Poiseuille流动,得到了与实验趋势相一致的结果.     

The linear problem of a vibrator in a subsonic boundary layer

The subsonic flow over a flat plate with a fitted to it triangular vibrator which effects harmonic oscillations is studied. The plate and vibrator are assumed heatinsulated, and the vibrator dimensions and oscillation frequency is such that the flow can be defined by equations of the boundary layer with self-induced pressure. The oscillation amplitude is assumed small, making it possible to linearize these equations. The solution is obtained by double application of the Fourier transform with respect to time and longitudinal coordinate. Inverse transformation is achieved by numerical methods. Analysis is carried out for the vibrator frequency ω lower than the critical ω_* predicted by the classical theory of stability. It is shown that vibrator-induced perturbations become rapidly damped upstream. Damping downstream is rapid for ω considerably lower than ω_* and slows down as ω approaches ω_*.     

Bio-inspired propulsion: Downstream travelling wave motion of a surface with finite & infinite extension

The thrust force on a surface that performs a fish-like travelling wave motion downstream to an oncoming flow is discussed. Unsteady potential flow, with vortex shedding from the trailing edge, is known to explain the generation of thrust. Contrarily, fish swimming has been related to the flow over an infinitely extended surface. To interlink both problems, the potential flow over the surface of finite length is considered in the limit of high wave numbers. It turns out that the leading order, space-periodic pressure does not contribute to thrust. Thus, the perturbation pressure is essential for propulsion. Besides, laminar flow is considered in the space-periodic setting. The present results reveal – in contrast to literature – that the surface force is always drag. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)