Group properties of utt=[F(u)ux]x

The group properties and the associated Lie algebra are developed for the subject quasilinear wave equation, for arbitrary f[fεC²(R), f > 0, f ≠ 0]. From the resulting information sets of explicit invariant solutions are constructed for wave propagation in gases and for the transonic equation.     

On the radiation properties of an asymmetric cylinder

The radiation properties of an asymmetric cylinder, formed by slicing a circular cylinder with a plane making an angle π/2 − Λ with the cylinder axis, are investigated as a model problem of relevance to noise emission by novel aeroengine intakes. We consider the scattering of incident duct modes and use asymptotic analysis in the limit Λ 1; the unsteady flow then comprises an inner incompressible region around the cylinder rim and an outer region comprising the rest of space, and the radiation in the outer region is determined using the method of matched asymptotic expansions. From this we are able to conclude that the correction to the radiation directivity is O(Λ), whilst the correction to the total integrated sound power reaching infinity is much weaker, and is in fact only O(Λ²).     

An application of Nagumo''s lemma to some singularly perturbed systems

The existence and asymptotic behavior as ε → 0⁺ of periodic, almost periodic, and bounded solutions of the differential system x = f(t, x, y, ε), Ωy′ = g(t, x, y, ε), are considered where x, f; are n-vectors, y, g are m-vectors and Ω = diag{ε^h₁}…, ε^h_m for integral h_i, h₁ h₂ …, h_m. The principal tools are a lemma of Nagumo which allows the construction of appropriate upper and lower solutions and the asymptotic theory of singularly perturbed linear differential systems.     

Uniformly valid approximations for non-linear oscillations with small damping

A uniformly valid zeroth-order approximation is obtained for the general equation y + εH(y)y + M(y)y = 0, where ε is a small parameter. The notion of multiple scaling is utilized to set up a systematic approximation scheme. Examples are given for simple polynomials for H(y) and M(y), which lead to results involving elliptic integrals. Further restrictions allow progress to be made in terms of gamma functions.     

Non-linear oscillations with multiple forcing terms

The problem of determining the transient response of a non-linear oscillator of the form ü + u = εƒ(u,u) + E(t) is studied by the method of multiple time scales, using the symbolic computation system MACSYMA. when the excitation E(t) consists of a finite number of harmonic forcing terms. Here ε is a small parameter and ƒ(u,u) is a non-linear function of its arguments. In particular, the Van der Pol and Duffing oscillators are studied in detail. It is found that when the forcing frequencies are not close to each other or close to the primary resonance of the system, then the response of the system is analogous to the behavior when only one forcing term is present. However, when the forcing frequencies are close to each other or close to the primary resonance, then the behavior is quite different, exhibiting certain oscillations not observed in the case of one forcing term.     

The near wake structure of a square cylinder

The near wake structure of a square cross section cylinder in flow perpendicular to its length was investigated experimentally over a Reynolds number (based on cylinder width) range of 6700–43,000. The wake structure and the characteristics of the instability wave, scaling on θ at separation, were strongly dependent on the incidence angle () of the freestream velocity. The nondimensional frequency (St_θ) of the instability wave varied within the range predicted for laminar instability frequencies for flat plate wakes, jets and shear layers. For = 22.5°, the freestream velocity was accelerated over the side walls and the deflection of the streamlines (from both sides of the cylinder) towards the center line was higher compared to the streamlines for = 0°. This caused the vortices from both sides of the cylinder to merge by x/d 2, giving the mean velocity distribution typical of a wake profile. For = 0°, the vortices shed from both sides of the cylinder did not merge until x/d 4.5. The separation boundary layer for all cases was either transitional or turbulent, yet the results showed good qualitative, and for some cases even quantitative, agreement with linearized stability results for small amplitude disturbances waves in laminar separation layers.     

Wavefront expansions for non-linear axial shear wave propagation

Wavefront expansions are employed to study non-linear axial shear waves propagating from a cylindrical cavity in an unbounded medium. Depending on the nature of the boundary condition, an acceleration front or a shock front propagates from the boundary of the cavity. For an acceleration front the coefficients in the wavefront expansion satisfy a sequence of transport equations which can be solved analytically. Numerical results based on a Padé-extended version of these expansions are presented for this case. For a shock front, a wavefront analysis gives approximate formulas for the wave speed, shock front and intensity of the various field variables at the front. In order to understand the usefulness and limitations of wavefront expansions, all problems analysed using these expansions are also solved numerically by MacCormick's finite difference scheme.     

Equations differentielles non lineaires: Differentiabilite des solutions par rapport aux conditions initiales et problemes de la stabilite

In this paper, we examine the problem of the asymptotic stability of the solutions of a differential equation (E), Y = X(Y, t), from the point of view of the differentiability of the solutions with respect to the initial conditions. The method allows us to deal with cases in which the mapping X is not differentiable everywhere, and in which the variational equation of (E) is not defined in the usual meaning. This method can be carried on for the research of periodic solutions. We give two examples.

Zusammenfassung

In diesem Bericht wird das Problem der asymptotischen Stabilität der Lösungen einer Differential Gleichung (E), Y = X(Y, t), vom Standpunkt der Differenzierbarkeit der Lösungen in Bezug auf die Anfangsbedingungen untersucht. Das Verfahren erlaubt die Fälle zu behandeln, worin die Abbildung X nicht überall differenzierbar ist, und worin auch die Variations Gleichung nicht im üblichen Sinne definiert ist. Das Verfahren kann erweitert werden zur Untersuchung periodischer Lösungen. Zwei Beispiele werden vorgeführt.

Abstract

B aдннoй paбaтe иccлeдyeтcя вoпoc oб accимeтpиХecкoй ycтoйХибocти peшeний диффepeHциaлънoгo ypaвнeния (E), Y = X(Y, t), c тoХки зpeния диффepeнциpyeмocти peшeний пo oтнoшeнию к нaХaлъным ycлoвиям. Иcпoлъзyeмый мeтoд дaeт вoзмoжнocтъ иccлeдoвaтъ cлyХaи, кoгдa пpeoбpaзoвaниe X нe вcюдy диффepeнциaлнoe. Bapиaциoннoe ypaвнeниe для oпpeдeляeтcя нecкoлъкo инaХe Хeм oбыХнo. Дaнный мeтoд мoжeт иcпoлъзoвaтъcя для иccлeдoвa*ncy;ия пepиoдиХecкиx peшeний. Дaютcя двa пpимepa.     

Acoustic scattering from baffled membranes that are backed by elastic cavities

An elastic membrane backed by a fluid-filled cavity in an elastic body is set into an infinite plane baffle. A time harmonic wave propagating in the acoustic fluid in the upper half-space is incident on the plane. It is assumed that the densities of this fluid and the fluid inside the cavity are small compared with the densities of the membrane and of the elastic walls of the cavity, thus defining a small parameter . Asymptotic expansions of the solution of this scattering problem as →0, that are uniform in the wave number k of the incident wave, are obtained using the method of matched asymptotic expansions. When the frequency of the incident wave is bounded away from the resonant frequencies of the membrane, the cavity fluid, and the elastic body, the resultant wave is a small perturbation (the “outer expansion”) of the specularly reflected wave from a completely rigid plane. However, when the incident wave frequency is near a resonant frequency (the “inner expansion”) then the scattered wave results from the interaction of the acoustic fluid with the membrane, the membrane with the cavity fluid, and finally the cavity fluid with the elastic body, and the resulting scattered field may be “large”. The cavity backed membrane (CBM) was previously analyzed for a rigid cavity wall. In this paper, we study the effects of the elastic cavity walls on modifying the response of the CBM. For incident frequencies near the membrane resonant frequencies, the elasticity of the cavity gives only a higher order (in ) correction to the scattered field. However, near a cavity fluid resonant frequency, and, of course, near an elastic body resonant frequency the elasticity contributes to the scattered field. The method is applied to the two dimensional problem of an infinite strip membrane backed by an infinitely long rectangular cavity. The cavity is formed by two infinitely long rectangular elastic solids. We speculate on the possible significance of the results with respect to viscoelastic membranes and viscoelastic instead of elastic cavity walls for surface sound absorbers.     

On a Korteweg–de Vries-like equation with higher degree non-linearity

We deal with a non-linear partial differential equation which has been widely investigated owing to its applications in quantum field theory, as well as plasma and solid-state physics. It is the matter of a third order KdV-like equation with higher degree non-linearity in the coefficient of the transport term; it can be derived from a Lagrangian or an Hamiltonian density. In the current literature specific attention has been devoted to the search for traveling-wave solutions, depending upon a positive parameter v, which assesses the speed of the solitary wave. The velocity v is always assumed to be constant, as its dependence on the wave-amplitude is neglected in the mathematical model. In this context, Coffey [On series expansions giving closed-form solutions of Korteweg–de Vries-like equations, SIAM J. Appl. Math. 50 (6) (1990) 1580–1592] exploits an algebraic recursive technique to obtain these solutions in closed form for particular values of v. The aim of this paper is to extend these results by showing that closed-form solutions are achievable for every value of v: to this purpose we supply a proper mathematical framework for these issues by taking into account a suitable special function, namely an elliptic function in the sense of Weierstraß. Furthermore we obtain two classes of the so-called kink solutions, see [M.W. Coffey, On series expansions giving closed-form solutions of Korteweg–de Vries-like equations, SIAM J. Appl. Math. 50 (6) (1990) 1580–1592; B. Dey, Domain wall solutions of KdV-like equations with higher order non-linearity, J. Phys. A 19 (1) (1986) L9–L12], as well as an exponential development of the general solution, for which we prove the convergence. Eventually we show how to implement the resulting functions by means of a symbolic manipulation program.     

Stress state of compound polygonal plate

The constructions made of bars and plates with holes, openings and bulges of various forms are widely used in modern industry. By loading these structural elements with different efforts, there appears concentration (accumulation) of stress whose values sometimes exceeds the admissible one. The durability of the given element is defined according to the quantity of these stresses. Since the failure of details and construction itself begins from the place where the stress concentration has the greatest value.

Therefore the exact determination of stress distribution in details (bars, plates, beams) is of great scientific and practical interest and is one of the important problems of the solid fracture.

Compound details (when the nucleus of different material is soldered to the hole) are often used to decrease the stress concentration.

In the present paper, we study a stress–strain state of polygonal plate weakened by a central elliptic hole with two linear cracks info which a rigid nucleus (elliptic cylinder with two linear bulges) of different material was put in (soldered) without preload.

The problem is solved by a complex variable functions theory stated in papers [Theory of Elasticity, Higher School, Moscow, 1976, p. 276; Plane Problem of Elasticity Theory of Plates with Holes, Cuts and Inclusions, Publishing House Highest School, Kiev, 1975, p. 228; Bidimensional Problem of Elasticity Theory, Stroyizdat, Moscow, 1991, p. 352; Science, Moscow (1996) 708; MSB AH USSR OTH 9 (1948) 1371].

Kolosov–Mushkelishvili complex potential (z) and ψ(z) satisfying the definite boundary conditions are sought in the form of sums of functional series.

After making several strict mathematical transformations, the problem is reduced to the solution of a system of linear algebraic equations with respect to the coefficients of expansions of functions (z) and ψ(z).

Determining the values of (z) and ψ(z), we can find the stress components σ_r, σ_θ and τ_rθ at any point of cross-section of the plate and nucleus on the basis of the known formulae. The obtained solution is illustrated by numerical example.

Changing the parameters A₁, m₁, e, A₂, and m₂ we can get the various contour plates.

For example, if we assume m₁=0, A₁=r, then the internal contour of L₁ becomes the circle of radius r with two rectilinear cracks (for the nucleus––a rectilinear bulges).

Further, if we assume a small semi-axis of the ellipse b₁ to be equal to zero (b₁=0), then a linear crack becomes the internal contour of L₁ (and the nucleus becomes the linear rigid inclusion made of other material). For m₂=0; A₂=R, the external contour L₂ turns into the circle of radius R.

The obtained method of solution may be applied and in other similar problems of elasticity theory; tension of compound polygonal plate, torsion and bending of compound prismatic beams, etc.     

Generalised vortex rings with and without swirl

Steady solutions of the Euler equations for flow of an inviscid incompressible fluid may be obtained by considering the process of magnetic relaxation to analogous magnetostatic equilibria in a viscous perfectly conducting fluid. In particular, solutions which represent rotational disturbances propagating without change of structure in an unbounded fluid may be obtained by this method. When conditions are axisymmetric, these disturbances are vortex rings of general structure, which may include a swirl component of velocity. This situation is analysed in some detail, and it is shown that the vortex is characterised by two functions: V(ψ), the volume within toroidal surfaces ψ = cst. and W(ψ), the toroidal volume flux inside the torus ψ = cst. For each choice of {V(ψ), W(ψ)}, satisfying appropriate limit conditions, there exists at least one vortex ring of steady structure.     

Further consideration of the shape-factor relationship for turbulent boundary layers

The lag-entrainment predictive scheme developed by Green et al. has been modified to include the pressure-gradient parameter Π₁. In the original model suggested by Green et al. the mass-flow shape factor H₁ is related to the common shape factor H, H₁ = f(H). In the present model H₁ is related to H, Reynolds number based on the local momentum thickness θ, and Π₁; thus H₁ = f(H, R_eθ, Π₁). The modified formula for H₁, is introduced into the original lag-entrainment integral model. Calculations are made to examine the present model for the predictions of the development of boundary layers approaching separation studied experimentally by the authors. Slightly improved predictions are obtained using the model developed by El Telbany et al. However, the present model proved to give an improved representation of the development of wall shear stress in cases the two-equation turbulence model proved to be unsuccessful.     

Permeability of periodic arrays of spheres

For periodic arrays of spheres the permeability is obtained numerically as a function of the dimensionless wave number kD in the flow direction, where D is the sphere diameter, k = 2π/λ is the wave number, and λ is the distance between the spheres in the flow direction. Our numerical results for the solids fraction of 0.45 show that for kD < 6.5 the permeability increases with increasing kD. But, it decreases for 6.5 < kD < 8.5 and reaches a local minimum at kD  8.5, and then increases again with increasing kD. Since the Fourier spectrum of the area fraction is zero for kD = 8.98, this result suggests that the area fraction plays an important role in determining the dependence of permeability on the distance between the spheres in the flow direction. For smaller solids fractions, the positions of the local maximum and minimum of permeability shift to slightly smaller kD’s.     

Temperature spectra from a turbulent thermal plume by ultrasound scattering

This paper reports the results of a study on temperature inhomogeneities conducted on a thermal plume by using ultrasound scattering as a non-intrusive measurement technique. The plume rises from a metallic disk which can be heated up to 800 °C. The working fluid is air at atmospheric pressure. In the measurement technique, an incoming ultrasound wave is emitted towards the thermal plume. The incident wave is scattered because of non-linear couplings with the flow instabilities present in the measurement region. The scattered wave carries information about those flow instabilities. The technique allows for the retrieving of this information. The shape of the obtained spectrum of temperature fluctuations as a function of wave vector modulus is consistent with previous theoretical analysis. Three qualitatively different regions were identified: first, a production region characterized by a q² law; secondly, a region with behavior as per q⁻³ associated with a buoyancy region and; finally, a dissipation region associated with a q⁻⁷ law. These spectral regions characterize the energy transfers mechanisms among the length scales of flow investigated here. A coefficient of anisotropy γ was defined to analyze anisotropic features of the flow.     

Double-diffusive natural convection in a porous enclosure partially heated from below and differentially salted

This paper reports numerical results of two-dimensional double-diffusive natural convection in a square porous cavity partially heated from below while its upper surface is cooled at a constant temperature. The vertical walls of the porous matrix are subjected to a horizontal concentration gradient. The parameters governing the problem are the thermal Rayleigh number (Ra=100 and 200), the Lewis number (Le=0.1, 1 and 10), the buoyancy ratio (−10N10) and the relative position of the heating element with respect to the vertical centerline of the cavity (δ=0 and 0.5). The effect of the governing parameters on fluid characteristics is analyzed. The multiplicity of solutions is explored and the existence of asymmetric bicellular flow is proved when the heated element is shifted towards a vertical boundary (δ=0.5). The solutal buoyancy forces induced by horizontal concentration gradient lead to the elimination of the multiplicity of solutions obtained in pure thermal convection when N reaches some threshold value which depends on Le and Ra.     

Hydromagnetic flow of a dusty fluid over a stretching sheet

Analysis of hydromagnetic flow of a dusty fluid over a stretching sheet is carried out with a view to throw adequate light on the effects of fluid-particle interaction, particle loading, and suction on the flow characteristics. The equations of motion are reduced to coupled non-linear ordinary differential equations by similarity transformations. These coupled non-linear ordinary differential equations are solved numerically on an IBM 4381 with double precession, using a variable order, variable step-size finite-difference method. The numerical solutions are compared with their approximate solutions, obtained by a perturbation technique. For small values of β the exact (numerical) solution is in close agreement with that of the analytical (approximate) solution. It is observed that, even in the presence of a transverse magnetic field and suction, the transverse velocity of both the fluid and particle G phases decreases with an increase in the fluid-particle interaction parameter, β, or the particle-loading parameter, k. Moreover, the particle density is maximum at the surface of the stretching sheet, and the shearing stress increases with an increase in β or k.     

Numerical study of the parametric evolution of bifurcations in the three-dimensional ring problem of N bodies

We study the dynamics of a massless particle in an annular configuration of N bodies, N − 1 of which have equal masses m and are located in equal distances on a fictitious circle and one has mass βm and is located at the center of the circle. Our interest is focused on the bifurcation points from planar to three-dimensional families of symmetric periodic orbits in the above problem. We study numerically the evolution of these bifurcation points with respect to the variation of the mass parameter β. In particular we investigate the continuous evolution of bifurcation points for values of β from 2 up to 1000. The two distinct cases of the system’s behavior at β = 2 and 1000 are examined comparatively and various conclusions are drawn regarding the overall dynamical evolution of the three-dimensional system as the relative mass of the central body grows.