Asymptotic Modeling of Nonlinear Wave Processes in Shock‐Loaded Elastoplastic Materials

Nonlinear wave processes in shockloaded elastoplastic materials are modeled. A comparison of the results obtained with experimental data and numerical solutions of exact systems of dynamic equations shows that the model equations proposed qualitatively describe the stressdistribution evolution in both the elasticflow and plasticflow regions and can be used to solve one and twodimensional problems of pulsed deformation and fracture of elastoplastic media.     

Investigation of the flow past wings of infinite span with a blunt leading edge in the vorticity interaction regime

An asymptotic analysis of the Navier-Stokes equations is carried out for the case of hypersonic flow past wings of infinite span with a blunt leading edge when 0, Re , and M . Analytic solutions are obtained for an inviscid shock layer and inviscid boundary layer. The results of a numerical solution of the problems of vorticity interaction at the blunt edge and on the lateral surface of the wing are presented. These solutions are compared with the solution of the equations of a thin viscous shock layer and on the basis of this comparison the boundaries of the asymptotic regions are estimated.deceasedTranslated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 120–127, November–December, 1987.     

Hyperbolicity of Steady‐State Equations of Gas Shear Flows in a Thin Layer

The steadystate threedimensional motion of an ideal gas in a thin layer of variable height is considered. In the longwave approximation, the equations of gas dynamics reduce to a system of integrodifferential equations. The generalized characteristics and hyperbolicity conditions of the obtained system are found.     

Exact Solutions of the Hydrodynamic Equations Derived from Partially Invariant Solutions

The paper proposes a heuristic approach to constructing exact solutions of the hydrodynamic equations based on the specificity of these equations. A number of systems of hydrodynamic equations possess the following structure: they contain a reduced system of n equations and an additional equation for an extra function w. In this case, the reduced system, in which w = 0, admits a Lie group G. Taking a certain partially invariant solution of the reduced system with respect to this group as a seed:rdquo; solution, we can find a solution of the entire system, in which the functional dependence of the invariant part of the seed solution on the invariants of the group G has the previous form. Implementation of the algorithm proposed is exemplified by constructing new exact solutions of the equations of rotationally symmetric motion of an ideal incompressible liquid and the equations of concentrational convection in a plane boundary layer and thermal convection in a rotating layer of a viscous liquid.     

Asymptotic theory of wave-propagation

A general method is presented for finding asymptotic solutions of problems in wave-propagation. The method is applicable to linear symmetric-hyperbolic partial differential equations and to the integro-differential equations for the electromagnetic field in a dispersive medium. These equations may involve a large parameter . In the electromagnetic case is a characteristic frequency of the medium. The parameter may also appear in initial data or in the source terms of the equations, in a variety of different ways. This gives rise to a variety of different types of asymptotic solutions. The expansion procedure is a ray method, i.e., all the functions that appear in the expansion satisfy ordinary differential equations along certain space-time curves called rays. In general, these rays do not lie on characteristic surfaces, but may, for example, fill out the interior of a characteristic hypercone. They are associated with an appropriately defined group velocity. In subsequent papers the ray method developed here will be applied to the analysis of transients, Cerenkov radiation, transition radiation, and other phenomena of wave-propagation.An interesting by-product of the ray method is the conclusion, derived in section 6.3, that the theory of relativity imposes no restriction on the speed of energy transport in anisotropic media.This research was supported by the Air Force Cambridge Research Laboratories, Office of Aerospace Research, under Contract No. AF 19(628)4065     

The asymptotic theory of semilinear perturbed telegraph equation and its application

I.IntroductionConsiderthefollowingsemilinearperturbedtelegraphequationuII-u., P'u==sj(t,x,u,ul,u,,e)(-ooo)(l.l)u(o,x)=u,(x,e)(-ooo,u=u(t,x),fuoandulsatisfycertainsmoo…     

Group Classification of Equations of the Form y″ = f(x,y)

The problem of classification of ordinary differential equations of the form y = f(x,y) by admissible local Lie groups of transformations is solved. Standard equations are listed on the basis of the equivalence concept. The classes of equations admitting a oneparameter group and obtained from the standard equations by invariant extension are described.     

Similar solutions and the asymptotics of filtration equations

In this paper we consider the asymptotic behavior of solutions of the quasilinear equation of filtration as t. We prove that similar solutions of the equation u _t = (u )_xx asymptotically represent solutions of the Cauchy problem for the full equation u _t = [(u)]_xx if (u) is close to u for small u.     

Channel flow of an anisotropically conducting medium in a zone of entry into a magnetic field

A considerable number of papers are devoted to the problem of the deformation of a plane flow of a conducting liquid moving through a channel |x| < , 0 y h=const in a zone of entry into a magnetic field B=(0, 0, B. (x)), where (x) is the Heaviside unit function((x)=0 for x < 0 and (x)=i for x < 0). Apparently the first paper in this direction was that of Shercliff [1, 2] in which the asymptotic (for x .o- )profile of a perturbed velocity was. determined for a flow of an isotropic conducting liquid in a channel with nonconducting walls. The flow considered by Shemliff takes place in magnetohydrodynarnic flowmeters. Complete calculation of the perturbed flow of an isotropie conducting liquid in the channel of a magnetohydrodynamic generator is carried out in [3]. Asymptotic velocity profiles in the channel of a magnetohydrodynamic generator, with ideally segmented electrodes and the flow of an anisotropically conducting medium along them, were found in [4]. General formulas for the calculation of the asymptotic velocity profile, from the known distribution of the perturbing forces along the channel, are presented in [5]. In [6, 7] the Green function is proposed for the solution of the equation for the stream function of the perturbed flow. Finally, in [8], the solution for the perturbed flow of an anisotropically conducting liquid in a channel with continuous electrodes is described by means of the Green function, and the asymptotic profiles of the velocity are calculated.In this paper the flow of anauisotropically conducting liquid is determined in a channel with ideally segmented electrodes. The solution is set up with the aid of the Fourier method. The resulting series, in which the slowly converging part can be related to the asymptotic profile [4] calculated from the solution of an ordinary differential equation, make it possible to determine the velocity field rapidly. A detailed deformation pattern of the velocity profile is set up. Certain general properties of the flow in a zone of entry into a magnetic field are noted; with the aid of these it is possible to discover the error in the calculations [8].     

The dynamic performance of the Weissenberg rheogoniometer III. Large amplitude oscillatory shearing — harmonic analysis

The harmonic content of the nonlinear dynamic behaviour of 1% polyacrylamide in 50% glycerol/water was studied using a standard Model R 18 Weissenberg Rheogoniometer. The Fourier analysis of the Oscillation Input and Torsion Head motions was performed using a Digital Transfer Function Analyser.In the absence of fluid inertia effects and when the amplitude of the (fundamental) Oscillation Input motion _I is much greater than the amplitudes of the Fourier components of the Torsion Head motion _Tn empirical nonlinear dynamic rheological propertiesG _n (, ⁰),G _n (, ⁰) and/or _n (, ⁰), _n (, ⁰) may be evaluated without a-priori-knowledge of a rheological constitutive equation. A detailed derivation of the basic equations involved is presented.Cone and plate data for the third harmonic storage modulus (dynamic rigidity)G ₃ (, ⁰), loss modulusG ₃ (, ⁰) and loss angle ₃ (, ⁰) are presented for the frequency range 3.14 × 10^–2 1.25 × 10² rad/s at two strain amplitudes, _CP ⁰ = 2.27 and 4.03. Composite cone and plate and parallel plates data for both the third and fifth harmonic dynamic viscosities ₃ (, ⁰), _S (, ⁰) and dynamic rigiditiesG ₃ (, ⁰),G ₅ (, ⁰) are presented for strain amplitudes in the ranges 1.10 _CP ⁰ 4.03 and 1.80 _PP ⁰ 36 for a single frequency, = 3.14 × 10^–1 rad/s. Good agreement was obtained between the results from both geometries and the absence of significant fluid inertia effects was confirmed by the superposition of the data for different gap widths.     

Flicker noise and self‐organized criticality in crisis boiling regimes

The paper describes an experimental study of thermal fluctuations during transition from bubble to film boiling of water on a wire heater and fluctuations of the shape of a superheated liquid jet discharged from a highpressure vessel. It is found that for a heattransfer crisis on the wire heater and for intense volume boiling of the superheated liquid jet, the fluctuation power spectrum has a lowfrequency component (flicker noise) that diverges under the law 1/f. This effect is due to nonequilibrium phase transitions in the system: the heattransfer crisis during transition from bubble to film boiling and a flow crisis during boiling of the superheated liquid jet.     

Stability of Thermal Vibrational Flow in an Inclined Liquid Layer Against Finite‐Frequency Vibrations

Stability of the flow that arises under the action of a gravity force and streamwise finitefrequency vibrations in a nonuniformly heated inclined liquid layer is studied. By the Floquet method, linearized convection equations in the Boussinesq approximation are analyzed. Stability of the flow against planar, spiral, and threedimensional perturbations is examined. It is shown that, at finite frequencies, there are parametricinstability regions induced by planar perturbations. Depending on their amplitude and frequency, vibrations may either stabilize the unstable ground state or destabilize the liquid flow. The stability boundary for spiral perturbations is independent of vibration amplitude and frequency.     

“Thermal layer” effect in MHD: Interaction of a parallel shock with a layer of decreased density

Interaction of a parallel fast MHD shock with a layer of decreased density is discussed using ideal MHD approach. This is an extrapolation of gas dynamic thermal layer effect on ideal MHD. Computer simulations show that a magnetic field of a moderate intensity ( 1) may change the character of the flow for intermediate Mach numbers (M 5) and a new raking regime may occur which is not observed in the absence of a magnetic field. Self similar precursor analogous to that in gas dynamics may develop in the case of highM and low density in the layer but magnetic forces essentially decrease its growth rate. This problem appears in connection with cosmical shock propagation where planetary magnetic tails play the role of the thermal layer, and it may also be observed in the laboratory when the shock is strong enough to heat the walls ahead of it.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

Stationary Simple Waves on a Barotropic Liquid Shear Flow

Stationary threedimensional flows of a barotropic liquid in a gravity field are considered. In the shallowwater approximation, the Euler equations are transformed into a system of integrodifferential equations by the EulerLagrange change of coordinates. A system of simplewave equations is obtained, for which the theorem of existence of a solution attached to a given shear flow is proved. As an example, a particular solution analogous to the solution of the problem of a gas flow around a convex angle is given.     

Anwendung des Übertragungsverfahrens bei stark abklingenden Lösungsfunktionen

Übersicht Bei stark abklingenden Funktionen wird die Übertragungsmatrix U() aufgespalten in die Anteilc U ₁() e^– und U ₂() e. Der zweite Term spielt am Rand = 0 keinc Rolle. Die unbekannten Anfangswerte sind über die Matrix U ₁(0) an die bekannten gebunden und eindeutig bestimmbar.

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Summary For strongly decaying solution functions the transfer matrix U() is splitted into the parts U ₁() e^– and U ₂() e. The second term does not influence at the boundary = 0. The unknown initial values are related by the matrix U ₁(0) to the known values and they can be uniquely determined.

     

Dispersion of Dynamically Passive Impurities in Non‐One‐Dimensional Liquid Flow

Equations that describe dispersion of a substance in a nononedimensional incompressible liquid flow through a plane channel are derived. The model under consideration extends the traditional Taylor model to the case where sources of the substance are present in the flow and relaxation transfer processes are taken into account. Additional conditions for the dispersion equations are obtained. The relation between the proposed model and the Taylor model is analyzed. Based on the equations obtained, the mass transfer between circulation regions in the flow is calculated and a system of cellularmodel equations for stagnant cavities is constructed.     

Deformed particle image pattern matching in Particle Image Velocimetry

The deformation of particle image patterns due to velocity gradients causes errors of velocity measurements and false velocity detections in PIV (Particle Image Velocimetry). A novel technique to overcome those limitations inherent in the conventional PIV by correcting the particle image pattern according to the local velocity gradients in two dimensional flows, i.e. u/x, u/y, v/x and v/y, is proposed and successfully applied to a water flow downstream of a backward facing step.     

Stochastic-Perturbation Analysis of a One-Dimensional Dispersion-Reaction Equation: Effects of Spatially-Varying Reaction Rates

We carry out a stochastic-perturbation analysis of a one-dimensional convection–dispersion-reaction equation for reversible first-order reactions. The Damköhler number, Da, is distributed randomly from a distribution that has an exponentially decaying correlation function, controlled by a correlation length, . Zeroth- and first-order approximations of the dispersion coefficient, D are computed from moments of the residence-time distribution obtained by solving a one-dimensional network model, in which each unit of the network represents a Darcy-level transport unit, and the solution of the transfer function in zeroth- and first-order approximations of the transport equation. In the zeroth-order approximation, the dispersion coefficient is calculated using the convection–dispersion-reaction equation with constant parameters, that is, perturbation corrections to the local equation are ignored. This zeroth-order dispersion coefficient is a linear function of the variance of the Damköhler number, (Da)². A similar result was reported in a two-dimensional network simulation. The zeroth-order approximation does not give accurate predictions of mixing or spreading of a plume when Damköhler numbers, Da 1 and its variance, (Da)² > 0.25 Da². On the other hand, the first-order theory leads to a dispersion coefficient that is independent of the reaction parameters and to equations that do accurately predict mixing and spreading for Damköhler numbers and variances in the range (Da)²/Da0.3     

Measurement of dynamic surface tension by the oscillating droplet method

An optical measuring method has been applied to determine the dynamic surface tension of aqueous solutions of heptanol. The method uses the frequency of an oscillating liquid droplet as an indicator of the surface tension of the liquid. Droplets with diameters in the range between 100 and 200 m are produced by the controlled break-up of a liquid jet. The temporal development of the dynamic surface tension of heptanol-water solutions is interpreted by a diffusion controlled adsorption mechanism, based on the three-layer model of Ward and Tordai. Measured values of the surface tension of bi-distilled water, and the pure dynamic and static (asymptotic) surface tensions of the surfactant solutions are in very good agreement with values obtained by classical methods.List of symbols a coefficient of intermolecular forces, Nm^-1 - B adsorption constant - c _o bulk concentration, mol m^-3 - D apparent diffusion coefficient, m²s^-1 - t time, s - T absolute temperature, K - R universal gas constant=8.314, J mol^-1 K^-1 - (, t) droplet contour function - _o droplet equilibrium radius, m Greek symbols maximum surface excess concentration, mol m^-2 - (t) droplet volume normalization function - azimuth of the polar coordinate system - density, kgm^-3 - surface tension, N m^-1 - (t) concentration in the subsurface, molm^-3 - droplet oscillation frequency Daimler-Benz AG, Produktion & Umwelt, D-89081 UlmOn leave of absence from the Institute of Fundamental Technological Research, Polish Academy of Sciences, PL-00-049 Warszawa     

Stochastic Analysis of Macrodispersion in a Semi‐Confined Aquifer

In this paper, the macroscopic dispersion resulting from one and twodimensional flows through a semiconfined aquifer with spatially variable hydraulic conductivity K which is represented by a stationary (statistically homogeneous) random process is analyzed using the spectral representation technique. Stochastic fluctuation equations of the steady flow and solute transport are solved to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of the leakage factor and input covariances describing the hydraulic conductivity in a semiconfined aquifer bounded by a leaky layer above and an impervious stratum below. The macrodispersivity tensor is studied using some convenient forms of the log hydraulic conductivity process. The sensitivity of the resulting macrodispersivity to the input covariances is discussed along with the influence of the leakage factor for both one and twodimensional flows. It is found that the longitudinal macrodispersivities are increased due to the presence of leakage, while the transverse macrodispersivities are reduced due to leakage.