Shock Waves in a Fluid with Bubbles Containing Evaporating Drops of a Liquefied Gas

The propagation and reflection of one-dimensional plane unsteady waves and pulses in a mixture of a fluid with two-phase bubbles containing evaporating drops is investigated. A significant effect of unsteady evaporation of the drops in the zone ahead of the shock wave on the wave propagation is demonstrated. The evaporation of the drops results in a pressure increase ahead of the wave and the shock wave as it were climbs to increasing pressure level. In contrast to bubbly fluids with single-phase bubbles, in a fluid with two-phase bubbles, at a fixed phase volume fraction, a decrease in bubble size results in an increase rather than a decrease of the oscillation amplitude. The wave reflection from a solid wall is essentially nonlinear and the maximum pressure attained at the wall is several times greater than the incident-wave intensity.     

On the sensitive dynamical system and the transition from the apparently deterministic process to the completely random process

The detailed analysis of the dynamical process of coin tossing is made. Through calculations, it is illustrated how and why the result is extremely sensitive to the initial conditions. It is also shown that, as the initial height of the mass center of the coin increases, the final configuration, i.e. head or tail, becomes more and more sensitive to the initial parameters (the initial velocity angular velocity, and the initial orientation), the coefficient of the air drag, and the energy absorption factor of the surface on which the coin bounces. If we keep the head upward initially but allow a small range for the change of some other initial parameters, the frequency that the final configuration is head, would be 1 if the initial height h of the mass center is sufficiently small, and would be clo to 1/2 if h is sufficiently large. An interesting question is how this frequency changes continuously from 1 to 1/2 as h increases. Detailed calculations show that such a transition is very similar to the transition from laminar to turbulent flows. A basic difference between the transition stage and the completely random stage is indicated: In the completely random stage, the deterministic process of the individual case is extremely sensitive to the initial conditions and the dynamical parameters, out the statistical properties of the ensemble are insensitive to the small changes of the initial conditions and the dynamical parameters. On the contrary, in the transition stage, both the deterministic process of the individual case and the statistical properties of the ensemble are sensitive to the initial conditions and the dynamical parameters. The mechanism for this feature of the transition stage is the existence of the long-train structure in the parameter space. The illuminations of this analysis on some other random phenomena are discussed.     

Indentation of the semi-infinite elastic solid by a concave rigid punch

A solution is obtained for the relationship between load, displacement and inner contact radius for an axisymmetric, spherically concave, rigid punch, indenting an elastic half-space. Analytic approximations are developed for the limiting cases in which the ratio of the inner and outer radii of the annular contact region is respectively small and close to unity. These approximations overlap well at intermediate values. The same method is applied to the conically concave punch and to a punch with a central hole. , , . , . . .     

An application of nonsymmetric Lax-Milgram lemma to nonassociated plasticity

Based on a general assumption for plastic potential and yield surface, some properties of the nonassociated plasticity are studied, and the existence and uniqueness of the distribution of incremental stress and displacement for work-hardening materials are proved by using nonsymmetric Lax-Milgram lemma, when the work-hardening parameter A>F/Q/–F/, Q/.     

Resonant generation of a solitary wave in a thermocline

The resonant generation of a second-mode internal solitary wave, resulting from a ship internal waves system damping in a thermocline, is studied experimentally. The source of the stationary internal waves was provided by an oblong ellipsoid of revolution towed horizontally and uniformly at the depth of the thermocline center. The ranges of the Reynolds and Froude numbers were 500Re=Ul/v 15000 and 0.3Fi=U/N _max D1.0, respectively. When the body's speed and the linear long-wave second-mode phase speed were equal, an internal solitary wave of the bulge type was observed. The shape of the wave satisfied the Korteweg-de Vries equation. The Urcell parameter was equal to 10.2.List of Symbols L, B, H towing tank length, breadth and height respectively - z vertical coordinate - D characteristic vertical dimension of the body - a minor semiaxis of an ellipsoid - b major semiaxis of an ellipsoid (maximum ellipsoid diameter D=2a) - l length of the body ( =2b) - U velocity of the body - t temperature - g acceleration due to gravity - i fresh water density at ith level - _4° fresh water density for temperature t=4°C - o water density at the center of the thermocline - _i density variation due to the temperature variation at the ith horizon - N Brunt-Väisälä frequency - N _max maximum value of Brunt-Väisälä frequency - Re Reynolds number - Fi internal Froude number - f _n eigenfunction of the boundary-value problem for the nth mode - _n nth mode frequency - k _{n n}th mode horizontal wavenumber - C _n limiting phase speed of a linear nth mode interval wave (= _n/k_n;k_n 0) - Ur Urcell parameter - v fresh water kinematic viscosity - conventional density - half-length of a solitary wave - ₀ solitary wave height - time This work was partially supported by the INTAS (grant no. 94-4057) and by the Russian Foundation of Basic Research under grant no. 94-05-17004-a.A version of this paper was presented at the Second International Conference on Experimental Fluid Mechanics, Torino, Italy, 4–8 July, 1994.     

Spectral Properties of Schrödinger Operators on Domains with Varying Orders of Thinness

We consider the Schrödinger operator on two types of domains depending on a small parameter : dumbbell domains and thin domains with varying orders of thinness. In both situations we compare the eigenvalues and eigenvectors of the Schrödinger operator with the corresponding eigenvalues and eigenvectors of a limit operator defined on the limit domain.     

Effect of Porous Coatings on Stability of Hypersonic Boundary Layers

The influence of ultrasoundabsorbing coatings on stability of hypersonic boundary layers is considered. Two types of coatings were used in experiments: feltmetal with a random porous microstructure and a sheet perforated by blind cylindrical microchannels. The experiments were performed in a wind tunnel at a Mach number M = 5.95 on sharp cones with a 7° apex halfangle. Evolution of natural disturbances and artificially induced wave packets in the boundary layer was studied with the help of hotwire anemometry. Spatial characteristics of artificial disturbances were obtained. It is demonstrated that such coatings exert a stabilizing effect on secondmode disturbances.     

Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions

In this paper, we show that the maximum principle holds for quasilinear elliptic equations with quadratic growth under general structure conditions.Two typical particular cases of our results are the following. On one hand, we prove that the equation (1) {ie77-01} where {ie77-02} and {ie77-03} satisfies the maximum principle for solutions in H ¹()L(), i.e., that two solutions u ₁, u ₂H¹() L() of (1) such that u ₁u₂ on , satisfy u ₁u₂ in . This implies in particular the uniqueness of the solution of (1) in H ₀ ¹ ()L().On the other hand, we prove that the equation (2) {ie77-04} where fH^–1() and g(u)>0, g(0)=0, satisfies the maximum principle for solutions uH¹() such that g(u)¦Du|{²L¹(). Again this implies the uniqueness of the solution of (2) in the class uH ₀ ¹ () with g(u)¦Du|{²L¹().In both cases, the method of proof consists in making a certain change of function u=(v) in equation (1) or (2), and in proving that the transformed equation, which is of the form (3) {ie77-05}satisfies a certain structure condition, which using ((v₁ -v ₂)⁺)ⁿ for some n>0 as a test function, allows us to prove the maximum principle.     

Filament stretching equation in Lagrangian coordinates

The general integral of the very simple equation ^21/n/() was found to describe the cross sectional area of filaments of isothermal power law fluids while in transient stretching where is time and is the initial location of fluid molecules at time = 0 given as the distance from a reference point fixed in space. Any such stretching transient given as a solution of the above equation is physically realizable subject to the restrictions > 0 and/ < 0.     

Beitrag zur Auswertung von Anlauf-Vorgängen in viskoelastischen Flüssigkeiten

Zusammenfassung In diesem Bericht wird die Kraftübertragung zwischen zwei unbegrenzt ausgedehnten und durch eineMaxwell-Flüssigkeit gekoppelten Platten bei beliebigem Zeitgesetz der Bewegung der oberen auf die elastisch aufgehängte untere Platte unter Einbeziehung der Trägheit von unterer Platte undMaxwell-Flüssigkeit berechnet.Diese Fragestellung führt mathematisch auf ein Randwertproblem, dessen eine Randbedingung die Form ³ u/x ² y + ₁ u + ₁ u/y – ₂ ² u/xy = 0 fürx = 0 annimmt, also gemischte Ableitungen nach 2 unabhängigen Veränderlichen enthält.Es wird gezeigt, daß sich die Lösung als unendliche Reihe darstellen läßt, deren 1. Glied aus abklingenden Exponentialfunktionen, alle weiteren hingegen aus gedämpften Schwingungen bestehen.Als spezielle Zeitgesetze für die obere Platte werden die impulsförmige Erregung, die zeitproportionale und die periodische Bewegung untersucht.Es wird abgeleitet, wie sich die Kenngrößen undG (Viskosität und Schubmodul) aus Eigenfrequenz und Dämpfung bzw. aus den beiden Exponenten dere-Funktionen in den beiden ersten Fällen berechnen lassen. Bei periodischer Beanspruchung werden undG aus der Resonanzfrequenz und der Halbwertbreite bzw. dem Phasenwinkel ermittelt.Abschließend wird gezeigt, daß und wie ein Rotationsviskosimeter auf den behandelten Fall ebener Platten zurückgeführt werden kann.

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Summary In this report the force transmission between two infinitely extended plates coupled by aMaxwell-Fluid is calculated. In this calculation the inertia of the fluid and the influence of the elastically suspended lower plate is included, while the upper plate can perform any time-dependent movement.Mathematically this question leads to a boundery value problem; one boundery condition has the form ³ u/ ² y + ₁ u + ₁ u/y – ₂ ² u/xy = 0 (x = 0) ( ₁, ₂, = constant), that means it contains mixed differential-quotients of two independent variables.The solution is demonstrated as an infinite series, the form of the first term is of two negative exponential functions, all the other terms are damped oscillations.The excitation from a single pulse, the uniform timeproportional and the periodic movement as timedependent laws for the upper plate are analysed.The values of andG (viscosity and shear modulus) are derived from eigenfrequency and damping-constant respectively from the values of the two exponentials. In the periodic case andG are calculated from the resonance frequency and halfwidth or phase-angle respectively.Finally the conditions of the rotation-viscometer are reduced to the case of parallel-plate-viscometer.

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Vorgetragen auf der Jahrestagung der Deutschen Rheologen vom 7.–9. Mai 1973 in Berlin.

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Mit 6 Abbildungen und 1 Tabelle     

Waves in a plasma with Hall dispersion

In a magnetohydrodynamic approximation, an investigation is made of the propagation of waves in a plasma, whose characteristic frequency is much less than the collision frequency of the electrons _e ^–1. It is assumed that the magnetic field is sufficiently strong so that the equality _ee1 will be satisfied, where _e is the cyclotron frequency of the rotation of the electrons. With large magnetic Reynolds numbers (R_m1), which are characteristic for many astrophysical problems, this latter condition leads to a need to take account of dispersion effects connected with Hall currents, in the absence of Joule dissipation. The dispersion equation for the propagation of small perturbations is analyzed in the limiting cases of weak dispersion and of a wave propagating along the magnetic field. In the case of weak dispersion, an equation is derived for nonlinear waves. The solutions are found in the form of stationary solitons. The region of such solutions is analyzed. A typical example of a medium with Hall dispersion is an interplanetary plasma, in which the parameter _ee is generally great.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 108–113, May–June, 1974.     

Design of a Beam with Controllable Force Elements

A method for solving the problem of design of an intellectual structure formulated for the pair optimal position of actuators, optimal control of actuators is developed. In the method proposed, physical and logical objects are treated as equivalent.     

Asymptotic theory for calculating heat fluxes near the corner of a body

A method is presented for calculating the distribution of the thermal fluxes, friction stresses, and pressure near the corner point of a body contour in whose vicinity the outer supersonic flow passes through an expansion wave. The method is based on a study of the asymptotic solutions of the Navier-Stokes equations as the Reynolds number R approaches infinity for the flow region in which the longitudinal gradients of the flow functions are large, invalidating conventional boundary layer theory. This problem was examined in part in [1], in which the distribution of the friction and pressure in a region with length on the order of a few thicknesses of the approaching boundary layer was obtained in the first approximation. The leading term of the expansion for the thermal flux to the surface of the body vanishes for a value of the Prandtl number equal to unity and for other values of the Prandtl number does not match directly with its value in the undisturbed boundary layer.The thermal-flux distribution is obtained for values of the Prandtl number approaching unity. For this purpose it was necessary to consider a more general double passage to the limit as 1 and 0 for a finite value of the parameter B=[(–1)/] [–ln ^1/4/]^1/4 characterizing the ratio of the effects of thermal conduction, viscous dissipation, and convection. The solution obtained previously [1] corresponds to the particular case B and therefore for actual values of R=10⁴–10⁶, ~ 0.7 overestimates considerably the effect of the dissipative term on heat transfer, although even in first approximation it describes the pressure distribution well and the friction distribution satisfactorily. For smooth matching of the solutions with the corresponding flow functions in the undisturbed boundary layer it was necessary to introduce a flow region with free interaction for the expansion flow. Equations and boundary conditions which describe the flow as a whole are presented. Examples are given of numerical calculations and comparison with experiment.     

Unsteady Radiative–Convective Heat Transfer in a Flow Emitting‐Absorbing and Scattering Medium around an Ablating Plate

A conjugate problem of radiative–convective heat transfer in a turbulent hightemperature gasdisperse flow around a thermally thin ablating plate is considered. The plate experiences intense radiative heating by an external source, which is a blackbody. The temperature fields and the distributions of heat fluxes along the plate under unsteady conditions are calculated. The data gained make it possible to examine the effect of the Stark number and phasetransition heat in the plate material on the time evolution of the thermal state of the boundarylayer medium and the plate itself being heated by a hightemperature radiation source.     

Lp-estimates for the nonlinear spatially homogeneous Boltzmann equation

This paper studies L^p-estimates for solutions of the nonlinear, spatially homogeneous Boltzmann equation. The molecular forces considered include inverse k^th-power forces with k > 5 and angular cut-off.The main conclusions are the following. Let f be the unique solution of the Boltzmann equation with f(v,t)(1 + ¦v²¦)^(s ₁ ^{+ /p)/2} L¹, when the initial value f ₀ satisfies f ₀(v) 0, f ₀(v) (1 + ¦v¦²)^(s ₁ ^{+ /p)/2} L¹, for some s₁ 2 + /p, and f ₀(v) (1 + ¦v¦²)^s/2 L^p. If s 2/p and 1 < p < , then f(v, t)(1 + ¦v¦²)^{(s s} ₁)^/2 L^p, t > 0. If s >2 and 3/(1+ ) < p < , thenf(v,t) (1 + ¦v¦²)^(s(s ₁ ^{+ 3/p))/2} L^p, t > 0. If s >2 + 2C₀/C₁ and 3/(l + ) < p < , then f(v,t)(1 + ¦v¦²)^s/2 L^p, t > 0. Here 1/p + 1/p = 1, x y = min (x, y), and C₀, C₁, 0 < 1, are positive constants related to the molecular forces under consideration; = (k – 5)/ (k – 1) for k^th-power forces.Some weaker conclusions follow when 1 < p 3/ (1 + ).In the proofs some previously known L-estimates are extended. The results for L^p, 1 < p < , are based on these L-estimates coupled with nonlinear interpolation.     

On the evolutions of the discontinuities of orders ⩾ m in solutions for quasi-linear systems of PDEs of order m,or for the linearizations of these systems

Summary One considers a system L[u]=0 of PDEs, quasi-linear (according to [1]) and of order m, which possesses a bicharacteristic line , as it happens in the hyperbolic case. For v=0, , –m (>0) let u(^v) be a discontinuity wave of order m+v that solves the system above and whose discontinuity hypersurface includes . The corresponding transport equations along are considered. Furthermore some interesting cases are pointed out, in which these equations turn out to be mutually equivalent in a suitable sense. Some theorems are stated to compare the transport equations for the discontinuities of the above kinds, that are connected with the systems d^hL[u]/dt^h=0 (h=0, , –m) and/or the linearization of the system L[u]=0 around any regular solution of it.

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Sommario Si considera un sistema L[u]=0 di equazioni alle derivate parziali, quasi lineare (secondo [1]) e di ordine m, il quale sia dotato di qualche bicaratteristica , come accade nel caso iperbolico. Per v=0, , –m(>0) sia u(^v) un'onda di discontinuità di ordine m+v risolvente il detto sistema e avente ipersuperficie di discontinuità contenente Si considerano le relative equazioni di trasporto lungo e si determinano casi interessanti in cui queste equazioni sono mutuamente equivalenti in senso opportuno. Si stabiliscono teoremi di confronto per il trasporto delle discontinuità del tipo suddetto, relative ai sistemi d^hL[u]/dt^h=0 (h=0, , –m) e/o alla linearizazione del sistema L[u]=0 attorno a qualche sua soluzione regolare.

     

An extension of Reiner's “Deborah Number” concept to a Wide field of rheological investigations

Summary Reiner defined a numeric, which he called theDeborah Number to represent the ratio of a relaxation time to a natural (observation) time. This implies aMaxwell model but is readily extended to complete relaxation spectra. Similar Numbers are proposed for retardation times and also for some conditions of coagulation thixotropy and for data from certain psychophysical experiments.     

Method of mass-average quantities for the boundary layer in an outer flow with transverse nonuniformity

An engineering method is proposed for calculating the friction and heat transfer through a boundary layer in which a nonuniform distribution of the velocity, total enthalpy, and static enthalpy is specified across the streamlines at the initial section x₀. Such problems arise in the vortical interaction of the boundary layer with the high-entropy layer on slender blunt bodies, with sudden change of the boundary conditions for an already developed boundary layer (temperature jump, surface discontinuity), and in wake flow past a body, etc.Notation x, y longitudinal and transverse coordinates - u,, H, h gas velocity, stream function, total and static enthalpy - p,,, pressure, density, viscosity, Prandtl number - , q friction and thermal flux at the body surface - r(x), (x) body surface shape and boundary layer thickness - V, M freestream velocity and Mach number - u⁽⁰⁾(x₀,), H⁽⁰⁾(x₀,), h⁽⁰⁾(x₀,) parameter distributions at initial section - u⁽⁰⁾(x,), h⁽⁰⁾(x,), h⁽⁰⁾(x,) profiles of quantities in outer flow in absence of friction and heat transfer at the surface of the body The indices v=0, 1 relate to plane and axisymmetric flows - , w, b, relate to quantities at the outer edge of the inner boundary layer, at the body surface in viscid and nonviscous flows, and in the freestream, respectively. The author wishes to thank O. I. Gubanov, V. A. Kaprov, I. N. Murzinov, and A. N, Rumynskii for discussions and assistance in this study.     

Calculation of heat transfer accompanying hypersonic three-dimensional flow of air over slender blunt-nosed cones

The effective length method [1, 2] has been used to make systematic calculations of the heat transfer for laminar and turbulent boundary layers on slender blunt-nosed cones at small angles of attack ( + 5° in a separationless hypersonic air stream dissociating in equilibrium (half-angles of the cones 0 20°, angles of attack 0 15°, Mach numbers 5 M 25). The parameters of the gas at the outer edge of the boundary layer were taken equal to the inviscid parameters on the surface of the cones. Analysis of the results leads to simple approximate dependences for the heat transfer coefficients.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 173–177, September–October, 1981.     

Hypersonic flow past blunt-edged delta wings

A method is proposed for calculating hypersonic ideal-gas flow past blunt-edged delta wings with aspect ratios = 100–200. Systematic wing flow calculations are carried out on the intervals 6 M 20, 0 20, 60 80; the results are analyzed in terms of hypersonic similarity parameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 175–179, September–October, 1990.