A class of three-level explicit schemes with higher stability properties for a dispersive equation ut=auxxx

In this paper, a class of three level explicit schemes for a dispersive equation u_t=au_xxx with stability condition |r|=|α|Δt/(Δx)³≤2.382484, are considered. The stability condition for this class of schemes is much better than |r|≤0.3849 in [1], [2] and |r|≤0.701659 in [3], and |r|≤1.1851 in [4].     

On group classification of the spatially homogeneous and isotropic Boltzmann equation with sources

In Nonenmacher (1984) [1] an admitted Lie group of transformations for the spatially homogeneous and isotropic Boltzmann equation with sources was studied. In fact, the author is Nonenmacher (1984) [1] considered the equation for a generating function of the power moments of the Boltzmann equation solution. However, this equation is still a non-local partial differential equation, and this property was not taken into account there. In the present paper the admitted Lie group of this equation is studied, using our original method developed for group analysis of equations with non-local operators (Grigoriev and Meleshko, 1986; Meleshko, 2005; Grigoriev et al., 2010 [2], [3], [4]). The Lie groups obtained are compared with Nonenmacher (1984) [1]. The deficiency of Nonenmacher (1984) [1] is corrected.     

Transient pressure effects in the evolution equation for premixed flame fronts

A nonlinear evolution equation for a scalar field G(x, t) is derived, whose level surface G ₀=const. represents the interface of a thin premixed flame propagating in a flow field. The derivation is an extended version of an equation already proposed by Markstein [1]. It was reconsidered by Williams [2] as a basis for theoretical and numerical analysis and takes, in addition to flame curvature and flame stretch time variations of the bulk pressure, heat loss and nonconstant transport coefficients into account. The equation is an extension of earlier analyses where a flame evolution equation was derived for slightly wrinkled flames such that the front can be described by a single-valued function of a normal coordinate. That formulation excluded situations where the mean flame front has an arbitrary shape in space. Here the more general situation is analysed by using a two-length-scale asymptotic analysis. The leading-order solution of this analysis is equivalent to the equation originally derived by Markstein [1]. In addition to nonconstant properties and heat-loss effects, that had already been considered by Clavin and Nicoli [3], the influence of transient changes of the bulk pressure is analysed. All these effects are combined into a unified formulation which will serve as a basis for a new flamelet concept for premixed turbulent combustion.     

System of equations for describing dynamical problems associated with the mechanics of lung parenchyma

In [1], which is an expanded version of the paper [2], the equations of conservation of mass and momentum are shown to be valid for dynamical problems of lung parenchyma. This system of equations is now closed by means of the heat flux equation. As in [1, 2], the heat flux equation is obtained on the basis of the methods of the mechanics of heterogeneous media [3] and anatomical data on the structure of lung parenchyma.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 21–29, May–June, 1988.     

The uniformly convergent difference schemes for a singular perturbation problem of a self-adjoint ordinary differential equation

In this paper, we construct a class of difference schemes with fitted factors for a singular perturbation problem of a self-adjoint ordinary differential equation. Using a different method from [1], by analyzing the truncation errors of schemes, we give the sufficient conditions under which the solution of the difference scheme converges uniformly to the solution of the differential equation. From this we propose several specific schemes under weaker conditions, and give much higher order of uniform convergence, and applying them to example, obtain the numerical results.     

On the Global Existence and Convergence to Steady State of Navier-Stokes Flow Past an Obstacle that is Started from Rest

A new approach to the existence theory for the Navier-Stokes equations, using a technique of Kato [15], further developed in combination with estimates for Oseen's equation by Kobayashi & Shibata [17] and Shibata [24], has made possible the solution of a long-standing open problem often referred to as Finn's “starting problem”. This paper provides the solution. (Accepted January 31, 1996)     

Experimental investigation of supersonic flow past blunt bodies with strong distributed injection

The entry of bodies into planetary atmospheres at high supersonic velocities is accompanied by intense evaporation of the surface due to radiative heat fluxes. A series of problems involving the conduction of investigations of such kind has been proposed by Petrov. In [1], in particular, the entry of a meteorite into an atmosphere was examined. The gasdynamic aspects of this problem have been approximately simulated by many authors by intense injection of gas in theoretical, e.g., [2–5], and experimental [6, 7] studies. The theoretical studies were based on two-layer [3, 4] or three-layer [5] schemes of gas flow between the shock wave and the surface of the body. The aim of the present work was an experimental investigation of the interaction of injection with a counter supersonic flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 84–95, May–June, 1978.     

A new look at the laminar flow of power law fluids through granular beds

Summary The paper deals with laminar flow of power law fluids through granular beds. A critical review of the assumptions concerning the capillary model of the bed, applied by various authors, led us to the conclusion that the derivation of the correlation eq. [13] given byChristopher andMiddleman was based on a too simplified model of the granular bed. Taking advantage of the approach presented in the classical works ofKozeny andCarman (which seems to be partly overlooked by some authors, including our own previous works) a modified correlation equation for power law fluids [21], a corrected formula for shear rate in the bed [29] and for Deborah number [32], as well as corrected correlation equation for fluids exhibiting memory effects [34] were presented.

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Zusammenfassung Diese Arbeit betrifft laminare Strömungen von Potenzgesetzflüssigkeiten durch Kornschüttungen. Eine kritische Prüfung der Annahmen, die von verschiedenen Autoren für das Kapillar-Modell der Schüttung gemacht worden sind, führt uns zu der Folgerung, daß die Herleitung der Korrelationsgleichung [13] nachChristopher undMiddleman auf einem übervereinfachten Modell der Kornschüttung basiert. Unter Nutzbarmachung der Annahmen, die in den klassischen Arbeiten vonKozeny undCarman dargestellt worden sind (sie wurden sowohl von manchen anderen Autoren als auch in unseren früheren Arbeiten nicht beachtet), werden nun eine modifizierte Korrelationsgleichung für die Potenzgesetzflüssigkeiten [21], eine korrigierte Formel für die Schergeschwindigkeit in der Schüttung [29], eine korrigierte Formel für die Deborah-Zahl [32] und eine korrigierte Korrelationsgleichung für Flüssigkeiten, die Gedächtnis-Effekte zeigen [34], angegeben.

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Notation A constant in eq. [9] - d _p effective particle diameterd _p = 6/a (wherea is the specific surface of the bed), m - f _BK modified friction factor, defined by eq. [1] - k power law parameter, N s ⁿ /m² - K Kozeny constant, defined by eq. [8] - K ₀ constant depending on the shape of the channel cross-section - K ₁ constant, defined by eq. [5] - l bed height, m - l _e channel length, m - n power law parameter - p pressure drop due to friction, N/m² - r _h hydraulic radius, defined by eq. [6], m - s bed permeability, defined by eq. [16], m² - v ₀ mean linear velocity related to an empty crosssection of the column, m/s - v _e mean linear velocity in the channel, m/s - shear rate at the wall of the channel, s^–1 - shear rate at the wall of the channel calculated according to the formula [29], s^–1 - bed porosity - characteristic time of the fluid, s - friction factor, defined by eq. [25] - µ dynamic viscosity of the fluid, N s/m² - parameter, defined by eq. [15], N s ⁿ /m¹⁺ⁿ - De Deborah number, defined by eq. [33] - De ^* Deborah number, defined by eq. [32] - Re _BK modified Reynolds number, defined by eq. [2] - Re _BK modified Reynolds number, defined by eq. [26] - Re _BK ^* modified Reynolds number, defined by eq. [23] - Re _CM modified Reynolds number byChristopher andMiddleman, defined by eq. [14] - Re _CM modified Reynolds number, defined by eq. [17] With 3 figures and 1 table     

Initial and boundary value problems of hyperbolic heat conduction

This is a study on the initial and boundary value problem of a symmetric hyperbolic system which is related to the conduction of heat in solids at low temperatures. The nonlinear system consists of a conservation equation for the energy density e and a balance equation for the heat flux , where e and are the four basic fields of the theory. The initial and boundary value problem that uses exclusively prescribed boundary data for the energy density e is solved by a new kinetic approach that was introduced and evaluated by Dreyer and Kunik in [1], [2] and Pertame [3]. This method includes the formation of shock fronts and the broadening of heat pulses. These effects cannot be observed in the linearized theory, as it is described in [4]. The kinetic representations of the initial and boundary value problem reveal a peculiar phenomenon. To the solution there contribute integrals containing the initial fields as well as integrals that need knowledge on energy and heat flux at a boundary. However, only one of these quantities can be controlled in an experiment. When this ambiguity is removed by continuity conditions, it turns out that after some very short time the energy density and heat flux are related to the initial data according to the Rankine Hugoniot relation. Received October 6, 1998     

On the general equations,double hormonic equation and eigen-equation in the problems of ideal plasticity

In this paper the outcome of axisymmetric problems of ideal plasticity in paper [39], [19] and [37] is directly extended to the three-dimensional problems of ideal plasticity, and get at the general equation in it. The problem of plane strain for material of ideal rigidplasticity can be solved by putting into double hormonic equation by famous Pauli matrices of quantum electrodynamics different from the method in paper [7]. We lead to the eigen equation in the problems of ideal plasticity, taking partial tenson of stress-increment as eigenfunctions, and we are to transform from nonlinear equations into linear equation in this paper.     

Kinematic analysis of the general spatial 7R mechanism

The displacement. velocity and acceleration analysis of the general spatial 7R mechanism is discussed in this paper. Based on the method proposed in Ref. [2], an input-output algebra equation of the 16th degree in the tan-half-angle of the output angular displacement is derived. The derivation process and computation are considerably simple. A program written in AI language is used to derive the coefficients of displacement equations: therefore the amount of manual work is greatly decreased. The results are verified by a numerical example. The researches of this paper and Ref.[5] found a base for establishing an expert system of spatial mechanism analysis in the future.     

Nonlinear waves on shallow water in the presence of a horizontal magnetic field

Benjamin [1] and Davis and Acrivos [2] derived an equation for long steady nonlinear internal waves in an infinitely deep stratified fluid when the density varies only in a layer whose thickness is small compared with the characteristic perturbation length. Ono [3] generalized this equation to the unsteady case. The resulting equation was subsequently called the Benjamin—Ono equation. Steady solutions of this equation were found by Benjamin and Ono in the form of solitons and periodic waves. In the present paper it is shown that long nonlinear waves on shallow water in the presence of a horizontal magnetic field can also be described by the Benjamin—Ono equation, and not the Korteweg—de Vries equation [4], as in the case when there is no field. Moreover, in contrast to a soliton in a stratified fluid a soliton on shallow water in a horizontal magnetic field moves with a velocity less than the velocity of infinitely long perturbations of small amplitude. The dependence of the parameters of a soliton and a periodic wave on the intensity and direction of the unperturbed magnetic field is investigated.     

Measurement of the velocity profile in liquid streams of large volumes using a laser Doppler velocimeter

The use of the laser Doppler velocimeter (LDVM) in fluid mechanics already has a solid past history and work on its development is proceeding rather intensively [1, 2], This method has been used successfully to measure various parameters of flows in channels or chutes of small size [3–6]. Fundamental difficulties due to possible local variations in the index of refraction along the path of the laser beam [7], in addition to technical difficulties, have not been eliminated for its use in other cases of practical importance, particularly in basins or reservoirs of large size. The aim of the present paper is the investigation of the velocity structure of streams in large volumes using an LDVM on the example of a turbulent submerged water jet. In order to estimate the effect of the thickness of the water layer on the applicability of the method we tested three schemes: based on direct [8] and back scattering [9] and a scheme with reflection of forward-scattered light from a mirror [10]. The results of the investigation of a submerged turbulent jet using an LDVM are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 170–173, September–October, 1977.     

Abhängigkeit der Wärmeübergangskoeffizienten vom Wandtemperaturverlaufund die Konsequenzen für die Berechnung von Wärmedurchgangskoeffizienten bei Wärmetauschern

In a recent article [2] it has been shown that the common equation in VDI Heat Atlas [1], which has been used for the calculation of overall heat transfer coefficients, satisfies the energy balance of the heat exchanger in seldom exceptional cases only. This equation has been corrected for constant wall temperature. In this article an equation for the calculation of overall heat transfer coefficients valid for any shape of the wall temperature will be developed at which it becomes obvious, that only in the case of identical shapes of wall temperature in measurement and real heat exchanger process the calculation according to VDI Heat Atlas [1] is valid. In all other cases this procedure leads to errors.     

Nonlinear boundary control of the unforced generalized Korteweg–de Vries–Burgers equation

In this paper, we consider the boundary control problem of the unforced generalized Korteweg–de Vries–Burgers (GKdVB) equation when the spatial domain is [0,1]. Three control laws are derived for this equation and the L ²-global exponential stability of the solution is proved analytically. Numerical results using the finite element method (FEM) are presented to illustrate the developed control schemes.     

A variational principle for the dynamic problem of linear coupled thermoelasticity

Summary A functional is constructed by whose stationarity the mixed problem for linear and dynamic thermoelasticity is obtained. In contrast with previous results[2], [4], [8], neither constrained variations nor previous transformations upon the equations of the problem are used. The unrestricted variational formulation is made possible by means of a convolutive bilinear form.

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Sommario Si costruisce un funzionale dalla cui stazionarietà si ricava il problema misto della termoelasticità lineare accoppiata. A differenza di precedenti risultati[2], [4], [8] non si ricorre né a variazioni bloccate né a preliminari trasformazioni del problema. La formulazione variazionale non ristretta è resa possibile dall'uso di una forma bilineare convolutiva.

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Work done in the sphere of activity of the Group for Mathematical Research of the (Italian) C.N.R.     

Generalization of the Krook kinetic relaxation equation

One of the most significant achievements in rarefied gas theory in the last 20 years is the Krook model for the Boltzmann equation [1]. The Krook model relaxation equation retains all the features of the Boltzmann equation which are associated with free molecular motion and describes approximately, in a mean-statistical fashion, the molecular collisions. The structure of the collisional term in the Krook formula is the simplest of all possible structures which reflect the nature of the phenomenon. Careful and thorough study of the model relaxation equation [2–4], and also solution of several problems for this equation, have aided in providing a deeper understanding of the processes in a rarefied gas. However, the quantitative results obtained from the Krook model equation, with the exception of certain rare cases, differ from the corresponding results based on the exact solution of the Boltzmann equation. At least one of the sources of error is obvious. It is that, in going over to a continuum, the relaxation equation yields a Prandtl number equal to unity, while the exact value for a monatomic gas is 2/3.In a comparatively recent study [5] Holway proposed the use of the maximal probability principle to obtain a model kinetic equation which would yield in going over to a continuum the expressions for the stress tensor and the thermal flux vector with the proper viscosity and thermal conductivity.In the following we propose a technique for constructing a sequence of model equations which provide the correct Prandtl number. The technique is based on an approximation of the Boltzmann equation for pseudo-Maxwellian molecules using the method suggested by the author previously in [6], For arbitrary molecules each approximating equation may be considered a model equation. A comparison is made of our results with those of [5].     

Équation énergétique locale pour la prevision des niveaux vibratoires et acoustiques instationnaires

Insights and additionnal clarifications concerning the energy diffusion model [4] are in concern herein. We showed in former times under steady state conditions, that the diffusion model is well suited for an uncorrelated plane wave dynamics [7], energy behaviour modelling. We will prove, in this short note, that the later is not enough when transient dynamics is in concern. We thus propose a new equation alternative to the diffusion equation.     

Conditions of applicability of the Boltzmann equation

The existence of certain characteristic times, introduced by Bogolyubov [1], is of fundamental importance for the derivation of the Boltzmann equation from the Liouville equations. In the present paper characteristic spatial scales are also introduced, which permit a more detailed study of the influence of spatial gradients and boundary conditions. A convenient formalism, which is a generalization of the formalism of [2], is used in this study. The following has been shown for a Boltzmann gas (compare [1–4]):

1. a)
   The Boltzmann equation is applicable for describing flows in which the condition of molecular chaos is satisfied and in which the characteristic dimension L (time T) is much greater than the diameter d (time τ_c) of molecular interactions.