Solving the Problem of Gravitational Compression of a Layered Sphere (by the Example of the Earth)

The problem of spherically symmetric, gravitational compression of an isotropic hyperelastic layered sphere which modeling the region of the Earth below the Mohorovii boundary is solved. The known mechanical characteristics of the Earth in the compressed state are used to find its characteristics in the unstrained state obtained by adiabatic or isothermal stress relief. The stress state differs significantly from the state of purely hydrostatic compression. The minimum bulk compression and maximum radial tension occur not on the boundary of the sphere but in depth at certain distances from the boundary.     

Relaxation of a Low‐Intensity Atomic Beam in a Quiescent Gas

Relaxation of a lowintensity atomic beam in a gas at rest is examined by means of numerical modeling with the method of test particles. Temperaturefield features in the mixing region are considered. A relation between the relaxation length and the initial velocity and mass of injected particles is obtained. Conditions are found under which the relaxation length is minimal.     

Experimental determination of the hydrodynamic moment associated with the nonsymmetric penetration of a disk into a compressible fluid

The nonsymmetric penetration of a disk (circular cylinder) into a compressible fluid is investigated. The results are obtained by physical modeling. A fluid with a low speed of sound (finely dispersed medium with gas bubbles, whose dimensionless equation of state coincides with the dimensionless equation of state of water [5]) was used as the working medium. The experiments were carried out at entry angles on the interval 54 < < 88, angles of attack on the interval –15 < < +15 and Mach numbers on the interval 0.002 M 0.2.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 88–94, September–October, 1990.     

Approximate First Integrals of a Galaxy Model

First-order approximate first integrals (conserved quantities)of a Hamiltonian dynamical system with two degrees of freedomwhich arises in the modeling of central part of a deformed galaxy [1] havebeen obtained based on the approximate Noether symmetries for resonances₁=₂, ₁=2₂ and 2₁=3₂. Furthermore,KAM curves have been obtained analytically and they have been compared with thenumerical ones on the Poincaré surface of section.     

A Two-Equation Analysis of Convection Heat Transfer in Porous Media

This paper presents a two-equation analysis on the convection heat transfer in porous media based on the modeling developed by Carbonell and Whitaker (1984). The porous system under consideration is bounded by two parallel walls and heated uniformly from one side surface. The Darcy flow is imposed and the fully developed heat transfer is assumed. General solutions, which take into account the additional convective and conductive terms, are obtained for the temperature fields and the Nusselt number. The detailed studies are presented for the porous systems characterized by consolidated and unconsolidated circular unit cells. The results show that, for the consolidated unit cell case, a prediction without the additional convective term overestimates the heat transfer, while for the unconsolidated unit cell case, this effect is negligible. The additional conductive terms are also examined and found to act conventionally as part of the conductive terms.     

Shock waves and turbulence in a hypersonic inlet

A numerical investigation for an axisymmetric hypersonic turbulent inlet flow field of a perfect gas is presented for a three-shock configuration consisting of a biconic and a cowl. An upwind parabolized Navier-Stokes solver based on Roe's scheme is used to compute an oncoming flow Mach numberM =8, temperatureT =216 K, and pressureP =5.5293×10³ N/m². In order to assess the flow quantities, the interaction between shock and turbulence, and the inlet efficiency, three different flow calculations — laminar, turbulent with incompressible and compressible two-equationk- turbulence models — have been performed in this work.Computational results show that turbulence is markedly enhanced across an oblique shock with step-like increases in turbulence kinetic energy and dissipation rate. This enhancement is at the expense of the mean kinetic energy of the flow. Therefore, the velocity behind the shock is smaller in turbulent flow and hence the shock becomes stronger. The entropy increase through a shock is caused not only by the amplification of random molecular motion, but also by the enhancement of the chaotic turbulent flow motion. However, only the compressiblek- turbulence model can properly predict a decrease in turbulence length scale across a shock. Our numerical simulation reveals that the incompressiblek- turbulence model exaggerates the interaction between shock and turbulence with turbulence kinetic energy and dissipation rate remaining high and almost undissipated far beyond the shock region. It is shown that proper modeling of turbulence is essential for a realistic prediction of hypersonic inlet flowfield. The performed study shows that the viscous effect is not restricted in the boundary layer but extends into the main flow behind a shock wave. The loss of the available energy in the inlet performance therefore needs to be determined from the shock-turbulence interaction. The present study predicts that the inlet efficiency becomes relatively lower when turbulence is taken into account.     

Hamiltonian Formulation for the Motion of a Two-Fluid System with Free Surface

In this work, a theoretical investigation is performed on modeling interfacial and surface waves in a layered fluid system. The physical system consists of two immiscible liquid layers of different densities ₁ > ₂ with an interfacial surface and a free surface, inside a prismatic-section tank. On the basis of the potential formulation of the fluid motion, we derive a nonlinear system of partial differential equations using the Hamiltonian formulation for irrotational flow of the two fluids of different density subject to conservative force. As a consequence of the assumption of potential velocity, the dynamics of the system can be described in terms of variables evaluated only at the boundary of the fluid system, namely the separation surface and the free surface. This Hamiltonian formulation enables one to define the evolution equations of the system in a canonical form by using the functional derivatives.     

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.     

Physical properties of the flow in the separation region for three-dimensional interaction of a boundary layer with a shock wave

In a supersonic stream we consider the three-dimensional flow in the plane of symmetry in the region of interaction of a boundary layer with a shock wave which arises ahead of an obstacle mounted on a plate. The principal characteristic of this flow is the penetration of a filament of the ideal fluid within the separation zone and the formation on the surface of the plate and obstacle of narrow segments with high pressures, high velocity gradients, and large heat transfer coefficients.Pressure distribution measurements were made, shadow and schlieren photos were taken, and photographs of the flow pattern on the surface were made using dye coatings and low-melting models. The basic physical characteristics of the separation flow are established. The independence of the separation zone length of the boundary layer thickness is shown. Local supersonic flows are detected in the separation region, flow regimes are identified as a function of the angle of encounter of the separating flow with the obstacles, characteristic flow zones in the interaction region are identified.Notation s coordinate of separation point on the plate - l length of separation zone - H obstacle height - d obstacle transverse dimension - u freestream velocity - velocity gradient on stagnation line of obstacle - b jet width - compression shock standoff from the body - p static pressure - p_* pressure at stagnation point on obstacle - density - viscosity coefficient - boundary-layer thickness - compression shock angle - effective angle of separation zone - setting angle of obstacle on plate - M Mach number - R Reynolds number - P Prandtl number     

Contribution to the theory of the high compression ratio gas ejector with cylindrical mixing chamber

Ways of improving the operation of a gas ejector with a high compression ratio are investigated. The conditions for obtaining the maximal compression ratio at the critical operating regime of the gas ejector are studied theoretically and experimentally with account for mixing of the supersonic injecting and subsonic ejected streams ahead of the choking section. The principles for the rational utilization of the effect of stream mixing in the ejector ahead of the choking section are indicated; the use of these principles permits a several-fold increase of the compression ratio of the supersonic ejector. A theory is given for the critical regime of the gas ejector with uniformly perforated nozzle, and the hydraulic parameters of the required wall perforationss are determined. It is shown that perforation as a hydraulic factor can improve significantly the parameters of the sonic ejector in the critical regime.The foundations of modem gas ejector theory were developed by Khristianovich [1, 2]. In these studies he established the relationship between the parameters of the flow at the end of the mixing chamber (section 3, p ₀ is the total pressure, is the reduced velocity) and the parameters of the ejecting (section 1, p ₀ ,) and the ejected (p₀₁,₁) flows with account for compressibility for the ejector with a cylindrical mixing chamber (Fig. 1a). The ejector theory [1, 2] (see also [3, 4]) is given in the hydraulic approximation: the flow at the end of the mixing chamber is assumed uniform, flow friction on the mixing chamber walls is neglected. The use of the gasdynamic functions [5–9] made it possible to obtain computational equations for the ejector in a convenient form and to extend them to the case of mixing of gases with different thermophysical properties. We note that for subsonic velocities of the ejecting and ejected flows the system of ejector equations [1, 2] is supplemented by the condition of equality of the static pressures p=P₁ at the stream contact section 1.The results of extensive experimental studies of subsonic ejectors are in good agreement with the results of this theory.For sonic or supersonic velocity of the ejecting gas (=1) the condition p=p₁ is not satisfied in the general case. Fundamental for the development of ejector theory was the establishment by Millionshchikov and Ryabinkov in 1948 of the existence of a critical operating regime of the supersonic ejector [7, 10]. They showed that the limiting operating regimes of the gas ejector for high pressure differentials ==p ₀ /p₀₁ are determined by the conditions for the choking of the ejected jet by the expanding supersonic ejecting flow. With the occurrence of the critical regime the velocity of the ejected jet at the choking section (section 2, Fig. 1a) reaches the speed of sound (=1); this limits the further increase of the pressure ratio and the ejector compression ratio =p ₀ /p ₀ for a given ejection coefficient k (k is the ratio of the ejected and ejecting gas flow rates). The relationships between these flow parameters at sections 2 and 1 supplement the system of ejector equations and permit determining its critical characteristics.Millionshchikov and Ryabinkov showed that for moderate values of the pressure ratio good agreement of the theoretical and experimental ejector characteristics are given by the assumption of constant static pressure p₂=const at section 2 (Fig. 1a).The limit of the applicability of the theory based on the condition p₂= = const, was studied experimentally by Lyzhin [10].The theory of the critical regime of the gas ejector was developed in 1953 in studies of Nikol'skii, Shustov, Vasil'ev, Taganov, and Mezhirov [10, 11]. Nikol'skii showed that the condition of constant static pressure at the choking section is not in agreement with the momentum equation.For a more rigorous theoretical determination of the critical ejector regime he proposed joining between sections 1 and 2 (Fig. 1a) the calculation of the ejecting jet using the method of characteristics and the hydraulic calculation of the ejected jet; example calculations were made by Nikol'skii and Shustov. Taganov and Mezhirov suggested a method for calculating the ejector critical regime using a linear distribution of the pressure in the supersonic ejecting jet (at the choking section 2).A simple and successful method for calculating the ejector critical regime was given by Vasil'ev, who used the hydraulic representation of the ejecting and ejected flows in the choking section; both flows are assumed uniform at section 2, the static pressures in these flows in the general case are different and are determined by the momentum equation. A similar theory for the ejector critical regime was developed independently in [12, 13], and the theory with account for the supersonic ejecting flow (ahead of the choking section) was developed using the method of characteristics in [14].It should be noted that the results of the calculations of the critical characteristics of the ejectors using all three of these methods were practically indentical and in good agreement with experiment for large and moderate values of the ejection coefficients. We emphasize that in the theories of the ejector critical regime the flow mixing between sections 1 and 2 is neglected.The critical regime theory imposes significant limitations on the possible characteristics of the gas ejector, first of all, on the achievable compression ratio =p ₀ /p ₀ . Thus, from the data of [10], even for a pressure ratio =1000 the maximal theoretical value of the compression ratio for the supersonic ejector does not exceed 40 (see in Fig. 2 the limiting ejector characteristics based on the critical regime theory); for the sonic air ejector (=1) the theoretical value of 3.5 (see Fig. 9b on p. 26). Therefore it is important to analyze the methods for influencing the critical regime parameters in order to determine ways to improve the operation of the gas ejector with a high compression ratio.     

Estimation of the Effective Absorptivity of Metals from the Observation of Their Martensitic Restructuring under Laser Radiation

A new experimental method is proposed for determining the effective absorptivity of a metal under pulsed laser radiation. The method is based on solving an axisymmetric boundaryvalue heatconduction problem for a halfspace with the use of metallographically measured sizes of the polymorphictransformation zone in the irradiated material. The method is tested on singlecrystal cobalt and St.45 steel samples.     

Irrotational outflow of liquid from a stream through a lateral hole of finite depth

An algorithm is constructed for numerical determination of the flow parameters and coefficient of contraction of a jet in the case of irrotational lateral outflow of liquid from a semiinfinite stream through a nozzle of finite depth situated at an arbitrary angle to the mainstream flow. The solution is based on the use of N. E. Zhukovskii's method and the Schwarz-Christoffel formula. The results of calculations for a nozzle situated at an angle = /2 ± , where = /6, are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 162–164, January–February, 1977.     

Numerical modeling of perturbation propagation in a supersonic boundary layer

The growth of two-dimensional disturbances generated in a supersonic (M = 6) boundary layer on a flat plate by a periodic perturbation of the injection/suction type is investigated on the basis of a numerical solution of the Navier-Stokes equations. For small initial perturbation amplitudes, the second-mode growth rate obtained from the numerical modeling coincides with the growth rate calculated using linear theory with account for the non-parallelism of the main flow. Calculations performed for large initial perturbation amplitudes reveal the nonlinear dynamics of the perturbation growth downstream, with rapid growth of the higher multiple harmonics.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 33–44. Original Russian Text Copyright © 2004 by Egorov, Sudakov, Fedorov.     

Dissipation effects on MHD free convection flow past a semi-infinite vertical plate

Viscous and Joule dissipation effects are considered on MHD free convection flow past a semi-infinite isothermal vertical plate under a uniform transverse magnetic field. Series solutions in powers of a dissipation number (=gx/c _p) have been employed and the resulting ordinary differential equations have been solved numerically. The velocity and temperature profiles are shown on graphs and the numerical values of ₁(0)/₀(0) (, temperature function) have been tabulated. It is observed that the dissipation effects in the MHD case become more dominant with increasing values of the magnetic field parameter (=M ²/(Gr _x /4)^1/2) and the Prandtl number.     

On the Discretization of Distributed-Parameter Systems with Quadratic and Cubic Nonlinearities

Approximate methods for analyzing the vibrations of an Euler--Bernoulli beam resting on a nonlinear elastic foundation are discussed. The cases of primary resonance ( _n ) and subharmonic resonance of order one-half ( 2 _n ), where is the excitation frequency and _n is the natural frequency of the nth mode of the beam, are investigated. Approximate solutions based on discretization via the Galerkin method are contrasted with direct application of the method of multiple scales to the governing partial-differential equation and boundary conditions. The amplitude and phase modulation equations show that single-mode discretization leads to erroneous qualitative as well as quantitative predictions. Regions of softening (hardening) behavior of the system, the spatial dependence of the response drift, and frequency-response curves are numerically evaluated and compared using both approaches.     

Theoretical derivation of tangential velocity profiles in a flat vortex chamber-influence of turbulence and wall friction

Summary As part of a study on the hydrodynamics of a cyclone separator, a theoretical investigation of the flow pattern in a flat box cyclone (vortex chamber) has been carried out. Expressions have been derived for the tangential velocity profile as influenced by internal friction (eddy viscosity) and wall friction. The most important parameter controlling the tangential velocity profile is = –u ₀ R/(v+ ), where u ₀ is the radial velocity at the outer radius R of the cyclone, the kinematic liquid viscosity and is the kinematic eddy viscosity. For values of greater than about 10 the tangential velocity profile is nearly hyperbolic, for smaller than 1 the tangential velocity even decreases towards the centre. It is shown how and also the wall friction coefficient may be obtained from experimental velocity profiles with the aid of suitable graphs. Because of the close relation between eddy viscosity and eddy diffusion, measurements of velocity profiles in flat box cyclones will also provide information on the eddy motion of particles in a cyclone, a motion reducing its separation efficiency.List of symbols A cross-sectional area of cyclone inlet - h height of cyclone - p static pressure in cyclone - p static pressure difference in cyclone between two points on different radius - r radius in cyclone - r ₁ radius of cyclone outlet - R radius of cyclone circumference - u radial velocity in cyclone - u ₀ radial velocity at circumference of flat box cyclone - v tangential velocity - v ₀ tangential velocity at circumference of flat box cyclone - w axial velocity - z axial co-ordinate in cyclone - friction coefficient in flat box cyclone (for definition see § 5) - ₁ value of friction coefficient for ₁<< ₂ - ₂ value of friction coefficient for ₂<<1 - = - ₁ value of for ₁<< ₂ - ₂ value of for ₂<<1 - thickness of laminar boundary layer - =/h - turbulent kinematic viscosity - ratio of z to h - k ratio of height of cyclone to radius R of cyclone - parameter describing velocity profile in cyclone =–u ₀ R/(+) - kinematic viscosity of fluid - density of fluid - ratio of r to R - ₁ value of at outlet of cyclone - ₂ value of at inner radius of cyclone inlet - _w shear stress at cyclone wall - angular momentum in cyclone/angular momentum in cyclone inlet - ₁ value of at = ₁ - ₂ value of at = ₂     

Approximate computation of the cavitational flow around small span wings

The linearized problem of a completely cavitating wing of the small span is considered. The system of singular integral equations of a cavitating lifting surface is reduced to one-dimensional equations by using the Lawrence approximation. The method of discrete singularities is used for the numerical solution of this system. Dependences of the lift coefficient and the cavern length on the cavitation number are presented for rectangular wings for 0.254.Translated from Izvestiya Akademiya Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 166–170, July–August, 1973.     

Dimensional analysis of pore scale and field scale immiscible displacement

A basic re-examination of the traditional dimensional analysis of microscopic and macroscopic multiphase flow equations in porous media is presented. We introduce a macroscopic capillary number which differs from the usual microscopic capillary number Ca in that it depends on length scale, type of porous medium and saturation history. The macroscopic capillary number is defined as the ratio between the macroscopic viscous pressure drop and the macroscopic capillary pressure. can be related to the microscopic capillary number Ca and the LeverettJ-function. Previous dimensional analyses contain a tacit assumption which amounts to setting = 1. This fact has impeded quantitative upscaling in the past. Our definition for , however, allows for the first time a consistent comparison between macroscopic flow experiments on different length scales. Illustrative sample calculations are presented which show that the breakpoint in capillary desaturation curves for different porous media appears to occur at 1. The length scale related difference between the macroscopic capillary number for core floods and reservoir floods provides a possible explanation for the systematic difference between residual oil saturations measured in field floods as compared to laboratory experiment.     

Mathematical modeling of Maklakoff’s method for measuring the intraocular pressure

The physical content of Maklakoffs tonometric (based on the loading of the cornea) method of measuring the intraocular pressure, widely used in medical practice, is discussed. For this purpose, we employ both the results of physical modeling of the eye described in the literature and the results of our own mathematical modeling based on the representation of the eyeball as a thin shell. The effect of the physical properties of the shell on the results of the modeling is investigated. Qualitative conclusions that follow from our study and may be of practical interest in measuring the intraocular pressure are discussed.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, 2005, pp. 24–39. Original Russian Text Copyright © 2005 by Bauer, Lyubimov, and Tovstik.     

Fatigue crack propagaion under mixed mode loading

Mixed model fatigue crack propagation is analyzed in this paper, using a centre cracked plate geometry, loaded under un-iaxial cyclic tension. Based on maximum principal stress criterion, a modified Paris expression of fatigue crack growth rate is derived in terms of ΔK and crack angle β_α for an inclined crack. It is also shown that it is more convenient to express the Paris equation by means of crack length projected on the x -axis, α_x rather than the actual length, α itself. The crack trajectory due to cyclic loading is predicted, β is varied from 29° to 90°. Experimental data on Type L3 aluminium agree fairly well with predicted values when β_α exceeds 30°.