Hydraulic fracturing of a saturated porous medium-II: Special cases

This investigation deals with the problem of steady state hydraulic fracture in an infinite isotropic fluid-saturated elastic porous medium induced by a uniform pressure applied to the crack surfaces. A quasi-static approach is employed in the study. A boundary value problem is formulated and then analyzed by means of the Fourier transform associated with the Wiener-Hopf technique. Stress intensity factor and potential energy release rate are found by asymptotic analysis and the superposition principle as functions of the speed of crack propagation. The material breakdown process at the crack tip is discussed based on Dugdale's model. Finally, a brief discussion of the effect of pressure drop on the hydraulic fracture process and the decrease in crack speed during crack extension is included.     

Continuous transformation groups and series solution of initial value problems

This paper details an efficient method of deriving séries solutions of initial value problems traditionally encountered in non-linear mechanics. The developed technique, based on the theory of continuous transformation groups, can be used to solve autonomous as well as non-autonomous systems of ordinary differential equations. Solutions to selected problems are presented for illustration. In addition, the notion of invariants of a transformation group provides information on the nature of trajectories of an initial value problem.     

Motion of a variable-volume sphere in an ideal fluid near a plane surface

The motion of a spherical cavity in a fluid is investigated. The radius of the sphere varies under the action of a constant pressure at infinity. The problems of the collapse of a cavity moving in an unbounded fluid and of the collapse of a cavity near a plane are solved in the exact formulation. The occurrence of an initial translational velocity or the presence of a solid surface, by contrast with the collapse of a sphere at rest in an unbounded fluid [1], yields a limiting radius at which the process of collapse ceases. A sphere initially at rest near a plane always comes into contact with the plane as a result of collapse. The radius and velocities at which the sphere arrives the plane are calculated for various initial distances from the latter. The possible mechanism of the action of a cavitation bubble on a solid surface is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 94–103, September–October, 1971.     

Determination of the body shape for minimum radiative heating along a flight trajectory

Space vehicles are subject to intense aerodynamic heating in planetary entry. According to estimates in [1], the heat shield mass for entry of a probe into the atmospheres of the outer planets can make up 20–50% of its total mass; here the radiative component predominates in the aerodynamic heating. It is therefore interesting to investigate methods of reducing the heat flux to the nose region of a vehicle. Analysis shows [2–6] that, for a given atmospheric composition, the heat-shield weight is determined by the trajectory, the body shape, the heat-protection method, and the chemical composition and the thermophysical and optical properties of the heat shield material. In such a general statement of the problem, optimization of the heat-shield mass depends on many parameters, and has not been solved hitherto. A number of papers have examined simpler problems, associated with reducing spacevehicle heating: optimization of the trajectory from the condition that the total heat flux to the body stagnation point should be a minimum for given probe parameters [2, 3], optimization of the characteristic probe size for a given trajectory [2–4], and optimization of the probe shape in a class of conical bodies at a given trajectory point [3, 5, 6J. In [7] a variational problem was formulated to determine the shape of an axisymmetric body from the condition that the radiative heat flux to the body at a given trajectory point should be a minimum for the entire surface, and an analytical solution was found for this in limiting cases. The present paper investigates a more general variational problem: determination of the shape of an axisymmetric body from the condition that the total radiative influx of heat to the body along its atmospheric trajectory should be a minimum. A solution has been obtained for a class of slender bodies for different isoperimetric conditions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 94–100, March–April, 1978.     

Rheologische Eigenschaften konzentrierter Polyisobutylenlösungen in Lösungsmitteln verschiedener thermodynamischer Güte

Zusammenfassung Die Konzentrationsabhängigkeit der viskoelastischen Eigenschaften von Polyisobutylenlösungen in Lösungsmitteln mit verschiedener thermodynamischer Güte wurde mit einem Weissenberg-Rheogoniometer untersucht. Es wurde festgestellt, daß die Spriggssche 4-Parameter-Theorie den Übergang von niedrigen zu hohen Schergeschwindigkeiten nicht zufriedenstellend beschreibt. Die Schergeschwindigkeitsabhängigkeit des Schermoduls zeigt in den verschiedenen Lösungsmitteln ein Maximum. Die Konzentrationsabhängigkeit des Maximums wird durch das Molekulargewicht des Polymeren und die Lösungsmittelgüte wenig beeinflußt. Die Schergeschwindigkeiten, die dem Maximum entsprechen, zeigen dagegen eine Konzentrationsabhängigkeit, die von der Lösungsmittelgüte abhängt. Die Konzentrationsabhängigkeit und der Einfluß der Lösungsmittelgüte bei niedriger Scherrate entsprechen nur qualitativ den Voraussagen der strukturrheologischen Theorien.

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Summary Concentration dependence of viscoelastic properties of polyisobutylene solutions in thermodynamically different solvents was investigated in the Weissenberg-Rheogoniometer. It was found, that the 4-Parameter Spriggs theory does not exactly describe the changes of rheological properties of these solutions by changing the shear rate from low to high.The shear modulus as a function of the shear rate has a maximum for the investigated solutions. The concentration dependence of this maximum shows a low sensitivity for the molecular weight of the polymer as well as for the thermodynamical properties of the solvent. The shear rate corresponding to the maximum value of the shear modulus is dependent on the concentration and the thermodynamical properties of the solvent. The existing structural rheological theories describe only qualitatively the concentration dependence and the influence of the solvent on the structural parameters even at low shear rates.

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Mit 11 Abbildungen und 4 Tabellen     

Simple stationary waves in an inclined magnetic field

One component of the solution to the problem of flow around a corner within the scope of magnetohydrodynamics, with the interception or stationary reflection of magnetohydrodynamic shock waves, and also steady-state problems comprising an ionizing shock wave, is the steady-state solution of the equations of magnetohydrodynamics, independent of length but depending on a combination of space variables, for example, on the angle. The flows described by these solutions are called stationary simple waves; they were considered for the first time in [1], where the behavior of the flow was investigated in stationary rotary simple waves, in which no change of density occurs. For a magnetic wave, of parallel velocity, the first integrals were found and the solution was reduced to a quadrature. The investigations and the applications of the solutions obtained for a qualitative construction of the problems of streamline flow were continued in [2–8]. In particular, problems were solved concerning flow around thin bodies of a conducting ideal gas. The general solution of the problem of streamline flow or the intersection of shock waves was not found because stationary simple waves with the magnetic field not parallel to the flow velocity were not investigated. The necessity for the calculation of such a flow may arise during the interpretation of the experimental results [9] in relation to the flow of an ionized gas. In the present paper, we consider stationary simple waves with the magnetic field not parallel to the flow velocity. A system of three nonlinear differential equations, describing fast and slow simple waves, is investigated qualitatively. On the basis of the pattern constructed of the behavior of the integral curves, the change of density, magnetic field, and velocity are found and a classification of the waves is undertaken, according to the nature of the change in their physical quantities. The relation between waves with outgoing and incoming characteristics is explained. A qualitative difference is discovered for the flow investigated from the flow in a magnetic field parallel to the flow velocity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 130–138, September–October, 1976.The author thanks A. A. Barmin and A. G. Kulikovskii for constant interest in the work and for valuable advice.     

Numerical investigation of nonequilibrium flows of mixtures of fusible metal particles and gases in a laval nozzle

There are numerous papers [1–11] on the determination of the parameters of condensed oxide particles which are formed during combustion of metallized fuels. The ambiguity, and sometimes the contradictoriness, of the test results obtained [3–5, 9–11] indicate the difficulties in conducting correct experimental investigations. In this connection, numerical studies using mixtures of calibrated liquid-metal particles and different gases are of practical interest. Different probes can be calibrated by using calibrated two-phase flows, the two-phase flow around models and probes can be studied, as can the interaction between liquid-metal particles and the front of an aerodynamic compression shock, their intrusion in different entraining media, the interaction between fine particles (particle-projectiles) and large size particles (particle-targets), etc. In many cases, the prehistory of the flow and the parameters of the gas mixture with the particles in the area of the nozzle exit section must be known to investigate the above-mentioned phenomena. The parameters of different nonequilibrium flows of mixtures of gallium particles and gases in a Laval nozzle are investigated numerically in this paper; the maximum diameter (upper boundary of the spectrum) of the particles (d_s = 30 ) which are not destroyed in the nozzle under the effect of the aerodynamic forces and are suitable for use in a calibrated two-phase stream is determined. The computations were carried out in a one-dimensional approximation according to [12–14].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 86–91, March–April, 1976.The authors are grateful to V. K. Starkov and U. G. Pirumov for discussing the results of the research and to N. M. Alekseev for aid in constructing the graphs.     

Heat exchange between a selectively emitting liquid and a laminar gas flow in the presence of an external source of radiation

An investigation is conducted in the solution of a number of practical problems of the radiative and combined heat exchange in nonuniform systems having widely different physical properties. The processes of thermal interaction between the ocean and the atmosphere have been treated in the paper [1], the effect of thermal radiation on the melting and solidification of semitransparent crystals has been investigated in [2], the flow of a selectively emitting gas around the lateral surface of an object evaporating under the action of radiative heating has been discussed in [3], and heat transfer from a jet to the molten mass of glass in a glassmaking furnace tank has been investigated in [4]. The radiative and combined heat exchange between a selectively emitting liquid and a transparent heat-conducting laminar gas flow in the case of a specified external thermal radiation field is discussed in this paper. The energy conservation equations are set up taking into account the heat transfer by radiation, convection, and molecular thermal conduction. A differential approximation is used in calculating the values of the radiation fluxes. A system of fundamental computational equations is solved by the method of finite differences and iterations and by the Runge-Kutta method. The results of the calculations are presented in the form of graphs.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 116–122, May–June, 1976.     

The elastic-plastic stress distribution near the flat surfaced notch with cycloidal tip in a work-hardening material

Summary Based on a deformation theory of plasticity, the stress distribution near an edge crack having finite root radius in strain-hardening material is given for the case of antiplane shear (Mode III). The approximate solution is derived in an analytical form by using Neuber's result and integral J by Rice.

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Übersicht Unter Zugrundelegung der Deformationstheorie der Plastizität wird die Spannungsverteilung in der Nähe eines Randrisses mit endlichem Radius für den Fall des nichtebenen Schubspannungszustandes (Mode III) und eines dehnungsverfestigenden Materials gegeben. Die angenäherte Lösung wird durch Verwendung der Ergebnisse von Neuber sowie dem J-Integral von Rice in analytischer Form erhalten.

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In honour of the 70th birthday of Professor H. Neuber.