#### Volume 92, №2

**SOLUTION OF ILL-POSED PROBLEMS ON THE PREDICTION OF NONSTATIONARY REGIMES OF WORK OF A REACTOR FOR FISCHER–TROPSCH SYNTHESIS**

On the basis of analysis of experimental data on the temperature in a reactor for Fischer–Tropsch synthesis, a
method for prediction of loss in thermal stability of such a reactor by extrapolating the reactor temperature in time
with the use of the first-order and second-order time derivatives of experimental data on this temperature has been
developed. Since, in this case, the calculation of derivatives is an incorrect procedure because of the rapid changes
in the reactor temperature and the accidental errors in its measurement, this problem was solved using the Tikhonov
stabilizing-functional method. An integro-differential equation of calculating the fi rst-order and second-order time
derivatives of experimental temperatures of the reactor, providing a minimum of the stabilizing functional, has been
derived. The possibility of predicting the behavior of the temperature in a reactor under the conditions of loss in
its thermal stability was demonstrated through the comparison of calculation and experimental data on the reactor
temperature. The method proposed can be used for development of an automated system for control over the working
conditions of a reactor for Fischer–Tropsch synthesis for prevention of its thermal explosion.

**Author:**
I. V. Derevich, V. S. Ermolaev, I. G. Solomonik, and A. Yu. Fokina

**Keywords:**
Fischer–Tropsch synthesis, thermal stability, temperature field, functional derivative

**Page:**
289

**THEORETICAL PRINCIPLES OF RADIOGEOCHEMICAL EFFECT**

Based on general physical laws, the theory of the radiogeochemical effect has been developed. Equations have been
obtained that describe the impurity fi elds, the solutions of which yielded expressions for calculating the space-time
distributions of the concentrations of radioactive substances.

**Author:**
A. P. Mikhailov, P. N. Mikhailov, and A. I. Filippov

**Keywords:**
fi ltration, chemical potential, Henry isotherm, impurity transfer velocity, gamma activity

**Page:**
300

**DEVELOPMENT OF ELEMENTS OF A REUSABLE HEAT SHIELD FROM A CARBON–CERAMIC COMPOSITE MATERIAL. 2. THERMAL TESTS OF SPECIMENS OF THE MATERIAL**

This paper presents the results of testing specimens of porous carbon–ceramic composite materials on the hightemperature gas-dynamic plasmatron VGU-4. The structure of carbon–ceramic composite materials of various
densities before and after tests has been investigated. It has been shown that the porosity of such a material infl uences
the process of its thermal destruction and ablation in a high-enthalpy gas fl ow

**Author:**
S. V. Reznik, A. F. Kolesnikov, P. V. Prosuntsov A. N. Gordeev, and K. V. Mikhailovskii

**Keywords:**
reusable heat shield, porous carbon–ceramic composite material, thermal tests, gas-dynamic stand, convective heating

**Page:**
306

**FINITE-ELEMENT MODELING IN THE TECHNOLOGY OF HARDENING LASER PROCESSING OF A METAL-CUTTING TOOL**

A problem on heating of a cutting tool by triangularly shaped pulsed laser radiation has been solved numerically in
a three-dimensional formulation by the fi nite-element method. The developed model takes account of the infl uence
of both the geometry of the tool and the spatial and temporal characteristics of laser radiation on the formation
of a temperature fi eld in the laser-processing zone. A study has been made of the temperature distribution on the
front face of the tool and over the depth of the laser-impingement point for different positions of the center of a
laser-processing spot relative to the major cutting edge. For different tool angles, the regularities of formation of a
temperature fi eld during the hardening by pulsed laser radiation have been established; processing regimes (energy
density, the multiplicity of processing, and the position of the center of the laser-processing spot relative to the tool′s
cutting edges) that ensure the best parameters of the laser-impingement point (maximum hardening depth and the
hardening-zone length) have been determined. A procedure to select technological parameters for the process of
hardening of the cutting tool has been developed

**Author:**
S. I. Yares′ko

**Keywords:**
modeling, fi nite-element method, temperature fi eld, laser hardening, cutting tool, cutter point

**Page:**
314

**DUAL SERIES METHOD FOR SOLVING A HEAT EQUATION WITH MIXED BOUNDARY CONDITIONS**

The paper is devoted to solving a nonhomogeneous nonstationary heat equation in cylindrical coordinates with a nonaxial symmetry. This equation is subjected to nonhomogeneous, mixed, and discontinuous boundary conditions of the second and third kinds that are specifi ed on the disk of a fi nite cylinder surface. In fact, the solution of the given problem is obtained by using a new type of dual series equations (DSEs) with a Bessel function of the fi rst kind. In particular, DSEs reduce the problem to a Fredholm integral equation of the second kind which can be solved numerically by several effi cient methods.

**Author:**
N. A. Hoshan, Y. A. Al-Jarrah, and A. A. Al-Habahbeh

**Keywords:**
nonstationary heat equation, dual series equations, mixed boundary conditions

**Page:**
326

**PARABOLIC PROFILE IN HEAT-CONDUCTION PROBLEMS. 2. SEMI-BOUNDED SPACE WITH A TIME-VARYING SURFACE TEMPERATURE**

Some approaches to the optimization of the exponent n in the parabolic temperature profi le for a semi-bounded space at a boundary condition with a time-varying surface temperature are considered. It is shown that solutions obtained using the scheme of T. Myers with minimization of the Langford norm are not even close to the optimum ones and that the minimization scheme based on the new error norm estimating the modulus of a temperature deviation is much more effi cient. New hybrid integral schemes convenient to use are proposed. Parabolic solutions have been obtained on the basis of a number of integral relations: the temperature-momentum integrals of single and double modifi cations and the temperature-function integral. On the basis of these solutions in combination with known integral relations, different variations of the hybrid integral method have been developed. It was established that the highest approximation accuracy is provided by the hybrid scheme combining the temperature-momentum integrals of single and double modifi cations. It is shown that the hybrid integral schemes proposed are much more accurate compared to the known integral methods, including those realized using the scheme with minimization of the Langford norm. In particular, for a semi-bounded space with a surface of constant temperature, the hybrid integral method variant combining the temperature-momentum integral of single modifi cation and the refi ned integral method makes it possible to obtain a simple cubic parabola representing a substantially better approximation solution compared to the temperature profi le obtained on the basis of the refi ned integral method with optimization of the exponent n by the scheme of T. Myers.

**Author:**
V. A. Kot

**Keywords:**
temperature-disturbance front, heat-balance integral method, temperature-function integral, hybrid integral method

**Page:**
333

**MODEL OF COOLING OF COMPACT SURFACES BY MICROCHANNEL RECUPERATIVE HEAT EXCHANGERS WITH A MATRIX OF FILAMENTARY SILICON SINGLE CRYSTALS**

An analysis has been made of a novel class of recuperative heat exchangers for cooling of compact surfaces in
electronic systems for various object-oriented purposes with a matrix of fi lamentary silicon single crystals grown
directly on the heat-liberating boundary. A mathematical model has been proposed based on the assumption of the
regime of ideal displacement of the heat-transfer agent in the form of a system of equations of two-temperature
convective heat transfer with volume averaging over the element in a porous skeleton. The solution of the system
has been obtained in analytical form. The experiment conducted on a pilot plant has confi rmed the adequacy of the
mathematical model

**Author:**
D. A. Konovalov, V. I. Ryazhskikh, I. N. Lazarenko, and N. N. Kozhukhov

**Keywords:**
microchannel heat-exchange element, fi lamentary silicon single crystals, matrix of quasi-cylinders, test bench, convective heat transfer

**Page:**
355

**REFRIGERATOR–RADIATOR FIN EFFECTIVENESS**

The paper presents a solution to the problem on stationary heat conduction in the refrigerator–radiator fi n that may be of use for engineering applications.

**Author:**
V. A. Nevrovskii

**Keywords:**
thermal conductivity, thermal radiation, refrigerator–radiator, fin

**Page:**
365

**MECHANISMS OF FORMATION OF NANOPARTICLES IN AQUEOUS SALT SOLUTIONS UNDER THE ACTION OF A HIGH-POWER PERIODIC LASER RADIATION**

The average temperature of a gas cloud formed in an aqueous solution of a gold salt under the action of a nanosecond laser-radiation pulse with a wavelength of 532 nm was calculated. It is shown that the average temperature of such a cloud reaches thousands of degrees for several nanoseconds, and the collisions of molecules in it cause the dissociation of all the chemical compounds in the cloud. As the cloud is cooled, the gold vapor is supersaturated, and solid gold clusters nucleate and grow in it. The repeated laser pulses heat not only the solution but also the gold nanoparticles formed in it, with the result that relatively large nanoparticles are evaporated, and the temperature of the particles with a radius of about 5 nm reaches only the melting point

**Author:**
J. A. Bobb, S. P. Fisenko, C. J. Rodrigues M. S. El-Shall, and K. M. Tibbetts

**Keywords:**
absorption coeffi cient, nucleation, melting, radiative and conductive heat exchange

**Page:**
369

**INFLUENCE OF THE NANOSTRUCTURED-CARBON-BASED PLASTICIZING ADMIXTURE IN A SELF-COMPACTING CONCRETE MIX ON ITS TECHNOLOGICAL PROPERTIES**

A study has been made of the infl uence of adding a nanostructured-carbon-based plasticizing admixture to a
self-compacting concrete mix on the properties of heavy-weight structural concrete produced from it. It has been
shown that this admixture in the concrete mix changes its structure and makes it possible to produce concrete with
exceptional physicomechanical properties. Furthermore, the plasticizing admixture added to a concrete mix ensures
the high degree of preservation of its placeability to provide a possibility of transporting this mix for a long period
and of using it in monolithic construction. The indicated admixture in the concrete mix reduces its water–cement ratio
to 0.2 and ensures good placeability of the mix, which guarantees the high rate of its hardening and considerable
improvement in the strength of concrete based on this mix

**Author:**
S. A. Zhdanok, E. N. Polonina, S. N. Leonovich, B. M. Khroustalev, and E. A. Koleda

**Keywords:**
heavy-weight structural concrete, plasticizing admixture, nanostructured carbon, self-compacting concrete mix, strength characteristics

**Page:**
376

**MODELING THE SYNTHESIS OF COMPOSITE TITANIAAND SILICA-BASED CORE–SHELL TYPE PARTICLES IN A PLASMACHEMICAL REACTOR**

One-stage synthesis of composite titania and silica nanoparticles in the working area of a plasmachemical reactor by a chloride method, which is based on separate oxidation of titanium and silicon tetrachlorides, has been modeled for the fi rst time. The infl uence of a number of parameters of the physicomathematical model on both the general size of the particles and the shell thickness has been investigated. Results of calculations for certain variants of the parameters have been given.

**Author:**
S. M. Aul′chenkoa, and E. V. Kartaev

**Keywords:**
titania, silica, composite particle, plasmachemical reactor, one-velocity multicomponent medium, homogeneous and heterogeneous reactions, coagulation

**Page:**
383

**SIMULATION OF HEAT AND MASS TRANSFER OF DROPLETS IN DRYING AN OVERHEATED LIQUID UNDER CONDITIONS OF COMBINED ENERGY EFFECT**

The results of simulation of heat and mass transfer of droplets in drying an overheated liquid under conditions of convective and infrared energy effect are presented. The infl uence of various parameters, in particular, of the liquid overheating temperature, infrared radiation fl ux density, and of the carrier phase velocity oscillations on the drop evaporation kinetics has been analyzed. The possibilities of reducing the duration of droplet drying by means of energy effect are shown.

**Author:**
P. V. Akulich

**Keywords:**
heat and mass transfer, drop evaporation, drying kinetics, spray drying

**Page:**
389

**EXPLOSION BREAKDOWN OF DROPS OF EMULSIONS OF WATER WITH PETROLEUM PRODUCTS AS A RESULT OF THEIR INTENSIVE HEATING**

The conditions and characteristics of intensive evaporation of very inhomogeneous drops of emulsions of water with petroleum products (gasoline, kerosene, residual oil, heavy crude, transformer oil, turbine oil) were experimentally investigated at relative volume concentrations of these products in emulsions varying in the wide range from 10 to 70% corresponding to a large number of high-temperature (1100 K) processes occurring with gas and vapor drops, such as the heat and fl ame purifi cations of water and the production of a synthetic gas from fuel emulsions and suspensions based on water. The conditions of explosion breakdown of emulsion drops in the process of their heating have been determined. Three regimes of breakdown of emulsion drops, differing in the number of small droplets formed and in their surface area, were separated. Proposals were made on the mechanisms of the physical processes leading to the transformation of the surface of emulsion drops and their explosion breakdown.

**Author:**
D. V. Antonov, I. S. Voitkov, R. S. Volkov, and O. V. Vysokomornaya

**Keywords:**
water, petroleum products, emulsion, inhomogeneous drop, intensive vaporization, explosion breakdown

**Page:**
398

**REFLECTION OF ACOUSTIC WAVE FROM THE BOUNDARY OF A TWO-FRACTION POLYDISPERSE GAS SUSPENSION**

The problem of normal incidence of an acoustic wave on the interface between a pure gas and a gas suspension is considered. The case of gas suspension consisting of two fractions, each of which is polydisperse and is described by a certain size distribution function of inclusions has been studied. The dependences of the phase velocity and refl ection coeffi cient on the frequency of perturbations are calculated. The infl uence exerted by the size distribution function of inclusions, as well as by the mass content on inclusions of the dependences of the refl ection coeffi cient on the frequency of perturbations has been established.

**Author:**
D. A. Gubaidullin, Yu. V. Fedorov, and R. R. Zaripov

**Keywords:**
polydispersity, gas suspension, refl ection coeffi cient, phase velocity

**Page:**
409

**NUMERICAL AND ANALYTICAL MODELING OF ONE GAS-SUSPENSION PROBLEM**

In the present paper, the authors have given nonsteady and numerical solutions to a certain initial boundary-value problem for dynamics equations of a two-phase two-velocity medium (Kh. A. Rakhmatulin′s model). To this end, a numerical method of the TVD type was developed and relevant testing calculations were carried out. A comparison of the calculation data and the analytical solution has been given. The comparison results have shown their satisfactory agreement, i.e., the effectiveness of the numerical scheme. The domain of hyperbolicity of the model under study on this solution has been found.

**Author:**
A. V. Fedorov, D. A. Tropin, and A. V. Panov

**Keywords:**
dynamics of a two-phase medium, analytical solution, numerical method, domain of hyperbolicity

**Page:**
414

**FRACTAL MODEL OF THE SURFACE FILM FORMED IN THE PROCESS OF DRYING OF LIQUID DISPERSE PRODUCTS ON SUBSTRATES**

This paper presents a model of the formation of the surface fi lm in the process of drying liquid dispersed products on solid substrates. The surface fi lm is modeled by constructing a fractal cluster formed on the basis of the algorithm of diffusion-controlled clustering with account for the accidental appearance of new particles within the limits of the contact line and calculating the formation kinetics of the modeled surface. We have obtained experimental values of the fractal dimension of the formed surface structures in the process of convective drying on substrates of a liquid post-liquor slop and a concrete plasticizer. The formula for calculating the probability of particle coalescence is given, it permits estimating the kinetic characteristics of the process of surface structure formation by drying

**Author:**
A. N. Pakhomov and N. Ts. Gatapova

**Keywords:**
drying, surface, fractal, model, kinetics

**Page:**
424

**ON THE MOTION OF PARTICLES IN THE VORTEX ZONE OF A CYCLONE–GRATE-FIRED CHAMBER**

The characteristic features of the motion of single particles in a swirled gas fl ow are considered. A dependence has been obtained for calculating the maximum height of particle ascent. An extreme dependence of the time taken by the particle to reach the chamber walls on its diameter has been established.

**Author:**
E. A. Pitsukha, Yu. S. Teplitskii, and A. R. Roslik

**Keywords:**
cyclone–grate-fi red furnace, vortex chamber, gas velocity, particle velocity, Archimedes number, Froude number, bottom blast, relaxation time, bubbling bed

**Page:**
432

**INTEGRAL SIMULATION OF OIL DISPLACEMENT BY WATER**

An integral model of the displacement of oil from a stratum by water has been developed, and, using it, solutions of the coupling equations defi ning nonstationary liquid fl ows in the stratum–well system with regard for the deformability of the stratum have been constructed. The results obtained allow one to determine the parameters of the waterfl ooding of a stratum with account of the dynamic coupling of the stratum–well system and the wellheard pressure of a production well in it.

**Author:**
É. M. Abbasov and T. S. Kengerli

**Keywords:**
integral model, stratum, well, oil, water, pressure pulsations, Laplace transform

**Page:**
441

**ON THE THEORY OF PRESSURE RELAXATION IN AN UNDERGROUND TANK WITH A DAMAGED WALL**

A study has been made of the process of pressure relaxation in a liquid-fi lled tank with impaired leakproofness, which
is found in the ground. Assuming that the leak rate is limited by fi ltration characteristics of the ground surrounding
the tank, the authors have constructed a theoretical model representing a linear integral equation. Here, the
obtained equation has an exact analytical solution. On the basis of a numerical analysis of this solution, the authors
have analyzed the infl uence of the scale of the damaged portion of tank walls, the permeability of the ground, and
also the rheological properties of the liquid on the rate of pressure relaxation (decay) in the tank after its sudden
pressure testing.

**Author:**
V. Sh. Shagapov and Z. R. Khakimov

**Keywords:**
pressure testing, pressure relaxation, fi ltration

**Page:**
450

**ON THE STABILITY OF OSCILLATIONS OF SOLUBLE GAS BUBBLES**

It is shown that the bubbles of a gas solved in a liquid and performing small radial-symmetric oscillations in an acoustic fi eld are amplitude unstable. An expression has been obtained for the increment that characterizes the rate of instability development

**Author:**
R. N. Khabeev and N. S. Khabeev

**Keywords:**
gas bubble, soluble gas, oscillations, stability

**Page:**
455

**INFLUENCE OF THE INLET VALVE DIAMETER, CONTAINER VOLUME, AND OF HORIZONTAL FINNING ON THE RATE OF FILLING A CONTAINER WITH UF6**

The paper presents a mathematical model of the process of fi lling a vertical immersible container with gaseous UF_{6.}
The mathematical model was program-realized and verifi ed on a test problem related to liquid fl ow in a cavity with
a mobile lid. The results of calculations of the dynamics of fi lling a container at different diameters of the inlet valve,
container volumes, and at different dimensions of a horizontal fi n are given. It is shown that the inlet valve diameter,
container volume, and the presence of horizontal fi nning substantially infl uence the dynamics of fi lling the container

**Author:**
A. A. Orlov, A. F. Tsimbalyuk, R. V. Malyugin, and D. A. Leont′eva

**Keywords:**
uranium hexafl uoride, diffusion, mathematical simulation, gas dynamics, fi nned containers, SIMPLE procedure

**Page:**
460

**NUMERICAL INVESTIGATION OF FLOW NEAR A ROUND EXHAUST CHANNEL SCREENED BY AN ANNULAR SWIRLED JET**

Based on the Reynolds equations closed with the aid of the model of shear stresses transfer and of the model with a correction for curvature and rotation, numerical simulation of air fl ow near a round exhaust hole screened by an annular swirled jet was carried out. The infl uence of the swirling intensity on the air fl ow velocity fi eld in the range of action of an exhaust device and its range of action at different fl ow rate ratios of the sucked and infl owing air and also at different distances of the circular hole from the axis has been determined. The results obtained will be of use in designing effective systems of exhaust ventilation of lowered energy capacity.

**Author:**
M. S. Gritskevich, A. K. Logachev, and K. I. Logachev

**Keywords:**
exhaust ventilation, vertical suction, Reynolds equations, model of shear stresses transfer, correction for curvature and rotation

**Page:**
468

**NUMERICAL SOLUTION OF THE PROBLEM OF FLOW PAST PROJECTED BODIES FOR DETERMINING THEIR AERODYNAMIC COEFFICIENTS**

The method of solving the problem of external fl ow past a body for calculating the aerodynamic characteristics of projected bodies of complex geometrical shape is presented. Two approaches to the solution of the problem are considered: a numerical solution of the Favre-averaged Navier–Stokes equations with the use of the turbulence model and simulation of large vortices on the basis of the Navier–Stokes equations, in which numerical dissipation acts as a subgrid model of turbulence. Testing the results of calculations for the problem of fl ow past bodies by various methods showed their good correspondence to each other and with experimental data. For a projected body of a characteristic shape (missile) numerical simulation is made for a wide range of parameters: Mach numbers, angles of attack, and rotation speeds. Approximation dependences of the coeffi cients of aerodynamic forces and moments are constructed that can be used further for calculating the trajectories of projected bodies.

**Author:**
A. M. Lipanov, I. G. Rusyak, S. A. Korolev, and S. A. Karskanov

**Keywords:**
problem of external fl ow past a body, Navier–Stokes equations, Favre averaging, turbulence model, simulation of large vortices, aerodynamic coeffi cients

**Page:**
477

**EXPERIMENTAL INVESTIGATION OF THE INFLUENCE OF THE SHAPE OF ICE OUTGROWTHS ON THE AERODYNAMIC CHARACTERISTICS OF THE WING**

The results of experimental investigations of the aerodynamic characteristics of the wing of NACA 0012 profi le subjected to icing are presented. The ice accreted to the leading edge was simulated by appendages. The form and roughness of the outgrowths corresponded to the icing regimes typical of the formation of a glassy ice. Experiments were carried out in a wide range of angles of attack α = –4–18^{o} in an AT-4 low-speed wind tunnel of the Kharkov Aviation Institute on a wing of rectangular shape with elongation λ = 5 at the Reynolds number Re = 0.67·10^{6} and fl ow velocity V_{∞ }= 53 m/s. It is shown that in the cases considered, a change in the aerodynamic shape of the wing as a result of icing leads to a decrease in the lift force of the wing down to 50%, to a decrease in the angle of fl ow stalling to a value of α ~ 6^{o} , and to a considerable (several times) increase in the drag

**Author:**
A. A. Prikhod′ko, S. V. Alekseenko, and V. V. Chmovzh

**Keywords:**
wing, icing, wind tunnel, aerodynamic coeffi cients

**Page:**
486

**INTENSE REGIME OF VIBRATORY COMBUSTION OF BIOFUEL**

Experimental investigation of the vibratory combustion of a solid biofuel (wood pellets) has been conducted. In some experiments, a signifi cant enhancement of acoustic radiation with vibratory combustion of the pellets was observed which was accompanied by the increase in their burnup rate. In the event of such a regime the fl ame temperature decreased, whereas the density of the heat fl ux on the combustion-chamber wall increased due to the turbulization of the boundary layer. An empirical-theoretical dependence to calculate heat transfer in intense regimes of vibratory combustion of the solid biofuel has been obtained.

**Author:**
Yu. V. Polezhaev, V. D. Geshele, and I. P. Raskatov

**Keywords:**
vibratory combustion, intense regime, thermoacoustic oscillations, fl ame temperature, heat transfer, ecology (environment), wood pellets

**Page:**
493

**NUMERICAL INVESTIGATIONS OF THE PROCESSES OF BURNING A GASEOUS FUEL IN THE FURNACE OF THE DE-10/14 STEAM WATER-TUBE BOILER WITH A SECONDARY TUBULAR RADIATOR**

The processes of burning a gaseous fuel in the furnace of the DE-10/14 steam water-tube boiler have been investigated numerically. The temperature and velocity distributions of the gases in the furnace space of the boiler, as well as the concentrations of combustion components have been determined. It has been shown that placing a secondary radiator along the horizontal axis of the burner in the furnace space of the boiler promotes the formation of a uniform temperature fi eld along the length of the furnace and a decrease in the temperature in its space, and the reverse fl ow of combustion products to the combustion front provides a decrease in the concentration of nitrogen oxides to 123–125 mg/m^{3} at the furnace outlet.

**Author:**
A. A. Red′ko, A. V. Davidenko, A. A. Pavlovskaya, N. V. Kulikova, S. V. Pavlovskii, and I. A. Red′ko

**Keywords:**
numerical investigations, burning processes, gaseous fuel, steam boiler, secondary radiator.

**Page:**
500

**INFLUENCE OF THE IRRADIATION WITH HIGH-ENERGY Xe IONS ON THE STRUCTURE AND PHOTOLUMINESCENCE OF SILICON AND SILICA WITH InAs NANOCLUSTERS**

The authors have given results of investigations into the structural and optical properties of InAs nanoclusters formed by the ion-implantation method in silicon and silica matrices. The infl uence of the heat treatment at a temperature of 900^{o} C and of the irradiation with Xe ions with an energy of 167 MeV and a fl uence of 3·10^{14} cm ^{–2} on the structure and photoluminescence of the formed systems has been found. In the case of the system ″InAs in SiO^{2}″ the irradiation with Xe ions leads to an enhancement of the intensity and a broadening of the luminescence spectrum in the visible spectral region (550–750 nm). The ordering of the nanoclusters and their stretching along the path of the Xe ions in the SiO_{2} matrix has been found.

**Author:**
F. F. Komarov, O. V. Milchanin, I. N. Parkhomenko, L. A. Vlasukova, N. S. Nechaev, V. A. Skuratov, and V. N. Yuvchenk

**Keywords:**
ion implantation, InAs nanoclusters, high-speed heavy ions, tracks, heat treatment, crystalline silicon, silica, Rutherford backscattering, transmission electron microscopy, photoluminescence

**Page:**
508

**TWO-DIMENSIONAL GENERALIZED MAGNETOTHERMOELASTIC DIFFUSION PROBLEM FOR A THICK PLATE UNDER LASER PULSE HEATING WITH THREE-PHASE LAG EFFECT**

A two-dimensional magnetothermoelastic problem for a thick plate with diffusion in the context of the generalized thermoelasticity theory with three-phase lag effect is considered. The upper surface of the plate is taken to be traction free and heated by a laser beam, and the lower surface to be thermally insulated. The chemical potential on the bounding surface is assumed to be a known function of time. An analytical solution of the problem is derived in the Laplace–Fourier transform domain. The numerical inversions for the stresses, displacement, chemical potential, etc. in the space–time domain are obtained

**Author:**
K. Paul and B. Mukhopadhyay

**Keywords:**
generalized thermoelasticity, thermoelastic diffusion, Laplace transform, Fourier transform, laser beam

**Page:**
516

**DIFFERENTIAL MODELS OF RHEOLOGICALLY NONSTATIONARY FLUIDS**

An analysis has been made of basic differential models of rheologically nonstationary (viscoelastic) fl uids as well as of their development and interrelation. The considered models cover a rich variety of viscoelastic media: polymer solutions and melts, natural formations (glaciers), and others. Among these models, a key role is played by the Maxwell upper convection model: it provides a theoretical basis for experimental determination of the dynamic characteristics of viscoelastic fl uids and for development of new rheological models. It has been shown that to improve the reliability of results obtained with the aid of a complex rheological model, it is expedient to ensure a possibility of reducing it to the existing models and thus fi nding analytical solutions for a number of the simplest fl ows. Examples of use of the models in question when results of rheometric investigations are approximated and viscoelastic-fl uid fl ows are calculated have been given. Special emphasis has been placed on an analysis of the correspondence of the derived solutions to the physical essence of the described processes, and also of the correctness of interpretation of some results or others.

**Author:**
O. M. Sokovnin, N. V. Zagoskina, and S. N. Zagoskin

**Keywords:**
mathematical models, viscoelastic fl uids, determination of dynamic characteristics

**Page:**
528

**INTEGRAL MODELS OF RHEOLOGICALLY NONSTATIONARY FLUIDS**

Consideration has been given to the status and prospects for development of integral models of rheologically nonstationary (viscoelastic) fl uids. The predominant fi eld of application of these models are media with a fairly long relaxation time (concentrated polymer solutions and melts, biological tissues, foams, and others), for which taking account of the preceding states assumes substantial importance. The emphasis has been placed on today′s integral models developed on the basis of a classical Kaye–Bernstein–Kearsely–Zapas model through singling out viscous and purely elastic components in the potential function of the medium′s stored energy (so-called factorization of the K-BKZ model). An analysis has been made of the thixotropy and rheopexy phenomena caused by the lag of the reaction of response of rheologically nonstationary fl uids to external action. The authors have shown the expediency of the loop test used in determining the integral parameters of motion of viscoelastic fl uids (velocity, fl ow rate, pressure difference, etc.), and also the possibility of using a fl ow curve representing the averaging of the ascending and descending branches of a hysteresis loop in the case of its small width.

**Author:**
O. M. Sokovnin, N. V. Zagoskina, and S. N. Zagoskin

**Keywords:**
mathematical models, viscoelastic fl uids, thixotropy, rheopexy

**Page:**
542