#### Volume 92, №6

**EFFECTS OF RANDOM MIGRATION ON THE GROWTH OF THE POPULATION OF A BIOLOGICAL SYSTEM**

The problem under consideration is an example of the anomalous behavior of systems with an explosive pattern of behavior in a random medium. The growth is modeled of a population of microorganisms as a result of their migration into a closed region with favorable conditions for regeneration. The initial number of microorganisms inside the region under consideration can be zero and the number of organisms outside the region is a random quantity. The rate of growth of the population as a result of reproduction is proportional to the squared size of the population, and the rate of extinction of the population is proportional to the population size. The microorganisms arrive inside the region due to diffusion, and the condition for mass transfer is assigned at the region′s boundary. As a result of averaging over the region′s volume, an equation has been obtained for the average size of the population. For a stationary number of microorganisms outside the region, a new class has been obtained of analytical solutions describing the change in the size of the population inside the region depending on the biological characteristics of the system and parameters of migration. The effects of random fl uctuations in the number of microorganisms outside the region under consideration are studied on the basis of two approaches. In the fi rst case, a closed-form equation was obtained for the probability density function of a random number of microorganisms inside the region. The effects of random migration on the population′s growth dynamics have been investigated on the basis of a closed system for the fi rst and second moments of the fl uctuation in the number of microorganisms. In the second case, direct modeling was implemented of the behavior of a random size of the population inside the region on the basis of solution of a stochastic ordinary differential equation. It has been shown that random fl uctuations may result in anomalous growth of the microbiological population size compared to the determinate case.

**Author:**
I. V. Derevich and A. A. Panova

**Keywords:**
microbiological system, migration, diffusion transfer, probability density function, moments of a random quantity, stochastic ordinary differential equation.

**Page:**
1375

**ON THE RELATIONSHIP BETWEEN THE TEMPERATURE COEFFICIENT AND THE RELATIVE VOLATILITY FOR THE PROCESS OF RECTIFICATION**

The relationship between the temperature coeffi cient and the value of relative volatility has been considered for the
process of rectifi cation. Various methods to calculate the temperature coeffi cient have been given. The obtained
results can be used for fi nding the optimum sequence of separation of a multicomponent mixture

**Author:**
A. M. Tsirlin and I. A. Sukin

**Keywords:**
rectifi cation, temperature coeffi cient, volatility, separation, multicomponent mixtures.

**Page:**
1385

**FORMATION OF A HYDRATE LAYER AT A GAS–WATER (ICE) INTERFACE**

The problem on the formation of a hydrate layer in a spherical particle with an ice core and on the surface of a gas
bubble coming to the surface of a water was solved numerically. As a limiting scheme of hydrate formation in these
cases, the diffusion mechanism, implying the diffusion of a gas (water) in the hydrate layer formed at the contact
line between a gas and an ice or a water to the surface of contact of the hydrate with the ice, the water, or the gas,
was adopted. A comparative analysis of the results of numerical calculations of the formation of a hydrate layer in
the indicated cases with the corresponding experimental data made it possible to estimate the reduced coeffi cients
of diffusion of a gas and water in such a hydrate layer. It is shown that the model of hydrate formation comprising
kinetic relations, obtained as a result of the solution of a quasi-stationary diffusion equation, allows one to defi ne
the transformation of a water or a gas into the hydrate state with the use of a single empirical parameter having
the dimensions of the diffusion coeffi cient. The qualitative and quantitative patterns of the formation of a hydrate
particle, obtained with the use of the indicated kinetic relations, agree with the corresponding experimental data and
theoretical calculations of other researchers, and by them, a large number of empirical parameters, which are not
known in advance but should be determined, can be estimated.

**Author:**
A. S. Chiglintseva and A. A. Rusinov

**Keywords:**
hydrate layer, gas bubble, ice particle, reduced diffusion coeffi cient

**Page:**
1396

**MATHEMATICAL MODEL OF THE EFFECT OF SELF-PRESERVATION OF GAS HYDRATES**

A diffusion model of dissociation of a plane layer of gas hydrate into ice and gas has been presented, which permits
modeling the effect of self-preservation of gas hydrates. In this model, the gas-hydrate dissociation into ice and
gas is described with account taken of the internal kinetics of the process and of the pore structure of the formed
ice layer. Calculated data obtained within the framework of a quasi-stationary approximation for the cases of
dissociation of plane layers of methane hydrate and carbon-dioxide hydrate into ice and gas have been given as an
example

**Author:**
V. A. Vlasov

**Keywords:**
gas hydrate, self-preservation effect, diffusion, chemical kinetics, mathematical modeling

**Page:**
1406

**ON THE THEORY OF EXTRACTION OF HIGH-VISCOSITY OIL FROM THE STRATUM UNDER THERMAL ACTION**

The authors have presented a theoretical model of the process of development of a stratum with high-viscosity oil
by the technology of double horizontal wells. In this model, a system of two wells is replaced by one reduced well
through which the heating of the stratum and the extraction of the oil are implemented simultaneously. Numerical
and analytical solutions of the problem have been obtained. The expenditure of heat to warm up the stratum,
the evolution of the well discharge, and the mass of pumped oil in a certain time interval have been determined.
The obtained solutions make it possible to select the mode of thermal action on the stratum that is the most effi cient
as far as power consumption is concerned.

**Author:**
V. Sh. Shagapov, Yu. A. Tazetdinova, and A. A. Gizzatullina

**Keywords:**
high-viscosity oil, heating of an oil stratum, fi ltration of oil

**Page:**
1415

**DYNAMICS OF ACOUSTIC WAVES IN A POROUS MEDIUM PARTIALLY SATURATED WITH A GAS HYDRATE**

The propagation of acoustic waves in a porous medium partially fi lled with a gas hydrate was investigated with regard
for the viscoelastic properties of the skeleton of this medium on the assumption that the gas hydrate is distributed
homogeneously in the thin layer at the inner surface of pores of the medium. A system of equations defi ning the
propagation of acoustic waves in a porous medium containing a gas hydrate is proposed. A dispersion relation
has been derived for investigating the infl uence of different parameters of a porous medium on the propagation of
acoustic waves in it

**Author:**
I. K. Gimaltdinov and V. L. Dmitriev

**Keywords:**
acoustic waves, porous medium, gas hydrate, dispersion relation

**Page:**
1423

**INTEGRAL MODELING OF THE PROCESS OF RESTORATION OF PRESSURE**

An integral model of the process of restoration of pressure has been constructed, and solutions of coupled differential
equations have been given. An analytical expression has been obtained that allows determining the dynamics of the
process of restoration of pressure with account taken of the stratum–well dynamic link.

**Author:**
É. M. Abbasov, Sh. A. Kerimova and N. A. Agaeva

**Keywords:**
pressure-restoration curve, integral model, liquid, infl ow, Laplace transformation

**Page:**
1432

**SPHERICAL AND CYLINDRICAL WAVES IN MULTIFRACTIONAL GAS SUSPENSIONS WITH POLYDISPERSE INCLUSIONS**

The propagation of plane, spherical, and cylindrical waves in multifractional gas suspensions with polydisperse
inclusions has been studied. The fractions differ in inclusion size and inclusion-size distribution functions, and also
in material. A dispersion relation and equilibrium and frozen velocities of sound have been obtained. Dependences
of the damping decrement at the wavelength on the dimensionless perturbation frequency have been constructed.
The infl uence of heat transfer on the dampings of the waves has been shown

**Author:**
D. A. Gubaidullin and R. R. Zaripov

**Keywords:**
spherical waves, cylindrical waves, polydispersity, gas suspension, dispersion relation

**Page:**
1439

**THE SECOND MOMENT OF THE DISTRIBUTION FUNCTION OF NANOPARTICLES IN A LIMITED REGION AND THEIR BROWNIAN DIFFUSION**

The change of the second moment of the distribution function of nanoparticles in a square under the infl uence

of Brownian diffusion has been studied numerically and analytically. It is shown that the second moment of the

distribution function of nanoparticles in an equilibrium system is independent of time, which is an important factor to

be taken into account in processing experimental data. It has been established that the second moment of the function

of nanoparticles distribution in a nonequilibrium spatially limited system depends on time, with this dependence

being nonlinear

**Author:**
S. P. Fisenko

**Keywords:**
Galerkin method, scalar coeffi cient of Brownian diffusion, method of variable directions, characteristic time

**Page:**
1443

**ANALYSIS OF THE CHEMICAL EQUILIBRIUM OF CARBON COMBUSTION PRODUCTS IN THE CASE OF OXYGEN DEFICIENCY**

Equilibrium states of a thermodynamic system representing a mixture of reactive products of combustion of carbon
and inert gas have been calculated at assigned pressure and temperature. The possibility has been shown of the
gaseous and condensed carbon phases appearing simultaneously in the products of a chemical reaction proceeding
in the system when the concentration of oxygen atoms is exceeded by the concentration of carbon atoms, i.e., in
the case of oxygen defi ciency. Numerical investigation has been performed under the assumption that a carbon
condensate is a suspension of ultrafi ne graphite particles. The infl uence of the inert gas on the shift of chemical
equilibrium in the thermodynamic system in the case of oxygen defi ciency has been analyzed. Conditions have been
determined under which a carbon condensate appears in the reaction products and the thermodynamic system,
homogeneous earlier, becomes heterogeneous

**Author:**
E. S. Prokhorov

**Keywords:**
combustion products, chemical equilibrium, carbon condensation

**Page:**
1446

**EXPERIMENTAL INVESTIGATION OF THE SUPPRESSION OF CROWN AND GROUND FOREST FIRES**

Results of experimental investigations on the suppression of the fl ame combustion and thermal decomposition of forest
combustible materials by aerosol fl ows of a pure water and aqueous solutions are presented. The characteristics of
some sprayers were determined and the densities of wetting of a fi re hotbed, provided by them, were calculated.
The times of extinguishing hotbeds of model crown and ground forest fi res were measured, and distributions of
temperatures and heat fl ows in them were determined. It is shown that the effi ciency of extinguishing a forest fi re is
mainly determined by the sizes of the aerosol droplets acting on it.

**Author:**
R. S. Volkov, N. P. Kopylov, G. V. Kuznetsov, and I. R. Khasanov

**Keywords:**
crown and ground forest fi res, thermal decomposition, fi re suppression, water droplets, aerosol.

**Page:**
1453

**MODEL OF THE DYNAMICS OF A POLYDISPERSE VAPOR–DROPLET MIXTURE WITH GAS-DYNAMICAL FRAGMENTATION OF DROPLETS**

A model of the dynamics of a multivelocity multitemperature polydisperse vapor–droplet mixture has been constructed
with account for gas-dynamical fragmentation of droplets on surpassing the critical Weber number. Calculations of
the dynamics of a vapor–droplet mixture of polydisperse composition on instantaneous insertion of droplets into
fl ow have been made. To describe the motion of a carrier medium, the system of Navier–Stokes equations for a
compressible heat-conducting gas was used. The dynamics of disperse fractions is described by systems of equations
that include continuity equations, as well as momentum and internal energy conservation equations. The equations
of motion of the carrier medium and of droplet fractions are written with account for the interphase momentum
and energy exchange. To describe the process of gas-dynamical fragmentation, a semiempirical model, well-known
from the literature, is applied allowing one to take into account the decrease in the radius, mean density, and
volume content of the fractions being split up. Changes in the dispersity and dynamics of the vapor–droplet mixture
occurring as a result of the fragmentation of droplets on instantaneous insertion of disperse phase into the fl ow are
analyzed

**Author:**
N. A. Tukmakova and A. L. Tukmakov

**Keywords:**
Navier–Stokes equations, equations of motion of a polydisperse vapor–droplet mixture, explicit McCormack scheme, gas-dynamical fragmentation of particles

**Page:**
1466

**OPTICAL MEASUREMENTS OF THE PRECIPITATION OF AN AEROSOL UNDER THE INFLUENCE OF EXTERNAL PHYSICAL FIELDS**

Results of experimental investigation of the evolution and precipitation of a cloud of pneumatically obtained
aerosol particles in a closed space under the infl uence of external physical fi elds are presented. The precipitation
is contributed to by imposing acoustic and electrostatic fi elds, which leads to an acceleration of the coagulation of
fi ne-aerosol particles. A method of searching for the aerosol′s dispersion characteristics is proposed, which has been
implemented in a laser measuring complex. Experimental dependences of the average volume–surface diameter and
of the relative concentration of aerosol particles of a model aerosol (talc) on time are presented.

**Author:**
M. Yu. Stepkina, A. A. Antonnikova, A. A. Zhirnov, O. B. Kudryashova, and S. S. Titov

**Keywords:**
aerosol particles, fi ne aerosol, closed space, ultrasonic treatment, electroprecipitation device, concentration of particles

**Page:**
1475

**ON THE LIMITING LAWS OF BUOYANT CONVECTIVE JETS AND THERMALS FROM LOCAL SOURCES OF A HEAT RELEASING IMPURITY**

By using the methods of scale analysis, dimensional analysis, and of the symmetry of differential equations, the
dynamics of buoyant convective jets connected with stationary, local sources of a weightless heat-releasing impurity
in a neutrally stratifi ed medium is investigated. An example of such an impurity is an aerosol that absorbs solar
radiation in the atmosphere (in particular, coal dust, and soot, which earlier was suggested to be used to actively
affect certain atmospheric processes). An analog of Ya. B. Zel′dovich′s theory of convective jets for laminar and
turbulent axisymmetric and plane jets has been obtained. The laws of motion of turbulent thermals connected with
instantaneous sources of the impurity are given that follow from the considerations of dimensionality and similarity.

**Author:**
L. Kh. Ingel′

**Keywords:**
convective jets, asymptotics, bulk heat release, heat releasing impurity, thermals, analysis of the symmetry of differential equations

**Page:**
1481

**NUMERICAL INVESTIGATION OF THE INFLUENCE OF SPECIAL STRUCTURES ON SUPPRESSION OF PRESSURE PULSATIONS IN THE DRAFT TUBE OF A HIGH-HEAD HYDRAULIC TURBINE**

The authors have performed numerical simulation of pressure pulsations in the draft tube of a hydraulic turbine
that are excited by large-scale vortex structures. Numerical simulation of unsteady transfer in the draft tube in
different operating regimes of the turbine has been carried out. It has been shown that the structure of fl ow in the
outlet diffuser corresponds to vortex-core fl ow. Calculated data have been compared with experimental results and
their agreement has been shown. To suppress pressure pulsations, a study has been made of the infl uence of various
structural inserts into the draft tube, such as fi ns, a cross piece, and a hollow cylinder in the draft-tube cone.
Consideration has been given to the infl uence of various constructions on the structure of fl ow behind the wheel
and on pressure pulsations on the draft-tube wall. A comparison has been made of the constructions and optimum
parameters have been selected to stabilize the fl ow in the diffuser of the hydraulic-turbine draft tube.

**Author:**
A. V. Sentyabov, A. V. Minakov, D. V. Platonov, D. A. Dekterev, A. V. Zakharov, and G. A. Semenov

**Keywords:**
mathematical simulation, turbulence, detached eddy simulation, pressure pulsations, vortex-core precession, hydraulic turbine

**Page:**
1489

**MATHEMATICAL MODELING OF THE SHOCK ACTION OF FRAGMENTS OF SPACE DEBRIS ON SPACECRAFT WINDOWS**

By the methods of continuum mechanics, the authors consider the problems of high-velocity (of the order of
10 km/s) interaction between aluminum and steel spherical particles of diameters no larger than 2·10–4 cm and
a thick fused-quartz (silica) target with account of the destruction and different states of aggregation of a substance,
and also of the dependence of the strength characteristics of materials on the internal energy. Mathematical
modeling is carried out within the framework of the Prandtl–Reuss elastic perfectly plastic model. The mechanism
of destruction is determined by successively developing processes of growth and coalescence of micropores in the
volumes under tensile stresses. To describe the behavior of a substance in wide ranges of variation in pressure
and temperature, use is made of an interpolation wide-range equation of state which includes a melting curve
and a two-phase region. The application-relevant region of interaction of the order of orbital (fi rst) and escape
(second) space velocities is diffi cult to experimentally investigate directly. In this connection, a theoretical analysis
and numerical simulation of high-velocity shock phenomena take on particular signifi cance.

**Author:**
M. V. Khabibullin, M. N. Krivosheina, and A. Yu. Sammel′

**Keywords:**
space debris, window, high-velocity interaction, mathematical modeling

**Page:**
1501

**VORTEX INTENSIFICATION OF HEAT TRANSFER IN CHANNELS AND PIPES WITH PERIODIC ELEMENTS OF DISCRETE ROUGHNESS**

A complex numerical investigation of vortex intensifi cation of heat transfer in turbulent inhomogeneous medium fl ow
in channels and pipes with periodic elements of discrete roughness has been carried out with the use of multiblock
computational technologies, realized in a specialized package VP2/3, and of a modifi ed model of transfer of shear
stresses. As roughness elements use was made of asperities and grooves of different geometries, with air and
transformer oil used as a working medium. The infl uence of the geometrical parameters of the indicated channels
and pipes as well as of the regime parameters of the fl ow in them on their thermal and hydraulic effi ciency has been
analyzed in detail.

**Author:**
S. A. Isaev, A. D. Chornyi, Yu. V. Zhukova, A. A. Vysotskaya, and V. B. Kharchenko

**Keywords:**
vortex intensifi cation, turbulent fl ow, channels, pipes, discrete roughness

**Page:**
1509

** INVESTIGATION OF PRESSURE PULSATIONS AND POWER LOADS IN THE COMPENSATOR WITH THE AIM OF REDUCING VIBRATION TRANSFER IN A PIPELINE WITH A LIQUID**

The authors have developed and investigated experimentally physical and mathematical models to calculate pressure
pulsations and dynamic loads produced by them, which determine vibration transfer through compensators of pipelines
with a liquid. The loads caused by pressure pulsations may substantially increase with frequency. Consideration has
been given to vibration transfer over the structure of the compensator′s elastic elements. The authors have shown
analytically and have confi rmed experimentally the presence of a broad frequency range, in which the compensation
of the forces transferred over the structure by the forces from pressure pulsations occurs. Vibration transfer may be
reduced ten or more times compared to the transfer over the structure of the compensator with a liquid. The revealed
phenomenon has been explained physically, and the ways of using it in practice have been proposed. At frequencies
exceeding this range, the presence of the liquid enhances vibration transfer through the compensator

**Author:**
A. V. Kiryukhin, O. O. Mil′man, A. V. Ptakhin, L. N. Serezhkin, and S. A. Isaev

**Keywords:**
pipeline, compensator, vibration, vibrational rigidity, dynamic force, oscillation frequency, working medium, pressure pulsation, elastic element, structural rigidity

**Page:**
1517

**A CHARACTERISTIC FEATURE OF PRESSURE FORMATION IN THE GAP BETWEEN ROTATING ECCENTRICALLY LOCATED CYLINDERS**

The results of theoretical and experimental investigations of pressure distributions in a shear laminar fl ow between
rotating eccentrically located cylinders are presented. It is shown that a change in the pressure in the slit gap is
attributed to two mechanisms: to the "wedge effect" from the eccentrically located rotating tenon in the bearing and
to the heat supply due to the adiabatic change of the volume, dissipative processes, and heat transfer.

**Author:**
V. D. Tyutyuma

**Keywords:**
The results of theoretical and experimental investigations of pressure distributions in a shear laminar fl ow between rotating eccentrically located cylinders are presented. It is shown that a change in the pressure in the slit gap is attributed to two mechanisms: to the "wedge effect" from the eccentrically located rotating tenon in the bearing and to the heat supply due to the adiabatic change of the volume, dissipative processes, and heat transfer.

**Page:**
1524

**DEVELOPMENT OF HEAT PIPES FOR COOLING THERMALLY STRESSED ELECTRONICS ELEMENTS**

An algorithm for the development of heat pipes involving the choice of the working fl uid, materials for the casing and
wick, and the methods of calculating the limits of heat-transmitting ability of heat pipes is suggested. This algorithm
has been applied successfully for creating different types of wicks and heat pipes on the basis of which a cooling
system has been created for a high-power light-emitting diode lantern.

**Author:**
K. I. Delendik, N. V. Kolyago, O. G. Penyazkov, and O. L. Voitik

**Keywords:**
heat pipe, wick, thermal resistance, light-emitting diode, cooling system, radiator

**Page:**
1529

**SIMULATION OF NONEQUILIBRIUM HEAT TRANSFER IN AN ANISOTROPIC SEMISPACE UNDER THE ACTION OF A POINT HEAT SOURCE**

Based on a novel analytical solution of the problem of wave heat transfer in an anisotropic semispace under the
action of a nonstationary point source of thermal energy, a study was made of a nonequilibrium thermal state of this
space originating as a result of its delay (for the time of heat fl ux density relaxation) from the temperature gradient
on the semispace surface. Based on the extended Vernotte–Cattaneo–Luikov hypothesis, a new law of wave heat
transfer has been formulated having the form of the Fourier law, and a wave equation has been derived for heat
transfer of parabolic type with a lagging argument in time equal to the relaxation time derived.

**Author:**
V. F. Formalev, É. M. Kartashov, and S. A. Kolesnik

**Keywords:**
anisotropic semispace, wave heat transfer, relaxation time, nonstationary temperature fi eld, boundary condition of the second type, thermal conductivity tensor

**Page:**
1537

**PARABOLIC PROFILE IN HEAT-CONDUCTION PROBLEMS. 4. SEMI-BOUNDED SPACE WITH A CONVECTIVE BOUNDARY CONDITION**

New approximate solutions of the heat-conduction problem for a semi-bounded space with a convective boundary
condition have been obtained with the use of the heat balance integral method, the refi ned integral method, and the
combined integral method as well as with new hybrid schemes involving a new generalized integral relation, the
temperature momentum integral, and the heat-fl ow momentum integral. The solutions obtained were represented as a
parabola with a time-dependent exponent n(t), and they are entirely explicit in form. It is shown that the optimization
of the exponent n(t) on the basis of minimization of the Langford norm EL provides, as a rule, the obtaining of
solutions with a low approximation accuracy. The new hybrid schemes proposed are more effi cient, because they
make it possible to obtain solutions in the explicit form with a fairly high approximation accuracy. It is proposed
to defi ne parabolic solutions of the heat-conduction problem with a convective boundary condition with the use
of new generalized variables and without recourse to the Biot number as an independent variable. A triple hybrid
scheme based on three different-kind integral relations was used for the fi rst time in solving the heat-conduction
problem with a convective boundary condition. All the solutions obtained with the hybrid schemes proposed are
characterized by good approximation properties. The new hybrid schemes can be used in solving different practical
problems

**Author:**
V. A. Kot

**Keywords:**
heat-conduction problem, semi-bounded space, convective boundary condition, parabolic solution, generalized integral relation, triple hybrid scheme

**Page:**
1548

**ADSORPTION OF CO2 AND ETHANOL BY A SPHERICAL ACTIVATED CARBON IN A HEAT PUMP**

A promising adsorbent representing a phenol resin activated with potassium hydroxide, which can be used in
innovative next-generation adsorption cooling and heating pump systems, is proposed.

**Author:**
K. Uddin, A. Pal, K. Thu, and B. B. Saha,

**Keywords:**
activated carbon, adsorbent, carbon dioxide, ethanol, heat pump, phenol resin, uptake

**Page:**
1575

**INFLUENCE OF MICROGRAVITATION ON VAPOR FILM COLLAPSE NEAR A WIRE IMMERSED IN SUPERFLUID HELIUM**

The processes of heat and mass transfer occurring in the course of vapor fi lm collapse on the surface of a wire heater
immersed in a volume of superfl uid helium are analyzed. Application of the results of a molecular-kinetic analysis for
calculating the intensity of vapor condensation on the interphase surface allows one to obtain the needed relation
for calculating the motion of the interphase surface to be used in turn in solving the Rayleigh equation with account
for surface tension. The calculation results at a different level of microgravitation are compared with experimental
data. An analysis of the infl uence of various constituents of the pressure difference in the fl uid on the speed and time
of vapor fi lm collapse is carried out

**Author:**
Yu. Yu. Puzina and A. P. Kryukov

**Keywords:**
helium-II, vapor fi lm, interphase surface, condensation, collapse, microgravitation, molecular-kinetic theory

**Page:**
1582

**HEAT AND MASS TRANSFER ON FREE PATH PORTIONS IN DRYING PAPER IN MULTICYLINDER CONTACT-CONVECTIVE UNITS**

Convective heat- and mass-transfer processes are considered that occur on free surfaces of a wet paper web in an
air medium with a single-row arrangement of drying cylinders. Calculated dependences are determined making it
possible to predict the duration of the process of drying the paper web in multicylinder contact-convective units.

**Author:**
E. N. Gromova, V. Yu. Lakomkin,† and A. G. Nikolaeva

**Keywords:**
drying of paper, heat and mass transfer, contact-convective drying, moisture content, kinetics of drying

**Page:**
1588

**MATHEMATICAL SIMULATION OF THE SWIRLING FLOW OF A DILATANT HERSCHEL–BULKLEY FLUID IN A CYLINDRICAL CHANNEL**

Results of an investigation of the swirling fl ow of a dilatant Herschel–Bulkley fl uid with a yield point in a cylindrical
channel are presented. It was established that the effective viscosity of such a fl uid at the input cross section of
a cylindrical channel increases with increase in the intensity of its vortex, which can cause plugging of the channel.
As the Rossby number of this fl uid increases, its effective viscosity in the region of unstable fl ow at the axis of the
channel decreases, and the opposite effect takes place in the near-wall zone of the channel. As the fl uid fl ow stabilizes
downstream, the effective viscosity of the fl uid in the near-wall region of the channel decreases, and the effective
viscosity of the fl uid near the channel axis increases, which leads to the formation of a quasi-solid fl uid fl ow

**Author:**
O. V. Matvienko, V. P. Bazuev, and A. E. Aseeva

**Keywords:**
rheology, dilatant fl uid, viscoplastic fl uid, Herschel–Bulkley fl uid, swirling fl ow, computational hydrodynamics

**Page:**
1593

**A COMPARATIVE ANALYSIS OF THE OSTWALD–DE WAELE AND ELLIS RHEOLOGICAL EQUATIONS IN SOLVING THE GRAETZ–NUSSELT PROBLEM**

The evaluation of the accuracy of the Ostwald–de Waele model in solving the Graetz–Nusselt problem is made
by comparing with the standard solution for the Ellis liquid. As a result of the analysis, it is shown that the problem
solution constructed on the exponential equation leads to substantial distortions of the heat transfer pattern

**Author:**
V. M. Shapovalov

**Keywords:**
mathematical simulation, Newtonian liquid, round pipe, Ostwald–de Waele model, Ellis model, temperature, Nusselt number, shear stress, viscous dissipation

**Page:**
1603

**CHANGE IN THE STRUCTURE AND PROPERTIES OF BLOOD CELL MEMBRANES AS A RESULT OF TEMPERATURE ACTION**

A study has been made of the structure and properties of erythrocyte and thrombocyte membranes in patients
with diabetes mellitus type 2 by the atomic-force-microscopy method. The infl uence of the temperature and the
incubation time on the local mechanical properties of blood cells and the aggregation properties of erythrocytes has
been assessed. The critical temperature and the incubation time for erythrocytes have been established, at which
irreversible changes in the properties of erythrocyte membranes occur. It has been shown that thrombocytes are
characterized by the higher thermally stable local mechanical properties of the membrane than erythrocytes.

**Author:**
G. B. Mel′nikova, T. N. Tolstaya, A. S. Petrovskaya, O. N. Shishko, E. É. Konstantinova, S. A. Chizhik, and T. V. Mokhort

**Keywords:**
atomic-force microscopy, local mechanic properties, erythrocyte sedimentation rate, erythrocytes, thrombocytes, temperature

**Page:**
1612

**ACCURATE CALCULATIONS OF THE HEAT CAPACITIES OF PURE METALS USING THE EINSTEIN–DEBYE APPROXIMATION**

We propose a new algorithm for accurate calculation of the temperature dependence of the heat capacity of pure
metals. The analytical approach is based on the combination of the Einstein and Debye models. The Einstein–Debye
approximation is applied successfully for evaluating the heat capacities of the metals Cu and Al which are often
used in electric motor technologies as winding materials and in other industrial engineering fi elds. The proposed
approach has been tested by comparing the obtained results with the available analytical and experimental data.
This approach can improve the evaluation of electric motor winding heat losses.

**Author:**
Zafer Doğan and Tural Mehmetoğlu

**Page:**
Einstein–Debye approximation, heat capacity, motor winding, Debye model

**MHD FLOW OF A WILLIAMSON FLUID OVER AN INFINITE ROTATING DISK WITH ANISOTROPIC SLIP**

An MHD fl ow of a Williamson fl uid over an infi nite rotating disk with the Soret and Dufour effects and an anisotropic
slip was investigated. The system of nonlinear partial differential equations governing this fl ow and the heat
and mass transfer in it was rearranged to the ordinary differential equations with the use of the von Kármán
similarity transformation. The ordinary differential equations were numerically solved using the MATLAB routine
bvp4c.

**Author:**
Najeeb Alam Khan and Faqiha Sultan

**Keywords:**
Williamson fl uid, rotating disk, anisotropic slip, Soret–Dufour effects

**Page:**
1625