#### Volume 93, №5

**ANALYTICAL AND NUMERICAL SOLUTION OF THE EQUATION FOR THE PROBABILITY DENSITY FUNCTION OF THE PARTICLE VELOCITY IN A TURBULENT FLOW**

A study has been made of the random motion of inertial particles in a homogeneous isotropic turbulent gas fl ow.
Fluctuations of the gas velocity along the particle path were modeled by the Gaussian random process with a fi nite
time of degeneracy of the autocorrelation function. A closed equation has been obtained for the probability density
function of the particle velocity, for which two methods of numerical solution have been proposed: using the fi nitedifference scheme and using one based on direct numerical modeling of an empirical probability density function.
The empirical probability density function was obtained as a result of the averaging of random particle paths,
which are a solution of a system of ordinary stochastic differential equations. The results of numerical calculation
have been compared with the analytical solution describing the dynamics of the probability density function of the
particle-velocity distribution.

**Author:**
I. V. Derevich and A. K. Klochkov

**Keywords:**
probability density function, stochastic ordinary differential equation, two-phase turbulence, difference scheme, autocorrelation function, Green′s function, direct numerical modeling

**Page:**
1043

**MATHEMATICAL SIMULATION OF THE HEAT AND MASS TRANSFER IN THE MOVEMENT OF LIQUID DROPLETS IN A GAS MEDIUM UNDER THE CONDITIONS OF THEIR INTENSE PHASE TRANSFORMATIONS**

A physical and mathematical model of the movement of a liquid droplet in a high-temperature air fl ow has been
formulated with regard for the main factors of this movement: the inertia of the droplet, its viscous friction,
the surface tension of the liquid in it, the air drag, the gravity, lift, and Magnus forces acting on the droplet, its
turbophoresis and thermophoresis, the convective heat fl ows inside the droplet, the turbulence and compressibility
of the carrying medium, the concentration of the dispersed phase in it, the nonstationarity of the movement of
the droplet, its phase transformations, the collisions of the droplet with the neighboring liquid droplets in the gas
fl ow, the partial fragmentation of the droplet, and its breakdown. Numerical and experimental investigations of the
infl uence of the indicated factors on the velocity of movement, the heating, and the intensity of evaporation of a liquid
droplet in a high-temperature gas medium have been performed. The results of these investigations were generalized
for determining the possible applications of the model developed.

**Author:**
D. V. Antonov, G. V. Kuznetsov, and P. A. Strizhak

**Keywords:**
heat and mass transfer, gas medium, liquid droplet, phase transformations, high-temperature heating, mathematical simulation

**Page:**
1055

**INFLUENCE OF COOLING MEDIUM FLOW CHARACTER ON THE OPERATION OF HEAT EXCHANGERS WITH STEAM CONDENSATION INSIDE TUBES**

The infl uence of the character of cooling medium motion in a heat exchanging apparatus on the process of steam
condensation in the latter is considered in the cases of cross fl ow past a tube bank in the apparatus, longitudinal fl ow
past tubes in the bank in the regimes of forward fl ow and countercurrent fl ow, superheated steam condensation, and
of nonuniform distribution of the cooling medium in different zones of the apparatus. It has been established that in
the case of cross fl ow past the fi rst rows of the tube bank, cooling medium zones with reduced heat transfer effi ciency
appear in the apparatus. The dependence of the pressure losses by the steam, which condenses in a heat exchanger
in regimes of countercurrent fl ow, forward fl ow, and cross fl ow, is investigated. Experimental data on the superheated
steam temperature have been obtained and generalized. The positive infl uence of the throttle on the line of removal
of the steam–air mixture to a steam–jet ejector in nonuniform cooling of the parallel channels of the condenser
has been established experimentally, and the presence of the optimal size of the throttle on the line of removal of
noncondensing gases has been confi rmed.

**Author:**
O. O. Mil′man, S. A. Isaev, A. V. Ptakhin, A. V. Kondrat′ev, A. Yu. Kartuesova, and V. S. Krylov

**Keywords:**
tube bank, steam condensation, superheated steam, nonuniform cooling, countercurrent fl ow, forward fl ow, throttle, noncondensing gases

**Page:**
1077

**FACTORS INFLUENCING THE CHARACTERISTICS OF WETTING OF PARTS OF A VAPOR CHAMBER**

A study has been made of conditions for the electrochemical deposition of copper on copper plates and of the infl uence
of the microroughness of their surface on the wetting of the plates. This permitted determining the optimum regimes
of deposition of coatings on elements of vapor chambers with the aim of creating such elements with controlled
wetting properties.

**Author:**
O. L. Voitik, K. I. Delendik, N. V. Kolyago, and L. Yu. Roshchin

**Keywords:**
microroughness, contact wetting angle, hydrophilic and hydrophobic coatings, wetting hysteresis, electrochemical deposition

**Page:**
1089

**CALCULATION OF THE THERMOHYDRAULIC EFFICIENCY OF POROUS DIRECT-FLOW STEAM-GENERATING CHANNELS**

Results of calculation of the effi ciency of porous direct-fl ow steam-generating channels in the transition region of
motion of a heat-transfer agent with boundary conditions of the fi rst kind have been presented. The obtained data
have been compared with the results of calculations for a smooth-walled tube. It has been shown that positive values
of geometric coeffi cients of effi ciency of such a channel can be obtained when water is used as the heat-transfer
agent. For the investigated parametric range of the indicated channels, no positive values of energy coeffi cients
of effi ciency were found. The dependences of the effi ciency coeffi cients of a porous model channel on its basic
operating-structural parameters have been analyzed.

**Author:**
A. P. Lukisha

**Keywords:**
thermohydraulic effi ciency, porous steam-generating channels, heat-transfer agent, water, boundary conditions of the fi rst kind

**Page:**
1096

**ON SELECTING A POINT FOR INJECTION OF A SEPARATED MIXTURE INTO A MULTICOMPONENT-RECTIFICATION COLUMN**

Consideration has been given to the problem on selecting a point for injection of a separated mixture into a
multicomponent-rectifi cation column in accordance with the dissipation minimum condition during the mixing of
refl ux and mixture streams. An estimate for the minimum entropy production in mixing has been found. The manner
in which the irreversibility of the process of mixing at the injection point affects the region of permissible regimes of
the column has been shown. An algorithm for calculating the concentration profi le of the mixture′s components over
the column height has been proposed.

**Author:**
A. I. Balunov, I. A. Sukin, and A. M. Tsirlinb

**Keywords:**
rectifi cation, optimization thermodynamics, entropy production, point of injection of a separated mixture, dissipation

**Page:**
1108

**FORMATION OF MONODISPERSE AND NARROW DISPERSE ENSEMBLES OF DROPLETS OF AQUEOUS ORGANIC SOLUTIONS IN THE VAPOR OF VOLATILE COMPONENTS**

The equilibrium state of solution droplets formed in the vapor of volatile components has been considered. The
criterion of equilibrium of the solution droplets and the vapor of volatile organic compounds has been given, on
whose basis the droplet size was estimated quantitatively as a function of the partial pressure of the components. In
accordance with a thermodynamic description, the droplet size was unambiguously determined by the composition
of a gas medium. The equilibrium state (radius and concentration) of droplets of ideal and real binary solutions has
been compared at the same composition of the gas medium. It has been shown that the droplet size of the real solutions
with a positive deviation from ideality exceeds the droplet size of the ideal solution, and the droplet size of the real
solutions with a negative deviation from ideality is smaller than that of the ideal solution. The described regularities
have been illustrated by microscopic photographs demonstrating the forming of an ensemble of secondary droplets

**Author:**
V. B. Fedoseev and E. N. Fedoseeva

**Keywords:**
disperse system, liquid–vapor equilibrium, droplets, ideal and real solutions, dimensional effect, condensation, evaporation

**Page:**
1116

**THERMAL INTERACTION OF A GAS WELL CLUSTER WITH PERMAFROST ROCKS OF THE MOST PROMISING DEPOSITS OF CENTRAL YAKUTIYA**

The results of a numerical experiment to determine the thermal impact of a gas well cluster on permafrost rocks
are given for the conditions of four deposits of Central Yakutiya, Middle-Vilyuy (Srednevilyuiskoe), Middle-Tyung
(Srednetyungskoe), Sobolokh–Nidzhili (Sobolokh-Nidzhilinskoe), and Tolon–Mastakh (Tolon-Mastakhskoe). An
approximate mathematical problem is solved in a conjugate formulation. In this case, the process of melting of
permafrost rocks is described within the framework of a three-dimensional Stefan problem, and the motion of real
gas in wells is described within the framework of tubular hydraulics. Use has been made of real data for temperature
well logging, formation pressure and temperature, well fl ow rate, gas composition, and thermophysical properties
of rocks. The melting of frozen rock depending on the depth and time has been calculated, and instants of time for
linkage of thaw zones of neighboring wells have also been computed.

**Author:**
V. A. Ivanov and I. I. Rozhin

**Keywords:**
permafrost rocks, gas well cluster, Stefan problem, mathematical modeling

**Page:**
1123

**GAS DYNAMICS OF THE PROCESS OF SEPARATION OF GRANULAR MATERIALS INTO FRACTIONS BY SPECIFIC WEIGHT**

A mathematical model of the process of separation of granular materials by specifi c weight in fl uidized beds has
been constructed on the basis of the conservation equations of mass and momentum of the heterogeneous media
mechanics. Computational and physical experiments on studying the separation process have been carried out. The
regime and design parameters of the apparatus that ensure the needed degree of granular material separation into
fractions have been determined.

**Author:**
F. G. Akhmadiev

**Keywords:**
mathematical model, gas dynamics, separation, specifi c weight, granular material, fl uidized bed

**Page:**
1131

**GENERATION OF A FINE AEROSOL IN A CAVITATION REGIME**

Consideration is given to the problem of atomization of fi ne aerosols using atomizers of special designs that
implement a cavitation regime. The formation of a fi ne aerosol is studied with the model of a pulse atomizer utilizing
HEM energy and the model of an atomizer with a special nozzle to create counterfl ows. For these atomizers, the
role of cavitation in obtaining a fi ne liquid aerosol is shown. A mathematical model is proposed which describes
the processes of genesis of an aerosol cloud. Estimates of the critical pressure are obtained for the development of
cavitation, the outfl ow velocity, and the resulting size of droplets as functions of the geometric parameters of the
atomizers, the pressure in the structure, and the physicochemical properties of the liquid. Experimental investigations
of the dispersion and concentration of aerosol particles are carried out using optical methods of measurement.
Results of measuring the dispersion parameters of an aerosol in the process of cavitation atomization of liquids are
presented.

**Author:**
O. B. Kudryashova, E. V. Muravlev, and B. I. Vorozhtsov

**Keywords:**
cavitation, fi ne aerosol, pulse atomizer, arc nozzle

**Page:**
1138

**SIMULATION OF THE PROCESS OF FILTRATION OF A GAS–LIQUID MIXTURE IN THE BED–WELL CONJUGATE SYSTEM**

An integral mathematical model of the process of nonstationary fi ltration of a gas–liquid mixture has been constructed,
and solutions of the obtained coupled equations have been given. The obtained analytical formula permits determining
the dynamics of bottomhole pressure and the reservoir productivity as functions of the parameters of a system.

**Author:**
É. M. Abbasov, T. S. Kengerli, and N. R. Abdullaeva

**Keywords:**
integral model, conjugate system, differential equations, fi ltration, Laplace transform, inverse transform, fluid.

**Page:**
1147

**RIEMANN AND SHOCK WAVES IN A POROUS LIQUID-SATURATED GEOMETRICALLY NONLINEAR MEDIUM**

Within the framework of the classical Biot theory, the propagation of plane longitudinal waves in a porous liquidsaturated medium is considered with account for the nonlinear connection between deformations and displacements
of solid phase. It is shown that the mathematical model accounting for the geometrical nonlinearity of the medium
skeleton can be reduced to a system of evolution equations for the displacements of the skeleton of medium and of
the liquid in pores. The system of evolution equations, in turn, depending on the presence of viscosity, is reduced
to the equation of a simple wave or to the equation externally resembling the Burgers equation. The solution of the
Riemann equation is obtained for a bell-shaped initial profi le; the characteristic wave breaking is shown. In the
second case, the solution is found in the form of a stationary shock wave having the profi le of a nonsymmetric kink.
The relationship between the amplitude and width of the shock wave front has been established. It is noted that the
behavior of nonlinear waves in such media differs from the standard one typical of dissipative nondispersing media,
in which the propagation of waves is described by the classical Burgers equation.

**Author:**
V. I. Erofeev and A. V. Leont′eva

**Keywords:**
porous medium (Biot medium), geometrical nonlinearity, evolution equation, Riemann wave, generalized Burgers equation, stationary shock wave

**Page:**
1156

**INFLUENCE OF THE SEQUENCE OF ROTOR APPARATUS CYLINDERS DISPOSITION ON CERTAIN LAWS GOVERNING THE APPARATUS OPERATION**

Theoretical and experimental investigations were carried out on determination of the effect exerted by the
geometrical parameters of the operational elements of a multicylinder rotor apparatus, regime parameters of
processing and fl ow rate of the processed medium on the disperse composition of the droplets of emulsions formed
in the radial clearance between the cylinders of the rotor and stator. It has been established that the power
dissipation in the radial clearance of the apparatus depends greatly both on the order of cylinders disposition in
the operational elements of the apparatus and on the fl ow rate of processed medium. The greatest effi ciency of the
process of emulsifi cation in the rotor apparatus is attained under conditions where the circumferential velocity
profi le has a large curvature near the rotor surface. It has been established that obtaining such kind of profi le is
enhanced by an increase in the rate of fl ow of the processed medium through the apparatus in the case if the inner
cylinder in the "rotor–stator" pair is the cylinder of the stator, and conversely by a decrease in the rate of fl ow if the
inner cylinder is the cylinder of the rotor

**Author:**
A. A. Kukhlenko and M. S. Vasilishin

**Keywords:**
rotor apparatus, emulsifi cation, fl ow in the clearance, power dissipation, diameter of droplets

**Page:**
1163

**ANALYTICAL ESTIMATION OF PRESSURE DELAY TIME IN OBSERVATION WELLS DURING PULSE TESTING OF FORMATIONS**

In using the method of pulse testing of formations, a pressure pulse is generated in an active well by changing
the regime of its operation; the pulse propagates across the formation and is registered in an observation well.
Based on the results of processing the obtained dependence of pressure in the well shaft on time equated with
formation pressure dependence on time at the point of location of the observation well, the formation parameters
are determined by comparison with certain analytical relations. It is taken into consideration that the pressure in the
well shaft of the observation well follows the variation of the formation pressure in the vicinity of the observation
well with some delay due to the processes of fi ltration mass transfer of liquid between the well shaft and its wellbore
area. Mathematical analysis of this mass transfer process presented in this paper shows that the delay time may be
4–5 h or more, depending on the parameters of the formation and formation fl uid, which determines the marked
difference between the curves of the pressure in the shaft of the observation well and the formation pressure. In
this investigation, we obtained an analytical expression of delay time making it possible to correctly reproduce the
dependence of formation pressure on time and, accordingly, to raise the reliability of data from the analysis of the
results of formation pulse testing.

**Author:**
A. M. Svalov

**Keywords:**
formation pulse testing, formation parameters, observation well, delay time, formation pressure

**Page:**
1175

**GASDYNAMIC AND ACOUSTIC CHARACTERISTICS OF A SUBSONIC JET-EDGE ROD GENERATOR OF ACOUSTIC RADIATION**

The design of a subsonic jet-edge rod generator of the Hartmann-type oscillator and a scheme of measuring its
acoustic parameters are presented. In this generator, unlike the Hartmann oscillator, not a cylindrical gas jet but
an annular gas jet, moving along the thin cylindrical rod positioned at the axis of the generator nozzle, is used. A
parametric investigation of the gasdynamic and acoustic processes arising in the tubular cavity of such a generator
as a result of the interaction of a subsonic gas jet with this cavity has been performed with consideration for the
nonstationary structure of the gas fl ow in the generator cavity. Recommendations on the choice of the design of a
jet-edge rod generator of acoustic radiation with regard for its practical application are given.

**Author:**
K. N. Volkov, V. N. Emel′yanov, A. V. Efremov, and A. I. Tsvetkov

**Keywords:**
jet-edge rod generator, Hartmann effect, acoustics, subsonic gas jet, generation of sound, amplitudefrequency characteristics

**Page:**
1179

**DYNAMICS OF DETONATION WAVES IN CYLINDRICAL CHANNELS OF VARIABLE CROSS SECTION**

A bubble liquid with a fuel mixture of gases is an explosive in which a detonation wave with an amplitude to
100 atm may appear on exposure to a pressure pulse of about 10–20 atm. The mass calorifi c value of such an
explosive is six or more orders of magnitude lower than that of regular solid, liquid, and gaseous explosive materials.
Furthermore, in fuel liquids containing curtains with vapor-air bubbles, sharp jerks during the transportation may
contribute to the formation of detonation waves leading to emergencies. Therefore, studying detonation waves in
bubble media is of interest from the viewpoint of ensuring explosion safety and from the viewpoint of the problem
of transmission of information in the liquid in the form of waves. In the present work, a numerical study is made of
the propagation of detonation waves in a cylindrical channel with a sudden expansion, which is fi lled with bubble
liquid. Possible scenarios of the dynamics of detonation waves once they pass into the channel′s expanding part
are analyzed. The infl uence of the volume content of the fuel gas and of geometric parameters of the channel on
the propagation and separation of a detonation wave is established. It is shown that there can be two regimes of
propagation of detonation on passage of a detonation wave into the expanding zone: continuous propagation of
detonation and separation of detonation

**Author:**
I. K. Gimaltdinov and E. Yu. Kochanova

**Keywords:**
bubble liquid, waves, two-dimensionality, channel of intricate shape

**Page:**
1191

**DYNAMICS OF THE FLOATING-UP OF A BUBBLE CLUSTE**

Results of experimental investigation of the fl oating-up of a cluster of air bubbles in a viscous fl uid by an optical
method are presented. A setup has been developed for formation of a compact cluster of monodisperse bubbles with
assigned diameter and volume concentration. It is shown that the velocity of the fl oating-up of a cluster of bubbles is
higher than the velocity of the fl oating-up of a single bubble and depends on their number, diameter, and the distance
between the bubbles. An empirical dependence of the resistance coeffi cient of a cluster of air bubbles fl oating up in
glycerin on the Reynolds number has been obtained.

**Author:**
V. A. Arkhipov, S. A. Basalaev, K. G. Perfi l′eva, and A. S. Usanina

**Keywords:**
bubble cluster, resistance coeffi cient, optical method, experimental investigation

**Page:**
1200

**SPACE AND TIME CHARACTERISTICS OF THE VELOCITY AND PRESSURE FIELDS OF THE FLUID FLOW INSIDE A HEMISPHERICAL DIMPLE GENERATOR OF VORTICES**

Results of experimental investigations of the space and time characteristics of the velocity and pressure fi elds of the
fl uid fl ow inside a hemispherical dimple on a plane plate are presented. Features of the formation and development
of vortex strictures in such a dimple and the interaction of these structures with the dimple surface over which
the fl uid fl ows and with the boundary layer of the fl uid fl ow were investigated. Integral, spectral, and correlation
characteristics of the oscillations of the fl ow velocity and the dynamic and near-wall pressures in the dimple have
been obtained. The velocity and direction of transfer of large-scale vortex structures and the frequency of their
rotation inside the dimple, the frequency of throw of these structures out of the dimple, the frequencies of the velocity
and pressure oscillations of the vortex fl ow in the dimple, the frequency of the self-vibrations of the vortex structures
in the shear layer of this fl ow, and the subharmonics and higher-order harmonics of the indicated frequencies were
determined.

**Author:**
V. A. Voskoboinik,a A. A. Voskoboinik,a V. N. Turik, and A. V. Voskoboinik

**Keywords:**
hemispherical dimple, vortex structure, velocity and pressure fi elds

**Page:**
1205

**MATHEMATICAL MODELING OF THE DYNAMICS AND DESTRUCTION OF MAGMATIC BOMBS FORMING DURING THE ERUPTION OF SUBMARINE VOLCANOS**

Theoretical investigations into the dynamics and destruction of magmatic bombs forming during the eruption
of submarine volcanos have been conducted. Continuum mechanics methods have been used for mathematical
modeling of the dynamics and destruction of objects under investigation. In this case, it has been assumed that
the mechanical behavior of the magmatic bomb material obeys the law of linear elastic deformation of a solid
body.

**Author:**
V. A. Andrushchenko, V. A. Goloveshkin, I. V. Murashkin, and N. N. Kholin

**Keywords:**
deep-sea eruption, magmatic bomb, mathematical modeling, linear elasticity, destruction

**Page:**
1221

**HYDRAULIC RESISTANCE OF PIPES WITH UNIFORM CONTINUOUS ROUGHNESS IN THE FORM OF A METRIC THREAD OF VARYING PROFILE AND AN INSERTED TWISTED TAPE**

Results of experimental investigations into the hydraulic resistance of pipes with a metric thread of triangular,
rectangular, and rounded profi les with an asperity height of 0.25 to 1.25 mm, which is cut inside, in air fl ow and
with an inserted twisted tape have been given. The degree of swirling varied in the range from 2.5 to 7. A large data
array has been obtained on the hydraulic resistance of pipes with a metric thread of varying profi le and an inserted
twisted tape. The infl uence of the Reynolds number Re, the degree of swirling, and the relative roughness height on
the hydraulic resistance has been analyzed in the laminar regime of swirling fl ow with macrovortices and in the
turbulent regime with macrovortices. Generalized dependences have been obtained for calculating the coeffi cient
of hydraulic resistance of pipes with a triangular profi le as a function of the degree of swirling and the relative
roughness height at Re = 3000–80,000.

**Author:**
A. V. Zlobin and S. É. Tarasevich

**Keywords:**
hydraulic resistance, metric thread, roughness, fl ow swirl, twisted tape

**Page:**
1226

**VELOCITY FIELD BEHIND A PLATE INSTALLED IN THE INNER REGION OF A TURBULENT BOUNDARY LAYER**

The authors have presented results of an experimental investigation into the velocity fi eld in a turbulent boundary layer behind a thin (0.00045 m) three-dimensional plate. The chord of the plate (streamwise length) was equal to 0.55δ (δ is the boundary-layer thickness), and its width, to 1.0δ. The plate was installed at a zero angle of attack at the center of a water channel at a distance of 0.09δ from the surface. Velocity-fi eld measurements have been performed by the Particle Image Velocimetry method at the Reynolds number Reh = 7750 calculated from the channel half-width and the velocity at the center of the channel. It has been shown that the average velocity increased in a logarithmic region of the boundary layer at a distance of its three thicknesses behind the plate. Longitudinal-velocity pulsations decreased in the buffer region of the boundary layer, but grew in the logarithmic region. Vertical pulsations only decreased to a distance of 0.8δ behind the plate, but downstream they were higher than in an unperturbed boundary layer. The high resolution of the velocity fi eld (50·10^{–6} m) has made it possible to determine shear stresses on the
wall from the velocity gradient in a laminar sublayer. Shear stresses on the surface behind the plate decreased in
the interval where a growth in the average velocity in the logarithmic region was noted. Maximum reduction in the
shear stresses occurred at a distance of 1.8δ and amounted to ~33%. The infl uence of edge effects was manifested in
the less intense reduction on shear stresses in the shorter interval behind the plate.

**Author:**
V. L. Zhdanov I. G. Kukharchuk, and V. I. Terekhov

**Keywords:**
water channel, turbulent boundary layer, shear stresses, PIV

**Page:**
1233

**DETERMINATION OF SEPARATION LOSSES IN THE ROTOR WHEELS OF RADIAL-AXIAL TURBINES**

The use of the basic hydrodynamics equations written for the portion of contracted fl ow at the inlet into the vane
channel of the rotor wheel of the radial-axial turbines at considerable angles of attack and for the portion of
subsequent expansion up to the reattachment of fl ow to the channel walls made it possible to obtain analytical
dependences (characteristics) of losses on the angles of fl ow inlet into the channel. The circulation at the inlet into
the channel exerts a noticeable infl uence on the actual angle of the fl ow that entered into the channel. The obtained
method of calculation of losses also makes it possible to take into account the thickness of the blades, on which the
region of insensitivity to the angle of attack depends, and to determine the corresponding angles of the fl ow entry
that bound this region. The characteristics of the rotor wheels of radial-axial turbines obtained with the use of the
developed method agree well with the known experimental data in a wide range of the angles of attack of fl ow

**Author:**
Yu. A. Grishin and V. N. Bakulin

**Keywords:**
radial-axial turbine, rotor wheel, angle of attack, circulation, separation losses, characteristics

**Page:**
1240

**EXACT SOLUTION FOR ISOTHERMAL FLOW BEHIND A SHOCK WAVE IN A SELF-GRAVITATING GAS OF VARIABLE DENSITY IN AN AZIMUTHAL MAGNETIC FIELD**

Similarity solution for the propagation of a spherical shock wave in a self-gravitating perfect gas with an azimuthal
magnetic fi eld in the case of isothermal fl ow is investigated. The density and azimuthal magnetic fi eld strength in the
ambient medium are assumed to vary and obey power laws. An exact similarity solution obtained using the McVittie
method in the case of isothermal fl ow is reported for the fi rst time. The obtained solutions show that the radial fl uid
velocity, density, pressure, magnetic fi eld strength, and the mass tend to zero as the point of symmetry is approached.
The effects of the changes in the values of the adiabatic exponent γ and the exponent w in the variation of an initial
density are considered in detail. It is shown that the magnetic fi eld strength and mass increase with γ, whereas an
increase in w exerts the reverse effect on these fl ow variables.

**Author:**
G. Nath, Mrityunjoy Dutta, and S. Chaurasia

**Keywords:**
shock waves, similarity solution, gravitation, perfect gas, magnetogasdynamics, exact solution

**Page:**
1247

**PARAMETRIC STUDY OF NATURAL CONVECTION INSIDE A PARTITIONED CAVITY IN THE PRESENCE OF A MAGNETIC FIELD **

Steady laminar natural convection fl ow is studied numerically. The fl ow domain is a differentially heated square
cavity with two partitions that is exposed to a constant horizontal magnetic fi eld. A fi nite volume-based code is
developed by the SIMPLER algorithm. A parametric study is carried out, using different values of the Rayleigh
numbers, partition positions, partition heights, and the Hartmann numbers (from zero to 200). It is found that the
Nusselt number is an increasing function of the Rayleigh number, but a decreasing function of the partition height
and Hartmann number. The position of the partitions affects the streamlines and isotherms, but has a minimal effect
on the mean Nusselt number. In addition, the results show that for low partition heights convective heat transfer
in a cavity is signifi cant, the braking effect of the Lorentz force is more pronounced, and the mean Nusselt number
decreases considerably with increasing magnetic fi eld strength.

**Author:**
M. Pirmohammadi and A. Salehi-Shabestari

**Keywords:**
magnetic fi eld, natural convection, partitioned cavity

**Page:**
1255

**IGNITION OF FOREST COMBUSTIBLE MATERIALS IN A HIGH-TEMPERATURE MEDIUM**

The ignition of typical forest combustible materials (birch leaves) at temperatures characteristic of a crown forest
fi re was investigated. The dependence of the delay time of ignition of a birch-leaf sheet on the temperature of the
surrounding medium was determined, and the mechanism of ignition of leaves in a high-temperature medium was
revealed. A mathematical model of classifi cation of forest combustible materials on the basis of experimental data
on their properties has been developed.

**Author:**
N. V. Baranovskii and V. A. Kirienko

**Keywords:**
ignition, forest combustible material, crown forest fi re, ignition delay time, experiment, high-temperature medium, mathematical model

**Page:**
1266

**THEORETICAL STUDY OF THE COAGULATION OF ASH PARTICLES IN WASTE GASES DURING THE VIBRATING COMBUSTION OF SOLID BIOFUEL**

Consideration has been given to the infl uence of acoustic oscillations on the process of coagulation of soot particles
suspended in the fl ow of combustion products of wood and peat. It has been shown that at the characteristic
parameters of vibrating combustion of these materials, the main factor of growth of the soot particles in the process
of their coagulation is the deposition of small particles well entrained by the gas on large ones that are streamlined
by the gas but are poorly entrained by it.

**Author:**
O. G. Stonik, V. D. Geshele, and S. A. Kovalev

**Keywords:**
vibrating combustion, coagulation, environment

**Page:**
1272

**NUMERICAL SIMULATION OF FRICTION AND HEAT EXCHANGE IN VISCOUS-GRAVITATIONAL FLUID FLOW IN A LOOP THERMOSIPHON**

Numerical simulation of velocity and temperature fi elds with laminar liquid fl ow in a laboratory experimental loop
thermosiphon (natural circulation loop) has been made. A vertical electrically heated pipe served as a lift section.
Calculations were carried out for the case of heating the entire length of the pipe under conditions of constant heat
fl ux density on the wall. The construction of the loop is considered, in which local pressure losses and friction losses
of pressure in the descending pipe are negligibly small in comparison with friction losses of pressure in the ascending
section.
The object of the analysis is the laminar mode of fl ow. In this regime, in the fl ows originating exclusively under
the action of thermogravitational forces, the skin-friction coeffi cient varies in the most sophisticated way along the
fl ow. Using the results of calculation of velocity and temperature fi elds, the longitudinal changes of the skin-friction
coeffi cients and of heat transfer coeffi cients are determined. As a whole, according to the results of 2D numerical
simulation, the skin-friction coeffi cients depend in the main on the change of the velocity gradient on the wall along
the fl ow as a result of practically continuous rearrangement of velocity profi le along the entire heating zone. The
form of the velocity profi les and the degree of their deformation depend in turn on the heat fl ow density on the wall
and on the hydraulic diameter. It is shown that in a single-phase loop of natural circulation (i.e., under conditions
where the liquid moves exclusively under the action of thermogravitational forces) the shear stresses on the wall
change along the heating zone in a complex way, and the skin-friction coeffi cient cannot be described by the simple
dependence of the form ξ = a/Reb for being used in one-dimensional calculations. In all of the calculated regimes,
including the one with the least (of those considered) heat fl ux density on the wall, the Nusselt numbers exceeded
the stabilized values under the conditions of forced fl ow with constant thermophysical properties. With increase
in distance from the entrance into the zone of heating, the Nusselt numbers fi rst decrease monotonically, attain
minimum values, and then begin to increase.

**Author:**
L. A. Sukomel, O. N. Kaban′kov, and V. B. Ankudinov

**Keywords:**
thermosiphon loop, viscous-gravitational fl ow, thermogravitational forces, natural circulation, heat exchange, friction, numerical simulation

**Page:**
1275

**NUMERICAL MODELING OF THE PROCESS OF DRYING BIOMATERIALS AFTER PULSED ELECTRIC FIELD TREATMENT USING A SYSTEM OF TEMPERATURE, MOISTURE, AND PRESSURE EQUATIONS**

A new method is described to determine the kinetic coeffi cients for temperature, moisture, and pressure potentials
based on experimental data of a curve for drying capillary-porous bodies. This method includes numerical modeling
of the problem of calculating the potential of transfer of temperature, moisture, and pressure in the process of drying
based on the procedure of using the fi nite element method in combination with the stepwise method of fi nite differences
for a partial differential system, and an inverse problem formulation in the form of minimizing the residual function
square from the obtained experimental curve of the moisture potential. Based on this method, the updated coeffi cients
of kinetic potentials were found for many materials after pulsed electric fi eld treatment and dependence curves were
restored for temperature and pressure potentials that were not observed in the experiment. The presented approach
is quite useful for the study of the mechanism of a drying process with pulsed electric fi eld pretreatment, and for the
explanation of the occurring effects, and the updated kinetic coeffi cients based on experimental data contribute to
the substantiation of the processes occurring in the object of drying

**Author:**
I. A. Shorstkii, V. S. Kosachev, and E. P. Koshevoi

**Keywords:**
kinetic coeffi cient, inverse problem, heat and mass transfer, drying, potential, numerical modeling, pulsed electric fi eld

**Page:**
1285

**INFLUENCE OF TEMPERATURE AND PRESSURE ON VISCOELASTIC FLUID FLOW IN A PLANE CHANNEL**

The hydrodynamics of a steady-state nonisothermal fl ow of a viscoelastic polymer medium in a plane channel and
heat transfer in it under boundary conditions of the fi rst kind have been investigated. Fluid fl ow with a low Reynolds
number and a high Péclet number was investigated, which made it possible to neglect the gravity and inertial forces,
as well as the longitudinal thermal conductivity of the medium. From the rheological viewpoint, the polymer melt
represents a viscoelastic fl uid; therefore the Phan-Thien–Tanner fl uid model was used as a rheological model of the
fl uid, with viscosity depending on temperature and pressure. A high-viscosity medium was considered; therefore a
dissipation term was included into the equation of the energy of its fl ow. With the use of the indicated rheological
model the velocity profi le of fl uid fl ow was obtained in an explicit form from the equation of fl uid motion. It has been
established that the dependence of the fl uid viscosity on temperature and pressure exerts a noticeable infl uence on
the distribution of the Nusselt number and of bulk temperature of the fl uid along the channel length. It is shown that
account for the temperature dependence of fl uid viscosity leads to a decrease in the role of energy dissipation of its
fl ow in the process of fl ow heating and that, conversely, the dependence of the fl uid viscosity on pressure considerably
enhances the dissipation effect. The problem has been solved numerically by the method of fi nite differences.

**Author:**
A. V. Baranov

**Keywords:**
viscoelastic fl uid, heat exchange, nonisothermal fl ow, dissipation

**Page:**
1296

**EVALUATION OF STRUCTURAL AND MECHANICAL PROPERTIES AND PHASE TRANSITIONS OF GAS CONDENSATE**

The physical and chemical properties of a multicomponent disperse system, viz. gas condensate, are presented.
The structural and mechanical properties of the sample and the phase transitions that occur when the pumping
temperature decreases at startup and steady fl ow regimes have been determined. The values of the activation energy
corresponding to the liquid and quasi-liquid types have been estimated. Optimum temperature values have been
obtained for transportation of gas condensate on the Urengoy–Purpe condensate pipeline section with account for
the fl ow regime

**Author:**
A. A. Boitsova and S. V. Strokin

**Keywords:**
rheology, gas condensate, structural and mechanical properties, phase transition, activation energy of a viscous fl ow, fl ow regime

**Page:**
1303