Breakup of an anomolously viscous liquid film in a centrifugal force field

An equation is obtained for the breakup radius with consideration of tipping moments and Laplacian pressure forces acting on the liquid ridge at the critical point.Notation K, n rhenological constants - density - surface tension - r current cup radius - R maximum cup radius - r_c critical radius for film breakup - ¯r=¯r=r/R dimensionless current radius - ¯r_c=r_c/R dimensionless critical radius - ₀, _c actual and critical film thicknesses - current thickness - R_r ridge radius - h₀ ridge height - h current ridge height - ₀ limiting wetting angle - current angle of tangent to ridge surface - angle between axis of rotation and tangent to cup surface - angular velocity of rotation - q volume liquid flow rate - v₁ and v meridional and tangential velocities - =4vv _lm/r,=4v_m/r dimensionless velocities - M moments of surface and centrifugal forces - M_v moment from velocity head - p_r pressure within ridge - P_vm pressure from velocity head - p_m, p_pm pressures from centrifugal force components tangent and normal to cup surface - deviation range of breakup radius from calculated value - ¯r_max, ¯r_min limiting deviations of breakup radius - _c angle of tangent to curve _c/°₀=f(¯r) at critical point - t random oscillation of ratio _c/_c Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 1, pp. 51–56, July, 1980.     

The variation of the index of refraction in a vortex tube

Three models of a vortex tube (isentropic, isothermal, and isochoric) are considered as optical inhomogeneities. Expressions relating the index of refraction to the coordinate are obtained. The possibility of obtaining Schlieren pictures of such flows is discussed.Notation a speed of sound - n index of refraction - p pressure - r radius - T absolute temperature - v linear velocity - angular deflection of the beam - x adiabatic exponent - density - angular velocity - ¯v, ¯p etc. dimensionless parameters - v₀, ₀ values at the core bounboundary - p, etc. stagnation values - n_c index of refraction at center of vortex     

Radiation of internal waves during vertical motion of a body through a nonuniform liquid

Energy losses to radiation of internal waves during the vertical motion of a point dipole in two-dimensional and three-dimensional cases are computed.Notation _o(z), p_o(z) density and pressure of the ground state - z vertical coordinate - v, p, perturbed velocity, pressure, and density - H(d 1n _o/dz)^–1 characteristic length scale for stratification - N=(gH^–1–g²c _o ^–2 )^1/2 Weisel-Brent frequency - g acceleration of gravity - c_o speed of sound - vertical component of the perturbed velocity - V vector operator - k wave vector - frequency - d vector surface element - W magnitude of the energy losses - (t), (r) (x)(y)(z) Dirac functions - v_o velocity of motion of the source of perturbations - d dipole moment of the doublet - _o,l length dimension parameters - _o intensity of the source Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 4, pp. 619–623, October, 1980.     

Calculation of the characteristics of countercurrent axisymmetric jets

On the basis of analysis and generalization of experimental data, a method is proposed for calculating the parameters of the interaction of two coaxial jets flowing in opposite directions from circular nozzles of different diameters.Notation d, D diameter of the nozzles for the small- and large-diameter jets - density - x, y coordinates - u, U averaged longitudinal component of velocity of the small- and large-diameter jets - L_ax distance between nozzles - q relative impulse - G mass rate of the medium - 2r width of the jet - _i, L_i lengths of the flow zones of the small- and large-diameter jets, where i=1, 2 - range of the jet - 2B distance between corresponding points of lines of maximum velocity of the large-diameter jet Indices 0 initial state - m maximum value - ax value on the axis - d large-diameter jet - y_0.5 distance from the axis to the point at which the velocity is equal to half the velocity on the axis Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 56, No. 5, pp. 725–729, May, 1989.     

Heat and mass transfer of a multiatomic gas in a cylindrical capillary with arbitrary Knudsen numbers

Processes of heat and mass transfer of a multiatomic gas in a cylindrical channel of circular cross section with arbitrary Knudsen numbers are considered on the basis of a model kinetic equation, taking account of the excitation of rotational and vibrational degrees of freedom of the molecules.Notation Kn Knudsen number - f, f^tr total and translational Eucken factors - R_o capillary radius - m molecular mass - k Boltzmann's constant - n, T numerical density and temperature of gas - v_i i-th component of the molecular velocity - h_ij perturbation function - E_i ^(r), e_j ^(v) energy of the i-th rotational and j-th vibrational levels - E_o ^(r), E_o ^(v) equilibrium values of the rotational and vibrational energy - P_i ^(r), P_i ^(v) probability of rotational and vibrational states of energy E _i ^r and E _j ^v - , logarithmic pressure and temperature gradients - T_o mean gas temperature - R rarefaction parameter of gas - C _V ^r , C _V ^v contributions of rotational and vibrational degrees of freedom of the molecule to the specific heat at constant volume - U macroscopic gas velocity - q^(t), q^(r), q^(v) components of the heat flux density due to translational, rotational, and vibrational degrees of freedom of the molecules - P, pressure and dynamic viscosity of the gas - l free path length of molecules - up velocity of Poiseuille flow - u_T rate of thermal creep - cross-sectional area of capillary - I_n, I_q numerical and heat fluxes averaged over the channel cross section - universal index characterizing the thermomolecular pressure difference - ^t, ^r, ^v thermal conductivities due to translational, rotational, and vibrational degrees of freedom of the molecules - mass density of the gas - D_rr, D_vv diffusion coefficients of rotationally and vibrationally excited molecules among the unexcited molecules - Z_r rotational collisional number Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 47, No. 1, pp. 71–82, July, 1984.     

Joule-Thompson effect in a disperse medium

An expression for the Joule-Thompson coefficient of a polydisperse medium subject to throttling is derived in the relaxation approximation of thermodynamics of irreversible processes, with both temperature and velocity relaxation in the phases taken into account.Notation A_qk, A_fk thermal and momentum interphase exchange affinities - _qk, _fk relaxation parameters - T, w temperature and velocity of a phase relaxation in the mixture - density of the mixture - T_o, T_k temperature of the carrier phase and of the k-th group of solid particles - p pressure of the carrier phase - h enthalpy of the mixture - W _o ² /2 specific kinetic energy of the carrier phase - _o, _k volume concentration of the carrier phase and of the k-th group of solid particles - _o, _k true density of the carrier phase and of the k-th group of solid particles - c_v and c_p constant-volume and constant-pressure specific heats of the mixture - c_k specific heat of the k-th group of solid particles - c_v, c_p constant-volume and constant-pressure specific heats, respectively, of the mixture referred to volume - _qk, _fk temperature and velocity relaxation times, respectively, of the k-th group of solid particles - t times - frequency in the Fourier series expansion - differential Joule-Thompson coefficient (adiabatic throttle effect) - N number of groups of particles in the mixture Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 37, No. 5, pp. 825–829, November, 1979.     

Effect of unsteady natural convection on the structure of a liquid flow in a horizontal mixing chamber

Results are presented of a numerical and experimental investigation of the effect of natural convection on the structure of a liquid flow in a horizontal mixing chamber with changes in the temperature of the liquid at the inlet.Notation t_o, t_in initial temperature and temperature at the inlet to the channel - dimensionless temperature - a heat conductivity - kinematic viscosity - coefficient of cubical expansion - density - P pressure - g acceleration due to gravity - d_equ equivalent diameter of porous body - time - v_o mean velocity at the inlet - X and dimensionless vertical and radial coordinates - U and V dimensionless vertical and horizontal components of velocity - H dimensionless height of channel - R radius of inlet to channel - W velocity of liquid - t temperature drop along channel height - Re = v_oR/ Reynolds number - Pe = v_oR/a Peclet number - Fr = v _o ² /g¦t_in–t_o R Froude number - Ho = v_o/R homochroneous number Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 4, pp. 603–610, October, 1980.     

Stirring the heat carrier in a heat exchanger by twisting the stream

A theoretical-experimental study was made of the effective diffusion coefficient in longitudinally streamlined helically twisted tubes with an oval profile, using as the basis a homogenized model of flow and applying the method of diffusion from linear heat sources of finite dimensions.Notation D_t effective diffusion coefficient - u stream velocity - d_e equivalent diameter - S pitch of the tube profile - d largest profile dimension - N_Re,b Reynolds number - q_v volume heat emission - density - x and r coordinates - P pressure - N_Pr,t Prandtl number for turbulent flow - hydraulic drag coefficient - c_p specific heat - T temperature - G mass flow rate of heat carrier - m porosity of the bundle of tubes relative to the heat carrier - L length of the bundle of tubes - r_h radius of heated tubes - r_c radius of the bundle of tubes - k dimensionless effective diffusion Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 40, No. 5, pp. 773–779, May, 1981.     

Coalescence of coarsely dispersed concentrated emulsions with turbulent agitation

The coalescence of drops, larger than microscale vortices, induced by velocity pulsations in concentrated emulsions is examined based on the theory of locally isotropic turbulence.Notation n number of drops per unit volume of emulsion - t time - ₀ and L micro- and macroscales of turbulence - _e, _c, and _d densities of the emulsion, the continuous, and the disperse phases, respectively - interphase tension - W concentration of the dispersed phase - D_T diameter of the agitating turbine - N rate of rotation of the turbine - D and H diameter and height of the mixing volume - d₃₂ averaged volume-surface diameter of the drops - d_m maximum size of drops that are stable with respect to fragmentation - f(W) concentration dependence - ¯v and ¯v(W) averaged magnitudes of the pulsation velocity in finely dispersed and coarsely dispersed emulsions - ¯gq and rates of collisions and coalescence of drops - t_c and _c time and rate of circulation - Re_M Reynolds number of mixer - _e effective viscosity of the emulsion - K_v constant describing the efficiency of collisions drops induced by velocity pulsations - F_v(W) and F_g(W) functional relations between the rate of coalescence and concentration of drops in the inertial and gradient regimes Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 42, No. 1, pp. 27–33, January, 1982.     

Calculation of the characteristics of a mixing CO2 gasdynamic laser with a nozzle unit of honeycomb construction

A mathematical model of a gasdynamic laser with parallel supersonic mixing of the components and its applications to the choice of the geometrical characteristics of the nozzle unit of honeycomb construction are presented.Notation T static temperature - u velocity - P pressure - molecular weight - e_i average number of vibrational quanta - k amplification ratio for a weak signal - I radiation intensity - _i molar fraction of the component in the mixture - ^* displacement thickness - r, d current radius and diameter of the nozzle - angle of inclination of the generating profile of the nozzle and characteristic temperature - h enthalpy - G flow rate - L characteristic length - A/A* expansion ratio of the nozzle - T_w wall temperature - ¯T_w=¯T_w/T_o temperature factor - =E _a /E_* nozzle efficiency - E stored vibrational energy - X=X/(r_a1 + r_a2) dimensionless coordinate Indices 1 radiating gas stream - 2 exciting gas stream - 0 stagnation value - a, *, values at the cut of the nozzle, in its critical cross section, and beyond the compression shock in the cross section of the nozzle cut Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 47, No. 1, pp. 53–59, July, 1984.     

Hydrodynamic focusing of a particle flux

Based on numerical integration of the equations of mechanics of multiphase media, an effect of focusing of a particle flux generated by a source located on the upper wall of a closed vessel has been revealed and investigated.Notation t time - x, y Cartesian coordinates - U₁ (u ₁, v₁),P, ₁ velocity, pressure, and density of the gas - ₂,U ₂ mean density and velocity of the dispersed phase - V _k , r _k velocity and radius vector of a macroparticle - g gravitational acceleration - e(0–1) gravity force vector - dynamic viscosity of the gas - f friction force of particles in the gas - Eu, Re Euler and Reynolds numbers - dimensionless time of the high-rate relaxation of particles Élektrogorsk Research Center, Russia; Institute of Mechanics and Biomechanics, Sofia, Bulgaria; Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 68, No. 3, pp. 355–360, May–June, 1995.     

The engulfment of foreign particles by a freezing interface

The interactions of second-phase particles, liquid droplets or gas bubbles with a solidification front form the basis of various materials synthesis and purification processes and the design of microstructures in cast metal-matrix composites, as well as frost heaving and biological cell interactions. The physical mechanisms of this interaction phenomenon are based upon surface thermodynamic factors, solidification parameters, and fluid dynamic effects such as fluid drag and buoyancy. An overview is presented of the role of various factors which determine the nature as well as the kinetics of foreign particle-solidification front interactions, and the current status and limitations of the various theoretical models of the phenomenon.Nomenclature V Critical velocity for particle engulfment - L Latent heat of fusion - a ₀ Atomic radius - Atomic volume - D ₁ Diffusion coefficient in the liquid - T Temperature - R Particle radius - S Entropy of fusion - _s Density of the solid - ₁ Density of the liquid - _p Density of the particle - k Boltzmann's constant - v Difference in the specific volumes of solid and liquid - G Temperature gradient - h ₀ Critical gap thickness - R _b Radius of surface bump on particle - _sl Surface energy of solid-liquid interface - _pl Surface energy of particle-liquid interface - _sp Surface energy of solid-particle interface - Viscosity of the melt - g Acceleration due to gravity - Density difference between particle and liquid - A Hamaker constant - B A/6 - K _p Thermal conductivity of the particle - K _l Thermal conductivity of the liquid - C Bulk concentration of the liquid - m _l Slope of liquidus line - K _c Partition coefficient - C _p Specific heat of the particle - C ₁ Specific heat of the liquid     

Concentration field of particles ejected into a nonhomogeneous atmosphere by a moving source

We obtain and analyze analytical expressions describing the space-time evolution of the density of solid particles ejected into the atmosphere.Nomenclature f velocity distribution function of particles (depends on position and time) - t time - x position vector referenced to an earth-fixed coordinate system - x₁ and x₂ horizontal plane with x perpendicular to the trajectory plane - x₃ vertical coordinate - x_s position vector of the aircraft (source of particles) - v particle velocity - a particle acceleration due to braking in the atmosphere - g acceleration due to gravity - reciprocal of the velocity relaxation time - H scale height of the atmosphere - F(v) initial velocity distribution function - B(t) defined by (8) - Q(t) mass flow rate of particles - (t, x) density of particles created by a continuous source - o(t, x0, x⁰) density of particles created by an instantaneous point source at the point x^u - angle of the trajectory to the horizontal - V_s velocity of the aircraft - V_a velocity of the exhaust gases at the nozzle exit plane - F initial distribution function for the transverse components of the velocity - R transverse dimension of the exhaust trail - t^* time up to which one can neglect the effect of gravity - v ₃ ^* characteristic value of the vertical velocity - v characteristic value of the initial transverse velocity - determined by (17) - L determined by (43) Translated from Inzhenerno-Fizicheskii Zhurnal,58, No. 4, pp. 561–566, April, 1990.     

A method of joint determination of the effective coefficients of horizontal mixing of large particles and of external heat exchange in an organized fluidized bed

A method is proposed for the joint determination of the coefficients of horizontal particle diffusion and external heat exchange in a stagnant fluidized bed.Notation c_f, c_s, c_n specific heat capacities of gas, particles, and nozzle material, respectively, at constant pressure - D effective coefficient of particle diffusion horizontally (coefficient of horizontal thermal diffusivity of the bed) - d equivalent particle diameter - d_t tube diameter - H₀, H heights of bed at gas filtration velocities u₀ and u, respectively - H_a height of active section - l width of bed - L tube length - l _o width of heating chamber - N number of partition intervals - p=H/H₀ expansion of bed - s_n surface area of nozzle per unit volume of bed - S_h, S_v horizontal and vertical spacings between tubes - t_c, t₀, t_s, t_n, t_w initial temperature of heating chamber, entrance temperature of gas, particle temperature, nozzle temperature, and temperature of apparatus walls, respectively - u₀, u velocity of start of fluidization and gas filtration velocity - y horizontal coordinate - ^*, coefficient of external heat exchange between bed and walls of apparatus and nozzle - ₁, ₁, ₂, ... coefficients in (4) - thickness of tube wall - _b bubble concentration in bed - ₀ porosity of emulsion phase of bed - _n porosity of nozzle - =(t_s – t₀)/(t_c – t₀) dimensionless relative temperature of particles - _n coefficient of thermal conductivity of nozzle material - f, _s, _n densities of gas, particles, and nozzle material, respectively - _be=_s(1 – ₀) (1 – _b) average density of bed - time - _max time of onset of temperature maximum at a selected point of the bed - R =l _o/l Fourier number - Pe = ₁ l ²/D Péclet number - Bi = /_n Biot number Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 41, No. 3, pp. 457–464, September, 1981.     

Temperature field in a plate containing a system of heat sources

The temperature field is determined in a circular plate with a system of thin extrinsic heat sources.Notation T temperature in the plate with the inclusions - r polar radius - polar angle - time - (r,) coefficient of thermal conductivity - (r,) heat transfer coefficient - C(r,) volume heat capacity - W(r,, ) specific intensity of the heat sources - half thickness of the plate - (x) Dirac's delta function - ¯T finite Fourier cosine transform of the temperature - p parameter for this transformation - T Laplace transform of the temperature - s its parameter - Iv(x) Bessel function with imaginary argument of order - K _v (x) the MacDonald function of order - and dimensionless temperature - Po Pomerantz number - Bi Biot number - Fo Fourier's number - dimensionless polar radius - b₁ ^* dimensionless radius of the circle on which the inclusions are placed - R^* dimensionless radius of the plate Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 40, No. 3, pp. 495–502, March, 1981.     

Electronic computer study of separation flow features around a vibrating cylinder

The nonstationary separation flow around a circular cylinder performing harmonic vibrations across-the stream by an incompressible viscous fluid is investigated in a numerical experiment.Notation d, y, yo diameter, transverse deflection, and amplitude of cylinder vibrations - l spacing between vortices - ₁,₂ angular location of the points of separation - U_o unperturbed stream velocity - V_y velocity of transverse cylinder motion - u velocity of vortex motion - f₁ cylinder vibrations frequency - f vortex shedding frequency - t time - kinematic velocity - Sh₁=f₁d/U_o dimensionless cylinder vibrations frequency, the kinematic Strouhal number - Sh=fd/U_o Strouhal number of vortex shedding - Re=dU_o/ Reynold number - =tU_o/d dimensionless time - ¯y=y/d, ¯y_o=yo/d, ¯l=l/d; /U_o, t₁ period of cylinder vibrations - T=t₁U_o/d dimensionless period of vibrations Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 47, No. 1, pp. 41–47, July, 1984.     

Some features of the thermally concentrated convective motion of a hardening binary melt and the impurity distribution

Some features of the thermally concentrated convective motion of a binary melt, hardening in a closed rectangular region with movable boundaries, and the impurity distribution are investigated numerically.Notation x₀ characteristic dimension - x_i (i=1, 2) a dimensional coordinate - l_i (i=1, 2) height and width of the crystallizer cavity - r_i, e_i (i=1, 2) dimensional coordinates of the phase transition in the Ox₁x₂ coordinate system - T, T₀, and T_K current temperature, initial temperature, and melt crystallization temperature - density of the melt - P, P_max, and P_min current pressure, maximum pressure, and minimum pressure in the system - c, c₀ current and initial impurity concentration - e₂ unit vector having the same direction as the direction as the force of gravity - ¯g acceleration due to gravity - coefficient of thermal expansion - diffusion broadening coefficient - ¯u velocity of convective motion - kinematic viscosity - k equilibrium impurity distribution coefficient - t current time - D diffusion coefficient - a thermal diffusivity - T=t₀-t_K initial overheating of the melt - _i=X_i/x₀ (i=1, 2) dimensionless coordinate - _i=L₁/x₀ (i=1, 2) relative height and width of the crystallizer cavity in the coordinate system O _{1 2} - R_i=r_i/x₀, _i=_i/x₀ dimensionless coordinates of the phase-transition boundary in the O _{1 2} coordinate system - =/u₀ dimensionless velocity of convective motion - Gr=¦¯g¦Tx ₀ ³ / ² Grashof hydrodynamic number - Gr_D=¦¯g¦c₀X ₀ ³ /₂ Grashof diffusion number - Fo=Dt x ₀ ² ) dimensionless time, Sm=/D, Schmidt number - Lu=D/a Lewis number Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 1, pp. 118–125, July, 1980.     

Effect of the elastic factor on the hydrodynamic stability of a structurally viscous medium

The effect of relaxation phenomena on the hydrodynamic stability of the plane gradient flow of a structurally viscous medium is investigated using linear theory.Notation _ij stress tensor deviator - U_i components of the velocity vector - x_i coordinates - t time - P pressure - =₀L/^*V plasticity parameter - _o limiting shear stress - andc dimensionless wave number and the perturbation frequency - Re=VL/* Reynolds number - density - F_ij deformation rate tensor Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 35, No. 5, pp. 868–871, November, 1978.     

Distribution of finely dispersed particles with respect to residence time in a vortex chamber

The motion of finely dispersed particles is described statistically with the use of the Fokker-Planck equation. An expression is obtained for the particle distribution function with respect to residence time. Results of the calculation illustrate the dependence of the average particle residence time in the apparatus on the process parameters.Notation A constant - C' parameter characterizing the intensity of random forces - d _p particle diameter, m - K drying rate coefficient - r radial coordinate of the particle, m - R ₀ radius of the outlet orifice, m - R radius of the chamber, m - u,u _in,u _eq instantaneous, initial, and equilibrium moisture contents of the particle, kg/kg - V _r radial gas velocity, m/sec - W tangential velocity of the particle, m/sec - x=r/R dimensionless variable - dynamic viscosity, Pa·sec - _p, _g density of the particles and gas, kg/m³ - time, sec - angular velocity of gas suspension, sec^–1 Academic Scientific Complex A. V. Luikov Heat and Mass Transfer Institute, Academy of Sciences of Belarus, Minsk, Belarus. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 68, No. 4, pp. 552–558, July–August, 1995.     

Heat transfer and the combustion temperature of coke particles in a fluidized bed

The temperature of carbon particles undergoing combustion in a fluidized bed is measured. Heat-transfer laws are ascertained.Notation a diffusivity of air - c heat capacity of air - D diffusion coefficient of oxygen in air - d₀, d initial and running diameters of carbon sphere - d_i diameter of inert particles - k rate constant for carbon monoxide combustion - q calorific value of carbon oxidation to CO₂ - T temperature difference between burning particle and fluidized bed - X, X_n oxygen concentration in the fluidized bed and on the surface of the burning particle - Z, Z_n running concentration of carbon monoxide and concentration on the surface of the burning particle - heat-transfer coefficient between fluidized bed and burning particle - _m maximum heat-transfer coefficient between fluidized bed and a stationary body submerged in the bed - masstransfer coefficient between fluidized bed and burning particle - thermal conductivity of air - kinematic viscosity of air - ₀, gr, ₄ density of oxygen, air, and inert material - relative thickness of burning gas layer - relative thickness of diffusion boundary layer Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 42, No. 1, pp. 21–27, January, 1982.