(Tohis80th Birthd?y)Tm 1 lth of September 1971 was the 80th birthday of Professor Alaksandr Savvich Predvoditeiev, Doctor of Physical and Mathematical
Sciences, Corresponding Member of the USSR
Academy of Sciences, State Prize Winner, Headof Molecular Physics Section at the Lomonosov
Moscow State University.
Aleksandr Savvich Predvoditelev was bornin 1891 in a poor peasant family in the villageof Bukrin, Pronsk District, Ryazan Province.
Before he joined the army Predvoditelev’s
father was a poor farmer, sharing a small land
plot with another male person of his family.
During his military service he was advanced tothe position of an orderly. After his military
service he worked as a surgeon’s assistant invarious towns of Russia. He died of tuberculosis
in 1907 at the age of 34. After her husband’s
death Alek~ndr’s mother worked at a factory
but her scanty earnings could not provide fornormal life of the family.
Aleksandr received his primary education ina village people school, then attended thePronsk secondary school. When his parents
moved to Ryaxan, be entered the Ryazan town
secondary school and upon finishing it, he wasadmitted to the grammar school (gymnasium).
Predvoditelev’s years in the gymnasium were
full of material hardships and, after his father’s
death, his tuition fee was paid for him by the
Ryazan So&e@ to Assist Poor ~~nasium
Students. In the upper classes he had to earn
his living, working as a tutor in rich families.
After successfully finishing the gymnasium,
AIeksandr entered the Erst course of the Physicaland Mathe~~~lDepartment of Moscow
University. In spite of poor financial position
Aleksandr took an active part in scientificresearch even in his first years at the University.
He also spent a lot of time and efforts to extend
his scientific knowledge. On graduating fromthe University, Predvoditelev, as one of thebest students, was left at the Department to
prepare to work as a professor.
From 1915 to 1919 Aleksandr Savvich worked
as a Junior Assistant for practical work in
physics at Moscow University, at the same time
he carried out scientific research on optics andmolecular physics.
In 1919 Aieksandr Sawich passed his master’s(.Ma~ster) examination and, after delivering
two test lectures, obtained the right to teach at
the Physical and Mathematical Department of
the Moscow University. From then on all his
subsequent life has been connected with the
University.
After 1919 he worked as a Senior Assistantfor practical work in physics and delivered anumber of special courses mainly in somebranches of optics and molecular physics. Beginning from 1928 the authorities of the University of the Physicaf and Mathematical Department promoted A. S. Predvoditelev, first, toVice-Chairman of the Subject Commission,then to Assistant of the Dean of the Physicaland Mathematical Department and, finally, tothe director of the Physical Research Instituteof Moscow University where he remained till
1946.
In 1929 A. S. Predvoditelev began to treat
Thermal Physics as a special field of physics, in1930 he was appointed as Professor and Headof the Chair of Thermal Physics which is now
called the Chair of Molecular Physics. He has
been the head of this chair up till now.
Being the Dean of the Physics Department at1729

1730A. V. LUIKOVMoscow University A. S. Predvoditelev carriedout extensive administrative and public activity
from 1937 to 1946. Those were the most difficult
years in the history of Moscow University. The
work of the whole body of the university had
to be rearranged for the needs of strengthening
the defence power of our country.
From 1920 to 1923 A. S. Predvoditelev combined his work at the University with teaching
physics at the Bauman Higher Technical School
where he delivered lectures and guided practicalwork. From 1919 to 1930 he was a senior
physicist (now senior scientific worker) at the
Research Institute of Physics and Biophysics.
From 1928 to 1932 A. S. Predvoditelev worked
at the physical laboratory of the All-Union
Institute of Labour Protection, and in 19299
1937 as a tutor and then as a scientific director
at the Physical and Engineering Laboratory of
the Dzerzhinsky All-Union Institute of ThermalEngineering. In 1938 A. S. Predvoditelev wasinvited to work at the Power Engineering
Institute of the USSR Academy of Sciences,where he arranged and headed the Problem
Laboratory of Combustion Physics up to 1967.
At all of the above research institutions A. S.
Predvoditelev organized either large researchlaboratories or groups of researchers on various
problems of modern physics.
A. S. Predvoditelev is a prominent Soviet
scientist whose activity is not restricted to anarrow sphere of a single scientific or engineering problem. A wide range of scientific interests,
high erudition, many-sided public activity anddevotion to high ideals are characteristic of hislife. The name of A. S. Predvoditelev is widely
known both in our country and abroad, and anumber of his brilliant and profound investigations in various fields of physical science havemade a valuable contribution to the treasure ofworld science.
Predvoditelev obtained his most fundamental
results in the solution of problems on gas andliquid state hydrodynamics, physics of combustion and explosion, physics of solids, physicalacoustics of small and finite amplitudes, physicaland V. A. BUBNOV

chemistry and physics of discontinuous processes.
The results of his experimental and theoreticalstudies are published in 200 monographs andpapers.
A. S. Predvoditelev is a founder and organizerof the scientific school that is active in many
research and educational institutions of thiscountry.
Two features are peculiar to Predvoditelev’s
school. First, all the problems put forward by
the interests of developing national industry
and technology are physically based. Their
practical study and solution are combined withtheoretical analysis of the physicai fundamentals of every problem considered. Investigations of physical and chemical processes inheterogeneous combustion carried out by Predvoditelev and his pupils, awarded with theState Prize in 1950, may serve as an example.
Later on many scientists from Predvoditelev’s
school founded new independent bodies, groups.
trends. Their activity has been progressing
under the influence of the ideas developed at
the laboratories directed by A. S. Predvoditelev.
Predvoditelev is a talented tutor of scientificworkers who always encourages his coworkers
to develop independent scientific views andapproaches to the problems studied. Broadvariety of his scientific ideas, high requirements
to his work and the work of his assistants,
ability to profound scientific analysis and highgeneral erudition have been favouring the
progresses of research groups directed by
Predvoditelev and his numerous pupils.
A. S. Predvoditelev has trained more than 120
scientific researchers. Some 30 of this number
are doctors and professors, two are corresponding members of the Academy of Pedagogical
Sciences of the USSR, one is an Academicianof the Byelorussian Academy of Sciences, one isa Corresponding
Member of the UkranianAcademy.
A. S. Predvoditelev has been paying serious
attention to the problems of history andmethodologyof physics. He has publishedALEKSANDRSAVVICHnumero~essays and papers concer~ng therole of Russian scientists in development offundamental ideas of modern physics. He was
the initiator of publication of “Essays on Historyof Physics in Russia” (1949) and contributed
some papers to the book. He is the Editor of
two volumes of “Development of Physics in
Russia” published in 1970.
A. S. Predvoditelev has greatly contributed
to publication of fundamental works on historyof physics by P. S. Kudryavtsev, B. I. Spassky
and others. He has taken great efforts to acquaintthe world scientific commute
with the achievements of Russian and Soviet scientists.
This is a general description of the life andwork of A. S. Predvoditelev who has walked
the way from a poor peasant to a prominent
physicist.
The Government has highly estimated hisscientific, pedagogical and organizing activity
and awarded him the Lenin Order, four Ordersof the Red Banner of Labour, the Order of Red
Star and numerous medals.

1731

PREDVODITELEV

1925
12. Gber der Wirklung der Schichtdicke auf den Photoelektrischen Effekt in Farbstoffen, Z. P/tys. 31, 718.
13. Absorption des Lichtes durch Ammoniak, Z. Phys. 6.
14. Zur Kinetik Photochemischer Prozesse in Farbstoffen.
Z. Phys. 32, 3.
15. Zur Theorie des Abklingens der Fluoreszenz. Z. Phys.
32, 11/12.
16. ifber die Snezifische W&me des Wasserstoffmolektil.
2. Phys. M-, 213.

17. Faber die Abhangigkeit des Absorptionskoeftizienten
van der Dicke der Absorbierten Farbschicht. 2. Phvs.
35. 1.
1926
Ermildung. Z.
18. Zur Frage nach der Phot~lekt~schen
Phys. 35, 10.
19. brdie Abhiingigkeit der Fliissigkeitsdichten von derTemperatur, Z. Phys. 36,7.
20. Der Einfluss des Absorbierten Gases auf die Griisse
des Photoelektrischen Effektes, Z. Phys. 38.4/S.
21. Zur Frage der Abhlngigkeit der Dichte von der Temperatur, Z. Phys. 40, 6.
1927

22. gber den Einfluss des Kristallwassers auf den Photoelektrischen Effekt in den Kristallhydraten (Teil I).
Z. Phys. 42, 1.
23. uber den Einfluss des Kristallwassers auf den Photoelektrischen Effekt in den K~s~llhydrat~(Teil II).
Z. Phys. 44,3.
24. On chemical reaction kinetics of solid and gaseouscomponents that result in formation of compounds.
Zh. Prikl. Fiz. 4. 3.

LIST OF THE MAIN SCIENTIFIC WORKS BY
A. S. PREDVDDMELEV
1918
1. Light dissipation in oblique media, Izv. Inst. Siofz.
2. L’indice d’absorption des millieux troubses. Arch.
Phys. SC.
1920
3. Necessary physical assumption forming the basis forderivation of the radiation law, Wsp. Fiz. Nauk II.
vyp. I.
4. Reaction rate kinetic theory, Usp. Fiz. Nauk 2, vyp. I.
5. On the theory of gas reactions, iJsp. Fiz. Nauk 2. vyp. I.
6. On thermal expansion of crystal solids, Vsp. Fiz. Nauk2, vyp. I.
7. Electronic theory of metals, Usp. Fiz. Nauk 2, 1.
8. Electric field and gas pressure effects on spectral lines.
Usp. Fiz. Nauk 2, 1.
9. New theory of opticai series, Usp. Fiz. Nauk 2. 1.
1923

10. One remarkable quantization case, Usp. Fir. Nauk 3. I.
11. Ammonia synthesis by the points of slow electrons.
Usp. Fiz. Nauk 3. 1.

1928
25. uber die Gesetze der Molektilstosse mit einer Oberfliiche mit welcher das Gas Reagiert, Z. Phys. 46, 5/6.
26. Das Maxwelliche Relaxationsgesetz und die Innere
Reibung der Fliissigkeiten, Z. Phys. 49, 3/4.
27. One possible method of measuring heating of equipment for chemical compounds which may dissociate invacuum, Zh. Prikl. Fiz. 5 (additional issue).
28. What is Light Recortling? Nauka i Tekhn., Izd. Mosk.
Rab.
29. uber die Absoluten Ges~hwindigkeiten der H,OMolekiile, welche bei der Dehydration von Kristallhydraten Herausfliegen, Z. Phys. 51, l/Z.
30. Zur Kinetik Chemischer Reaktionen zwischen einer
Festen und einer Gasformigen Komponente, die zur
Bildung Komplexer Verbindungen Fiihren. Z. Phys..
Chem. 132, l/2.
1929

31. Berichtigung zur Arbeit iiber die Absoluten Geschwindigkeiten der H,O-Molektile, welche bei der Dehydration von Kristallhydraten
Herausfliegen. Z.
Phys., Chem. Bd. 54, l/2.

of hydrogen atom
32. One geometric interpretation
structure following Rutherford-Bohr,
2h.R.F.Hz.U.
61, 5.

1732

A. V. LUIKOV

33. On possibility of relation between Bohr postulates andde Broglie-Schroedinger
equation. Zh.P.F.Kh.0.
61. 5.
34 On measuring heat of dehydrationin vacuum by torsion balance (with co-author L. SELIVANOV).Zh. Prikl.
Fiz. 5. 3-4.
1933
35. On the theory of fluids and viscosity of associatedmolecules, Zh. Eksp. Teor. Fiz. 3. 3.
36. Viscosity of liquids and gases from the viewpoint ofcyclic motion. Zh. Eksp. Tear. Fiz. 3. 3.
1934
37. Stability of mechanical motions in atomic physics. Zh.
Eksp. Teor. Fiz. 4, 1.
Zh. Eksp. Tear. Fiz. 4, 1.
38. Heat of strong electrolytes,
39. Relation between thermal conductivity,
heat capacityand viscosity of liquid bodies, Zh. Eksp. Teor. Fiz. 4, 1.
40. Thermal
conductivityof solid heat insulators.
Zh.
Eksp. Teor. Fiz. 4, 8.
41. Thermal conductivityof electroconducting
bodies, Zh.
Eksp. Teor. Fiz. 4, 8.
1935
42. A contribution
to the theory of the Bunsen flame.
Techn. Phys. USSR 11,4.
43. A contribution
to the problem of the decay of a gas
jet discharging
from a pipe, Techn. Phys. USSR 11, 4.
44. Theoryof gas reactions
in high-frequency
electric
discharge, Zh. Fiz. Khim. 6, 4.
45. Jet attenuation
at gas flow out of tube (with co-author
E. V. STUPOCHENKO), Zh. Tekh. Fiz. 5, 8.
46. On the Bunsen flame theory, Zh. Tekh. Fiz. 5, 8.
47. Solid admixture
effect on flame propagation
velocityin fuel gas mixtures (with co-author
S. I. GRIBKOVA).
Zh. Tekh. Fir. 7, 18-19.
1940
48. Burning of coal channel walls at forced oxygen diffusion (with co-author 0. A. PUKHANOVA), Zh. Tekh. Fiz.
10, 13.
49. Burning of coal particle in a gas flow, Zh. Tekh. Fiz.
10, 16.

and V. A. BUBNOVof statisticalsystems, Vesm,k54. Possible classification
MGU. 7.
55. The Arrhenius law in chemical kinetics of gas reactions.
Dokl. Akad. Nauk SSSR, Otd. Khim.. 51. 7.
56. Gasification
at coal burning in a bed. Izv. Akad. Nauk
SSSR, ONTI, 10.
57. Quasi-steadystate of chemical gas systems. Ve.ytnik
MGCJ, IO.
1948
in statistical systems, Vestnik MGU. 4.
58. Fluctuations
59. Theory of acoustic dispersion,
Vestnik MGU. 5.
60. On statistical assembly of material points of variablemass, Vestnik MGU, 6.
61. On molecular-kinetic
foundationsof hydrodynamic
equations, Izv. Akad. Nauk SSSR, ONTI, 4.
62. Some invariant
quantities
in the theory of thermal
conductivity and viscosity of fluids, Zh. Fiz. Khim. 22. 3.
1949
L. N. KHI?‘RIN. 0. A.
63. Carbon Burning (with co-authors
TSUKHANOV, KH. I. KOLQD~SEV and M. K. GRODZOVSKY). Izd. Akad. Nauk SSSR, Moscow-Leningrad.
64. Some considerations
on the operator method in wace
mechanics,
Vestnik MGU, 2.
65. On thermal motion in condensed
media and theirstate equations,
Vestnik MGU. 3.
66. Modificationof Zhukovsky’s
equation for disturbed
motions of statistical systems, Vestnik MGU, 5.
67. Some applicationsof Zhukovsky’s
equationfor disturbed motions of statistical systems, Vestnik MGU. 6.
68. Some remarks
on natureof elementary
particles.
Vestnik MGU, 8.
1950
69. On molecularheat transfer in liquids, Dokl. Akad.
Nauk. Otd. Fiz. Khim. 72. 2.
70. Clausius theorem of mean ergal and stable statisticalsystems, Vestnik MGU. Otd. Fiz.. 6.

1953

Zh. Tekh. Fiz. 11, 10.

71. The state of art in the theory of homogeneous-heterogeneous combustion.
Izv. Akad. Nauk SSSR. Otd.
?ekh. Nauk. 5.

1945
51. Experience of constructing
kinetics of sulphur dioxide
oxidationin electric discharge.
Uch. Zap. MGU 75.
kn. 2. ch. 2.

1956
72. Thermal Conducrivity and Viscosily oj’ Liquids andCompressed Gases. Collected Papers, volume dedicatedto P. P. LAZAREV, Izd. Akad. Nauk SSSR, Moscow.

50. Theoryof coal channel

52. Physical problems
Inst. 9-10.

1941
burn-out,

1946
of heat engineering,Izv. Vses. Tepl.

1947
53. Theory of heterogeneous
combustion
and mechanismof solid fuel burning in a bed (with co-author
Kh. I.
KOLODTSEV). Proc. World Power Engng Conference,
Hague Econom. Congr.. Gosenergoizd..
Moscow.

1957
Functions oJ Air (with
73. Tables of Thermodynamic
co-authors
E. V. SXJPOCHENKO, I. P. STAKHANOV.
E. V. SAMUILOV, A. S. PLESHANOV and I. B. ROZHDESIVENSKY).Izd. Akad. Nauk SSSR, Moscow.
74. Generalpropertiesof discontinuous
processes
andacoustic effects in nonhomogeneous
solids, in Application of Ultra-acoustics to Substance Investigation. Izd.
MOPI, vyp. 4.

ALEKSANDRSAVVICH1958
15. One method of acoustic effect description. Appijcatjonof Uhraacoustics
to Subsrance Investigation.
Izd.

MOPI, vyp. 6.
76. On the so-called Rao rule, ibid., 7.
Il. On the possible method of describing structural nonuniformities in liquids, ibid.

78. Theoretical examination of vibratory movement of
the flame front in closed vessel, Trans. 7th International Symposium on Combustion and Explosion.
19. On automodelling processes in chemically active
media, ibid.
80. On spin detonation, ibid.
1959
81. On the theory of adsorption wave. DokI. Akad. Nauk.
Otd. Fiz. k%im 127, No. 3; 121, No. 4.
of Combustion
zone as Hydrodyn~~cInhomogeneity. Collected Papers, Volume Dedicated to

82. Movement

G. M. KRWIZHANOVSKY.
83. Hydrodynamic inhomogeneities in combustion andexplosion theory, GasdynamicsandCombustion
Physics.
Izd. Akad. Nat& SSSR, Moscow.
84. Dispersion of acoustic waves in raretied gases (Paperl), Application of Ultraacoustics to Substance Investigation. Izd. MOPI, vyp. VIII.

1733

PREDVODITELEV

94. On condensed media state equation (TheoreticalPortion), In&. Fiz. Zfr. 5, No. 8.
95. On condensed media state equation (Experimental
Portion), Inzh. Fiz. Zh. Zh. 5, No. 11.
96. On conservation laws at the shock-wave front, PhysicalGas Dynamics,
Heat Transfer and Gas Thermodynamics at High Temperatures, Izd. Akad. Nauk
SSSR, Moscow.
97. On one inconsistency of the Zjuge Theory on regularmotion of shock breaks with spherical and cylindricalsymmetry, ibid.
98. On molecular-kinetic consideration of the ChapmanZjuge first rule, ibid.
1963
99. On critical phenomena of liquids and gases. Inzh. Fiz.
Zh. VI, 4, 123.
100. The teaching of space and time, modem science.
History and Methodology of Natural Sciences. Izd.
MGU.
101. On helium state equations, Inzh. Fiz. Zh. VI, No. 6.54.
102. On kinetics of statistical motion, Inzh. Fiz. Zh. VI, No.
9, 87.
103. On state equations of carbon monoxide and dioxide,
Inzh. Fiz. Zh. VI, No. 12, 100.
1964
On state equations of xenone and methane, Inzh. Fiz.
Zh. VII, Nb. 1, 93.

85. Dispersion of acoustic waves in rarefied gases (Paper2), ibid.
86. Combustion of spherical particles, Proc. Higher School
Conf. on combustion. Izd. Odess. Univ. lSO,6.
87. Front evaporation and sublimation of sphericalparticles, ibid, 150, 7.
88. Velocity of combustion front propagation in two-phasemixtures, ibid., 150, 7.

105. On the general theory of wave processes, Collected
Papers, Physical Gas Dy~rn~~ and Gas Properties at
High Temperatures. Izd. Akad. Nauk SSSR.

1961
89. On chemical reaction rates in turbulent flows. Inzh.
Fiz. Zh. 4, 1.
90. On thermal motion type in liquids, Inzh. Fiz. Zh. 4,

1967
107. To the theory of two-phase mixtures, Collected papers.
Gas Properties at High Temperatures, ENIN.
108. On conditions of ignition in fuel gas mixtures, ibid.

6; 7; 8.
91. On transverseand longitudinal wave dispersion inrelaxation media, Vestnik Mosk. Univ., ser. III, No. 3.
92. On acoustic wave dispersion in relaxation media, Vestnik Mosk. Gos. Univ., ser III, No. 4.
93. On aer~ynami~s of rarefied gases and problems ofheat transfer. int. f. Heat Mass Transfer 3.

1965

106. Mathematical computation
tory and Methodology

and our knowledge. HisSciences. Izd. MGU.

ofNatural

1970

109 Human Being and Reality, History and Methodology
of Natural Sciences. Izd. MGU.
110. On physical basis of h~r~lic
differential equations,Uch. Zap. Krasaodarsk. Politechn. Inst.
111. Theory of heat and Riemannian manifolds. Collected
Papers, Heat and Mass Transfer Problems, Izd.
Energia.

1734

ON A. S. PREDVODITELEV’SINVESTIGATIONSINTO HYDRODYNAMICS
AND HEAT TRANSFER
A. V. LUIKOVHeat and Mass Transfer

Institute

AND V. A. BUBNOVof the BSSR Academy

Hydrodynamic
motions represent a wide
range of natural processes. Therefore, hydrodynamic methods which allow transition frommicroscopic description of the natural processes
to macroscopic one are widely used in various
fields of physics.
Different degree of accuracy of the descriptionof the above transition corresponds to different
forms of hydrodynamic continuum equations.
The ideal liquid model described by hydrodynamic equations in the Euler form proved to
be valid for a narrow range of the naturalprocesses. Presumably, the incompleteness ofthese equations was originally expressed in thed’Alembert paradox.
The discovery of internal friction in liquids
and gases allowed the hydrodynamic equationsin the Euler form to be supplemented andspecified. For this we are indebted to Navier,Poisson, Saint-Venant and Stokes. The tirst of
the above scientists derived equations forviscous fluid from the molecular-kinetic theory
based on various considerations of molecularinteraction. However, Navier did it as far backas 1822, while Poisson seven years later. Alternative method for deriving the above equations
was originally suggested by Saint-Venant in1843 and Stokes in 1845. Now this method isreferred to as phenomenological.
Stokes has succeeded to prove that theconditions of adhesion exist at the liquid-solid
wall boundary and thus he was first to formulate
the boundary conditions and put an end to theargument lasting from Euler’s and Bernoulli’s
times.
At present the hydrodynamic equations of
this form are named the Navier-Stokes equations and constitute the basis of classical hydrodynamics.

of Sciences,

Minsk,USSR

In 1866 the memoirs On Dynamic Gus Theory
by CI. K. Maxwell appeared concerning themethod which may be used to obtain all heatand mass transfer equations.
In this paper Maxwell discovered the molecular-kinetic structure of Navier-Stokes equations, in particular, he has shown that in the
Navier-Stokes continuum the visible (macroscopic) and thermal (microscopic) motions
coexist.
Hagen and Osborne Reynolds were the first
who revealed incompleteness of the NavierStokes equations while studying water motion
along the tubes of various diameters. Reynolds
has shown that at certain conditions defined bya dimensionless number later named after him,
the flow pattern begins changing sharply andexperimental
data no longer agree with
Poiseuille solution. Later this phenomenon wasreferred to as turbulence and defined by somecritical Reynolds number.
For a long time the hydrodynamic NavierStokes equations did not rise any doubts under
the so-called laminar flow conditions below thecritical Reynolds numbers. This was partly
accounted for by huge mathematical difficulties
involved in general solutions of the above
equations.
However, reported developments in mathematical physics and aircrati engineering and newmaterials revealed inadequacy of the NavierStokes equations even in the laminar flow
region. The necessity arose to refine the above
equations. This problem was most sharply
stated by the American scientist Truesdell in“Modem Crisis in the Kinetic Gas Theory”,
1958.
Professor Predvoditelev’s hydrodynamic investigations are connected with this problem.

ALEKSANDRSAVVTCH

1. Fundamental equation of kinetic theory

Nowadays the hydrodynamic equations arederived from solutions of the Boltzmann integrodifferential equation by the Chapman-Enskog
method. The idea of this method was known toBoltzmann; it was presented in his lectures on
the gas theory. However, in the greater portionof the book studying transfer processes Boltzmann deals with Maxwell’s method.
He wrote :*“Up till now we followed the brilliant methodof Maxwell which has been also used byKirchholf and others. The essence of thismethod is that it not at all connected withcalculation of the function f(x, y, z, 5, q, c, t)
defining the velocity distribution. There exists
an alternative method: it just proceeds from
calculation of this function. Though this method
did not attract any attention, nevertheless, I
would like to touch upon it in some words as
we shall need this very function f when calculating entropy”.
In his hydrodynamic
investigations A. S.
Predvoditelev has never used the Boltzmann
equation. He expressed his view on it in thepaper submitted to the 2nd All-Union Heat andMass Transfer Conference, Minsk, 1964. By
that time a number of experimental data had
been accumulated which came in conflict withthe solutions of Boltzmann’s integro-differential
equation.
In this paper Predvoditelev supposed that if
the methods of solving the Boltzmann equation
are quite perfectly right then the contradictions
spoken of may be attributed to incompletenessof this equation. Further, he analyzed physicalsignificance of the Boltzmann equation as
follows :“As is known, two operations at the molecule
distribution function with respect to heat velocities (f) are equated. The first operation is used
to calculate change of the function with respect* L. Boltzmann, Lectures on Gas Theory, Translationfrom German, ed. B. I. Davydov. Gostekhizdat,
Moscow,
pp. 225 (1956).

PREDVODITELEV1735

to time, coordinates and velocities per unit time.
This change may be written down as follows :
Df
_=Dt

af
af
fug”-+
at

af
ay

af
wau+xzaf+yaf+zaf
auaw
here the equalities are implied :au-=x!!=,w=z

at

’ at

?at

.

The second operation is used to calculate thechange (per time unit) of the same function due
to molecular collision. If velocity componentsof the first type of molecules are designated bytl, qr, cl and those of the second type of molecules before collision by t2, q2, c2, then theprobability measure for their approaching eachother may be taken equal to fifi. Subscripts
indicate the components of heat velocities ofwhich the distribution function must be taken.
Upon collision, velocity components of thefirst and second type will take the values

and the productf; f; will serve as a probability
measure of increasing distances between them.
Thus each collision of one molecular group withanother is characterized by the difference
f;f;

-f1f2

= Sfifi

Summation of these differences over the whole
phase volume gives the change off function pertime unit due to molecular collision. Thus we
have
Df
-=
Dt

..

”

Sfl fi Vb db dw, d$.

Here the product Vb db do, d4 denotes aunit phase volume where I/ is the relative
velocity of colliding molecules; b is the shortest
distance between straight-line trajectories ofmolecules; 4 is the angle between the plane oftrajectories of relative motion of the first type of1736

A. V. LUIKOV

molecules and some uniform plane crossing fi:do, is the product dc, dq, d[,. The question
arises whether the operations discussed areequal. The first operation may give continuous
values of the phase space coordinates. The same
is not evident for the second operation. I am
inclined to believe that the second operation
should lead to discontinuous values, as thevariation SfiJZ may be not only infinitesimal.
This implies that equating the operations discussed to each other is principally wrong. As is
known, these are equallized without thinking
over, i.e. it is taken for granted that the variation
SfifZ is a continuous function of the phasespace coordinates. I believe that validity of such
equalizing will increase with the density of gas”.
The criticism however was not substantiatedenough. At this time Predvoditelev completed
the generalization of the Navier-Stokes equations and these generalized equations did not
follow from the solutions of Boltzmann’s equation. Moreover, in this paper experimental
check of these equations was made in the caseof acoustic wave dispersion. In particular, thefollowing formula was obtained for the soundabsorption coefticient in a rarefied gas based on
Predvoditelev’s equations discussed(0.7~ + 0.23),/r@= (r + 0.575)Z~(r + 0,575)’
Here r is the known dimensionless quantity inacoustics which is the ratio of pressure to the
product of viscosity by frequency.
Further, it appeared that Predvoditelev’s
formula agrees better with Greenspan’s observations of acoustic wave dispersion in xenon andkrypton compared to other available formulae.
Returning again to Boltzmann’s collision
integral, it should be noted that discontinuityof the variation Sfifi emphasized by Predvoditelev is caused by discontinuous behaviour ofthe distribution function at the moment ofmolecular collision,
Boltzmann realized that and in his last years
in the discussion of F. Klein and A. Hefler’sand V. A. BUBNOV

paper at the Colloquium of the Vienna Philosophical Society he expressed very curious ideas
concerning statistical description of physicaleffects. He considered that in a statistical system
of material points the trajectory of position 01
the points and trajectories of their velocity arecontinuous functions that had no derivatives atany point. This means that for a given trajectoryof position, calculation of velocity is impossible.
and positions of a material point of a statisticalsystem cannot be determined from the given
trajectory. Thus, for any physical system wecannot but have two sets: a set of positions anda set of velocities. Moreover, we should choosesuch a calculation method that would allow
analytical relationships to be found which
would not violate the above principle ofBoltzmann.
In his recent work “The Science of Heat andthe Riemannian
Manifolds”
Predvoditelev
further developed the idea of the velocity
distribution function as continuous and differentiable over the free molecular path length atturning points of their trajectories. First of all,in this work he developed the following representation of a statistical system :“We shall concentrate on a certain moleculeof gas. In space we shall draw sections of themolecule path between two collisions within avery long time. As a result our limited space will
be filled almost completely with molecule
trajectory. Also turning points of the molecule
after its collision with others will till the space
almost completely. If we draw a curve overthese points, it will form a tangle. The curve will
be continuous and nondifferentiable and willalmost completely coincide with the trajectoryof molecule position plotted. This tangled curve
will be a locus of molecular velocity deviations
from a mean velocity. Density of points on thisline is nonuniform. A certain heat velocity of amolecule with the same direction as that of the
trajectory will correspond to each point.”
Here Predvoditelev is trying to prove that
this physical image of a statistical system does
not contradict Maxwell’s conception. With this

ALEKSANDRSAVVICH

aim we shall take some continuo~ and differentiable function with heat velocity of a molecule as an argument. We assume that for eachpoint of the curve drawn through the turning
points, a single-valued correspondence of thevalues of the above function may be found, thefunction being denoted byf.
As the turning point is the place of collisionof two molecules, another trajectory passes
through it which corresponds to another colliding molecule. The distribution function for heatvelocities of the points of this trajectory isdesignated by F.
If we denote the value of the function f afterturn of the first molecule trajectory by fi, thenthe function jump isfi -f:
According to Dirichlet, the value of the samefunction at the break point may be assumed
equal tofi +f
2 .
If gas is in a steady state, then the relation holds

2(fl -f) =const

cfl +f1.

1737

PREDVODITELEV

Pr~v~itelev,
however, did not restrict himself by the Maxwell equation (1.2). In the said
work he tries to obtain a special method fordescription of statistical effects without theconcept of velocity distribution function. With
this aim, we preliminarily suppose that crosspoint of intersection of colliding molecules
forms a continuous surface in the space, noproperty of the system being possible to change
continuously in transition through this surface.
Existence of discontinuity surfaces should not
be necessarily understood as collision of molecules. It may be supposed that in these sites
sharp jumpwise change of some properties ofthe substance occurs and the cause of suchchanges may be considered the behaviour ofthermal motion in the substance.
The said discontinuo~ surface is denoted by
S. Let partial derivative of this surface withrespect to time be equal to the total energy withthe reverse sign; partial derivatives with respect
to the coordinates are equal to the corresponding
momenta. These properties of the surface allow
the Helmholtz equation to be written for it

as

i

at+2m

c

as2

8%

u

=

*

(1.3)

This equation implies that molecular collisionoccurs follow~g the energy conservation law.
In a similar way it is possible to find the relation
The function which determines the state of afor the function F also:
statistical system before and after collision isdenoted by ‘I+,, x2, xjr t). Let it possess theconsifollowing properties: the first derivatives with(Fl + F).
respect to coordinates are equal to momenta ;Here we see that the minus sign appears since
the first derivative with respect to time is equalincrease of the function f corresponds toto a certain value that has the energy dimension.
decrease of the function F.
Such a function of the state should undergo
Further we can write
discontinuitiesof all the first and second
derivatives in transition through the surface S.
fiJ--~,(1.1) Besides, the very function that in a sense deterf1+f
Fl+F’
mines the structure of the assembly should also
From this equality the famous functional Maxundergo discontinuity as molecular collisions
well equation may be easily obtained
should be accompany
by disturbance ofstructural properties of statistical assembly as
well.

2(F1
-’ =

A. LtJIKOV

1738

and V. A. BUBNOVGeneralizing the conditions of commensurability of Hugoniot-Hadamard
for the cases
when discontinuities are observed that change
in time and space, Predvoditelev showed thatfor the case considered the following commensurability conditions should be valid

~H=A.
J

Here H is the first differential invariant of thefunction S; A,, I,, I2 are arbitrary factors referred to as discontinuity parameters of thecorresponding order.
Substitution of equation (1.4) into equation(1.3) yields

equation. It becomes completely identical withthe latter if we assume
A=2;;
Thus, Predvoditelev has reduced searching forthe distribution function of a statistical system
to searching for the integral in equation (1.6).
Predvoditelev’s method just discussed reveals
quite a new approach to searching for velocity distribution functions and functions that
determine structural peculiarities of statisticalsystems.
2. On Navier-Stokes equation generalization
N. P. Kasterin’s research on vortex motions

became a starting point for Predvoditelev in his
attempt to generalize Navier-Stokes equationr.
Kasterin was the first in the Soviet Union who
discussed the necessity to revise Euler equations
with reference to vortex flows. He was probably
the first who discovered that vortex motion
initiation is connected with discontinuities
appearing in hydrodynamic velocity field.
With assumption 2, = 69, the above equality is
However, in derivation of the Euler equations
symmetrical with respect to operation 6:
smooth change of velocities within physicallysmall volume is postulated. That is the reason
why the Helmholtz theorems based on the
Euler equations disagree with experimental
observations of vortices.
orThis view of Kastering contradicted thegenerally adopted opinion that vortices arising
USY.
(1.5) may be attributed to viscous force effect.
In 1937 Kasterin made an attempt to include
In the equality obtained all factors at operadiscontinuous change of velocities in hydrotion 6 are continuous. Therefore. a reverse
dynamic equations for the ideal liquid. For
operation may be applied to equality (1.5). Then
example, for potential flows Kasterin’s equation
is of the form
AZ+
UY + F(x,, x2. x3. t).
av- - grad:= - bgradp.
(2.1)
atHere arbitrary and continuous function F(x. t)
may be assumed equal to zero. Then
The only difference from the Euler equation is adifferent sign before the gradient operation in
Ag+&Y=
U!P.
(1.6) the left-hand side of equation (2.1).
In June 1947 in his joint researches with
Equation (1.61 is the same as the Schrijdinger
A. K. Timiryasev and T. M. Sviridov on condi-

Adg+g6A2*=$,r =ALEKSANDRSAVVICHPREDVODITELEV

1739

tions for sandstorms to arise in the laboratory,
Kasterin experimentallyconfirmed the saidchange of the sign.
A. S. Predvoditelev
decided to create amolecular-kinetic
basis for the sign change.
With this aim he made use of Maxwell’s method.
If Q is the qualitative expression of some ofthe properties transferred by moving molecules,
then Maxwell’s equation assumes the form

Maxwell identified visible velocity of moving
gas with mean velocity of thermal molecularmotion. Thus, he assumed that within thedomain of validity of Navier-Stokes equations
visible and thermal motions coexist and there isno gradient of hydrodynamic velocities withina physically small volume.
Such a flow structure is not always the case.
At rather large velocities such a motion incontinuum may be disturbed near the wall or
dwhen vortices are formed. This hypothesis+&<QM
becomes doubtful in the cases also when flow
dimensions are of the same order as mean+$QN)+$
CQN)=gN.
(2.2)
distances between molecules that occurs inmotion of rarefied gases.
Here
In 1948 Predvoditelev paid attention to thisweak point of Maxwell’s hypothesis and formulated some new conditions for transition tocontinuum.
is the collision integral
If subscripts 1 and 2 indicate transfer rates ofthe first and second molecules, respectively,$ N = (Q’ - Q) Vb db d$& fi do, do,.
then following Maxwell, we obtain the following
formula for the stress tensor in a very general
(2.3)
case
The remaining notations are those generally
pr2=p+$-p(U:-u:)
adopted.
2
However, to obtain hydrodynamic NavierStokes equations from equation (2.2) Maxwell
used two hypotheses.
Hypothesis A. Molecular motion is caused by
V.4)
central forces. Molecular interaction force is arepulsive force which is inversely proportional
to the fifth power of the distance between4+
j-p-442~2%VI)
interacting molecules.
J
2
Hypothesis B. Transfer rates of two colliding
where constants A, and A, are defined by themolecules are equal within a physically smallinteraction law.
volume over which averaging is carried out.
Definition of hydrodynamic velocity must be
Maxwell applied the above interaction law torelined. For this we assume that vector V withsolution of a number of molecular-kinetic
projections u, v, w is superimposed on a certain
problems, and in 1905 Langevin confirmed his
physically small volume with molecules involved
calculations starting from any law of molecularin heat motion of which transfer velocities areinteraction forces.
different. Then projection of the material point
The second hypothesis of Maxwell is relatedwith respect to a fixed coordinate system may
to the problem on determination of hydrobe expressed as
dynamic velocity in a moving gas. Within thedomain of validity of the said hypothesis,
V, = u + 5, r/; = u + q, v, = w + i. (2.5)

1A. V. LUIKOV

1740

and V. A. BUBNOV

Different transfer velocities of all the molecules within a physically small volume createhydrodynamic velocity gradient, therefore vector V may be treated as velocity of the gravitation centre of this volume.
Equations (2.4) and (2.5) allow the following
set of equations to be obtained from equation(2.2) :

[PG

+

-

4)l

L
[P(U,U, - Ul%)l
dY

+

12.7)

(z 1

where x0, y,, z. are gravitation centre coordinates of a physically small volume. Similar
formulae are implied for the components u,. ~1~.
Wi) W2.
Now instead of equation (2.6) we have
Predvoditelev’s equationdV
x + $(I - ,!I)grad I” - jW div V+ (1 - j?)rot VxV = - igrad

p + vVzVP+ v(2 - y)graddivV.

P$ f $2

+

z-

CP($

ay

-

4Jl + ;

LP(U2 w, -u1 WI)]”
TZ.Z---tpap

JY

V%+(2--)zdivV

LPhzW2

.
aY

L

-

us

I

+%)I

where y is the index of Poisson adiabatic curve.
Conditions for transition to continuum presented by Predvoditelev may be reduced to thefollowing hypothesis that expresses transfer
velocities of colliding moIecules in terms ofhydrodynamic velocity vector(2.8)

j3 designates the quantity /? = 2AA,j3A2 thatwas later called by Predvoditelev a nonideal
continuity parameter.
Predvoditelev’s equation (2.8) is more generalthan the Navier-Stokes equation which Maxwell tried to justify in terms of molecularkinetic theory.
In fact, if we assume p = 0 and y = 5 that
corresponds to monoatomic gas, then equation(2.8) becomes identical to the Navier-Stokes
equation, But this is not always the case.
Further, Maxwelt did not notice that his
limitation impj~citly involved adiabatic equation
for monoatomic gas, therefore instead of (2 - y)
he obtained +.
Now it may be easily seen that Predvoditelev’s
equation becomes Kasterin’s equations if [i = 2,
div V = 0, v = 0.
In his investigations on acoustics Predvoditelev expressed nonideal continuity parameterin terms of Knudsen and Mach numbers
p=$KnM(2.9)

and demonstrated that equation (2.8) described
well Greenspan’s experiments in a wide rangeof Knudsen’s number.

ALEKSANDRSAVVICH

3. On turbulentflows
Hagen was probably the first who observeda special flow pattern that was later called
turbulent flow by Lord Kelvin. Hagen foundthat up to a certain velocity limit liquid flow
consists of jets which have the form of infinitely
thin rods, the flow dimension being prescribed.
After the limit is reached, motion becomes
immediately nonuniform; jets disintegrate andbegin to fluctuate. Hagen also paid attention to
the fact that the transition from jet flow toturbulent flow depends on viscosity and dimensions of the flow. However, he did not discover
general principles that may define commencement of transition.
The honour of this discovery belongs to
Osbom Reynolds, Maxwell’s pupil who demonstrated that transition of one type of flow to theother is characterised by a special dimensionlesscriterion referred to as the Reynolds number.
He reported this discovery in his work published in 1894 in which he also presented
statistical method for writing the equation of
turbulent flow.
Reynolds realized that the dynamic theory of
Maxwell is based on geometrical division of thetrue molecular velocity into transfer and thermal
ones, transfer velocity being mean velocity ofthermal molecular motion which coincides withhydrodynamic
velocity within the NavierStokes continuum.
Mean value of thermal velocity is zeroaccording to Maxwell’s conception.
This geometrical basis of Maxwell’s method
was adopted by Reynolds for physical representation of turbulent flow. For this he suggested
to divide the velocity at any point, obtained
from formulae similar to those of kinetic gas
theory, into mean, referred to now as averaged
velocity and relative velocity
u = ii + u’, u = fi + v’, w = w f d.

(3.1)

Here u, u, w are projections of hydrodynamic
velocity vector at the axis x, y, z.
Further, Reynolds supposed that turbulent
motions are 1Periodical motions. that mav_I be

1741

PREDVODITELEV

periodical either in space or in time. In accordance with these hypotheses on types of periodicity, averaging rules may be different.
I. The averaging rules for motions periodic in
space

Let us separate out a physically small volume
S. If u is the corresponding
projection ofvelocity vector of any point inside S and p isliquid density, then upon summation over thevolume S, we shall get the formula for determination of average velocity

-

&(PU)

u=p,

(3.2)

Similar formulae exist for projections E and E.
Now in correspondence with formulae (3.1)
velocities ii, U,W may be taken for projections of
the velocity vector of the centre of gravity ofvolume S. Velocities u’, v’, U, are the velocities ofrelative movement of liquid particles around thecentre of gravity. It is obvious that at any
moment of time these velocities will be functionsof coordinates x, y, z. The centre of gravity ofthe separated volume travels in space. Let these
movements be represented by Ax, Ay, AZ,. Thenthe average velocity can be represented asfollows

U = ii, +Ax (g)O+ AY(;>,+ AZ(;),+ ....

(3.3)

0

Now we shall put formula (3.3) into equation(3.1) multiply both parts of the resulting
equality by density p, sum up over volume S
and, taking into account thatc(S) Mu - Uo)l= 0,

2 MWI
G.+)

= 0 (3.4)

1742

A. V. LUIKOV

we shall get the formulaof U’ fromfor the average

and V. A. BUBNOV

value

we have(PU’)__- = - ;;

g

c (P)

(t - o2(3.10)Let us assume that within a singled out interval+ ($)OI:(Az)z]

(3.5)

+,...

If we assume that inside of separated

volume

S

($(

= ($J

= ($X

= const.

= (E)O= cg)O= const.

(3.6)

Similar(PU’)= 0.

considerations-for v, wgive tinall,yg (PU’) = & (PV’) = & (PMJ) = 0.

(3.8)

ii = ii, + (t - t’)of time t’ we shall get

0d; f+;(t - t’)2(3.9)
Takinginto account5 Mt - t’)l = 0,

(0

= 0.

(3.12)(0(3.7)

II. Averaging rules for motions periodic in time
If we choose a time interval r which would
include time IT: and carry out summationforthis interval,
we shall have for the averagevelocity the following formula :For any other moment(0

Formulae(3.7) and (3.12) allow formulationof the universal
averagingrules which arecorrect both for the periodic movements
withrespect to space and for the periodic movementswith respect to time:then we getg(3.11)

Then we havec (pu’) = 1 (pv’) = c (pd)

rg10

r

that

5 [PC%- u)l = 0

Here barred quantities
mean averaging operation either with respect to space or with respect
to time.
This universalityof formulae (3.13) allowed
Reynolds to make no distinction
between timeand space periodicity
but to call turbulent
motions merely periodic motions.
In the same work published in 1894 Reynolds
conducted
averagingof the Navier-Stokes
equations following rules (3.13). Thus he included
constant velocity gradients within a physicallysmall volume and for a physically short time
into turbulent motion equations.
Reynolds’
considerations
discussed
above
probably served for Predvoditelev
as the basis
in his attempt to describe turbulent
flows interms of hydrodynamic
equations corrected fornonideal continuity.
Moreover.
Reynolds’ averaging
rules allow
introductionof constant velocity gradients intohydrodynamics
equations
within a physicallyALEKSANDRSAVVICH

small volume and Predvoditelev’s hypothesis(2.7) includes any gradients.
In his work “On Turbulent Flows” preparedfor publication as far back as in 1950* Predvoditelev suggested a hypothesis that turbulent flow
appears because of the existence of a special
kind of a discontinuous process due to compressibility of any liquid or gas.
This approach to a turbulent flow reduces tofinding such a discontinuity front which would
allow a fluid flow to possess all properties
characteristic of turbulent flows.
To solve this problem, Predvoditelev madeuse of hydrodynamic equations such as (2.8) and
Hugoniot-Hadamard
method.
In case of monoatomic gas Predvoditelev’s
equation for a viscous vortex-free fluid flow isof the form

PREDVODITELEV

1743

equations (3.14)-(3.16), Predvoditelev has arrived at the following formula for velocityp= P - P wm4
PB

x(1 - [(l-&/3]

P - P @wP)

and

PB

(3.14)It should be completed by the state equation
in a general form
p = 4 (P, 7).

(3.15)- L(4v grad div V).

’ -’Bmust be sought from experiment, thus it is convenient to express them in terms of velocity at
the wall V, and at the axis I’,,.
Now Predvoditelev has written formula (3.17)
asTo describe the condition of heat input, we
assume that expansion work is compensated byheat conduction and work of dissipative forces
p div V = kV2T - p(VV2V)(3.17)

where 0 is the angle between the velocity vector
V and the normal to the front surface.
Formula (3.17) shows that at the wall where
the discontinuity front touches it, the flow
velocity Y is ditferent from zero but has a jumpwise change. Consequently, turbulent flow begins when the flow separates from the wall.
Near the wall no definite value can be
ascribed to /3 as determination of flow dimensions at the wall is impossible. Therefore theexpressionsdii
p dt = - grad p + pV2V + 1 grad div Y+ pfl($ grad V2 + V div V).

cos%}

4 = J(l

-

(1

!“n)

cos2 0)t3.18)

where(3.16)

Further Predvoditelev suggested that turbulent flow lluctuations are characterized by sucha discontinuity front that first and second
derivatives of the flow velocity, as well as first
derivatives of pressure and density change
sharply while passing this front. As concerns thetemperature, it changes continuously together
with all its derivatives.
In operation of passing through the front
following Hugoniot-Hadamard’s
method in* This work of Predvoditelev will soon be published in
Problems of Physical Hydrodynamics, ed. by A. V. Luikov

Having defined the equation of the discontinuity front surface as a family of ellipses, Predvoditelev has presented the following formula forthe angle 0

cos 8 =

(1 - Y2)
1 - e2y2’

Here e is the eccentricity of the ellipse and y is
the ratio of a local radius to the tube radius.

A. V. LUIKOV

1744

and V. A. BUBNOVThus formula (3.18) becomesd, =y = vy/v. He labelled this curve “old”. Then
Nikuradse plotted the straight line 4 = q andlabelled it “laminar”. This line appeared to crossthe “old” curve. That meant that some measured
values appeared higher than they would be if
the flow were not turbulent but laminar. As it
was not acceptable, Nikuradse supposed that his
measurements were not as accurate as hethought. If a laminar layer exists then the curve
which corresponds to a turbulent region of theflow should be tangent to the straight line for alaminar flow at the boundary of the layer.
To obtain a curve for a turbulent flow region
tangential to the straight line, Nikuradse added
seven units to each pl value, thus every point
being displaced to the right by seven units.
But this displacement was merely a distortionof the data to make them compatible with the
hypothesis.
This fact seems to be rather an evidence thatno Prandtl laminar sublayer exists but there isvelocity discontinuity at the wall.
Thus Predvoditelev’s views on the nature of
turbulent motions stated above do not disagreewith experimental data and are waiting for
further investigation.

_____&)2)
1
JL i)@--_-.. .-_-+o(1------~.
- Y2)(3.19)

1 - (I - 4

Comparisonof the above formula withexperimental data of Nikuradse made by Prevoditelev has demonstrated that the formula
gives a correct description of liquid flow incircular tubes.
In modern science on turbulent motions
Prandtl’s hypothesis on a laminar sublayer isis adopted. Later a concept of a transition layerthat isolates laminar layer from a turbulent core
was introduced.
Karman adopted this three-region model andused it as the basis for his formula of velocity
distribution in turbulent flows which was considered for many years to be very useful and toagree fairly well with Nikuradse’s experiments.
However, in 1949 the American worker Miller
reported the work “Hypothesis of a Laminar
Film”. Here the author, using experimental dataof Nikuradse convincingly proved that thelatter matched his results so that they would
agree with the above hypothesis.
Miller found that in Nikuradse’s paper there
vyas a disagreement between Table 2 that furnished results of direct measurements and Table
3 reporting experimental data expressed in adimensionless form that was necessary for comparison with Karman’s hypothesis.
In December of 1946 a discussion was held
between Rose, Bethler and Professor Martinelli.
Rose suggested to approach Prandtl, who
directed the work of Nikuradse, concerning theabove disagreement.
Prandtl replied that Nikuradse compiled anumber of tables and plotted I$ = V/V vs.

REFERENCES
I. A. S. PREDVODI~ELEV,
On molecular-kinetic basis ofhydrodynamic equation, Izv. Akad. Nmk SSSR. Otd.

Tekhn. Nauk, No. 4 (1948).
2. A. S. PREDVODIIELEV,
On rarefied gas aerodynamics andheat transfer problems, Heat and Mass Transfer, vol. 3,edited by A. V. LUIKOVand B. M. SMOLSKY. Energiya(1963).
3. A. S. PREDVOD~TELEV,
Theory of heat and Riemannian
manifolds, Heat Transfer-&v. Res. 3, No. 1, 1 IO (1971).
4. Osborne Reynolds and Engineering Science Today edited
by D. M. MCDOWELL and 1. D. JACKSON.Barnes andNoble (1970).
5. W. MILLER, Hypothesis on laminar film, Trans. ASME

No. 4 (1949).