# H. A. RAHMATULIN: A THEORY OF A DOUBLE VELOCITY FOR THE BOUNDARY LAYER IN A HOMOGENEOUS LIQUID WITH TWO COMPONENTS

Document Type:

Collection:

Document Number (FOIA) /ESDN (CREST):

CIA-RDP80T00246A003500230002-6

Release Decision:

RIPPUB

Original Classification:

K

Document Page Count:

12

Document Creation Date:

December 22, 2016

Document Release Date:

July 31, 2009

Sequence Number:

2

Case Number:

Publication Date:

April 18, 1958

Content Type:

REPORT

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CIA-RDP80T00246A003500230002-6.pdf | 267.12 KB |

Body:

USAF review
completed.
Ey the work of rrandtl and his school the theory of boundary layers
has become one of the universal methods in the mechanics of the
continua during th,u last decades. This theory, however, to-day is
a.,plied for solving several questions which are not directly
connected with the problem of motion of viscous fluids.
In the SSSF: the boundary layer theory has bu:en applied during the
last time for investigations of the following kind :
V
(U,-. Cernij).
2) Calculation of instationary filter processes (r.Ja. Kocin,
3) Calculation of instationary heat transfer for a heat-transfer
coefficient depending on the temperature (Ju.IT. Dem'janov).
The aim of this report is to call the researcher's attention
still to another field of application of the boundary layer theory.
"ere I mainly think of the application of the boundary layer theory
to the so-called "double-velocity hydro- and aerodynamics"
according to Landau's terminology. In order to clear the problem
I :!rant to give a survey on some concrete tasks of double-velocity
hydra- and aerodynamics which were formulated corresponding to the
boundary layer problems :
1) The flow of a homogeneous fluid over a porous surface. In this
case the task can be formulated in such a way that a part of the
ideal fluid with the mean density y1 and the speed V~ flows
through the surface, while an2other part with the mean density 2
and the speed V2 flows along. If it is put = p1 + K2 as real
density, then the boundary layer conditions are evidently
(1.1) k VIr = p
(1.2) V2n = 0
furthermore for
(1 .3) V1 V 2 -a Vo
x + y2 + z' oo it holds :
Although such a double-velocity state is everywhere theoretically
possible, it really occurs in that range only which lies directly
on the porous wall. Fur ideal or viscous fluids the boundary layer
near the wall must be investigated. This is, however, not the classi-
cal boundary layer problem for which the flow is given in the layer,
but a case in which the boundary layer and the external flow mutually
influence each other - similar to the case of high supersonic veloci-
ties. )'hereby the flow around a porous plate in supersonic must be
solved as the first task.
2) The problem of porous cooling. In its character this problem does not
differ from the first one; only the condition (1.1) must be replaced
by a corresponding other one.
3) The problem of the boundary-layer control. After the things mentioned
above it is not necessary to explain this in detail.
4) Furthermore the double-velocity hydrodynamics is of importance for the
motion of such liquids and gases which are saturated with macroscopic
dust particles. Here the question is essentially a generalization of
the filter theory.
The last case shall be discussed somehat more detailed.
The deduction of the differential equations for the boundary layer
in a liquid with double velocities :
I think that we can consider with good reason liquids and gases
saturated with macroscopic dust particles as a mixture of several
continuous media. (Footnote Prof. Voronec, '3elgrad called my
attention to the possibility to consider the mentioned media as
continuous. This is most suitable also in the sense of a verdict
of Zukovskij : "Mechanics is the art to set up integrable differentil
equations".)
It is evident that the tension and deformation relations are valid
for each of the media taking part in the motion. (Footnote: see
Pahma_tulin : "Foundations of gas dynamics of mutually penetrat-
ing motions of compressible media", Fri'-:lad.IMIat.L1ech.20, 1956.
A su:.i:-ary of the results of this paper was published in the reports
of the Congress for Theoretical and Applied Llechanics in Brussels.
- u i div Vi
correspondingly iy
oui dv.
i=/`i( )
xy d y x
171
correspondingly r iyz and ?ixz
under the penetration of two media the index i runs from
1 to 2. We introduce the expressions :
f1 f
41w 2 ~2w
where ~1 and ~2 are the mean densities and 41w and ~ 2w
the real ones of the two media.
The following equations of :-potion are obtained with the aid of these
expressions :
dui
fi - + a f1 ,, . 1 (2
at c)x d x
- 3 div Vi)
3y
(correspoudinzcly also an equation with vi)
u (Qiui) G(9 y )
+ = U
ax N y
Yfi = 0
if the densities q and e2 are assumed to be constant in the
equations of motion so tnat the condition f1 + f2 = 1 is satisfied,
then six equations with five unknowns remain over. Therefore the
mean densities Q1 and ~)2 are not allowed to be assumed as constant
in a double-velocity fluid, each component of which is incompressible.
Nevertheless this assumption for the motions in the boundary layer
does not lead to a redundancy in determination of the ;)roblem, since
in this case two equations drop out. Namely, if it is put Q1 = const
and const and, if it is estimated for the motions in the
G
b.,urdary layer, then it remains
F2 du2 = - f2= + i Y2 2 + I{(u,i - u2)
dt 1' x --y
2
du 1~ dt1 f1 L'" + f1 /41 3 2r + IC(u2 - u1)
u,
V
0
Pnus one obtains four equations for four unknowns.
If the pressure gradient vanishes, then the integral relations of
v.:Kurman can be applied and one obtains according to Fohlhausen
r1 [. S1 - (32 )3 i for 0 < y 17
r3 _ 1,y )3
J ~~ t2 3' '2 J
0 E y < 2
u2 = U for 0 2 \< y ,<
d -_ 1
_ 2 + 22?2(5-6A + Ili) with Il =2
?1
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It ccn be shown that for 2< 'i it always holds
2/-2+ ' 2c2(5 - 6 1+3 )
p1(r1 - 2)
d;
with 2 = 2 1
dc' 1
= f(c1) with
iure 1 shows a picture of I
the isoclines. From this it ~zl
2 = 2(.1) lies below a
straight line. A detailed cal-
culation shows that 2 very
quickly tends to C1 and
both tend to a value a which
corresponds to the boundary-
layer thickness of a liquid
with the density +
If a pressure gradient exists, then u1 and u2 and consequently
also - 1 and 2 are different from each other in every arbitrary
coint. Therefore a double-velocity theory in this case surely gives
results which differ from the corresponding results in a one-
comnonent medium.
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In his former paper on "The origin of vortices in ideal fluids"
Ludwig Prandtl directed to the possibility and suitability of
investigating the origin and further development of spiral sur-
faces of discontinuity on shelves, particularly the similar flows
of this kind.
Investigations of this form of motion which I shall denote as
"second form of motion", have become particularly necessary during
the last time, since the forces acting on thin wings of small
aspect ratio for an afflux in fluids or gases under nonvanishing
angle of incidence are mainly determined by the second form of
motion. Furthermore the flow around obstacles by impact waves even
depends on the second form of motion.
If the body with borders or edges which is assumed to be two-
dimensional rests in the moment t = 0 and moves for t) 0
according to the law v = vo(t), then the second form of motion
develops on the edges for
t
t