hypothesis). It is obvious from the example we have given that the weak closing relations can be satisfied under much less stringent assumptions than the traditional closing relations. Moreover, as can be seen from the foregoing, they follow from the requirement of consistency of the system of basic equations (the hierarchy of Fridman--Keller equations) with regard to similarity solutions and, essentially, reflect a certain self-consistency of turbulent similarity fields. LITERATURE

CITED

1. A. A. Townsend, Structure of Turbulent Shear Flow, Cambridge University Press (1956). 2. B. G. Newman, "Turbulent jets and wakes in a pressure gradient," in: Fluid Mechanics of Internal Flow, Elsevier, Amsterdam (1967). 3. E. Naudascher, "Flow in the wake of self-propelled bodies and related sources of turbulence," J. Fluid Mech., 22, No. 4 (1965). 4. G. I. Barenblatt and Ya. B. Zel'dovich, "Intermediate asymptotic behaviors in mathematical physics," Usp. Mat. Nauk, 2_6_6, No. 2 (1971). 5. V. A. Sabel'nikov, "On some features of turbulent flows with zero excess pressure," Uch. Zap. TsAGI, 6, No. 4 (1975). 6. G. Birkoff and E. H. Zarantonello, Jets, Wakes and Cavities, Academic Press, New York (1957). 7. A. S . G i n e s k i i , Theory of Turbulent J e t s a n d Wakes [ i n R u s s i a n ] , Mashinostroenie, Moscow ( 1 9 6 9 ) . 8. H. T e n n e k e s a n d J . L . L u m l e y , A F i r s t Course in Turbulence, MIT P r e s s , Cambridge (].972)~ 9. G. E. M e r r i t t , "Wake g r o w t h a n d c o l l a p s e i n a s t r a t i f i e d f l o w , " AIAA J . , 1 2 , 940 ( 1 9 7 4 ) . 1 0 . J . A. S c h e t z a n d A. K. J a k u b o w s k i , "Experimental studies of the turbulent wake behind self-propelled bodies," AIAA J., i_33, 1568 (1975). ii. A. I. Korneev, "Degeneracy of turbulent axisymmetric wakes," Nauchn. Tr. Nauchno-Issled. Mekhan. MGU, No. 31 (1974). 12. A. I. Korneev, "Similarity hypotheses in the theory of turbulent wakes," in: Turbulent Flows [in Russian], Nauka, Moscow (1977). 13. M. L. Finson, "Similarity behavior of momentumless turbulent wakes," J. Fluid Mech., 7__ii, No. 3 (1976).

EQUATIONS OF LAMINAR BOUNDARY LAYER IN A TWO-PHASE MEDIUM V. P .

Stulov

UDC 5 3 2 . 5 2 9 533.697.4

The motion of a two-phase medium in which the carrier component has low viscosity is considered. The equations obtained in [i], to which the viscous stress tensor in the fluid is added, are used. The boundary layer method [2] makes it possible to obtain asymptotic equations for the wall region. These equations have different forms depending on the characteristic values of the dimensionless determining parameters.

i. The proper derivation of equations of motion of many-phase systems is one of the difficult problems of modern mechanics. Usually, these equations are obtained on the basis of ideas about continuous media that interact and penetrate each other; the system of particles is replaced by a continuum. Such equations were obtained by means of the mass and momentum conservation laws in, for example, [3] in the special case of a two-phase medium. After special averaging over volumes containing sufficiently many particles,, the equations take the form

--+divpV=0,0t Moscow. 1, pp. 51-60,

0P" +divpsVs=00t

' P[OV+(v'v)V] =div(E-R)-F-~-

,

p~

+(V,-V)V,

Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti January-February, 1979. Original article submitted May 30, 1977.

0015-4628/79/1401-0037507.50

9

Plenum Publishing

Corporation

=div(E,-R,)+F i Gaza,

No.

37

F=a(p)

(V-V,)+o.C(p)

(V-V,)+

~o

.div(E-R),

E=-pU+~,

(1.1)

E,=-p,U+a,, ( p ~ / p , ~ 1 7 6

Here, E is the a v e r a g e d s t r e s s tensor, R is the s t r e s s t e n s o r that takes into a c c o u n t the departure of the local v e l o c i t y field f r o m the a v e r a g e v e l o c i t y field, F is the force of the i n t e r a c t i o n b e t w e e n the phases, U is the unit tensor, pO is the p h y s i c a l p h a s e d e n sity and the s u b s c r i p t s r e f e r s to the p a r t i c l e s ; the o t h e r n o t a t i o n is standard. In Eqs. (I.i), F c o n t a i n s t h r e e terms. The first d e s c r i b e s the d r a g of a p a r t i c l e , this b e i n g g i v e n by S t o k e s ' s law in the c a s e of small r e l a t i v e v e l o c i t i e s of the f l o w a r o u n d the p a r t i c l e . The s e c o n d term c o r r e s p o n d s to the a p p a r e n t m a s s of the particle, and the t h i r d takes into a c c o u n t the e f f e c t of l a r g e - s c a l e c h a n g e s in the s t r e s s e s in the carrier p h a s e s u c h as a r e p u l s i v e f o r c e in the p r e s e n c e of a p r e s s u r e gradient. In [i], in w h i c h an i n v i s c i d c a r r i e r p h a s e is c o n s i d e r e d , the s e c o n d term in F is ignored, and in the third it is a s s u m e d that E=--pU, R ~ O . In the p r e s e n t paper, the same a s s u m p t i o n s are m a d e about the f o r c e F. In addition, as in [I], it is a s s u m e d that the m e d i u m of the p a r t i c l e s is rarefied, i.e., E s = R s ~ 0, and that the local v e l o c i t y field of the fluid d i f f e r s l i t t l e f r o m the a v e r a g e d v e l o c i t y field, i.e., R s 0. N o t e that the v i s c o u s t e n s o r o in the third t e r m of the e x p r e s s i o n for F is o m i t t e d on a c c o u n t of the i n s u f f i c i e n t p h y s i c a l j u s t i f i c a t i o n for this term, a l t h o u g h the c a r r i e r p h a s e is a s s u m e d to be viscous. Thus, in the p r e s e n t p a p e r we s t u d y s t a t i o n a r y f l o w around b o d i e s by m e a n s of the s y s t e m of e q u a t i o n s

divpV=0,

divp:V,=0,

P" + po P---= I ,

vp, The u s u a l N a v i e r - - S t o k e s tensor o in (1.2); ~ is the spherical particle.

p(V-V)V=divE-F,

formulas viscosity

p,(V,.V)V,=F

E=-pU+a,

(1.2) a = 4 . 5 ~ / (a2p, ~

are adopted for the components of the viscous of the carrier p h a s e , an d a i s t h e r a d i u s o f

stress a

We s h a l l c o n s i d e r p l a n e m o t i o n o f t h e medium d e s c r i b e d by E q s . ( 1 . 2 ) a r o u n d s o l i d surfaces. In the formulation of the problem, it must be borne in mind that in the general c a s e o f f l o w a r o u n d b o d i e s when t h e v i s c o s i t y of the carrier phase is ignored the contour of the body around which the flow takes place cannot serve simultaneously as a streamline of the carrier p h a s e and t h e medium o f t h e p a r t i c l e s . This circumstance, w h i c h was n o t e d in [4], obtains a perspicuous physical interpretation if Eqs. (2.1) for a = 0 are written in the natural coordinate system (of the streamline s an d t h e n o r m a l n ) o f t h e c a r r i e r phase. The p r o j e c t i o n s o f t h e momentum e q u a t i o n s onto the normal take the form

1

w z OTOs+ p--gOP'

Ow,

p, p

Ow,

w COS(T--T~)sin(~--X')"~'S- w " s i n i ( T - - T " ) O n

W ZCOSi(.~_T,)OT,+wisin(T_T,)COS(T_.~) OT~ On

i

Op

p O On

+ a w , sin(~--T,) = 0

-(1.3)

S u p p o s e that T=T, on a s t r e a m l i n e n = const, i.e., a s t r e a m l i n e of the c a r r i e r p h a s e and the p a r t i c l e s s i m u l t a n e o u s l y . F r o m (1.3), we o b t a i n

T ~ w:0 +__0P=0,

w,' O~ + t Op -0

p~

p~

Os

On.

Os

On

(1.4)

Equations

(1.4) are satisfied simultaneously i n o n l y two c a s e s : p~176 or i.e., the considered section of the streamline is rectilinear. In other 'words, in the general case of a curvilinear surface around which the flow occurs the condition of impenetrability for the medium of the particles cannot be satisfied by s o l u tions of the system (1.2). This property can also be assumed to hold in the case of low viscosity of the carrier phase.

OT/Os=Op/On=O,

Suppose that a homogeneous two-phase mixture with V flows over a solid surface. We go o v e r t o d i m e n s i o n l e s s variables, taking as scales the following quantities: for x and y, t h e c h a r a c t e r i s t i c l e n g t h 1 o f t h e p r o b l e m , f o r V a n d Vs t h e q u a n t i t y V~, for P a n d Ps t h e q u a n t i t y pO, a n d f o r p , t h e q u a n t i t y p~ We w r i t e E q s . ( 1 . 2 ) i n t h e c u r v i linear coordinate s y s t e m (xy) f o r m e d by t h e c o n t o u r o f t h e body, x, and t h e n o r m a l t o i t , 38

y, at some p o i n t O:

R O p u + O p v + pv = 0 , R+y 6x Oy R+y R Ou Ou uv - - u ,+v + R+y Ox Oa R+y 1

t

[

( R + u ) ~ Ox ~-

Ov

vRR"

(R~-y) ~- ax

(R+y)

Ov Ov +v Ox Oy

0

R+y

oy ~

+ -(-R- ~+ y )

Oa

R+y

yRR" Ou I

R+y Ox u -+

@

i

(R+y) ~

t

r

R~

02u

7 x ] + -p ~3 tie [ ~ - ~ x 2 +

vRR'__~ yRR' Oa ] (R+y) ~ (R+y) ~ Ox t

ne

]~z

-

Ozv

V

t

[o2v ,

net

uRR'

Ox

yRR'

O~

b = --'po

Here, R is the radius of curvature We s h a l l a s s u m e t h a t R ~ d R / d x ~ i . At i n f i n i t y in front and the particles and the we s p e c i f y t h e c o n d i t i o n s

Oy

+

t Ov R au, v ] p 3 R e [ 0y~ R+y 0x0y R+y 8y ( R + y ) ~ 8x (R+y) 2 R _ _ 0 ~ + + O~+~ u+v~ ~ (~-++)=-b R Op R Or+ , Or+ a, 2 R+g u+ Ox v+ Oy R+g Re R+~ Ox ' R+g u~'-~x• Oy R+g po l2 po p~ p=l,bp~,

R

R+y

av

Ov ]

_( B _+ y ) ~ + (R+y)~ ~ ( R 4 y ) ~ ' - a x

( B + y ) ~- ax ~

l

Oy

+

R c)~-v 4 R Ov R+y Ox 8y (R+y) ~ Ox R ~ u R+g

Ox

Re p 0 2 a + O~u_f_ 1

R ~"

~'-fi-~ 2R

F - - T - - =

R+y

of

7 = 4 . 5 -~7 ~ o ,

the

body at

Re --

the

~ (v-v,)=-baP Re

~

point

with

coordinates

a-~ (1.5) (x, 0).

o f t h e b o d y , we s p e c i f y t h e v e l o c i t y V~ o f t h e c a r r i e r phase density p~ of the particles. On t h e s u r f a c e o f t h e s o l i d b o d y , of impenetrability and adhesion for the carrier phase: a=v=0, y=0

(1.6)

A shortcoming of this formulation of the problem is the use of Stokes's law i n t h e c o m p l e t e r e g i o n o f t h e f l o w r i g h t up t o t h e w a l l . Some d a t a g i v e n i n t h e b o o k [5] i n d i c a t e t h a t when a v i s c o u s f l u i d f l o w s a r o u n d a s p h e r e t h e i n t e r a c t i o n force is changed near the wall. In addition, the approximation of particles of irregular form by s p h e r e s also deteriorates near the wall, where particles enter a flow with large velocity gradient. Therefore, the analysis which follows should be regarded as an initial approximation; t h e p o i n t s t o w h i c h we h a v e j u s t d r a w n a t t e n t i o n require further investigation. We s h a l l o b t a i n we c a n c o n s i d e r f o u r

an asymptotic solution different cases.

of

the problem

(1.5)-(1.6)

a s Re § ~ .

Here,

2. S u p p o s e Re § ~ . We d e n o t e ~2 = 1 / R e " Equations (1.5) have a small parameter as coefficient of the highest derivatives. F o r ~ = 0, a d i s c r e p a n c y arises in the boundary condition of adhesion of the fluid. We c o n s t r u c t a solution t o t e r m s os o r d e r e. In the zeroth approximation, a smooth solution i s d e s c r i b e d b y E q s . ( 1 . 5 ) f o r z = 0. They d e s c r i b e m o t i o n s of t h e f l u i d and p a r t i c l e s , which are coupled through the displacem e n t os t h e f l u i d f r o m t h e v o l u m e o c c u p i e d b y a p a r t i c l e . F o r b = 0, t h e m o t i o n s o f t h e phases take place completely independently, and the particles move a l o n g s t r a i g h t lines. We s e e k t h e

complete

zeroth

approximation

u=a~(x,y)+~*(x,~), tions

Analyzing the original equations for the remaining flow variables

in the

form

v=v~(x,y)+ew*(x,~),

G=y/s

( 1 . 5 ) , we c a n show t h a t are identically zero.

the

(2.1) boundary-layer

correc-

Substituting (2.1) in (1.5) and (1.6), going over to the variable ~, t a k i n g i n z o account the equations for the smooth part of the solution, expanding the smooth functions in series i n E, a n d i g n o r i n g t e r m s o f o r d e r ~ a n d h i g h e r , we o b t a i n t h e s y s t e m o f e q u a t i o n s

39

and boundary

conditions

0

.Ox [pe(x'O) . . u*]+pe(x'O) . .

aw* Oq

for the boundary-layer

oa*

0,

Pe(X,

For the functions:

0) c o r r e s p o n d s

transition

u, au~(x,O)

[ae(x, 0 ) + u * ] - -ax +

to

to

the

the

w*=0,

(2.4)

in

boundary

(2.2),

a[p~(x, 0) a]

8x layer

Here and in as for the

what follows, variables of

we

~=0;

layer

[

av~(x,O) ] Oa*

w*+q

u*~0,

smooth part

U*(X,~)=U(X,~)--ae(x,O), Substituting

+

Ox

a*=-a~(x, 0), Here,

corrections:

of the

equations,

.

Oy

t

8~

o~a*

(2.2)

pc(X, 0) 0~ 2

~

(2.3)

solution

on the wall.

we make a c h a n g e o f

the unknown

W*(X,~)=V(x,~)--~ OVe(x,O)/ay

(2.4)

obtain

Ov

Oa Oa Op~(x,O) u--+v--+-ax ~T1 Ox

+ pdx, O) --~q= O,

the same notation is the original problem.

used

for

02a

t

pdx, 0) a~ 2 the

variables

of

the

boundary

These equations describe a boundary layer in an incompressible fluid with variable density Pe(X, O) on the outer boundary. The influence of the particles on the structure of the boundary layer is manifested in the present case only through the function Pe(X,

0).

3. S u p p o s e Re § ~ a n d y § ~ i n s u c h a way t h a t y / R e = c o n s t . As b e f o r e , when ~2 = 1/He = 0 t h e o r i g i n a l system of equations becomes degenerate and there is a discrepancy in the boundary condition of adhesion of the fluid. The e q u a t i o n s o f t h e s m o o t h p a r t o f the solution contain a term of the form (7/Re)(V-V,), i.e., the viscous interaction between the phases occurs in the complete flow field. The c o m p l e t e z e r o t h a p p r o x i m a t i o n to the solution of the original problem can be sought in the form

u=a,(x, y)+a*(x, ~l), v=v~(x, y)+ew*(x, ~1), p=p,(x, y) (3.1) p=p~(x, y), a , = a ~ ( x , y)+ea~*(x, ~1), v~=v~,(x, y), p~=p,~(x, y ) The c o r r e c t i o n for the longitudinal component us of the particle quantity, since a discrepancy for this variable is absent on account boundary condition. In addition, i t i s h e r e a s s u m e d t h a t vs,(x, 0 ) ~ 0 c o m m e n t s made i n t h e f o r m u l a t i o n of the problem. layer

Applying the corrections

transformations

above,

"x 0 0 w *

.

,

Ox

+ --

~ p.(z,0)

u* -

-

Re p~(x, 0)

conditions

for

-

1

p,(x, 0)

Eqs.

(3.2)

,

ou,* O) ~ Oq

v. (z,

have the

for

the boundary-

A change of variables

0 Ov Ox [Pe(x'O)a]+Pc(x'O)aq--

reduces

=0,

Eqs.

Ou Ox

u - - + v

-

,~

(3.2)

u*-=O

Re

form

u'-~0, a,*-~0, ~-~oo

the functions u* a n d w* a r e d e t e r m i n e d f r o m t h e f u n c t i o n Us* i s f o u n d b y a s i n g l e i n t e g r a t i o n :

I~s*(X, ~) =

first

(3.2).

two e q u a t i o n s

Re v (x 0 ) : a* d~l (3.2)

Ou a~l

~

to the

Opo(x, O) ax

u.. (x, O)a tt..ox(X,O) "4- v,. (x, O) Oa~"+ b

40

equations

Ovo(x,O) + w * ] Ou* +

aLt*

o~u* - 0~I~

u'=-ue(x, 0), w*=O, X--0, Initially, this, the

the

0n

oue(x,O)

The b o u n d a r y

After

we o b t a i n

:

0 ox a*

described

velocity is a small of the absence of a by v i r t u e o f t h e

equations

~--

the boundary

,{

p,o(x, O)

Re

p~(x,0) [ a - - U , e ( Z ,

ap. (x, o) O:

of

'y

layer:

i

02u

0) l = - - - -

Re [a--tt.. (x, O) ] -:-0

p,(x, 0) 0~ z (3.3)

u'(x, ~)=a(x, n)-a~(x, 0), w'(x, ~)=v(x, ~)-~Ov.(x, O) lay,

W(x, ~)=a,(x, n)-nou,.(x, o) l Oy

The first two e q u a t i o n s of (3.3) c a n be s o l v e d i n d e p e n d e n t l y o~ the last equation. The u s d i s t r i b u t i o n is f o u n d by a s i n g l e i n t e g r a t i o n of the v e l o c i t y p r o f i l e of the fluid. The r e s u l t i n g s o l u t i o n is v a l i d w i t h i n the b o u n d a r y layer, s i n c e the i n t e g r a l d i v e r g e s as y § ~.* 4. S u p p o s e Re § ~ and y § ~ in such a w a y that (y/Re) § ~. We d e n o t e E = i/Re, y / R e = l/~ (i.e., y = l/a2). W e t r a n s f o r m the o r i g i n a l s y s t e m of e q u a t i o n s , e l i m i n a t i n g from the third and f o u r t h e q u a t i o n s the terms w i t h the v i s c o u s i n t e r a c t i o n of the p h a s e s by m e a n s of the f i f t h and s i x t h e q u a t i o n s . I n t r o d u c i n g the small p a r a m e t e r ~ in t h e s e equations, we obtain

R R+y

Ou Ox

Ou "': Oy

R

av

- -

R+y

Here,

av R+y

(I)~, (1)v

~--

Ox

Ov + ~

R Op R+y Ox

u~

Oy

p~l R p ~ R+y

- -

R+y

--:-. + ~ +

8y

p

s t a n d for the v i s c o u s

The o t h e r e q u a t i o n s o b t a i n e d for ~ = O:

R

~ + v , ~ +

Ou. Ox

au~ Oy

a.v, R+y

Or.

Or.

u. ~

~W47v ~. --~x + ~,

Oy

R+y

--J-~a'\ +b

The e q u a t i o n s

R OpeaL+O.peve pev~ +. =0,. R+y Ox Oy " R+y

As i s

shown i n

[i],

Eqs.

phase medium with density

R Op~ea~ ap.~v~+ p,~ve + ~ =0 R+y ax Oy R+y

c o v e + Ov~

which

(4.z)

o

a. 2 \

~-'v-R+y; +

(4.2) .can be reduced

p~+p~,

(4.1)

ay t -- --9 0 ~

for the s m o o t h s o l u t i o n are

(o,+p,,) ~R-4--# ~ e -~x + . o y . --ff~. y l . R + y Ox I R

e -

terms.

remain unchanged.

(Oe+O,,) ~--U.~~

R Op \ 8 ]= -@~ R+y -~x p

+b

Oy

to

is c o n s t a n t

=o,

the

0e=1-b0,,

equations

of motion

of a single-

a l o n g each s t r e a m l i n e .

It can be seen f r o m Eqs. (1.5) and (4.1) that in the z e r o t h a p p r o x i m a t i o n the b o u n d a r y - l a y e r c o r r e c t i o n s for the v e l o c i t y c o m p o n e n t s of the f l u i d and the p a r t i c l e s are i d e n t i c a l l y equal. T h e r e f o r e , the c o m p l e t e z e r o t h a p p r o x i m a t i o n m u s t be sought in the form

u=u.(x, y)+u*(x, n), v=v~(x, y)+~%w'(x, ,1), p=pe(x, y)+p'(x, ~1), p~=O~(x, y)+p,*(x, n), ~q=Y/ ~ The u s u a l

transformations

give

the

equations

for

the boundary-layer

corrections:

0 Op*w* ~x[~.(z, 0)o*+p.(x,o)~*+o*~*l+ (0*+n ~ t o~.(x, o) ~ , . ( x , o )Ow* ~+-= Oy 0~1 " 0~1 ( +oo .o. 0 Ore (x, O) Fp,e(x, O)

o

ul 1 I

c'x [u'(x'O)p**+p*e(x'O)u*+p'*a*]+

Og

O~q

Ou*

0

O~

Ou~(x,O)

(p*+p.*) u.(z, O) ou~(z,O)ox ~[pe(x, O)+o*+ p.e(z, o) +~*] (~.(z, o) +u*)-z-+ +~~* ~z

(4.3)

(w,+no~(x,O) ~ O~*l o~* Oy

A c h a n g e of v a r i a b l e s

Opu + apv .= O, Ox a~

r e d u c e s Eqs.

/'-~-~ 1- 01"12'

(4.3)

Op,u + ......... Op,v O, Ox a~

P*=-b9**

to the b o u n d a r y - l a y e r

au 4-v a ~ (v+v.)

~ ax

equations

ap~(x,O) a~u

' ~ / + . . ax . . . . . O.qz .

o

*K. Ya. B a s k a k o v , who is a s t u d e n t at the M o s c o w S t a t e U n i v e r s i t y , has a n a l y z e d this c a s e w i t h a l l o w a n c e for the v i s c o u s s t r e s s t e n s o r in the e x p r e s s i o n (I.i) for the i n t e r a c t i o n force. The c h a n g e s in Eqs. (3.2) are net of a f u n d a m e n t a l nature, since they r e d u c e to a c e r t a i n c h a n g e of the s m o o t h c o e f f i c i e n t s in the s e c o n d e q u a t i o n of (3.2) and the addit i o n of a t e r m w i t h 02u'/0~~ in the third equation.

41

p=l-bp+,

u+=a, v + = u ,

a v, (x, o)

u*(X, t l ) = a ( X , ~ l ) - t 4 ( x , O ) ,

w*(x, t l ) = v ( x , ~ l ) - t 1 - -

(4.4)

ay

p'(x, n)=p(x, ~)-p~(x, o), p;(x, n)=p~(x, n)-p,~(x, O) The s y s t e m (4.4) can also be r e d u c e d to a form c o r r e s p o n d i n g m o t i o n of a s i n g l e - p h a s e m e d i u m w i t h d e n s i t y p+p~.

to d e s c r i p t i o n

of the

5. In the cases c o n s i d e r e d above, it was i m p l i c i t l y a s s u m e d that the normal component of the v e l o c i t y of the p a r t i c l e s at the wall is n o n z e r o in the s m o o t h a p p r o x i m a t i o n . If this c o n d i t i o n is not satisfied, the s o l u t i o n g i v e n in Sec. 3 must be reconsidered, since i f v,~(x, 0 ) = 0 we o b t a i n from the third equation of (3.2) the result a*~0. I t f o l l o w s from Eqs. (1.4) that in the general case the condition ~=~,e entails a~/as=O, i . e . , the flow takes place around a rectilinear section of the wall. Under suitable conditions in the initial section the condition T~=~,, i . e . , v ~ = 0 may b e r e a l i z e d on the rectilinear section. The simplest flow of this type is evidently a longitudinal homogeneous two-phase flow around a plate; in this case, the smooth solution corresponds to constant values of the parameters. Thus, suppose Re § ~ and y § ~ in The complete zeroth approximation must

u=u~(x, y) + a*(z, ~),

such a way that y/Re = const be sought in the form

v=v~(v,y) +ew* (x, ~),

a~=u~e(x, y)+as*(x, ~),

vs=v~e(x,y)+ew~*(x, ~),

The usual transformations os the original for the boundary-layer corrections. We r e c a l l of the zeroth approximation all terms of order 0

!

- - [ue(x,O)p*+pc(x,O)a*+p*u*]+ / ax

_

(

O [u~(x,O)p~*+p~c(x,O)u~*+p~*a~*]+ ax

x

[~(z, 0)+u*]

Oa" + [ ~ -ave(x,O) -+w* ax ay

[uAz, O) - u~ (z, O) ] + [a~,(z,O)+ a,*]

au,* ax

+

p=p~(x,y),

Re +

av~e(x,O)

~

ay

p=p~(x, y) +p*(x, n)

0)

(5.1)

aw*

ap*w* +--=0 On On

+ psi(x, 0)

aw~* a11

ap~*w,* + - -

Oae(x,O) ~ e [ p~(x,O)+p,* ax p~ (x, o) + p*

u * - - +

p~e(X, 0) + p~*

[ av,~(x,O) n ay

y = 0.

equations (1.5) for R § ~ give the equations that in the construction of the equations ~ and higher are ignored:

CO,*)

] Ca* + aq

v,~=0 f o r

p,=p~,(x,y)+p~*(x,n)

ap* \ av~(x,O) -~'~-) - - + p ~ ( x , P*+n @

P,*+n

and

p~(z,O)+ p*

( u * - u,*) =

t

aq

0

p,e(x, 0) l )< p~ (x, 0)

(5.2)

a2u *

p~(x,O)+ p* aq 2

] au,* aU,e(z,O) w~* ~ + a~* ax

"~ Re (a*-a~*) = 0,

p*=-bps*

The boundary conditions f o r u* a n d w* c o i n c i d e with (2.3). For the remaining variables, the boundary conditions consist of the requirement that p*, a,*, p~* h a v e t h e f o r m o f a l o c a l function, a n d w~* t h a t o f a l o c a l function plus a constant. Of course, boundary conditions must be imposed on all the unknown functions in the initial section of the rectilinear part. Note that the number of unknown functions is one greater (5.2). The missing equation must be obtained from the fourth all the terms of which have order e on the functions (5.1).

than the number of and sixth equations

equations of (1.5),

Of course, to argue rigorously about the construction of an approximate solution to the original p r o b l e m we m u s t s p e c i f y an iterative process for constructing any term of the expansion, prove the convergence of this process, and estimate the proximity to the exact solution [2]. To u n d e r s t a n d why it is necessary to use in the zeroth approximation equations that give a small discrepancy on arbitrary functions (5.1), we a n a l y z e the first approximation. We r e c a l l that in the flow of a single-phase fluid the projection of the momentum equation onto the y axis in the first approximation contains only a correction for the pressure; the remaining terms can be expressed in terms of the already known corrections (and smooth functions) of the zeroth approximation [6]. An a n a l o g o u s situation obtains in the problem considered here except for the important difference that there a r e now t w o e q u a t i o n s of this t y p e w i t h a common p r e s s u r e for the two phases.

42

Elimination corrections To imation:

of of see

the the

this,

pressure zeroth we

from these approximation.

write

down

the

a=ae+u*+eal~+ea~*,

two

solution

The projections the zeroth and

ave ue

Re p~

Ov~ + ve

Ox

(

9~ Ov~e

Ovs~e U,se ~

Substituting transformations terms of order

ay

)

P~

-]- 1 , l s i e -

Ox

the

the smooth

Og

be

Ox

+

ay

3

+v*+vie(x,O)

p~(x, 0) 1 [ (Ov~(x,O) p~(x, 0) 3 Oy

, [ ps~(x, 0)+p~* Re p~(x,0) + p*

the

first

the

approx-

t~te -

-

Ox

@~e Oie

+

Oy

ay

~-

Ovs~

Ov+~

~

8p~

Ox

8y

Re

~!/

(5.4)

Opie

0!1

~ +

onto the y axis have the form

OV~e 4- U e

Ovse -~ ~sie

(5.3)

and particles of the solution OYe

9~

Re

#g

in the fourth and sixth equations Eqs. (5.4), we obtain the principal the following form (after canceling

[u~(x, 0 ) + ~ * l - 0 x

in

for

ps=pse+ps*+epsle+epsi*

av~ ,

Ovste 4:- Us e - -

connection

problem

fluid part

of

Oy

Ox

missing

p=pe+ept~-e2p~ *, p=pe+p*+sp~+ep~*

(5.3) using ~) in

the

original

Opr (ve - v s e ) ~ - ~

Re p~

[

equations for the

~ 9se

"+----

ay

gives

Ys~Use+EV*s~EUsfe~E2~st*~

the momentum approximations

of

first

of

v=v~+ev*~ev~+~aw~ ~,

~s=~se~s*--E~ste~E~si*~

in

equations

of (1.5) parts of ~):

Oxdg - +

3

and making the these equations

- Ox

u*+

av~e(x,O) ) ] ~ ay +(vl~(x,O)- v ~ ( z , 0)) + Re

Oy

9~(x,O)+p~* p~(x,0) + p*

usual (the

v*+

(u,_vj)= (5.5)

0~

p~(x,0)+p* F

Z + 3 0H

[ 11

azv~(x,O) -

ox

+ 0H

, [u~(x,0)+u~,]--+0x

Ov~,~(x,O) ,

-

+

Ox ay

Ox

-

l

Ov~(z,O) u~*+

11

+v~*+v~(x,0'!,

oy

"f

0~

+

Op,*

- - v ~ * - - - ( v * - u ~ * ) = - b - -

Og

Re

a~

It is easy to see that if both Eqs. (5.5) refer to the first a p p r o x i m a t i o n we o b t a i n two e q u a t i o n s f o r t h e o n e u n k n o w n P l * " At t h e same t i m e , t h e p r i n c i p a l part of one of the original equations (1.5) does not vanish, i.e., one cannot construct an approximate solution of the complete problem. Thus, eliminating @~*/03 f r o m E q s . ( 5 . 5 ) , we o b t a i n the missing equation for the system (5.2). The c h a n g e o f v a r i a b l e s

8v~(x, O)

u * = u - u ~ ( z , 0), w*=v enables

us to

q - Oy

reduce

Eqs.

~p~ Opv --+--=0, Ox 03

, p*=p-p~(x, 0), a~*=a,-a~(x, 0), w~*=v~-3

(5.2)

to

the boundary-layer

ap~a, ap,v, O~ Oa 7 P, --+~, =0, u--+v--+----(u--us) Ox 03 Ox 03 Re p Ott, OLt, 7 Op~(x,O) u, + v, (a-a,)=- b~ , Ox 03 Re Ox

+v,

0x

(v-vf=b

0~

Re

u - - + v - - + - - - - ~ v - v ~ )

Ox

a3

Oy

, 9~*=p~-p~(x,O)

(5.6)

equations

We o b t a i n t h e m i s s i n g e q u a t i o n o f t h e b o u n d a r y - l a y e r from Eqs. (5.5), eliminating f r o m t h e m Op~*/Oj. We o b t a i n

~

ov,~(x, O)

Re p

The difficulties noted here in the formulation of two-phase medium on a plate (or rectilinear section of takes place) find their reflection in the literature -with regard to closure of the equations. For example,

ap~(x, O)

l

OZu

b--.--

Ox

9 O~2 (5.7)

p=t-bp, by m e a n s o f

-

the

_ _ +

P-

032

formulas

+

(5.6)

(5.8)

3 03

the boundary-layer equations of a a surface around which the flow there is no common point of view it is assumed in [7] that the

43

densities in the boundary layer are constant; then the need for the additional equation (5.8) disappears. In [8], a two-phase medium is considered under the condition b = 0 and the system of equations is closed by means of Eq. (5.8) (for b = 0), which coincides in this case with the projection onto the y axis of the momentum equation of the particles. In [9], it is assumed that there is no lagging of the transverse velocity component of the particles in the boundary layer on the plate, i.e., v z Vs; in this case, the need for Eq. (5.8) disappears. Finally, in the book [i0] the analog of Eq. (5.8) for b = 0 is also not considered.

I thank

A.

N.

Kraiko

for

discussing

the

work.

LITERATURE CITED 1. A. N. Kraiko and L. E. Sternin, "Theory of flows of a two-velocity continuous medium w i t h solid and fluid particles," Prikl. Mat. Mekh., 29, No. 3 (1965). 2. M. I. Vishik and L. A. Lyusternik, "Regular degeneracy and boundary layer for linear differential equations with a small parameter," Usp. Mat. Nauk, I_22, No. 5 (1967). 3. T. B. Anderson and R. Jackson, "A fluid mechanical description of fluidized beds," Ind. Eng. Chem. Fundam., 6, No. 4 (1967). 4. Kh. A. R a k h m a t u l i n and N. A. Mamadaliev, "Two-velocity theory of flow around a thin profile," Zh. Prikl. Mekh. Tekh. Fiz., No. 4 (1969). 5 S. J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics, Prentice-Hall, Englewood Cliffs (1965). 6 L. A. Chudov, "Higher approximations in the boundary layer," in: Some Applications of the Mesh Method in Gas Dynamics, Vol. 2 [in Russian], MGU, Moscow (1971). 7 H. H. Chiu, "Boundary layer flow with suspended particles," Princeton Univ. Dept. Aeronaut. Eng. Rept., No. 620 (1962). 8 R. E. Singleton, "The compressible gas-solid particle flow over a semi-infinite flat plate, Z. Angew. Math. Phys., 16, No. 4 (1965). 9 F. E. Marble, "Dynamics of a gas containing small solid particles," in: Combustion and Propulsion, Pergamon Press, Oxford (1963). i0 Soo Sao-lee, Fluid Dynamics of Multiphase Systems, Blaisdell, Waltham, Mass. (1967).

END EFFECT

IN TWO-PHASE

FLOW THROUGH A ~ D I U M

WITH LOW INITIAL WATER SATURATION P. M. Guseinov

and P. A. Yanitskii

UDC 5 3 2 . 5 4 6

A study is made of the nature of the increase in the water saturation at the end of a fairly long stratum (end effect [i, 2]) without allowance for the interaction with the displacement front and in the case when the initial water saturation S O only slightly exceeds the irreducible water saturation S.. Asymptotic expansions are constructed with respect to the small parameter = S O -- S.. The resulting expansion is compared with a numerical solution. The good agreement between the results indicates that the approach to the construction of the asymptotic behavior is well chosen.

I.

Formulation

of

the

The Rapoport--Leas by water in a homogeneous (for example, [i])

Problem equation, porous

os+~,(s) Ot

which describes medium, becomes

+8

~(s)/,(s)1"(s)

Baku and Tyumen'. Translated from i Gaza, No. I, pp. 61-66, January-February, 1977.

44

0015-4628/79/1401-0044507.50

one-dimensional after reduction

Izvestiya 1979.

9

=o,

Akademii Original

linear displacement of to dimensionless form

x-

,

L

t = - -

oil

(1.1)

Lm

Nauk SSSR, ~ekhanika Zhidkosti article submitted September 6,

Plenum Publishing

Corporation