1. Art and Diseño

S T R U C T U R E OF TIIE L A M I N A R B O U N D A R Y L A Y E R OF A D I S P E R S E
MEDIUM ON A FLAT PLATE
A. N.
UDC 532.529
Osiptsov
A study
is made of the stationary
flow over a semi-infinite
flat plate
of a
two,phase
medium
consisting
of solid particles
and a gas, which
has a low
viscosity.
The aim of the paper
is to investigate
the influence
of a difference
between
the velocities
of the phases
on the structure
of the flow in
the boundary
layer.
A similar
problem
was considered
in [i-5], but the obtained
solutions
do not make
it possible
to follow
clearly
the development
of the flow in the boundary
layer
or the
change
in the coefficient
of friction
along
the complete
length
of the plate.
In [i, 2,
4] no use was made of the projection
of the momentum
equation
for the particles
at right
angles
to the flow,
and additional
assumptions
were made:
constancy
of the averaged
density
of the particles
in the boundary
layer
in [i] and equality
of the transverse
components
of the gas and particle
velocities
in [4].
In the monograph
[2], only some oK
the flow features
were considered;
these
included
the deceleration
of the particles
along
the surface
of the plate.
In [3, 5], the solution
is constructed
by an expansion
in power
series
in positive
and negative
powers
of the longitudinal
coordinate,
it being
assumed
that far from the leading
edge of the plate
the averaged
density
of the particles
in the boundary
layer
is constant.
Below,
the method
of matched
asymptotic
expansions
is used to construct
boundarylayer
equations
for the flow region
with velocity
disequilibrium,
and the asymptotic
behavior of the solution is found at large distances from the leading edge.
A numerical
solution is used to study the distribution of the parameters of the phases and the change
in the coefficient of friction over the complete length of the plate.
I. The Boundary-Layer
Equations
The behavior of two-phase systems is usually studied in the framework of a model
of two interpenetrating continuous media [6].
In the present paper, we restrict ourselves
to the case oK an incompressible carrier phase and negligibly small bulk concentration of
the particles.
We assume that all particles are spheres of equal radius o, that the
physical density of the matter of the particles is much greater than the density of the
gas, Ps >> p ' and that the particles have no Brownian motion and do not influence each
other.
The Stokes force is used to describe the interaction between the continuous
media in the entire flow region, including the boundary layer.
As the estimates of
[3] show, the other components of the interaction force between the continuous media
must be taken into account when the physical densities of the particles and the carrier
phase are nearly equal.
We place the origin of a Cartesian coordinate system xy at the leading edge of the
plate, and direct the x axis along its surface.
In dimensionless form, the system of
equations describing the motion of the suspension takes the form
div V=O, div p:V:=O, (V' V) V§
§
L
l
L
ap:(V-V,) =-eAYL , (V:- V) V~ =--/-(V-V:)
(1.1)
IIere, the index s is appended to the parameters of the particle continuum, e =
~/v~p ~ l=mv~/6~G~, (m is the mass of a particle, v~ is the flow velocity at infinity,
is the viscosity of the carrier phase), a=ps~/p ~ and L is the characteristic length
Moscow.
4, pp. 4G-54,
512
Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti
July-August, 1980.
Original article submitted December lO, 1979.
0015-4628/80/1504-0512907.50 9
1981
P l e n u m Publishing
Corporation
i Gaza,
No.
c h o s e n to m a k e the c o o r d i n a t e s d i m e n s i o n l e s s .
T h e r e m a i n i n g n o t a t i o n is standard.
As
c h a r a c t e r i s t i c s c a l e s for m a k i n ~ the o t h e r q u a n t i t i e s d i m e n s i o n l e s s , we take v~ for the
o2
v e l o c i t y c o m p o n e n t s of the two phases, p v ~ for the pressure, and Ps~ for the d e n s i t y of
the p a r t i c l e c o n t i n u u m .
On
the
surface
of
the plate,
we m u s t
specify
the n o - s l i p
condition
for
the c a r r y i n g
phase:
k=v=O
In the unperturbed
(1.2)
flow,
u=~,=p,=l,
v=v,=O,
p=p~
(1.3)
~ h e n one is c o n s i d e r i n g f l o w over a plate, the c h a r a c t e r i s t i c l e n g t h is u s u a l l y
t a k e n to be equal to the d i s t a n c e from the edge of the p l a t e to the c o n s i d e r e d region.
S i n c e the task of the p r e s e n t i n v e s t i g a t i o n is to a n a l y z e the i n f l u e n c e of the d i f f e r e n c e
b e t w e e n the p h a s e v e l o c i t i e s on the s t r u c t u r e of the flow in the wall region, we set
L = Z in our d e r i v a t i o n of the b o u n d a r y - l a y e r equations, s i n c e i c h a r a c t e r i z e s the dec e l e r a t i o n l e n g t h of a p a r t i c l e w h i c h e n t e r s gas at rest w i t h v e l o c i t y v~.
W e shall a s s u m e that the v i s c o u s l e n g t h u/v~p ~ is m u c h less than i, i.e., ~ << I.
We c o n s t r u c t an a s y m p t o t i c s o l u t i o n of the s y s t e m (i.I) as ~ + 0.
The highest derivatives have a small p a r a m e t e r , and the s o l u t i o n of the d e g e n e r a t e p r o b l e m for s = 0
c a n n o t s a t i s f y the n o - s l i p c o n d i t i o n u = 0 on the p l a t e b e c a u s e of the r e d u c t i o n in the
o r d e r of the e q u a t i o n s .
In a c c o r d a n c e w i t h the m e t h o d of m a t c h e d a s y m p t o t i c e x p a n s i o n s
[7], we find first the e x t e r i o r e x p a n s i o n , i.e., an a s y m p t o t i c s o l u t i o n of (l.1) v a l i d
in the l i m i t E § 0 for f i x e d x and y.
In a c c o r d a n c e w i t h the p h y s i c a l m e a n i n g , the ext e r i o r e x p a n s i o n is a u n i f o r m h o m o g e n e o u s f l o w
9u = u ~ = p ~ = l , v=v~=O, p = p ~
We
To c o n s t r u c t the i n t e r i o r expansion, we
seek the i n t e r i o r e x p a n s i o n in the f o r m
introduce
(1.4)
the
stretched
k(x, ~ ) + . . . , k~(x, ~ ) + . . . , p~(x, ~ ) +
p(x, ~)+ .... Y~u(x, ~)+ . . . . ~ ( z ,
Substituting
ary-layer
(1.5)
in
(!.X)
Ok
Ov
Ox
a~I
Op~k~
=0,
Ox
...
(1.5)
~)+...
the principal
is
to
necessary
to
the parameters
4
ap,v,
o~1
--O,u
ok
+V
Ox
terms,
we o b t a i n
the bound-
O~l2
impose
the boundary
conditions
of the exterior
flow:
(1.2)
equations
can be
of boundary-layer
,
Op
0~1
=0
(1.6)
a----~=k--k~, k~--d-Z+v~ a q =v-v~
k,~,p~l
Similar
by the method
the particles.
o~I
O2a
+ ap,(k--u,)= ~
Or+
+v,
0~,
Ok
Or+
k,
It
q = ya -I/2.
equations
--+
matching
and r e t a i n i n g
variable
as
obtained
from
the
corrections
with
on
the
plate
and
require
~
(1.7)
boundary-layer
neglect
of the
equations
of [8] derived
volume
concentration
of
To construct
a unique
solution
to the system
(1.6),
which
is parabolic,
we must
specify
the initial
profiles
of the functions
u, Us, Os , v s . These
functions
must
satisfy
the conditions
of matching
to the asymptotic
solution
(i.I)
in the neighborhood
of the leading
edge
of the plate
(L - ~/v~p~
i.e.,
where
the assumptions
under
which
the boundary-layer
equations
are derived
are violated.
We write
down
the equations
for
the principal
terms
in the expansion
of the solution
in the neighborhood
of the leading
edge.
For this,
we set L = D/v~p
in (i.I)
and let ~ tend
to zero.
We obtain
the system
divV=O,
mining
It can be seen
the parameters
that
in
of the
divp~Vs=O, ( V . V ) V + V p = A V ,
the neighborhood
phases
separate:
(V~.V)V,=O
of the leading
edge
the
for the carrier
phase,
problems
we obtain
for
deter-
513
b
J
I
/
/
0.5
u, us 0
u,t 0
0.5
Fig.
O.,J v,~
0.5 u %
I
0
0.2 v,~
Fig.
Y
0
0.1 v,v~
2
Navier--Stokes
equations
with
Reynolds
number
equal
to unity;
for the particles,
taking
into account
the conditions
of matching
to the external
flow,
we obtain
Ps = Us = 1 and
v s = O.
As follows
from
the analysis
of the first
approximation,
the first
nonvanishing
term
in the expansion
of v s in this
region
has the order
c.
The solution
to the problem
for the carrier
phase,
as in a homogeneous
viscous
fluid,
is matched
in parabolic
coordinates
to the Blasius
solution
of the Prandtl
equations.
Thus,
for the boundary-layer
equations
(1.6)
it is necessary
to specify
the Blasius
profile
as the initial
conditions
for u, and for the particles
u s = Ps = 1 and v s = 0.
This
result
has a perspicuous
physical
interpretation:
at a distance
of the order
of the viscous
length,
interaction
of the
two phases
cannot
be manifested,
since
the deceleration
length
of a particle
is by hypothesis much
greater
than
the viscous
length.
2.
Results
of
For
pa__rabolic
~xvs,
X
Numerical
the numerical
coordinates
q:p,as, the system
oz
Solution
x
2
solution,
it is convenient
and C = ~/~
and introduce
(1.6)
then
becoming
x
ox
to rewrite
the
the new unknown
system
(1.6)
in
functions
w~v,
2
0 +~
(
the
w~:
)
02t~
(2.1)
Ow. + ( 2w. ~a~~ Ow.
x ( ~ - ~ ) , xu~ ~x
The boundary and initial conditions
:9~
u~w~ + x(w-w.)
! O~ '~=
2
are
u=w=O, ~=0, x~>0
u~=z~=q=l, ~=o% x~>0
zt~=q=l, w~=O, u=r
ttere, ~(~)
is the
boundary
conditions:
Blasius
function
[9],
which
x=O
satisfies
2~"'+~"=0
~(0) =~'(0)=0,
(2.2)
the
following
equation
and
(2.3)
r
=1
(2.4)
The system
(2.1)
with the boundary
conditions
(2.2)
was solved
numerically
on a
BESM-6 computer.
An i m p l i c i t
difference
scheme with order
of approximation
Ax + AG 2 w a s
use to approximate
the equations
on a rectilinear
mesh.
The sweep method was used to
solve
the difference
equations
in each x layer.
The final
calculations
were made with
mesh parameters
Ax = 0 . 0 0 1 a n d A~ = 0 . 0 1 .
As the results
of the numerical
calculations
show, the flow structures
for different
values
of ~ are of the same kind.
In Figs.
1 - 3 , we s h o w t h e r e s u l t s
of calculations
of
the parameter
profiles
for both phases
in the boundary
layer
for a = 3.
Figure
1 shows
the development
of the profiles
of the longitudinal
velocities
of both phases
with respect
to the coordinate
x, while
the broken
line
shows the Blasius
profile
in a gas without
particles.
Figure
2 shows the profiles
of the vertical
components
of the velocities
of
the phases.
The graphs
a), b), and c) correspond
to the values
x = 0.2,
1, !2, and the
514
indices
1 and 2 label
the profiles
of the carrying
phase
and the disperse
phase,
respectively.
Over
the complete
thickness
of the boundary
layer,
the longitudinal
velocity
of the particles
exceeds
that
of the gas,
and for x < 1 it is nonzero
on the surface
of
the plate.
The relaxation
of the longitudinal
velocities
of the phases
is terminated
effectively
at x = 5.
At small
x, there
exists
a range
of 5 values
for which
v s < v, i.e.,
the particles
intersect
the streamlines
of the gas along
the direction
toward
the plate.
At large
x,
we have
v s > v over
the complete
thickness
of the boundary
layer.
It should
be noted
that
the relaxation
of v and v s occurs
over
a much
greater
length
than
that of u and u s .
Figure
3 shows
the formation
of the density
profile
of the disperse
phase
with
the development
of the flow
along
x~ curves
1-3 corresponding
to the values
x = 0.2,
I, 12.
For
x < i, the density
of the disperse
phase
increases
monotonically
as the plate
is approached,
and on the wall
reaches
a value
equal
to Psw = i/(i
-- x).
For x ~ i, Ps tends
to infinity
as the plate
is approached.
With
the completion
of the relaxation
of the phase
velocities,
a strongly
inhomogeneous
profile
of Ps is formed,
and there
exists
a region
of 5 values
for which
Ps < I.
As can be seen
in Fig.
i,
both
phases
become
self-similar.
Blasius
profile
of the argument
equations
for homogeneous
fluid
In Fig.
4,
cf for the values
have
plotted
we show
~ = 0,
at
large
x the profiles
This
limiting
profile
z = ~(i + ~)I/2,
i.e.,
with
increased
density
of the longitudinal
velocities
of
(see Sec.
3) coincides
with
the
with
the solution
to the Prandtl
O = ! + ~.
the results
of calculations
of the
3, I0, 20 (curves
1-4,
respectively).
local
coefficient
of
Along
the ordinate
friction
we
Here, T w is the local t a n g e n t i a l s t r e s s on the wall, and Re is the local R e y n o l d s
number, in w h i c h the c h a r a c t e r i s t i c l e n g t h is t a k e n to be the v a l u e of the d i m e n s i o n a l
c o o r d i n a t e x.
For
the f l o w of a h o m o g e n e o u s
viscous
fluid
over
a plate,
cf=0.332/u
For Eqs.
(1.6),
this
case corresponds
to the absence
of an influence
of the particles
on the motion of the carrier
phase,
i.e.,
~ = O.
As can be seen in Fig.
4, the coefficient
of friction
for different
values
of a changes with increasing
x from the values
corresponding
to the Blasius
solution
in a gas without
particles
(the asymptotic
behavior
o f c f a s x § O) t o t h e v a l u e s
corresponding
to the Blasius
solution
in a homogeneous gas
with density
p = 1 + ~, i.e.,
to values
cj=0.332 (l+~)'~/~Re
(2.5)
(the asymptotic
behavior
of cf as x §
In the entire
flow region,
except
of very
small
x, the coefficient
of friction
exceeds
the value
corresponding
flow,
i.e.,
corresponding
to the Blasius
solution
for homogeneous
fluid
with
density.
3.
Asymptotic
Distances
Behavior
from
the
of
the
Leading
Solution
for
for the region
to frozen
increased
Large
Edge
To find
the asymptotic
behavior
of the solution
of the system
(1.6),
we construct
boundary-layer
equations
valid
in a region
far downstream
of the velocity
relaxation
region.
For this,
we set L >> l in the system
(I.i).
We denote
I/L = 6(s
with
6 § 0
as c § 0.
In what
follows,
we shall
label
all the variables
in this region
by the index
layer
By analogy
with
variable~l~YJ(86)
We
seek
the
the method
'~.
solution
within
set
the
forth
in
boundary
Sec.
layer
u,(x,, n,)+...., ~,(x,, ~,).+..., p,(x,, n,)+...
Substituting
ary-layer
equations
(3.1)
in
(i.I)
and
retaining
i,
we
in
introduce
the
a stretched
boundary-
form
(e6)'%(z,, n,)+...., (e6)'~v,,(x,, ~,)+...
the
principal
i.
terms,
we
obtain
the
(3.1)
bound-
515
0
!
2
I
t
%
g.5
t
2
Fig. 3
J
.r-
Fig. 4
Oui + Ovi = 0 ,
v . , 0p+t
--+v,
0p,t : 0 ,
O.5
0
Fig. 5
( l + a p ~ , ) (+, Ou, +
Oa, \ -------,
02ui
u+t=a,,
v+i~-u,
(3.2)
The b o u n d a r y c o n d i t i o n s
a r e a n a l o g o u s t o ( 1 . 2 ) and ( 1 . 7 ) .
Note that the coordinates
x 1 and x p l a y t h e p a r t o f e x t e r i o r
and i n t e r i o r
coordinates,
respectively,
for the flow
region that is not in equilibrium
with respect
to the velocities;
x 1 = 6x.
The s o l u tion to the system (3.2) for x 1 + 0 must be matched to the solution
(1.6) as x + ~, it__
/
i
being convenient
to carry out the matching in the variables
5 and x, since
~=~/]x=~],/~x~,
i.e.,
one moves from the region of nonequilibrium
flow along the parameters
0/v~x = c o n s t .
The f o r m u l a t i o n
o f t h e p r o b l e m ( 3 . 2 ) d o e s n o t c o n t a i n 1, an d t h e r e f o r e
the solution
must be self-similar.
We i n t r o d u c e
the variable
~=~/u
and s e e k t h e f l o w f u n c t i o n
in
the form ~ ( ~ x l ) = ~ , ( ~ )
.
From
(3.2), using the boundary conditions,
we obtain
(i+a)~cp~"+2~'"=O,
The boundary conditions
We introduce the function
we obtain f r o m
(3.3)
take the form
~,(0) = ~ / ( 0 ) = 0 ,
For F(z),
p~=l
q,'(~) =i
(3.4)
F(z)=(1+~)v'~,(~), z=~(l+~) '~.
(3.3) and (3.4)
the Blasius
law
(2.3)-(2.4).
For the velocity profile ul(~) we have
a,(~)=~,'(~)=F'(z)
(3.5)
Therefore, the solution Ul(5) to the system (3.2) is identical to the self-similar
Blasius solution of the variable z, i.e., to the solution of the Prandtl equations in a
homogeneous fluid with density 9 = 1 + ~.
The coefficient of friction in the considered
region is found f r o m (3.5) and has the form of (2.5).
The numerical calculations of the profiles of the longitudinal velocities of the
phases and the coefficient of friction in the nonequilibrium region of the flow confirm
the asymptotic solution that we have found;
thus, already for x = 12 the profile u(~)
in Fig. 1 virtually coincides with (3.5).
The self-similar solution to the system (3.2) with constant density is obviously
not uniformly applicable in the considered flow region, since an essentially inhomogeneous
profile of the disperse phase in the wall region follows from the results of the numerical
calculations (Fig. 3).
This fact indicates the existence of a "thinner" (in the asymptotic sense) region in which the solution of (3.2) with P s l = 1 is not valid.
We shall
call this region the nonself-similar boundary layer.
We shall denote the flow parameters
--V2 "~14
in this region by the index 2. We introduce the new stretched coordinate ~2--y~6
6 .
We shall
seek the solution in this region in the form
6 ~ ( x , , ~D+..., 6v'u~(x,,
516
~ ) + . . . . p~(x,, n ~ ) + . . .
~'~'6~
qD+...,
~v~6~
~)+ ...
(3.6)
T h e orders of m a g n i t u d e
the other regions.
Substituting (3.6)
we obtain the system
__
are chosen to be consistent
in (l.1) and retaining
~v~
Ou2.+
=0,
Ox~
0~2
matching
Taking
of
into
u 2 to
0p~2 ,
a2~vo
Ox~
account
the vanishing
u I as ~2 § ~' we find
~=~,(0)q~(t•
and m a t c h the solution in
the equations
0p~z
=0,
" 0~
02~
O~zz
of u 2 and v 2 on
from
(3.7)
--0,
the
for the principal
az=u~2,
plate
and
follows
from the
last
the
equation
in
(3.8)
that
the value
of
Ps2 i s
(3.7)
condition
ap,~
4x~ 0 ~
axr
It
v2~v~2
v2=~,,(O)~2~(t+a),~!xc~/4, ap~2 ~. ~
'
terms,
of
0
(3.8)
conserved
along
the curves
~2x71/4~
= const.
The value
of Ps2 must
satisfy
the condition
of matching to ~
in the region
of flow with
velocity
disequilibrium.
Since
the
matching
is conveniently
done
in the variables
@ and x as x I § 0 and x + ~, respectively.
Thus,
the density
profile
calculated
from
(1.6)
in the variables
~ and x must
go over
into
the self-similar
profile,
which
is the initial
condition
for Ps2'
from
the nonselfsimilar
boundary
layer.
This
result
is confirmed
by the numerical
calculations.
In Fig.
5, we show
the profile
i/Ps(@ ) calculated
for a = 5 with
x = 12;
for x > 12, the results
of the calculations
are virtually
coincident
with
the plotted
graph.
~2x]I/t=~x-'~=O,
Note
that
the influence
of an inhomogeneous
at large
x does
not have
a strong
influence
on
coefficient
of friction
in the region
in which
ended
can be calculated
approximately
from
the
particles.
density
profile
in the boundary
layer
the coefficient
of friction,
i.e.,
the
the velocity
relaxation
has effectively
Blasius
solution
for frozen
motion
of the
In the case
when
the mass
concentration
a of the particles
is negligibly
small,
all
the results
simplify
significantly
(it is sufficient
to set a = 0 in all the obtained
expressions).
It is found
that
the particles
do not affect
the motion
of the gas,
and
the Blasius
solution
in a gas without
particles
holds
for the carrier
phase.
I thank
V.
P.
Stulov
for
interest
in
the
work
and
helpful
recommendations.
LITERATURE CITED
1.
2.
3.
4.
5.
6.
7.
8.
9.
H. H. C h i n , " B o u n d a r y l a y e r f l o w w i t h s u s p e n d e d p a r t i c l e s , "
Princeton Univ. Dept.
A e r o n a u t . E n g . R e p t . , No. 620 ( 1 9 6 2 ) .
Soo S a o - L e e , F l u i d D y n a m i c s o f M u l t i p h a s e S y s t e m s , B i a i s d e l l ,
Waltham, Mass. (1967).
R. E. S i n g l e t o n ,
"The c o m p r e s s i b l e g a s - s o l i d
particle
flow over a semi-infinite
flat
plate,"
Z. Angew. M a t h . P h y s . , 16, No. 4 ( 1 9 6 5 ) .
F . E. M a r b l e , " D y n a m i c s o f a g a s c o n t a i n i n g
small solid particles,"
in: Combustion and
Propulsion,
Pergamon Press, Oxford (1963).
L. S. Soo, " N o n - e q u i l i b r i u m
fluid dynamics-laminar
flow over a flat plate,"
Z. Agnew.
M a t h . P h y s . 1_99, No. 4 ( 1 9 6 8 ) .
R. I . N i g m a t u l i n , F u n d a m e n t a l s o f t h e M e c h a n i c s o f H e t e r o g e n o u s M e d i a [ i n R u s s i a n ] ,
N a u k a , Moscow ( 1 9 7 8 ) .
J . D. C o l e , P e r t u r b a t i o n
Methods in Applied Mathematics, Blaisde!l,
Waltham, Mass. ( 1 9 6 8 ) .
V. P. S t u l o v , " E q u a t i o n s o f a l a m i n a r b o u n d a r y l a y e r i n a t w o - p h a s e m e d i u m , " I z v . A k a d .
Nauk SSSR, Mekh. Z h i d k . G a z a , No. 1 ( 1 9 7 9 ) .
H. S c h l i c h t i n g ,
B o u n d a r y L a y e r T h e o r y , M c G r a w - H i l l , New Y o r k ( 1 9 6 8 ) .
517