~kcAmi~ci K
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
TECHNICAL MEMORANDUM 1275
THE SOLUTION OF THE LAMTNARBOUNDARYLAYER EQUATION FOP
THE FLAT PLATE FOR VELOCITY AND TEMPERATURE FIELDS
FOR VARIABLE PHYSICAL PROPERTIES AND FOR THE
DIFFUSION FIELD AT HIGH CONCENTRATION
By H. Schuh
SUMMARY
In connection with Pohlhausen's solution for the temperature field
on the flat plate, a series of formulas were indicated by means of
which the velocity and temperature field for variable physical charac
teristics can be computed by an integral equation and an iteration
method based on it. With it, the following cases were solved: On the
assumption that the viscosity simply varies with the temperature while
the other fluid properties remain constant, the velocity and tempera
ture field on the heated and cooled plate, respectively, was computed
at the Prandtl numbers 12.5 and 100 (viscous fluids). A closer study
of these two cases resulted In general relations: The calculations
for a gas of Pr number 0.7 (air) were conducted on the assumption that
all fluid properties vary with the temperature, and the velocities are
low enough for the heat of friction to be discounted. The result was
a thickening of the boundary layers, but no appreciable modification
in shearing stress or heattransfer coefficient. The effects of
density and viscosity or density and heat conductivity have opposite
effect for velocity and temperature field and almost cancel one
another. Formulas allowing for the heat produced by the friction were
indicated, but no calculations were carried through in view of the
already existing report by Crocco. The methods of solution developed
here were finally applied also to the case of diffusion of admixtures,
where at higher concentration finite transverse velocities occur at the
wall.
*"Uber die Losung der laminaren Grenzschichtgleichung an der
ebenen Platte fur Geschwindigkeits und Temperaturfeld bel
veranderlichen Stoffwerten und fur das Diffusionsfeld bel hoheren
Konzentrationen.4 Zentrale fir wissenschaftliches Berichtswesen der
Luftfahrtforschung des Generalluftzeugmei sters (ZWB) BerlinAdlershof,
Forschungsbericht Nr. 1980, August 18, 1944.
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I. INTRODUCTION
The laminarboundarylayer equation for the flat plate lined up
with the flow and with constant fluid properties was solved by Prandtl
(reference 1) and Blasius (reference 2) for the flow field and by
Pohlhausen (reference 3) for the temperature field. Since temperature
and velocity field coincide when kinematic viscosity (V) and tempera
ture conductivity (a) are identically equal Pr= y = ), Pohlhausen's
formula for the temperature field contains a solution for the velocity
field also in the form of an integral equation. Piercy and Preston
(reference 4), proceeding from a rough approximation, indicated that,
with the aid of this integral equation and an iteration method, the
wellknown Blasius solution can be obtained in a few steps. This
method of solution has the advantage of being simple and requiring
relatively little time. It is as is shown in the following 
particularly suitable for boundarylayer calculations involving
variable fluid properties, because a first, and usually fairly close,
approximation, is already available in the solution for constant fluid
properties.
Crocco (reference 5) and von KArmAn and Tsien (reference 6) (see
also reference 9, 10) computed velocity and temperature field for
variable fluid properties. In both reports, the differential equations
are put in a different form from the elsewhere conventional boundary
layer calculation by changing to new variables. Crocco obtains two
simultaneous differential equations of the second order which he solves
for a gas with the Prandtl number Pr = 0.725 (air). Von Karmnn and
Tsien treat the case of Pr = 1 and have to solve only one differential
equation, since then the temperature is related in a simple manner to
the velocity.
In the following, it is shown that a number of boundarylayer
problems for the flat plate can be solved in a comparatively simple
manner, involving merely quadrature, by means of the cited integral
equation and an iteration method.
II. SOLUTION OF BOUNDARYLAYER EQUATION FOR VARIABLE PHYSICAL PROPERTIES
The boundarylayer equations for velocity and temperature field at
the flat plate at variable density read (reference 3)
Cu bu = / ou
p aru +p (1)
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c (ou) + Cp = (la)
ax 3Y
pT v (r)r (2)
with u and v the speed in flow direction and at right angle to it,
T, the temperature, x the distance from the plate leading edge,
y the distance from the wall, o the density, P the viscosity, Cp the
specific heat, and X the heat conductivity. In the equation for the
temperature field, the heat produced by friction is, at first, not
taken into account; as long as the speeds are not excessive and the
temperature differences not too small, this is justified.
For constant density, equations (1) and (2) can be reduced to an
ordinary differential equation (reference 3) on the assumption that u
and T are a function merely of the one dimensionlesss)
i
coordinate g = Since the density depends only on the tempera
ture, the idea suggests itself that the same simplification is possible
also for variable density. We put
U T T) 1 (o (3)
where U is the velocity at the edge of the boundary layer,
To and T1 the wall temperature and the temperature at the edge of the
boundary layer, respectively. The quantity Vk in the dimensionless 5
denotes the kinematic viscosity for the fixed temperature Tk, for
which in suitable manner the wall temperature (k = 0), or the tempera
ture at the edge of the boundary layer (k = 1), is chosen. The
boundary conditions for flow and temperature field read
y=0 =0 m=0 0 =
(4)
y > = 1 0 = 1
Putting
p( x (5)
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where the subscript k denotes the density at temperature Tk, gives
by (la)
pv out ou dt (6)
hence, by (1), after introduction of (5) and (6), the more suitable
form
_Ld (/W f = 2j dt (7)
From (7), regarded as differential equation for the quantity cp and
f temporarily as a known function of e, the following expression for
a) is derived
J(f) 1 0 d (8)
J(m) = O, T
This disposes of the integration constant from consideration of the
boundary condition (4). Likewise, there Is afforded for the dimen
sionless temperature 0 the expression
K(E) Prk I
S= K) e di (9)
K(0) X
where Prk is the Prandtl number with the density at temperature Tk.
For constant density (rp = = X = 1), velocity and temperature
field are independent of each other and (9) gives the Pohlhausen
expression (reference 3) for the temperature field, which represents
the solution for the velocity field at Pr = l(V = a). When the
velocity field is known, the solution for the temperature by (9) is
obtainable by simple quadrature. But the calculation of the velocity
field runs into difficulties, at first, because in (8) the still
unknown velocity appears on the righthand side In the expression
for f. The methods of solution by Piercy and Preston proceed from a
random approximation for a with which f and J(t) in (8) are
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computed. The improved value m obtained forms the starting point for
the next step, etc. Figure 1 represents the several steps of this
approximation method. The intentionally rough approximation c = 1
over the entire boundary layer was chosen as original solution; the
corresponding first approximation c is given by the error integral.
After the third approximation, the shearing stress shows a mere
difference of h.5 percent from the exact value. Instead of continuing
the process mechanically, the final solution to be expected was
estimated from the variation of the previously computed approximations
and utilized as basis for the subsequent step; the solution wu
contained but a percent error in shearing stress.
2
With this method of solution, the improvement effected by each
step can be estimated according to order of magnitude. The
equations (8) and (9) are identical for constant density and Pr = 1.
Assuming that the approximate solution for w was such that for each
individual value w the corresponding t coordinate differed by a
constant factor e from the E coordinate of the exact solution, the
effect of factor X is then obviously just as great as that of
quantity Pr for the temperature field. Pohlhausen found, on the
basis of his numerical calculations, that the heattransfer coefficient
is proportional to \Pr, thus the shearing stress at the wall is
afflicted at each new step by an error of only about onethird of the
error of the preceding step.
For variable density, the discussed solution steps of "mathe
matical nature" can be combined with the steps of physical nature :
Step 1: as starting point the known solutions for constant
density are assumed:
(a) The Blasius solution (reference 2) for the velocity
profile
(b) Pohlhausen's method for the temperature field
Step 2:
(a) Calculation of velocity profile by (8), the temperature
variation being based on the density of the tempera
ture profile according to step l(b)
(b) Calculation of temperature field by (9) with the velocity
profile according to step 2(a); relation of density to
temperature as in step 2(a)
lit took a subsidiary worker 10 hours to reach the final solution of
the velocity field in figure 1.
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The process is repeated till the final solution is sufficiently
exact, usually requiring three to four steps.
A few general remarks about the influence of the temperature
variability of the physical properties. The flow with constant
physical properties can be regarded as first approximation, and the
problem is then to ascertain the differences which are produced by
variable physical properties. The quality of this approximation
depends, of course, on the temperature assumed for the physical
properties at the isothermal" flow. Choosing the wall temperature or
the temperature at the edge of the boundary layer as reference tempera
ture for the isothermal flow so results on the basis of physical
point of view as well as on the basis of the equations that an increase
of the viscosity or density inside the boundary layer is accompanied by
an increase in the resistance; similarly, an increase in heat con
ductivity and density effects a greater heat transfer. But the
magnitude of the effect of variability of the separate physical prop
erties is contingent upon the ratio of the boundarylayer thickness of
the temperature and velocity field. (The ratio of both is proportional
according to Pohlhausen.)
This is illustrated by the following case, which is, at the same
time, of practical importance. The temperature boundary layer is
assumed very small compared to the flow boundary layer; consequently,
the variation of the physical properties within the thermal boundary
layer can be disregarded for the shearing stress and the latter
computed as if the temperature at the edge of the boundary layer
reaches to the wall. The same holds for the velocity profile, with the
exception of a small area within the thermal boundary layer, where the
velocity profile by the viscosity variation is deformed correspondingly.
But for the temperature profile this area is exactly decisive.
From the equality for the shearing stresses the velocity gradients
at the walls are:
[l o1 for Pr ' (10)
the subscript 11 denotes the isothermall" flow with the physical prop
erties at temperature Tl. The variability of density is noneffective for
the field of flow, in this instance. It can be mathematically derived from
the formulas (8) and (9). The ratios for the temperature field are
discussed in the next chapter by means of the two examples.
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TTI. FLOW AND TEMPERATURE FIELD FOR VISCOUS FLUIDS
In accordance with the physical properties of viscous fluids, the
velocity and temperature field were computed on the assumption that
only the .viscosity should change with the temperature by the following
formula
(Tkk Tcb
where b and Tc are constants, chosen so as to reproduce the tem
perature variation as closely as possible. The subscript k is to
be 0 or 1, depending upon the choice of the physical properties
in the dimensionless t The choice was b = 3 (viscous lubricating
oil) and the two cases of a heated and cooled plate computed with
 and 8 and Pr = 12.5 and 100; it thus concerned identically
great temperature differences of the same fluid, since Pr is
for the present formed with the physical properties at wall temperature.
Choosing To as reference temperature gives by (11)
= .M.= 3 (12)
o 1 +i
The result of the calculation by the iteration method of the preceding
section is seen in figures 2 and 3. In both graphs, the dimensionless
wall distances o and l1, formed with o and ~1, are plotted to
the scale 1:V and V:1, respectively, so that the actual wall
distance y is the same for both abscissas. Besides the solution m,
which took three steps to compute, the isothermal velocity profiles
(0)) and (Um)1 at constant density at temperature' T and T1 are
shown plotted against the dimensionless coordinates o and 1".2
In the subsequent compilation To and a denote the shearing stress
at the wall and the heattransfer coefficient ( = )o );
2
For the isothermall" temperature profiles (e) and (0) the
Prandtl numbers at temperatures T and T1 must be inserted. For
example, in figure 2: Pr = 12.5 and Pr = 100.
o 1
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(To and (a)) are the corresponding values in isothermal flow with
the viscosity at temperature T1 and T .
Table I
oo oC
(T0 a P( / d0 \
Heated wall 0.841 1.20 12.5 1.58 1.84
Cooled wall 1.08 0.98 100 0.255 3.01
Although in both cases the thermal boundarylayer thickness is far
from small compared to the flow boundary layer, the shearing stress
can still be computed satisfactorily by the isothermal formula with the
viscosity of the wall temperature. The assumptions to equation (10)
are thus shown for Pr > 10.
The conditions are more complicated for the heattransfer
coefficient; from (9), it follows that the heattransfer coefficient
a is proportional to a(Pr) ~\ wherein, according to Pohlhausen,
a is, with high accuracy, assumed as 0.664 V ;. Bearing in mind that
Pr = V it follows that the heattransfer coefficient is inversely
a)
proportional to the sixth root of the viscosity. Since all physical
properties except the viscosity have been assumed constant, there
results, when it is referred once to the wall temperature, the other
time to the temperature at the edge of the boundary layer
()o 1/6 (13)
A comparison with the foregoing tabulation indicates that (a)1
supplies a poorer approximation for the heattransfer coefficient
than (a)o; this is readily explained by the variation of the velocity
profile (figures 2 and 3). It is to be expected that the conditions
are similar at higher Prandtl numbers.
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Another reference point for the heattransfer coefficient is found
in the velocity gradient at the wall; by (10) and allowing for (3),
there follows
o I1 (14)
pL 11 Po oL 1 o
the subscripts io and il denoting isothermal flow at tempera
ture To and T1. These relations are confirmed in figures 2 and 3.
With these formulas, limits can be indicated for the heattransfer
coefficients (figures 2 and 3). One is given according to (13)
by (a)l, because the velocity profile (wu) yields at all points
higher velocities at cooled and lower velocities at heated wall. The
other limit is given by a velocity profile of isothermal form, where
the abscissa scale is so modified that its gradient at the wall agrees
with the actual velocity distribution. From the remark about the
convergence of the method of solution in II, it follows then that the
heattransfer coefficient is proportional to the third root of the
velocity gradient at the wall; for this extreme value, the second
equation of (14) gives: ( (a)o. Summed up, the limits of the
heattransfer coefficients, by a change in viscosity, are
(a.) a (15)
the upper signs applying to heated, the lower to cooled wall.
Hence, the following approximate rule for viscous fluids (Pr>10):
For computing the resistance, the physical properties are referred to
the temperature at the edge of the boundary layer; for heat transfer,
to the wall temperature.
IV. FLOW AND TEMPERATURE FIELD AT Pr = 0.7 (AIR)
FOR TEMPERATURE VARIABILITY OF EVERY PHYSICAL PROPERTY
In the 500 to 1400 temperature range the physical properties
of the air can be represented by the following formulas
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= ITO .780
p = K T1
2
X = K T0"821
3
T = temperature in absolute degrees. With
T T1
T1
there results
= = [ +(i 078
and similar expressions for r and X.
1 1
The calculations for a heated plate and = and 2 showed
only moderate differences in the velocity and temperature field from
the form for isothermal flow (Table 2). For the investigation of the
conditions at higher temperature differences, the case T1 = 20
and To = 6200 C was computed. The velocity and temperature fields
already exhibit, according to figure 4, appreciable differences from
the form for constant physical properties; to and l1 are formed
with the physical properties at temperatures To and T1, respectively.
This results in a substantial thickening of the boundary layer for both
fields; nevertheless, wall shearing stress and heattransfer coefficient
indicate only minor departures from the values for constant physical
properties.
Table 2
_/cku \0 \e 'o '''o a a,
Heating ( d) To T a a
1 0.575 0.90 1.02 1.00 1.01 .00o
S=1 .514 .420 1.05 1.00 1.02 1.00
To = 620 C
.286 .235 1.11 .93 1.03 .96
T1 = 200 C
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The explanation for it is that in air the growth of the viscosity with
the temperature acts in the sense of a resistance increase, the drop
in density in the sense of a resistance decrease, and both effects
practically cancel one another at Pr = 0.7, where thermal and flow
boundarylayer thickness are about equally great. The conditions for
the temperature field are almost identical, because the heat con
ductivity and the viscosity are similarly affected by the temperature.
The frictional heat can be allowed for in similar manner;
equation (2) contains then an additive term 4 C
side, and the solution reads
S1+B()Ag() B(S )
A()
A(t) = f leR() d
Pr, AT,, tT
B(t) = 2 Prk e '
(Tk To) T
R() = Prk dt
k 0 X
on the righthand
~. e'2 ep( d e R() d
e 2cp
(16)
The iteration method can be applied again, although a little more
paper work is involved. For constant physical properties, equation (16)
reduces to Eckert's solution (reference 8). The thermometer problem
(vanishing temperature gradient at the wall) can also be solved by
suitable variation of the integration constant. In view of Crocco's
calculations for a gas with Pr = 0.725, it was decided not to cal
culate any model problems by the new method.
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V. APPLICATION TO A DIFFUSION PROBLEM
The concentration field for the problem of diffusion at the flat
plate can be calculated in the same manner as the temperature field.3,4
The differential equation reads
3c + v = k 2c (17)
u vx ,y 6y2
where k is the diffusion factor and c the concentration which is
defined as quantity of gas or vapor per unit volume. The physical
properties are regarded as constant, but it is also taken into account
that for greater concentrations the velocity v at the wall no longer
disappears, as already pointed out by Nusselt (reference 7). When
fluid from a wall is vaporized, say by a gas such as gas flowing along
a wetted wall, substance passes continuously into the flow.
Hence v(O)>0 at the wall. When, on the other hand, vapor condenses
at the wall or when air containing ammonia, for example, passes over
blotting paper impregnated with hydrochloric acid, it results
in v(0)<0., The boundary conditions for v are according to the
equations (100) and (101) of reference (7);
k Jc 1 v(0) (18)
co\ io 2 1
Po
where c is the concentration of the gas or vapor, for which the wall
is permeable, co the concentration at the wall, p the corresponding
partial pressure, and p the total pressure.
Introduction of the flow velocity U and the dimensionless 5
results in
(vj 1 k cl c dC c co (19)
U 2 co  x c co
\Po
3Eckert reported a solution of this problem at the 1943 meeting
of the VDT committee for heat research in Bayreuth, where an approxima
tion method similar to Pohlhausen's method for the flow boundary layer
was used.
4 "
Damkohler's estimate for the present problem was published in
Z. fur Elektrochemie, 1942, p. 178.
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where co and cl are the concentration at the wall and at the edge
of the boundary layer. Similarly to (9), it gives
vU v 2]
E w d [W 2
and similarly to (9), the solution for the concentration field
(f( )M)dt
L(6) = 0
(20)
f(E) = 2 ud
J0 U
where 1 is a quantity analogous to the Prandtl number.
k
velocity w, simply put 1 = 1 in equation (21).
k
The concentration gradient at the wall is contained
can, in the first instance, be solved for any M values
N = l o computed with the aid of the value
o The velocity and concentration
the solution for %.) The velocity and concentration
*t 0
calculated M
dt
(21)
To obtain the
in M; but (21)
and the quantity
obtained from
fields for the
and N values are represented in figures 5 and 6, the
concentration gradient at the wall, in figure 7. M >0 denotes evapora
tion at the plate; M<0, condensation and absorption at the plate; the
value 0.6 chosen for the quantity 1 is applicable in good approxima
k
tion for the diffusion of water and ammonia in air. Strictly speaking,
for the specified higher concentrations, the density and viscosity of
the two fluids are dependent on the concentration; and the diffusion
factor, on the temperature. Cases of that kind can be calculated
with the aid of the described method. If the diffusion is
5According to Ten Bosch: Die Warmeubertragung, Berlin 1936,
pp. 189 and 257.
M= k cl co LCo
Vc o o d
S 
0 I
k
LFT
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accompanied by a heat transfer, the solution for the concentration field
can equally be applied to the temperature field with good approximation.
In the same way, the heat transfer can be derived from the solution for
the concentration field, when air is exhausted or blown at the plate
with transverse velocities at the wall corresponding to equation (20).
CONCLUDING NOTE
After completion of the calculations the writer received knowledge
of a report by Schlichting and Bussmann (reference 11) about the
velocity profile at the flat plate for exhaustion where the transverse
velocity at the wall was expressed by
(x) = 
Between the present value M and C the following relation exists
(see also (19) and (20)).
C =M
The present velocity distributions agree to about 1 percent with those
calculated by Schlichting (by a different method), with exception of
M = 1, where the writer plainly chose too few approximation steps and
the differences are therefore a little greater. For the present
calculation three steps were usually sufficient.
Translated by J. Vanier
National Advisory Committee
for Aeronautics
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REFERENCES
II
1. Prandtl, L.: Ueber Flussigkeitsbewegung bel sehr kleiner Relbung.
Verhandl. d. III. Intern. Math. Kongr. Heidelberg 1904, p. 484.
(Available as NACA TM 452.)
I!
2. Blasius, H.: Grenzschichten in Flussigkeiten mit kleiner Reibung.
Z. Math. Phys. Bd. 56, 1908, p. 1,. (Available as NACA TM 1256.)
3. Pohlhausen, E.: Der Warmeaustausch zwischen festep Korpern und
Flussigkeiten mit kleiner Reibung und kleiner Warmeleitung.
ZAMM Heft 2, 1921, p. 115.
4. Piercy, N. A. V., and Preston, J. H:: A Simple Solution of the
Flat Plate Problem of Skin Friction and Heat Transfer. Phil.
Mag. May 1936, p. 995.
5. Crocco, L.: Sullo strato limited laminare nel gas lungo una parete
piana. Rend. Circ. Mathem. Palermo Bd. LXITI, 1940/41.
6. Von Karman, Th., and Tsien, H. S.: Boundary Layer in Compressible
Fluids. Jour. Aero. Sciences, Vol. 5, No. 6, 1938, p. 227.
7. Nueselt, W.: Warmeubergang, Diffusion und Verdunstung. ZAMM
Band 10, Heft 2, 1930, p. 105.
8. Eckert, E., and Drewvtz, 0.: Der Warmeubergang an eine mit grosser
Geschwindigkeit langs angestromte Platte. Forsch. Ing.Wes.
Bd. 11, 1940, p. 116. (Available as NACA TM 1045.)
9. Hantzsche, W., and Wendt, H.: Zum Kompressibilitatseinfluss bei
der laminaren Grenzschicht der ebenen Platte. Jahrb. 1940 der
Dtsch. Luftfahrtforschung, p. I 517.
10. Hantzsche, W., and Wendt, H.: Die laminare Grenzschicht der ebenen
Platte mit und ohne Warmeubergang hunter Berucksichtigung der
Kompressibilitat. Jahirb. 1942 der Dtsch. Luftfahrtforschung,
p. I 40.
11. Schlichting, H., and Bussmann, K.: Exakte Losungen fi die lamlnare
Grenzschicht mit Absaugung und Ausblasen. Schriften der Dtsch.
Akademie der Luftfahrtforschung, Heft 2, 1943, P. 25.
NIACA TM 1275
SL0,
126
/.0 2.0 3.0
Figure 1. The several approximations for computing the velocity
distribution at the flat plate by the method of Preston and Piercy
(constant physical quantities).
/.0 2.0 3.0 = YF vo
2 0 vx
Figure 2. Velocity and temperature distribution at a heated plate
for variable viscosity. Viscosity exponent b = 3. (u)o, (e)o'
and (u)1, ()1 isother,.al velocity and temperature distributions,
v and v1 kinematic viscosity at wall temperature To and
temperature T1 at the edge of the boundary layer.
U
/.0
0.6
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02 0.4 0.6 = o.
0 A onA
Figure 3. Velocity and temperature distribution at the
Viscosity exponent b = 3. (u)o, (e)o, and (w)1, (e)l
velocity and temperature distributions, vo and v,
viscosity at wall temperature To and temperature
the edge of the boundary layer.
cooled plate.
isothermal
cinematic
T1 at
I.O 2.0o _?o v,
I 2 xv,
Figure 4. Velocity and temperature distribution at a heated plate
for Pr = 0.7 (air); all physical quantities constant with
temperature.
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b. 2.0
0.6 ______
2 0 51,____ __A 
Figure 5. Velocity field at diffusion with higher concentrations, where
finite transverse velocities occur at the wall (see text for equations (18)
to (20)).
to (20)).
Figure 6. Concentration distribution to figure 5.
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2.0 /.
AM dC k v p
\ k c,co (C
Mo
dC . ...
d ^?'0
1.0 0. 5
(dC
I/ / I3 4 5 6 7
/.o 0.5
Figure 7. Concentration gradient at the wall and quantity M plotted
against N (see text for equations (19) and (20)).
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