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Heat or mass transfer from single spheres in a low reynolds number flow
ht. 1 En@g Sci Vol. 20, No. 7, pp. 817-822, 1982 Printed in Great Britain.
CU2~7225/82/07l?417&$4l3.lWO @ 1982 Pergamon Press Ltd.
HEAT OR MASS TRANSFER FROM SINGLE SPHERES IN A LOW REYNOLDS NUMBER FLOW P. 0. BRUNN Department of Chemical Engineering and Applied Chemistry, Columbia University, New York, NY 10027, U.S.A. Ahatraet-The problem of forced heat or mass transfer from a single sphere in low Reynolds number flow is studied. In Stokes flow the Nusselt number N can depend only upon the Peclet numher P. For P < 1 a general “master-formula” is derived which involves Stokes resistance F,. On the other hand, the P B 1 limit requires different formulas, depending upon whether the tangential velocity vanishes at the sphere surface (rigid sphere) or not (slip flow and droplets). Although these differences do not disappear when first-order inertia effects are included, it is found that inertia enhances the heat or mass transfer in either case.
1. INTRODUCTION THE PROBLEMof
heat or mass transfer from a sphere into a low Reynolds number flow is of importance in many extraction processes: spray drying, aerosol scrubbing, meteorological studies and diffusive intake of swimming bacteria. Consequently, numerous theoretical studies have been concerned with that problem [l-5]. As long as the particles are small (< 100 p) so that the particle Reynolds number Re is small, inertia effects will be negligible. Then the Nusselt number N can depend only upon the P&let number P, defined as being equal to the product Reu, where o denotes the Prandtl number in case of heat transfer or the Schmidt number in case of mass transfer. Although for P < 1 the problem is diffusion dominated, a regular perturbation expansion around the pure diffusion limit P = 0 does not apply, and one has to resort to a singular perturbation expansion. This has been done for a rigid sphere [l], a rigid sphere with slip [3] and for a sphere immersed in an inviscid fluid[4]. With inertia neglected, the velocity field obeys the Stokes equations of creeping motion, for which Lamb[6] gives the most general solution for translation of a sphere. Lamb’s solution not only embraces each of the flow fields used in the studies just cited, but applies to other situations (e.g. droplets) as well. Consequently, the question arises as to whether we could not use Lamb’s solution to obtain for small Peclet numbers a single “master formula” for the N-P relation, which would cover all the possible cases. Actually, the same question can be posed for arbitrary Peclet numbers. However in the limit P + m, which can be investigated by means of a boundary layer theory, the results for a rigid sphere[l, 21 show an N-P relation totally different than the one found for droplets [7]. Consequently, it is important to understand the transition from one formula to the other. Since the neglect of inertia requires that Re +O, the high Peclet number limit P 9 1 is of rather academic interest, since it requires u+m. In order to weaken this restriction, we extend the P < 1 analysis into the region where Re is small but non-zero. 2. FORMULATION
Steady
state
fluid is governed
heat or mass
transfer
OF THE PROBLEMS
from an isolated
stationary
sphere
to an incompressible
by the equation
V2T= Pv;
T,
for which we adopt the boundary conditions T=l
T+O
atr=l,
(2a)
as r+m.
(2b)
817
818
P.O.BRUNN
Here T denotes a normalized temperature or concentration and P a P~clet-num~r based on the sphere radius a. The position vector I, measured from the sphere center is scaled with Q, while the velocity v has been made dimensionless by means of the free stream velocity. For the case in which the fluid at infinity is streaming past the sphere in the negative z-direction as t+w,
W-ii,
(3a)
In addition, we know that
-&v=o u, = 0
for t> 1, at r= 1.
(3c]
We can use Lamb’s genera1 solution[41 to deduce the Stokes stream function for a Newtonian fluid (4) where FO is the Stokes resistance of the sphere, made dimensionless by the factor 6ar)a, such that FO= 1 for a rigid sphere. Other cases which are of interest are
If h-r denotes the viscosity of the drop relative to the suspendi~ expressed in terms of the viscosity ratio h is
fluid the force FO
l+;A Fo=
l+h.
(5)
(b) Sfip pow If /I denotes Lamb’s coefficient of sliding friction [6] and (Ythe dimensionless parameter $a& measuring the extent of slip (a = 0 for the adherence case, a +m for perfect slip), then[8]
(c) Potential flow In this case the condition F. = 0
(7)
holds. Although this leads to d’Alembert’s paradox, the potential flow approximation does have some justification in the context of heat transfer for low Prandtl number tluids such as liquid metals [4]. In order to obtain the (d~ensio~ess) heat or mass transfer rate from the sphere, i.e. the Nusselt number
we have to solve eqn (11, which we can only do analytically for the limits P-0 respectively.
and P *co,
3.THE LIMIT P+O
Acrivos et al. [l] were one of the tirst to realize that a regular perturbation scheme could not be used to treat eqn (1) for low Peclet numbers. Although developed for a rigid sphere in the no
Heat or mass transfer from single spheres
819
slip limit (FO= l), their method of solution can immediately be used for our case since the velocity field retains the general form (4) no matter what value F0 actually assumes. This being the case, we merely list the final result (9a) with
and y = 0.5772156.. . the Euler-Mascheroni constant. Equation (9) reduces for F0 = 1 to the Acrivos-Taylor expression[ 11. In case of potential flow all the terms in In P drop out and we recover Sano’s result[4]. For slip flow, our result differs from Taylor’s[3]. We feel confident that our result is correct, the more so since (a) Taylor deduces from his results an incorrect potential flow limit and (b) eqns (9) have been obtained twice by totally independent ca1culations.t Besides being applicable in known cases eqn (9) can also be used for new ones, e.g. for droplets. Figure 1 illustrates the predictions. While for potential flow the increase of N with P is always less than linear, for droplets (including gas bubbles and rigid spheres) it is less than linear for extremely low flow speeds but larger than linear for P in the neighborhood of one. 4. THE LIMIT P+m
In this case the temperature or concentration gradients will be large near the sphere surface and a boundary layer theory seems appropriate. Thus, putting r=r-1,
(10)
the boundary layer approximation of (1) reads
a2 $T=sin8
-P [
a*aT *aT --aga@ at32 3’
(11)
with
(12)
Fig. 1. N as a function of P for various values of the resistance; solid line: perfect fluid (FO= 0); dotted line: gas bubble (FO= 2/3); dashed line: rigid sphere (FO= 1). tI am indebted to Mr. D. Bhaumik for independently evaluating the function g&F).
820
P. 0. BRUNN
Putting (13) we deduce by well established methods [9] the similarity solution [ -(2!K)~$(z-d’-x)].
(14)
Here I denotes the y function and x the similarity variable x
=
[
$pp,
q-= 7(@)
s
I
8
de’ sin O’H,.
(15)
0
Knowing the temperature or concentration the quantity N given by (8) is readily extracted
(16)
Only at this final stage does one need to know F&(8) and we explicitly have -
K.
Making use of (12), (13) and (15)
(3a)2’3(FoP)1’3= 0.6246(FoP)1’3, for F. = 1,
(174
otherwise.
(17b)
N = 2 “‘](1 - Fo)P]“* = 0.7979[(1- Fo)F]“*, 0
Actually the distinction in F. = 1 and F. # 1 is too stringent for the limit P +CQ under consideration and eqn (17a) should hold as long as the condition P Q Fi( 1 - Fo)-3 is met while the requirement Fi(l - FJ3 ti P, is needed for (17b) to be valid. Equation (17a) is well known[l] for rigid spheres. Its extension towards lower Peclet numbers reads [9] N = 0.6246P”3 + 0.461 + . . . .
(18)
On the other hand, eqn (17b) hardly is ever mentioned although for droplets it was obtained and found to agree with experiments some time ago[7]. In contrast to the low P&let number regime, different formulas have to be reckoned with for P s 1. The reason being that for P % 1 only the velocity field in the immediate vicinity of the sphere surface matters.t And the details of the flow field differ if the fluid adheres to the particle or not, i.e. whether ue, the tangential component of v, is zero at the sphere surface (rigid sphere) or not (droplets). And with a nonzero ue one expects the convective diffusion to be more intense than for the ue = 0 case, in perfect agreement with eqns (17). In Fig. 2 eqns (9) and (17) (actually (18) for F. = 1) have been plotted for the extreme cases F. = 1 (rigid sphere) and F. = 2/3 (gas bubble, perfect slip). It becomes apparent from these figures that one can actually interpolate between the asymptotic relations for large and small Peclet numbers with comparatively little uncertainty. tFor this reason the potential flow approximation F0 = 0 cannot he used for P s 1.
Heat or mass transfer from single spheres
821 /
IO’-
lOa”_ N
/
/
/
/
/
/
I’
IO’
/
I N
/
‘4F /
IO0
/
I
/
I
/I
I
Icf
104
IO0
10-Z
IO2
Icr
,
I IO+
P
lob
I IO2
P
(a)
(b)
Fig. 2. The rate of transfer N from a sphere for Stokes flow and arbitrary P&let number P; (a) rigid sphere, (b) gas bubble; solid line: eqn (9a); dashed line eqn (18) for a rigid sphere and eqn (17b) for a gas bubble.
5. INERTIAL
EFFECTS
The results of the previous chapters applies to the case of small isolated particles, for which the particle Reynolds number Re is small enough for inertia to be negligible. In order to explicitely see the validity of the approximation involved it is important to know how a small but nonzero Reynolds number actually effects the results. Unfortunately, for P Q 1 no unambiguous answer can be extracted from corresponding studies. For example, O’Brien 10 lists the result JV=l+iP[l-$F,(F,-$(l+i&)Re],
(19)
which, with the exception of a gaseous bubble (PO= 2/3) implies a decrease in heat or mass transfer with Re. Putting Re = P/a leads to the restriction u*O(llnPll’) for validity of (9a) on o. For P < 1 one sees that u may very well be of order one. On the other hand Gupala et al. [ 111, considering a rigid sphere, reports the result [ 1l]
with
(20) Since f(u) not only is positive but also is larger than the function g( 1) (defined by eqn (9(b)) eqn (20) predicts an increase in the net transfer due to inertia, in complete contradiction to eqn (19). Since O’Brien merely cites eqn (19) (as the only equation appearing in her article), it is hard to pinpoint the reason for the discrepancy. Although she does criticize the asymptotic matching employed by Acrivos-Taylor and consequently Gupala et a/.) it is hard to accept that the “intermediate” matching of O’Brien can really lead to a totally different formula. Further study clearly is needed to settle the issue. No such ambiguity exists for the high Peclet number limit. In this case, only the velocity or stream function in the immediate vicinity of the particle need to be known. And it is easy to
822
P. 0. BRUNN
establish from the results of Taylor et al. for droplets [ 123that for 0 < Re < 1 eqn (12) now reads
1,
1 + i Re( 1 - cos @)
for A = 0,
Re (l-~~cos63~],
(21)
otherwise.
Consequently, we again can utilize eqn (13), although H,, has to be replaced by H = H(B), H(8) = d/3 sin 8 [ 1 + 5 Re(1 - cos 6)]“2,
‘h H(8)=j1+Asm
K =-,
i
for A = 0,
* 2 Q [ l+, 3*Re(l-~$cos63~], l+*
K=O,
otherwise.
(22)
But this implies that the similarity solution (14) is still valid. Denoting by F the dimensionless drag on the droplet (such that the Re +O limit of F is the Stokes drag PO)we know [12]
In terms of F the Nusselt number N as given by eqn (16) becomes N = 0.6246(PP)“3, N = 0.7979(y
for A = 0, F)“2P112,
Wa) otherwise.
Wb)
As before, we should actually expect (24a) to be valid if P 6 Ffj (1 - P,J3(F-‘) but (P * 1), while (24b) requires the restriction P % Fi(l - P,,)-3(F’). But no matter which formula has to be used for any particular situation, for P 9 1 inertia always enhances the heat or mass transfer. The discrepancies, first noted by Friedlander[2] between experimental observations and theoretical Re = 0 predictions (i.e. eqn (17)), may solely be due to this effect. Acknowledgemen&I thank Mr. B. Guzman for preparing the graphs. REFERENCES [1] A. ACRIVOS and T. D. TAYLOR, Phys. Fluids 5,387 (1962). [2] S. K. FRIEDLANDER, AICh.E J. 3,43 (1957). [3] T. D. TAYLOR, Phys. Fluids 6,987 (1963). [4] T. SANO, J. En&g M&h 6, 217 (1972). [S] P. 0. BRUNN, J. Biomech. Engng ASME 103,32 (1981). [6] H. LAMB, Hydrodynamics,6th Edn. Cambridge (1932). (71 V. G. LEVICH, Physiochemical Hydrodynamics.Prentice-Hall, Englewood Cliffs, New Jersey (1962). [8] P. BRUNN, Rheol. Acta 15, 104(1976). [9] A. ACRIVOS and J. D. GODDARD, 1. JWd Mech. 23,273 (1965). ;lO] V. O’BRIEN, Phys. Fluids 6, 1356(1963). [ll] Y. P. GUPALA and Y. S. RYAZANTSEV, Chem. Engng Sci 27,61 (1972). :12] T. D. TAYLOR and A. ACRIVOS, J. Fluid Mech. 19,466 (1964). (Received 28 September 1981)
CU2~7225/82/07l?417&$4l3.lWO @ 1982 Pergamon Press Ltd.
HEAT OR MASS TRANSFER FROM SINGLE SPHERES IN A LOW REYNOLDS NUMBER FLOW P. 0. BRUNN Department of Chemical Engineering and Applied Chemistry, Columbia University, New York, NY 10027, U.S.A. Ahatraet-The problem of forced heat or mass transfer from a single sphere in low Reynolds number flow is studied. In Stokes flow the Nusselt number N can depend only upon the Peclet numher P. For P < 1 a general “master-formula” is derived which involves Stokes resistance F,. On the other hand, the P B 1 limit requires different formulas, depending upon whether the tangential velocity vanishes at the sphere surface (rigid sphere) or not (slip flow and droplets). Although these differences do not disappear when first-order inertia effects are included, it is found that inertia enhances the heat or mass transfer in either case.
1. INTRODUCTION THE PROBLEMof
heat or mass transfer from a sphere into a low Reynolds number flow is of importance in many extraction processes: spray drying, aerosol scrubbing, meteorological studies and diffusive intake of swimming bacteria. Consequently, numerous theoretical studies have been concerned with that problem [l-5]. As long as the particles are small (< 100 p) so that the particle Reynolds number Re is small, inertia effects will be negligible. Then the Nusselt number N can depend only upon the P&let number P, defined as being equal to the product Reu, where o denotes the Prandtl number in case of heat transfer or the Schmidt number in case of mass transfer. Although for P < 1 the problem is diffusion dominated, a regular perturbation expansion around the pure diffusion limit P = 0 does not apply, and one has to resort to a singular perturbation expansion. This has been done for a rigid sphere [l], a rigid sphere with slip [3] and for a sphere immersed in an inviscid fluid[4]. With inertia neglected, the velocity field obeys the Stokes equations of creeping motion, for which Lamb[6] gives the most general solution for translation of a sphere. Lamb’s solution not only embraces each of the flow fields used in the studies just cited, but applies to other situations (e.g. droplets) as well. Consequently, the question arises as to whether we could not use Lamb’s solution to obtain for small Peclet numbers a single “master formula” for the N-P relation, which would cover all the possible cases. Actually, the same question can be posed for arbitrary Peclet numbers. However in the limit P + m, which can be investigated by means of a boundary layer theory, the results for a rigid sphere[l, 21 show an N-P relation totally different than the one found for droplets [7]. Consequently, it is important to understand the transition from one formula to the other. Since the neglect of inertia requires that Re +O, the high Peclet number limit P 9 1 is of rather academic interest, since it requires u+m. In order to weaken this restriction, we extend the P < 1 analysis into the region where Re is small but non-zero. 2. FORMULATION
Steady
state
fluid is governed
heat or mass
transfer
OF THE PROBLEMS
from an isolated
stationary
sphere
to an incompressible
by the equation
V2T= Pv;
T,
for which we adopt the boundary conditions T=l
T+O
atr=l,
(2a)
as r+m.
(2b)
817
818
P.O.BRUNN
Here T denotes a normalized temperature or concentration and P a P~clet-num~r based on the sphere radius a. The position vector I, measured from the sphere center is scaled with Q, while the velocity v has been made dimensionless by means of the free stream velocity. For the case in which the fluid at infinity is streaming past the sphere in the negative z-direction as t+w,
W-ii,
(3a)
In addition, we know that
-&v=o u, = 0
for t> 1, at r= 1.
(3c]
We can use Lamb’s genera1 solution[41 to deduce the Stokes stream function for a Newtonian fluid (4) where FO is the Stokes resistance of the sphere, made dimensionless by the factor 6ar)a, such that FO= 1 for a rigid sphere. Other cases which are of interest are
If h-r denotes the viscosity of the drop relative to the suspendi~ expressed in terms of the viscosity ratio h is
fluid the force FO
l+;A Fo=
l+h.
(5)
(b) Sfip pow If /I denotes Lamb’s coefficient of sliding friction [6] and (Ythe dimensionless parameter $a& measuring the extent of slip (a = 0 for the adherence case, a +m for perfect slip), then[8]
(c) Potential flow In this case the condition F. = 0
(7)
holds. Although this leads to d’Alembert’s paradox, the potential flow approximation does have some justification in the context of heat transfer for low Prandtl number tluids such as liquid metals [4]. In order to obtain the (d~ensio~ess) heat or mass transfer rate from the sphere, i.e. the Nusselt number
we have to solve eqn (11, which we can only do analytically for the limits P-0 respectively.
and P *co,
3.THE LIMIT P+O
Acrivos et al. [l] were one of the tirst to realize that a regular perturbation scheme could not be used to treat eqn (1) for low Peclet numbers. Although developed for a rigid sphere in the no
Heat or mass transfer from single spheres
819
slip limit (FO= l), their method of solution can immediately be used for our case since the velocity field retains the general form (4) no matter what value F0 actually assumes. This being the case, we merely list the final result (9a) with
and y = 0.5772156.. . the Euler-Mascheroni constant. Equation (9) reduces for F0 = 1 to the Acrivos-Taylor expression[ 11. In case of potential flow all the terms in In P drop out and we recover Sano’s result[4]. For slip flow, our result differs from Taylor’s[3]. We feel confident that our result is correct, the more so since (a) Taylor deduces from his results an incorrect potential flow limit and (b) eqns (9) have been obtained twice by totally independent ca1culations.t Besides being applicable in known cases eqn (9) can also be used for new ones, e.g. for droplets. Figure 1 illustrates the predictions. While for potential flow the increase of N with P is always less than linear, for droplets (including gas bubbles and rigid spheres) it is less than linear for extremely low flow speeds but larger than linear for P in the neighborhood of one. 4. THE LIMIT P+m
In this case the temperature or concentration gradients will be large near the sphere surface and a boundary layer theory seems appropriate. Thus, putting r=r-1,
(10)
the boundary layer approximation of (1) reads
a2 $T=sin8
-P [
a*aT *aT --aga@ at32 3’
(11)
with
(12)
Fig. 1. N as a function of P for various values of the resistance; solid line: perfect fluid (FO= 0); dotted line: gas bubble (FO= 2/3); dashed line: rigid sphere (FO= 1). tI am indebted to Mr. D. Bhaumik for independently evaluating the function g&F).
820
P. 0. BRUNN
Putting (13) we deduce by well established methods [9] the similarity solution [ -(2!K)~$(z-d’-x)].
(14)
Here I denotes the y function and x the similarity variable x
=
[
$pp,
q-= 7(@)
s
I
8
de’ sin O’H,.
(15)
0
Knowing the temperature or concentration the quantity N given by (8) is readily extracted
(16)
Only at this final stage does one need to know F&(8) and we explicitly have -
K.
Making use of (12), (13) and (15)
(3a)2’3(FoP)1’3= 0.6246(FoP)1’3, for F. = 1,
(174
otherwise.
(17b)
N = 2 “‘](1 - Fo)P]“* = 0.7979[(1- Fo)F]“*, 0
Actually the distinction in F. = 1 and F. # 1 is too stringent for the limit P +CQ under consideration and eqn (17a) should hold as long as the condition P Q Fi( 1 - Fo)-3 is met while the requirement Fi(l - FJ3 ti P, is needed for (17b) to be valid. Equation (17a) is well known[l] for rigid spheres. Its extension towards lower Peclet numbers reads [9] N = 0.6246P”3 + 0.461 + . . . .
(18)
On the other hand, eqn (17b) hardly is ever mentioned although for droplets it was obtained and found to agree with experiments some time ago[7]. In contrast to the low P&let number regime, different formulas have to be reckoned with for P s 1. The reason being that for P % 1 only the velocity field in the immediate vicinity of the sphere surface matters.t And the details of the flow field differ if the fluid adheres to the particle or not, i.e. whether ue, the tangential component of v, is zero at the sphere surface (rigid sphere) or not (droplets). And with a nonzero ue one expects the convective diffusion to be more intense than for the ue = 0 case, in perfect agreement with eqns (17). In Fig. 2 eqns (9) and (17) (actually (18) for F. = 1) have been plotted for the extreme cases F. = 1 (rigid sphere) and F. = 2/3 (gas bubble, perfect slip). It becomes apparent from these figures that one can actually interpolate between the asymptotic relations for large and small Peclet numbers with comparatively little uncertainty. tFor this reason the potential flow approximation F0 = 0 cannot he used for P s 1.
Heat or mass transfer from single spheres
821 /
IO’-
lOa”_ N
/
/
/
/
/
/
I’
IO’
/
I N
/
‘4F /
IO0
/
I
/
I
/I
I
Icf
104
IO0
10-Z
IO2
Icr
,
I IO+
P
lob
I IO2
P
(a)
(b)
Fig. 2. The rate of transfer N from a sphere for Stokes flow and arbitrary P&let number P; (a) rigid sphere, (b) gas bubble; solid line: eqn (9a); dashed line eqn (18) for a rigid sphere and eqn (17b) for a gas bubble.
5. INERTIAL
EFFECTS
The results of the previous chapters applies to the case of small isolated particles, for which the particle Reynolds number Re is small enough for inertia to be negligible. In order to explicitely see the validity of the approximation involved it is important to know how a small but nonzero Reynolds number actually effects the results. Unfortunately, for P Q 1 no unambiguous answer can be extracted from corresponding studies. For example, O’Brien 10 lists the result JV=l+iP[l-$F,(F,-$(l+i&)Re],
(19)
which, with the exception of a gaseous bubble (PO= 2/3) implies a decrease in heat or mass transfer with Re. Putting Re = P/a leads to the restriction u*O(llnPll’) for validity of (9a) on o. For P < 1 one sees that u may very well be of order one. On the other hand Gupala et al. [ 111, considering a rigid sphere, reports the result [ 1l]
with
(20) Since f(u) not only is positive but also is larger than the function g( 1) (defined by eqn (9(b)) eqn (20) predicts an increase in the net transfer due to inertia, in complete contradiction to eqn (19). Since O’Brien merely cites eqn (19) (as the only equation appearing in her article), it is hard to pinpoint the reason for the discrepancy. Although she does criticize the asymptotic matching employed by Acrivos-Taylor and consequently Gupala et a/.) it is hard to accept that the “intermediate” matching of O’Brien can really lead to a totally different formula. Further study clearly is needed to settle the issue. No such ambiguity exists for the high Peclet number limit. In this case, only the velocity or stream function in the immediate vicinity of the particle need to be known. And it is easy to
822
P. 0. BRUNN
establish from the results of Taylor et al. for droplets [ 123that for 0 < Re < 1 eqn (12) now reads
1,
1 + i Re( 1 - cos @)
for A = 0,
Re (l-~~cos63~],
(21)
otherwise.
Consequently, we again can utilize eqn (13), although H,, has to be replaced by H = H(B), H(8) = d/3 sin 8 [ 1 + 5 Re(1 - cos 6)]“2,
‘h H(8)=j1+Asm
K =-,
i
for A = 0,
* 2 Q [ l+, 3*Re(l-~$cos63~], l+*
K=O,
otherwise.
(22)
But this implies that the similarity solution (14) is still valid. Denoting by F the dimensionless drag on the droplet (such that the Re +O limit of F is the Stokes drag PO)we know [12]
In terms of F the Nusselt number N as given by eqn (16) becomes N = 0.6246(PP)“3, N = 0.7979(y
for A = 0, F)“2P112,
Wa) otherwise.
Wb)
As before, we should actually expect (24a) to be valid if P 6 Ffj (1 - P,J3(F-‘) but (P * 1), while (24b) requires the restriction P % Fi(l - P,,)-3(F’). But no matter which formula has to be used for any particular situation, for P 9 1 inertia always enhances the heat or mass transfer. The discrepancies, first noted by Friedlander[2] between experimental observations and theoretical Re = 0 predictions (i.e. eqn (17)), may solely be due to this effect. Acknowledgemen&I thank Mr. B. Guzman for preparing the graphs. REFERENCES [1] A. ACRIVOS and T. D. TAYLOR, Phys. Fluids 5,387 (1962). [2] S. K. FRIEDLANDER, AICh.E J. 3,43 (1957). [3] T. D. TAYLOR, Phys. Fluids 6,987 (1963). [4] T. SANO, J. En&g M&h 6, 217 (1972). [S] P. 0. BRUNN, J. Biomech. Engng ASME 103,32 (1981). [6] H. LAMB, Hydrodynamics,6th Edn. Cambridge (1932). (71 V. G. LEVICH, Physiochemical Hydrodynamics.Prentice-Hall, Englewood Cliffs, New Jersey (1962). [8] P. BRUNN, Rheol. Acta 15, 104(1976). [9] A. ACRIVOS and J. D. GODDARD, 1. JWd Mech. 23,273 (1965). ;lO] V. O’BRIEN, Phys. Fluids 6, 1356(1963). [ll] Y. P. GUPALA and Y. S. RYAZANTSEV, Chem. Engng Sci 27,61 (1972). :12] T. D. TAYLOR and A. ACRIVOS, J. Fluid Mech. 19,466 (1964). (Received 28 September 1981)