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Sedimentation in a dilute emulsion
ChemicalEngineering Science. 1973, Vol. 28, pp. 1447-1453.
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Sedimentation in a dilute emulsion E. WACHOLDER Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, England (Received 1 August 1972) Abstract- In the present work the method of Batchelor [ l] is adopted to calculate the statistical mean speed of sedimentation in a dilute dispersion of identical spherical droplets. The mean settling speed is shown to be
u,
(
l-
[
e+K(o)]c)
where U,, is the settling velocity of a single fluid sphere falling under gravity through an unbounded quiescent fluid, m is the ratio of the viscosities /.L’/Pwhere the upper prime indicates the disperse phase, c is the volume fraction of the spheres, and K(u) is a numerical factor whose values are displayed in Table 1. An exact evaluation of K(u) requires knowledge of the solutions for the drag force components acting on two fluid spheres in the directions of their line of centre and perpendicular to it. Such a solution for the latter case is not yet available, hence, an approximate solution based on the ‘method of reflections’ is used. This causes some error in the numerical results for K(u), the magnitude of which is examined and discussed. 1. INTRODUCTION
of particles in a dispersion represents an area of interest in many fields of science and technology. An extremely wide range of problems is involved and, as would be expected, the number of possible variables is also large. We consider a dilute dispersion consisting of a mixture of Newtonian fluid and identical Newtonian fluid spheres which are randomly located and are falling through the ambient fluid under gravity. The spheres are of such small size that the Reynolds number of the fluid motion is small and inertia forces can be neglected. The number of the spheres dispersed is large so that there is only a minute effect of the container walls. Under these conditions the mean speed of fall of a particle is proportional to its excess weight and is otherwise a function primarily of the volume fraction of the particle phase. As it is well known this speed of fall is less than that for a single particle and the phenomenon is often referred to as ‘hindered settling’. There have been many contributions to the problem of determining the effect of concentration on the settling speed of small rigid spheres SEDIMENTATION
(see Happel and Brenner[Z] Chap. 8). However, not until Batchelor’s [ l] work was a systematic and rigorous theory established. He used probability methods and showed how one can overcome the familiar difficulty presented by the occurrence of integrals which are not absolutely convergent. This general method was used to show that the mean speed of fall in a dilute dispersion of identical rigid spheres is UO( 1 - 6.55~) where U,, is the velocity of a single sphere in an unbounded quiescent fluid and c is the volume fraction of the spheres. We shall follow the same general method and show how it may be applied to the case of spherical fluid particles. Detailed expositions which may be found in Batchelor’s [l] work are omitted throughout the present work. The emphasis here is on the aspects of the calculations associated with the fluid nature of the particles. The method requires knowledge of the solutions for the drag forces acting on two fluid spheres, falling under gravity, both along and perpendicular to their line of centres. The solution of two fluid spheres falling along their line of centres was recently solved by Wacholder
1447
E. WACHOLDER
and Weihs[4]. A solution of two fluid spheres falling perpendicular to their line of centres is not yet available. Thus an approximate expression based on the solution by the ‘method of reflections’ which was recently derived by Hetsroni and Haber[3] has been used. This introduces some inaccuracy in the results, the magnitude of which is shown to be small with respect to the mean speed of fall. 2. THE METHOD
OF SOLUTION
We consider a statistically homogeneous dispersion of identical spherical droplets* of radius a, viscosity p’, and density p’ falling under gravity through a continuous phase of fluid of viscosity p and density p. The fluids are immiscible, incompressible and Newtonian. Inertia forces in the fluid spheres and the ambient fluid are assumed to be negligible such that the velocity and pressure fields are governed by the linear Stokes equations (creeping motion equations). It is further assumed that the interfacial tension T acting at the junction between each droplet and the continuous fluid is large enough (T/p& P 1) to maintain them in a spherical shape and that the velocities and stresses across the interface match. The volume fraction of spheres in the dispersion is 4 c = - ?ra% 3
,2(p'-p)a%
u
O9
P
l+(+ 213 + u
(2.3)
where u is the ratio of viscosities ~//II. and g is the gravitational acceleration. If this fluid sphere, whose centre is located at G, is immersed in an unbounded non-uniform flow field u(x), the additional contribution to its settling velocity is (see Hetsroni et a1.[7]):
(2.1) V(x,, x0+ r) = u(x,, x0+ r)
where n is the uniform average number density of spheres. The emulsion considered is dilute, that isc Q 1. Following Batchelor [ l] we write the statistical mean speed of a test particle in the emulsion as
w = NJ,)+ 09
particles in the dispersion, and W represents the effect of the image systems in the boundaries of all other particles at the surface of the test particle. The angle brackets denote ensemble averages but when the dispersion is spatially homogeneous, ensemble averages may be replaced by spatial averages. In case of dilute dispersion (c G 1) Batchelor[l] showed how these averages may be reduced to integrals over the position of only one sphere of the configuration which surrounds the test particle. We shall follow his method precisely, the only modification being that (U),(V) and ( W) are to be evaluated for a dispersed phase which is fluid rather than solid. The settling velocity of a single fluid sphere (the test sphere) in an unbounded uniform fluid is given by the well known Hadamard-Rybczynski [5,6] expression,
+
w
(2.2)
where U,, is the translational velocity of the particle falling under gravity as if it were a single particle in an infinite quiescent fluid, V is the contribution to its translational velocity owing to the flow field caused by the presence of the other *The term droplets is used throughout this work, even though the solution is also applicable to spherical bubbles.
+~a22~3~{VZu(x,xo+r)}~,~.
(2.4)
The meaning of this notation is that V(x,,, x,,+ r) is the value of V at a point x,, in the dispersion (in either phase) when there is a particle at % + r. W represents the effect of the image system and is detined as W=W(xo,xo+r)
=U(xo,x,+r) -Uo-V(h,x,+r)
(2.5)
where U (q,, q, + r) is the velocity of a fluid sphere with centre at x, falling in an unbounded quiescent fluid in the presence of a second fluid sphere with centre at q,+r. This latter velocity has the
1448
Sedimentation in a dilute emulsion
general form r.U, = X1r7+X2
U(x(),q+r)
uo-r7 r ‘“O
(2.6)
where A1 and AZ are the inverse resistance coefficients for motions parallel and perpendicular to the line-of-centres, respectively. In the case of spheres of equal size they are functions of cr and the separation distance r only. The exact expression for A1 was obtained in a closed form by Wacholder and Weihs [4] :
Haber[3] by the method of reflections. The errors incurred in using this approximate form are discussed in Section 3. We turn now to the evaluation of the averages of the three contributions (2.3)-(2.5). The averaging process requires knowledge of two probability densities. These are (i) the probability density P (~0 + r) that a sphere centre lies at the point x, + r in the field, and (ii) the corresponding conditional probability density P(x, + r 1x0) given that there is a sphere whose centre is at %. In a homogeneous suspension we have P(x,+r)
4 ’ +O-sinh ff 5
Al-'(r/u,
u) = 3 2/3
= P= II
(2.9)
n=1
X
,2nnJ;h:A
+(2n+l)
_
Ij
everywhere in the field. Batchelor[ l] has given reasons for the belief that under usual conditions the conditional probability density for a dilute suspension of identical rigid spheres is
{o[2 sinh (2n + 1)~
sinh2a-4sinh2
(n++)~~+(2n+l)~ P(x,+r(x,)
x sinh2 a] + 4 [ (2n + 3) eza + 4e-t2n+1)n
- (2n-
l)e+]}/{u[2
Xsinh2a]+2[cosh
sinh (2n+ l)a+
(2n+ 1)
(2n+l)a+cosh2ar]}. (2.7)
where (Yis related to the separation distance by the relation OL= cash-’ (r/a). An exact formula for A2 is not yet available, but its far field asymptotic form is given by A,(r/a, (+) =
3
This expression is the series representation to terms of O(a@ of the inverse of the resistance coefficient k2 (see Eq. (A.2) of the Appendix) which was recently derived by Hetsroni and
if if
r 2 2u (2.10) r<2u
correct to order c. These results represent uniform probability for all physical accessible positions of one rigid sphere relative to another. These seem likely to apply also to a dispersion of identical fluid spheres which are initially randomly dispersed since two identical fluid spheres will also fall under gravity with the same velocity, no matter how close they are. The first expression on the r.h.s. of Eq. (2.2) is obviously (U,) = U, since U, is a constant vector. The evaluation of (V) involves some difficulty because of the slowness with which the velocity disturbance in an unbounded fluid due to a falling particle decreases with distance from the particle. Thus a straight-forward integration of V over r results in an integral which is not absolutely convergent. However, Batchelor [ l] has given a method by which, with the aid of known exact mean values involving all the spheres in the configuration, this difficulty can be overcome. Using the conditions that the velocity distribution and the divergence of the deviatoric stress tensor, defined in both phases, are station-
1449 CI!&Vd.28,N0.1-P
= Q= n =o
E. WACHOLDER
ary random functions of positions with zero mean, the average of V is given by: @) = I,,,.
u(x,,, xo+ r)
+ ,>O;a2& I
[Q- P] dr
Uo-~E?r~ >
r.U,
for
r c a
(2.12)
for
r 2 a
(2.13)
u(r) = (jj$+$)U,+(i$-F)ry
where D=+&
1
=_I
I-
u’(r)dr-n
JoCr42u
u(r) dr
(2.14)
and from Eq. (2.2) we have
J
u(x,,x,+r) allspace
[Q-P]
dr= -Uoc
8+ llu 2(1+(T) * (2.18)
A helpful relation in carrying out the integration is
where S is a sphere of unit radius and I is the idemfactor (unit tensor). The next integral on the r.h.s. of Eq. (2.11) vanishes. This can easily be verified by straightforward integration of V,“u of Eq. (2.16) between the limits r = (I and r = 2~. The force, f, exerted by the deviatoric stress at the surface of a sphere equals two-thirds of the total drag force. This is a general result valid for a rigid or fluid sphere submerged in an unbounded arbitrary Stokesian flow field, it may be easily shown by using Lamb’s general solution of the creeping motion equations. When the droplet fails through an unbounded quiescent fluid this force is given by
41+u*
2a3 1+u
The corresponding in both fluids are:
Jrra
dr
(2.17)
where f is the total force exerted by the deviatoric stress on the surface of a fluid sphere with centre at x0, immersed in a flow field u(x). In the problem at hand u(x) is to be interpreted as the flow field at x that obtains when there is a fluid sphere with centre at x+ r falling through an infinite fluid with velocity Uo. The flow field has been found by Hadamard and Rybcyznski[5,6] to be
G
r) [Q -PI
P
(2.11)
E,IIL
+
[Q-PI dr
Wu(x,xo+r)L.,
B=2,213+~ 2 1+u
u(xo, x0 allspace
z-n
-n$22+4,flp
n’(r)=
I
expressions
v, 2u’ = - 2EUo 3B r.U V, 2u = B-!+U,--Tr--$
for for
213 + u f = - 47r/.LaU,I+cr
for V,.%tat points
r S a r 5 a.
(2.15)
Consequently (2.11) is
Upon substitution of Eqs. (2.9), (2.10), (2.12) and (2.13) it is found that
the last term on the r.h.s. of Eq.
1 --na2&f/p=~UOc&. 2
(2.16)
(2.19)
(2.20)
We shall now calculate the mean value of W (~0, % + r) . Again, according to Batchelor[ l] 1450
Sedimentation in a dilute emulsion
this is given by
Table 1. Values of the integral K(a) of equation (2.23) as a function of the ratio of viscosities u = p’/p
(W> = 1 W(r)P(x,+
r 1x0) dr = n J,,_WW
dr
(2.21) where in view of Eqs. (2.4), (2.9, (2.16) W(r) can be written as
w = /L’//.l
W(r) = U(r) -Uo-Uo(~~+$)-r~
0 0.1
0.84 0.91 0.99 1.09 1.21 1.32 1.40 1.43 1.48 1.52 1.53 1.53 1.53
0.25 0.5 1.0 2.0 3.5 5.0 10.0 50.0 loo.0 1ooo~0 m
x(pE)-;.2_&U+~~)
(2.22) with U(r) given in Eq. (2.6). Upon substitution of Eq. (2.6) into (2.22), the integral of W(r) over a spherical surface of unit radius is then obtained in the form
.
Substitution of Eqs. (2.18), (2.20) and (2.23) into Eq. (2.1) show that the mean translational velocity of fluid spheres in a dispersion with volume concentration c( c Q 1) is
The average of W follows from Eq. (2.2 1) as (W(r))=cUOIm
)I
(2.13) and
[A1+2A2-3(l+te)] (U) = U,,(l-[E+K(o)]c)
2
x(;rd(;)
=-c&K(a)
(2.23)
(2.24)
correct to order c.
where A, and A, are given in Eqs. (2.7) and (2.8) respectively. At larger values of r/a the integrand is given with good accuracy by its asymptotic form, which is readily found by the method of reflections (Eqs. (A. 1) and (A.2)) to be
Thus the numerical integration of Eq. (2.23) has been carried out to some large value of r/a, say R/a and the value of the analytical integral of the asymptotic form over the range r/a a R/a was then added to it. This procedure was repeated for increasing values of R/a until a numerical agreement was achieved. The results for K(o) are displayed in Table 1.
3. DISCUSSION
It may be seen that the internal circulation in the disperse phase reduces the changes in the mean settling speed due to the hydrodynamic interactions. In Fig. 1 the dimensionless mean settling speed, defined as (U, - (U) )/U,c is plotted vs. the ratio of viscosities o. This quantity, when multiplied by the volume concentration c, describes the effect of the hydrodynamic interactions in the dispersion on the mean speed of fall. Its magnitude becomes smaller as u decreases. The reason for this is clear from the following: the downward volume flux of fluid consists of a volume flux associated with the fall of the spheres cU, and a volume flux of fluid dragged down in the inaccessible shells between r = a and r = 2a surrounding the spheres
1451
E. WACHOLDER
6.6
L
Fig. 1. Dimensionless mean speed of fall as function of the ratio of viscosities fs = ~‘/,LL.
&UO[ (Q+ (+)/ ( 1 + a)]. This flux is accompanied, in a homogeneous dispersion with zero mean flux at each point, by a corresponding net upward flux. The upward volume flux in the part of the fluid that is accessible to the centre of the test sphere changes its mean settling speed by an amount of -cU,,(l+#[($+u)/(l+o)] c.f. Eq. (2.18). This is equal and opposite to the rate at which fluid volume is dragged down, and likewise becomes smaller as v decreases. The gradients of fluid velocity generated in the continuous phase by the motion of the spheres cause a net change of +*[a/( 1 + a)]cU, (see Eq. (2.20)) in the mean settling speed. This change, although positive, is smaller than the previous one such that their sum is -cU[(4+Sa)/(l+a)] which still decreases with o. Finally, when the test sphere is near one of the other spheres in the dispersion, the interaction between these two close spheres gives further change in the mean settling speed equal to -K(a)cU,, which also decreases with u as can be seen in Table 1. Therefore the smaller u the smaller is the resultant change in the mean settling speed due to the hydrodynamic inter-
actions. An overall reduction of 25 per cent is found as (+ changes between infinity and zero with a corresponding increase in the mean settling speed, (U). Since the mean settling speed is proportional to U,, it will also increase between these limits of u by a factor of 3/2 due to a larger settling speed of an isolated bubble (u = 0) in comparison with a rigid sphere (u 4 a). Consider now the magnitude of the error associated with the use of the approximate & in the calculation of K(u). As shown in Table 1 the limiting value obtained for K(u) as u + CQ(rigid spheres) is 1.53. This differs by only 0.02 (- l-5 per cent) from the more accurate value of 1.55 obtained by Batchelor [ 11. The fractional error in (U) for u + m is obviously much smaller. We believe that the error in the numerical integration of K(u) for other values of u < m is always smaller than O-02. This may be justified as follows: the missing terms of higher order in the expression for AZ (Eq. (2.8)) are obviously the source of the error incurred in the calculation of K(u) . By checking qualitatively their character we realized that they consist, in general, of powers of (g+ u)/( 1 + u) and u/( 1 + u). These terms achieve their maximum value when u + m causing an error of 0.02 and decrease as u become smaller, with a corresponding decrease in the error. Thus it seems that the formula given for the mean settling speed (Eq. (2.24)) covers the whole range of viscosities ratio u s m with a maximum error in (U, - (U))/U,c which does not exceed O-02. Acknowledgement-This work was done while the author was in receipt of a Research Fellowship from the Royal Society-Israel Academy of Sciences and Humanities exchange program. Assistance from the Yad Avi Ha-Yishuv fund is also gratefully acknowledged. The author is indebted to Professors G. K. Batchelor and N. F. Sather for their helpful advice.
REFERENCES HI BATCHELOR G. K.,J. Fluid Mech. 1972 52 245.
PI HAPPEL J. and BRENNER H., Low Reynolds Number Hydrodynamics. Prentice Hall, N.J. 1965. [31 [41 PI WI r71
HETSRONI G. and HABER S., Report No. 1 Dept. Mech. Eng., Technion, Haifa, Israel Nov. 197 1. WACHOLDER E. and WEIHS D., Chem. Engng Sci. 1972 27 1817. HADAMARD J. S., Coopt. Rend. Acad. Sci. Paris 1911152 1735. RYBCZYNSKI W.,Bull.Acad. Sci. Cracooie (ser A) 40 1911. HETSRONI G., WACHOLDER E. and HABER S., ZAMM 19715145.
1452
Sedimentation in a dilute emulsion
APPENDIX
A
Hetsroni and Haber[3] have recently used a ‘method of reflections’ solution to derive a general expression for the drag forces of two spherical droplets of different size and different viscosities falling in an unbounded quiescent fluid in a gravitational field. The results for two identical droplets having radius a are given as follows: The drag force in the direction of the line of centres
---
1 a(2+3a) 2 (l+(r)*
+&(er](;r).
(A.1)
The drag force perpendicular to the line of centres
=-u,t.
1453
(A.2)
PC~~UIIOII heSS.
priottd
~IIChat
Britain
Sedimentation in a dilute emulsion E. WACHOLDER Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, England (Received 1 August 1972) Abstract- In the present work the method of Batchelor [ l] is adopted to calculate the statistical mean speed of sedimentation in a dilute dispersion of identical spherical droplets. The mean settling speed is shown to be
u,
(
l-
[
e+K(o)]c)
where U,, is the settling velocity of a single fluid sphere falling under gravity through an unbounded quiescent fluid, m is the ratio of the viscosities /.L’/Pwhere the upper prime indicates the disperse phase, c is the volume fraction of the spheres, and K(u) is a numerical factor whose values are displayed in Table 1. An exact evaluation of K(u) requires knowledge of the solutions for the drag force components acting on two fluid spheres in the directions of their line of centre and perpendicular to it. Such a solution for the latter case is not yet available, hence, an approximate solution based on the ‘method of reflections’ is used. This causes some error in the numerical results for K(u), the magnitude of which is examined and discussed. 1. INTRODUCTION
of particles in a dispersion represents an area of interest in many fields of science and technology. An extremely wide range of problems is involved and, as would be expected, the number of possible variables is also large. We consider a dilute dispersion consisting of a mixture of Newtonian fluid and identical Newtonian fluid spheres which are randomly located and are falling through the ambient fluid under gravity. The spheres are of such small size that the Reynolds number of the fluid motion is small and inertia forces can be neglected. The number of the spheres dispersed is large so that there is only a minute effect of the container walls. Under these conditions the mean speed of fall of a particle is proportional to its excess weight and is otherwise a function primarily of the volume fraction of the particle phase. As it is well known this speed of fall is less than that for a single particle and the phenomenon is often referred to as ‘hindered settling’. There have been many contributions to the problem of determining the effect of concentration on the settling speed of small rigid spheres SEDIMENTATION
(see Happel and Brenner[Z] Chap. 8). However, not until Batchelor’s [ l] work was a systematic and rigorous theory established. He used probability methods and showed how one can overcome the familiar difficulty presented by the occurrence of integrals which are not absolutely convergent. This general method was used to show that the mean speed of fall in a dilute dispersion of identical rigid spheres is UO( 1 - 6.55~) where U,, is the velocity of a single sphere in an unbounded quiescent fluid and c is the volume fraction of the spheres. We shall follow the same general method and show how it may be applied to the case of spherical fluid particles. Detailed expositions which may be found in Batchelor’s [l] work are omitted throughout the present work. The emphasis here is on the aspects of the calculations associated with the fluid nature of the particles. The method requires knowledge of the solutions for the drag forces acting on two fluid spheres, falling under gravity, both along and perpendicular to their line of centres. The solution of two fluid spheres falling along their line of centres was recently solved by Wacholder
1447
E. WACHOLDER
and Weihs[4]. A solution of two fluid spheres falling perpendicular to their line of centres is not yet available. Thus an approximate expression based on the solution by the ‘method of reflections’ which was recently derived by Hetsroni and Haber[3] has been used. This introduces some inaccuracy in the results, the magnitude of which is shown to be small with respect to the mean speed of fall. 2. THE METHOD
OF SOLUTION
We consider a statistically homogeneous dispersion of identical spherical droplets* of radius a, viscosity p’, and density p’ falling under gravity through a continuous phase of fluid of viscosity p and density p. The fluids are immiscible, incompressible and Newtonian. Inertia forces in the fluid spheres and the ambient fluid are assumed to be negligible such that the velocity and pressure fields are governed by the linear Stokes equations (creeping motion equations). It is further assumed that the interfacial tension T acting at the junction between each droplet and the continuous fluid is large enough (T/p& P 1) to maintain them in a spherical shape and that the velocities and stresses across the interface match. The volume fraction of spheres in the dispersion is 4 c = - ?ra% 3
,2(p'-p)a%
u
O9
P
l+(+ 213 + u
(2.3)
where u is the ratio of viscosities ~//II. and g is the gravitational acceleration. If this fluid sphere, whose centre is located at G, is immersed in an unbounded non-uniform flow field u(x), the additional contribution to its settling velocity is (see Hetsroni et a1.[7]):
(2.1) V(x,, x0+ r) = u(x,, x0+ r)
where n is the uniform average number density of spheres. The emulsion considered is dilute, that isc Q 1. Following Batchelor [ l] we write the statistical mean speed of a test particle in the emulsion as
w = NJ,)+ 09
particles in the dispersion, and W represents the effect of the image systems in the boundaries of all other particles at the surface of the test particle. The angle brackets denote ensemble averages but when the dispersion is spatially homogeneous, ensemble averages may be replaced by spatial averages. In case of dilute dispersion (c G 1) Batchelor[l] showed how these averages may be reduced to integrals over the position of only one sphere of the configuration which surrounds the test particle. We shall follow his method precisely, the only modification being that (U),(V) and ( W) are to be evaluated for a dispersed phase which is fluid rather than solid. The settling velocity of a single fluid sphere (the test sphere) in an unbounded uniform fluid is given by the well known Hadamard-Rybczynski [5,6] expression,
+
w
(2.2)
where U,, is the translational velocity of the particle falling under gravity as if it were a single particle in an infinite quiescent fluid, V is the contribution to its translational velocity owing to the flow field caused by the presence of the other *The term droplets is used throughout this work, even though the solution is also applicable to spherical bubbles.
+~a22~3~{VZu(x,xo+r)}~,~.
(2.4)
The meaning of this notation is that V(x,,, x,,+ r) is the value of V at a point x,, in the dispersion (in either phase) when there is a particle at % + r. W represents the effect of the image system and is detined as W=W(xo,xo+r)
=U(xo,x,+r) -Uo-V(h,x,+r)
(2.5)
where U (q,, q, + r) is the velocity of a fluid sphere with centre at x, falling in an unbounded quiescent fluid in the presence of a second fluid sphere with centre at q,+r. This latter velocity has the
1448
Sedimentation in a dilute emulsion
general form r.U, = X1r7+X2
U(x(),q+r)
uo-r7 r ‘“O
(2.6)
where A1 and AZ are the inverse resistance coefficients for motions parallel and perpendicular to the line-of-centres, respectively. In the case of spheres of equal size they are functions of cr and the separation distance r only. The exact expression for A1 was obtained in a closed form by Wacholder and Weihs [4] :
Haber[3] by the method of reflections. The errors incurred in using this approximate form are discussed in Section 3. We turn now to the evaluation of the averages of the three contributions (2.3)-(2.5). The averaging process requires knowledge of two probability densities. These are (i) the probability density P (~0 + r) that a sphere centre lies at the point x, + r in the field, and (ii) the corresponding conditional probability density P(x, + r 1x0) given that there is a sphere whose centre is at %. In a homogeneous suspension we have P(x,+r)
4 ’ +O-sinh ff 5
Al-'(r/u,
u) = 3 2/3
= P= II
(2.9)
n=1
X
,2nnJ;h:A
+(2n+l)
_
Ij
everywhere in the field. Batchelor[ l] has given reasons for the belief that under usual conditions the conditional probability density for a dilute suspension of identical rigid spheres is
{o[2 sinh (2n + 1)~
sinh2a-4sinh2
(n++)~~+(2n+l)~ P(x,+r(x,)
x sinh2 a] + 4 [ (2n + 3) eza + 4e-t2n+1)n
- (2n-
l)e+]}/{u[2
Xsinh2a]+2[cosh
sinh (2n+ l)a+
(2n+ 1)
(2n+l)a+cosh2ar]}. (2.7)
where (Yis related to the separation distance by the relation OL= cash-’ (r/a). An exact formula for A2 is not yet available, but its far field asymptotic form is given by A,(r/a, (+) =
3
This expression is the series representation to terms of O(a@ of the inverse of the resistance coefficient k2 (see Eq. (A.2) of the Appendix) which was recently derived by Hetsroni and
if if
r 2 2u (2.10) r<2u
correct to order c. These results represent uniform probability for all physical accessible positions of one rigid sphere relative to another. These seem likely to apply also to a dispersion of identical fluid spheres which are initially randomly dispersed since two identical fluid spheres will also fall under gravity with the same velocity, no matter how close they are. The first expression on the r.h.s. of Eq. (2.2) is obviously (U,) = U, since U, is a constant vector. The evaluation of (V) involves some difficulty because of the slowness with which the velocity disturbance in an unbounded fluid due to a falling particle decreases with distance from the particle. Thus a straight-forward integration of V over r results in an integral which is not absolutely convergent. However, Batchelor [ l] has given a method by which, with the aid of known exact mean values involving all the spheres in the configuration, this difficulty can be overcome. Using the conditions that the velocity distribution and the divergence of the deviatoric stress tensor, defined in both phases, are station-
1449 CI!&Vd.28,N0.1-P
= Q= n =o
E. WACHOLDER
ary random functions of positions with zero mean, the average of V is given by: @) = I,,,.
u(x,,, xo+ r)
+ ,>O;a2& I
[Q- P] dr
Uo-~E?r~ >
r.U,
for
r c a
(2.12)
for
r 2 a
(2.13)
u(r) = (jj$+$)U,+(i$-F)ry
where D=+&
1
=_I
I-
u’(r)dr-n
JoCr42u
u(r) dr
(2.14)
and from Eq. (2.2) we have
J
u(x,,x,+r) allspace
[Q-P]
dr= -Uoc
8+ llu 2(1+(T) * (2.18)
A helpful relation in carrying out the integration is
where S is a sphere of unit radius and I is the idemfactor (unit tensor). The next integral on the r.h.s. of Eq. (2.11) vanishes. This can easily be verified by straightforward integration of V,“u of Eq. (2.16) between the limits r = (I and r = 2~. The force, f, exerted by the deviatoric stress at the surface of a sphere equals two-thirds of the total drag force. This is a general result valid for a rigid or fluid sphere submerged in an unbounded arbitrary Stokesian flow field, it may be easily shown by using Lamb’s general solution of the creeping motion equations. When the droplet fails through an unbounded quiescent fluid this force is given by
41+u*
2a3 1+u
The corresponding in both fluids are:
Jrra
dr
(2.17)
where f is the total force exerted by the deviatoric stress on the surface of a fluid sphere with centre at x0, immersed in a flow field u(x). In the problem at hand u(x) is to be interpreted as the flow field at x that obtains when there is a fluid sphere with centre at x+ r falling through an infinite fluid with velocity Uo. The flow field has been found by Hadamard and Rybcyznski[5,6] to be
G
r) [Q -PI
P
(2.11)
E,IIL
+
[Q-PI dr
Wu(x,xo+r)L.,
B=2,213+~ 2 1+u
u(xo, x0 allspace
z-n
-n$22+4,flp
n’(r)=
I
expressions
v, 2u’ = - 2EUo 3B r.U V, 2u = B-!+U,--Tr--$
for for
213 + u f = - 47r/.LaU,I+cr
for V,.%tat points
r S a r 5 a.
(2.15)
Consequently (2.11) is
Upon substitution of Eqs. (2.9), (2.10), (2.12) and (2.13) it is found that
the last term on the r.h.s. of Eq.
1 --na2&f/p=~UOc&. 2
(2.16)
(2.19)
(2.20)
We shall now calculate the mean value of W (~0, % + r) . Again, according to Batchelor[ l] 1450
Sedimentation in a dilute emulsion
this is given by
Table 1. Values of the integral K(a) of equation (2.23) as a function of the ratio of viscosities u = p’/p
(W> = 1 W(r)P(x,+
r 1x0) dr = n J,,_WW
dr
(2.21) where in view of Eqs. (2.4), (2.9, (2.16) W(r) can be written as
w = /L’//.l
W(r) = U(r) -Uo-Uo(~~+$)-r~
0 0.1
0.84 0.91 0.99 1.09 1.21 1.32 1.40 1.43 1.48 1.52 1.53 1.53 1.53
0.25 0.5 1.0 2.0 3.5 5.0 10.0 50.0 loo.0 1ooo~0 m
x(pE)-;.2_&U+~~)
(2.22) with U(r) given in Eq. (2.6). Upon substitution of Eq. (2.6) into (2.22), the integral of W(r) over a spherical surface of unit radius is then obtained in the form
.
Substitution of Eqs. (2.18), (2.20) and (2.23) into Eq. (2.1) show that the mean translational velocity of fluid spheres in a dispersion with volume concentration c( c Q 1) is
The average of W follows from Eq. (2.2 1) as (W(r))=cUOIm
)I
(2.13) and
[A1+2A2-3(l+te)] (U) = U,,(l-[E+K(o)]c)
2
x(;rd(;)
=-c&K(a)
(2.23)
(2.24)
correct to order c.
where A, and A, are given in Eqs. (2.7) and (2.8) respectively. At larger values of r/a the integrand is given with good accuracy by its asymptotic form, which is readily found by the method of reflections (Eqs. (A. 1) and (A.2)) to be
Thus the numerical integration of Eq. (2.23) has been carried out to some large value of r/a, say R/a and the value of the analytical integral of the asymptotic form over the range r/a a R/a was then added to it. This procedure was repeated for increasing values of R/a until a numerical agreement was achieved. The results for K(o) are displayed in Table 1.
3. DISCUSSION
It may be seen that the internal circulation in the disperse phase reduces the changes in the mean settling speed due to the hydrodynamic interactions. In Fig. 1 the dimensionless mean settling speed, defined as (U, - (U) )/U,c is plotted vs. the ratio of viscosities o. This quantity, when multiplied by the volume concentration c, describes the effect of the hydrodynamic interactions in the dispersion on the mean speed of fall. Its magnitude becomes smaller as u decreases. The reason for this is clear from the following: the downward volume flux of fluid consists of a volume flux associated with the fall of the spheres cU, and a volume flux of fluid dragged down in the inaccessible shells between r = a and r = 2a surrounding the spheres
1451
E. WACHOLDER
6.6
L
Fig. 1. Dimensionless mean speed of fall as function of the ratio of viscosities fs = ~‘/,LL.
&UO[ (Q+ (+)/ ( 1 + a)]. This flux is accompanied, in a homogeneous dispersion with zero mean flux at each point, by a corresponding net upward flux. The upward volume flux in the part of the fluid that is accessible to the centre of the test sphere changes its mean settling speed by an amount of -cU,,(l+#[($+u)/(l+o)] c.f. Eq. (2.18). This is equal and opposite to the rate at which fluid volume is dragged down, and likewise becomes smaller as v decreases. The gradients of fluid velocity generated in the continuous phase by the motion of the spheres cause a net change of +*[a/( 1 + a)]cU, (see Eq. (2.20)) in the mean settling speed. This change, although positive, is smaller than the previous one such that their sum is -cU[(4+Sa)/(l+a)] which still decreases with o. Finally, when the test sphere is near one of the other spheres in the dispersion, the interaction between these two close spheres gives further change in the mean settling speed equal to -K(a)cU,, which also decreases with u as can be seen in Table 1. Therefore the smaller u the smaller is the resultant change in the mean settling speed due to the hydrodynamic inter-
actions. An overall reduction of 25 per cent is found as (+ changes between infinity and zero with a corresponding increase in the mean settling speed, (U). Since the mean settling speed is proportional to U,, it will also increase between these limits of u by a factor of 3/2 due to a larger settling speed of an isolated bubble (u = 0) in comparison with a rigid sphere (u 4 a). Consider now the magnitude of the error associated with the use of the approximate & in the calculation of K(u). As shown in Table 1 the limiting value obtained for K(u) as u + CQ(rigid spheres) is 1.53. This differs by only 0.02 (- l-5 per cent) from the more accurate value of 1.55 obtained by Batchelor [ 11. The fractional error in (U) for u + m is obviously much smaller. We believe that the error in the numerical integration of K(u) for other values of u < m is always smaller than O-02. This may be justified as follows: the missing terms of higher order in the expression for AZ (Eq. (2.8)) are obviously the source of the error incurred in the calculation of K(u) . By checking qualitatively their character we realized that they consist, in general, of powers of (g+ u)/( 1 + u) and u/( 1 + u). These terms achieve their maximum value when u + m causing an error of 0.02 and decrease as u become smaller, with a corresponding decrease in the error. Thus it seems that the formula given for the mean settling speed (Eq. (2.24)) covers the whole range of viscosities ratio u s m with a maximum error in (U, - (U))/U,c which does not exceed O-02. Acknowledgement-This work was done while the author was in receipt of a Research Fellowship from the Royal Society-Israel Academy of Sciences and Humanities exchange program. Assistance from the Yad Avi Ha-Yishuv fund is also gratefully acknowledged. The author is indebted to Professors G. K. Batchelor and N. F. Sather for their helpful advice.
REFERENCES HI BATCHELOR G. K.,J. Fluid Mech. 1972 52 245.
PI HAPPEL J. and BRENNER H., Low Reynolds Number Hydrodynamics. Prentice Hall, N.J. 1965. [31 [41 PI WI r71
HETSRONI G. and HABER S., Report No. 1 Dept. Mech. Eng., Technion, Haifa, Israel Nov. 197 1. WACHOLDER E. and WEIHS D., Chem. Engng Sci. 1972 27 1817. HADAMARD J. S., Coopt. Rend. Acad. Sci. Paris 1911152 1735. RYBCZYNSKI W.,Bull.Acad. Sci. Cracooie (ser A) 40 1911. HETSRONI G., WACHOLDER E. and HABER S., ZAMM 19715145.
1452
Sedimentation in a dilute emulsion
APPENDIX
A
Hetsroni and Haber[3] have recently used a ‘method of reflections’ solution to derive a general expression for the drag forces of two spherical droplets of different size and different viscosities falling in an unbounded quiescent fluid in a gravitational field. The results for two identical droplets having radius a are given as follows: The drag force in the direction of the line of centres
---
1 a(2+3a) 2 (l+(r)*
+&(er](;r).
(A.1)
The drag force perpendicular to the line of centres
=-u,t.
1453
(A.2)