Creeping flow over a composite sphere: Solid core with porous shell

Ciwnical E,,&eeri,,g Science, Vol. 42, No. 2. pp. 245-253, Printed in Great Brhin.

CREEPING

1987.

OOW-250!9/87 Pergamon

FLOW OVER A COMPOSITE SPHERE: CORE WITH POROUS SHELL

E3.00 + 0.00 Journals Ltd.

SOLID

JACOB H. MASLIYAH Department of Chemical Engineering, University of Alberta, Edmonton, Alberta, Canada GRAHAM NEALE Department of Chemical Engineering, University of Ottawa, Ottawa, Ontario, Canada K. MALYSA+ and THEODORUS G. M. VAN DE VEN Pulp and Paper Research Institute of Canada, McGill University, Montreal, Quebec, Canada

(Received 26

November

1985;

in revised form 20 May 1986)

Abstract-Creeping

flow past a solid sphere with a porous shell has been solved using the Stokes and The dimensionless solid core and shell radii, normalized by the square root of the shell permeability, are the two parameters that govern the flow. In the limiting cases, the analytical solution describing the flow past the composite sphere reduces to that for flow past a solid sphere and a homogeneous porous sphere. The settling rates of a solid sphere with attached threads are measured experimentally. This system can be considered a model for rigid linear molecules anchored or adsorbed onto a colloidal particle. The analytical solution for the composite sphere is in excellent agreement with the experimental results. Brinkman

equations.

INTRODUCTION

THEORETICAL

Fluid flow relative to floes of loosely packed fine particles is important in the study of floe sedimentation. Such floes can be described by an effective porous sphere. An understanding of fluid flow relative to a porous sphere can consequently lead to a better appreciation of flow past floe of fine particles. Theoretical and experimental studies are found in the literature dealing with sedimenting homogeneous porous spheres and porous shells. Theoretical studies include Brinkman (1947), Ooms et al. (1970), Sutherland and Tan (1970), Neale et al. (1973), Jones (1973) and Nandakumar and Masliyah (1982) and experimental studies include Matsumoto and Saganuma (1977) and Masliyah and Polikar (1980). One can generalize the flow relative to a homogeneous floe to the situation of a floe with a solid central core. A similar physical situation is present for the settling of a solid sphere with adsorbed long chain polymers (Lee and Fuller, 1985). One can model such a system by considering fluid flow past a composite sphere having a central solid core and an outer porous shell. This type of modelling has been recently made by Sasaki (1985). In this paper, we first present the governing flow equations, develop the appropriate solution and describe the solution for flow past a composite sphere having a solid core and a porous shell. In the second part of the paper, the settling of a solid sphere with attached threads is experimentally measured. The drag force on such a composite sphere is compared with the theoretical analysis of the first part of the paper.

The

Polish Academy

of Sciences,

30-239

employed

is that of an isolated

composite

outer radius b, comprising a solid impermeable sphere of radius a, surrounded by a shell of homogeneous and isotropic porous material of porosity E and permeability k. We will consider steady, axi-symmetric, creeping flow of a Newtonian fluid of viscosity p, relative to this composite sphere. For convenience, we consider the sphere to be stationary having its centre at the origin of spherical coordinates [r, 8,&l, with the fluid approaching in the + z direction at velocity U,, as illustrated in Fig. 1. The principal objective of the ensuing analysis is to determine the hydrodynamic resistance experienced by such a composite sphere. The analysis which follows parallels closely that presented by Neale et al. (1973) for the case of an homogeneous porous sphere.

sphere

tPresent address: Institute of Catalysis and Surface

Chemistry, Poland.

model

Krakow,

245

Defining

of

equations

Within the unobstructed fluid outside the composite sphere, the Stokes and continuity equations describe

the prevailing flowfield. Thus, Newtonian creeping flow: pV2u=Vp

b
for

incompressible (1)

and V-u=0

b
(2)

where II = [u,, ug, u+] denotes the fluid velocity vector, and p the fluid pressure (referred to a reference level). The corresponding equations describing the Aowfield within the permeable shell region have received considerable attention in the literature. The equation of motion used to describe Newtonian creeping flow through an isotropic porous medium is the extension

JACOB

246

H.

MASLIYAH

et al.

agreement with experimental data than when setting P* = ,u. Identifying p* with p is thus in keeping with established practice.

f

Boundary

conditions

The various boundary conditions which have been postulated to describe flow characteristics at the boundary between a porous medium and a free fluid or between a porous medium and a solid surface have received considerable attention in the literature. These conditions have been reviewed and discussed in detail by Neale et al. (1973) and Koplik et al. (1983) in relation to both Darcy’s Law and the Brinkman equation, and are not therefore reviewed again here. The physically realistic and mathematically consistent boundary conditions which describe the present problem are as follows (see Neale et al., Ooms et al., Brinkman and Koplik et al.) I

I

I

I

I

I

I

Approaching

Fluid

Fig. 1. Coordinate system for ax&symmetricflow relativeto an isolated composite sphere. of Darcy’s equation: +*

Law,

better known

+p*Vz”*

= VP*

as the Brinkman

a
(3)

with the incompressibility condition here assuming the macroscopic form, Lundgren (1972) and Saffman (1971): V-u*

=O

a
(4)

where * denotes any macroscopically averaged quantity pertaining specifically to the porous medium region. The Brinkman equation was used in preference to the Darcy equation in order to accommodate for the boundary conditions at the free fluid and the porous media. Full discussion on the merits of Brinkman equation over Darcy equation can be found in Neale et al. (1973), Ooms et al. (1970), Brinkman (1947) and Joseph and Tao (1964). Here p* denotes an effective viscosity in the porous medium which can differ from p. Equation (3) has received theoretical justification from Lundgren (1972), Saffman (1971), Slattery (1969), Tam (1969) and Childress (1972). Lundgren actually attempted to predict c(* (for a dilute swarm of stationary solid spheres) and demonstrated that p* + p as E ---, 1. This indicates that it is in order to assume P * = p for high-porosity porous media which is exactly the situation in the present problem. In fact, Brinkman favoured putting p* = ,u because this choice provided the closest agreement between his predictions for the permeability of a swarm of solid spheres and experimental data. This has been confirmed more recently by Koplik et al. (1983) who, although putting forward arguments that p* # p, observed poorer

uF(a, e) = 0

060<27t

(5)

ue*(a,@ = 0

0<8<2r

(6)

u:@,@

= u,(b,B)

0<6<2a

(7)

G(b.8)

= Ue(b.0)

0<0<2n

(8)

r$ (b, 0) = r,r tb. 0)

0<8<2n

(9)

s,” (b, 0 I= z,e (b, 0)

0<0<2n

(10)

0Ge<2n

(11) (12)

u,(r,‘O) =

-

U,

cos 19

limit r-00 ( z+(r, 0) = U, sin 0

o
where r,,[= p-2p(du,/&-)] and T,@ denote the normal and tangential components of the stress tensor, respectively. Equations (5) and (6) indicate that there is no slip at the solid surface, r = a. Equations (7+(10) are statements of continuity of the normal and tangential velocity and stress components across the permeable interface at r = b, while eqs (11) and (12) indicate that the velocity field becomes uniform far from the composite sphere. It is straight forward to show that when p* is equated to p, eq. (9) becomes algebraically equivalent to continuity of pressure, Koplik et al. (1983), provided that eqs (7) and (8) are satisfied simultaneously; i.e. p* (b, @) = p(b, 0). Method

(13)

of solution

Due to axisymmetry, we introduce a stream function I++related to the velocity field by -1 uI =pr’sin6,

al,& a6,

(14)

and 1 ati UB= p-_ r sin 8 dr

(15)

Similarly, a stream function I,Q*is also introduced for the porous region. Taking the curl of the Stokes eq. (1) and that of the Brinkman equation (4), one obtains,

Creeping flow over a composite sphere respectively E4rl/ =O

h
(26)

E = 2B+2p3F

(27)

D=O

(28)

(16)

and a
c=

(17)

247

F = (G cash a + H sinh a)/3a

-1

(29)

A=P3-f12B+E+/33F

where

+ (cash p - p sinh fi)G

+ (sinh fi - j? cash /3)H.

E”=$+

in spherical coordinates and where p//l* The general respectively

solutions

= 1.

of

eqs

(16)

and

(17)

are,

$+B<+C<2+D<4 1 x sin’ 0

/3 4 < < co

(30)

The drag force, F, experienced by the composite sphere is calculated by integrating the normal and tangential stress distributions over the surface thereof, Happel and Brenner (1965), viz.

F = 2xb2

u [r, s 0

cos 8 - Tr, sin 131,= b sin 0 de.

(31)

(19) The result obtained is: F = 6npUU,,bR

+G

cash < --sinfir 5

+H

sinh < p-cash <

where 5

>I

n = 2B/3jK

sin’ 0

a < r
(20)

a=a/,,h,~=b,,/%. where 5 = r/d, In order to satisfy the required set of boundary conditions stipulated in eqs (5+(12), the arbitrary constants A, B, C, D, E, F, G and H appearing above take the following specific values: B=-

BO

.

(324

(21)

2 (a smh B - cash a} J

(32b)

The drag force can actually be more easily evaluated using the Payne and Pell (1960) relationship and it leads to the same results shown here. The ultimate expression for Q is obtained by substituting for B from eq. (21). The algebra leading to eq. (21) is straightforward but exceedingly tedious. However the correctness of this equation may be confirmed by examining a number of limiting cases for which analytical solutions are already known.

where B, = 3 (a4 + 2018~ + 3a2) cash a

Case I: as a -+ j? (i.e. for a solid sphere of radius b) we obtain a --+ 1 and F + 67~~ U, b. In other words, Stokes Law is retrieved as required.

Limiting

+ 9a2 (cash /? - /? sinh fi - a sinh a)

+3coshA[(a3+2fi3+3a) -a

cash j?-pcosh

(aj?sinh/?

a)+ 3a’Psinh

a]

Limiting Case II: as /? + a (i.e. for a solid sphere of radius a) we obtain R - a/b and F - 6zpU,a. Again, Stokes Law is correctly retrieved.

+3sinhA[(~~+2/3~+3a)cosha + 3a2 (ap sinh j3 - a cash fl-

sinh a)]

(22)

and

Limiting

xcoshA+3(a2-l)sinhA with

Case

radius b), we required.

J = -6a+(3a+3fi+a3+2fi3)

as k - 0 (i.e. for a solid sphere of obtain R --+ 1 and F + 6npU, b, as

III:

(23)

A=fl-a

H = 3(a3+2fi3)cosha+9a(cosha-asinha-coshB+Psinhfi) ~ J

G = 3a - (a cash /3 - sinh a)H a sinh fi - cash c(

(25)

(24) Limiting Case IV: as k -+ 03 (i.e. for a solid sphere of radius a), we obtain Q-a/B and F + 67cpU,a, as required.

248

JACOB

H. MASLEYAH

Limiting Case V: as a - 0 (i.e. for a homogeneous porous sphere of radius b) we obtain:


This result is identical to that for an isolated homogeneous porous sphere predicted previously by Brinkman (1947, 1949), Ooms et al. (1970) and Neale et al. (1973).

Ib)

Theoretical

ef al.

cY=o.o01

-60

/3=0.0026

-

8.0

ff=l

_4.0:_

results

The flow streamlines are shown in Fig. 2(ab(d)

for

various values of a( = a/>) and /?( = b/s). In all cases the ratio of /I/o! was fixed at 2.6 where B/a = b/a; i.e. the ratio of outer shell radius to the inner core radius. The stream function was dimensionalized using In order to appreciate the effect of Y = ti/kU,. varying a it is best to think of fixing the inner core radius a and varying the shell permeability k. Consequently a low value of a signifies a high permeability porous shell. By fixing /?/a, we are in effect fixing the thickness of the porous shell surrounding the inner core. For a low value of a = 0.061 the streamlines become almost identical to those of flow over an isolated solid sphere of radius a and thus the permeable porous shell has little effect on the flow streamlines. However as c(is increased, whereby the permeability of the porous shell decreases, the streamlines shift away from the central core. At a = 10 the streamlines (u’ > 0.05) lie above the shell outer surface whereby very little flow passes through the porous shell. In this case the flow resembles that of fluid flow over a solid sphere of radius b. The angular velocity u,,/U, variation along the line 0 = x/2, is shown in Fig. 3 for various values of a with /l/a = 2. For the case of a = 0.01 the variation of is identical to the case of flow over a solid %0/U, sphere of radius a. Here, the porous shell has little effect on the fluid flow due to the large permeability (at fixed value of a). For a = 0.01, the curve use/U, vs r/a is smooth at the outer edge (r = b = 2a) of the porous shell. As a is increased, the porous shell offers more

Fig. 2. Contours of the stream function.

r/a Fig. 3. Variation of ugO/U,

.o

@=2.6

with radial distance at 0 = n/2.

Creeping flow over a composite sphere resistance to flow inside the porous shell. For c( = 10, the flow velocity is fairly low within the porous shell and the velocity profile approaches that of flow over a solid sphere of radius b. The variation of the drag force experienced by a composite sphere is shown in Fig. 4. The drag force, F, is normalized with that of a solid sphere of radius b, F,,. As the drag force of a composite sphere, F, is always less than that of a solid sphere of radius b, then all = 1. It is in the curves for F/F, lie below the line F/F, limit of c(+ B that the value of F/F, is unity. This is the case where the solid core radius is the same as the outer shell radius, i.e. no porous shell is present and one is dealing with the case of a solid sphere of radius b. This upper bound can also be reached when p -+ co, i.e. the shell permeability k - 0 and the composite sphere acts as a solid sphere with radius b due to infinite resistance to flow offered by the porous shell. A lower bound for F/F, exists and it represents the case of a + 0. In other words, the inner solid core is absent and the resistance is due to a homogeneous porous sphere of radius b, i.e. the porous shell extends all the way to the origin. The variation of F/F, with fl is shown for various values of do.It is most convenient to imagine that each constant o! curve represents a constant solid core of radius a and a constant permeability k. The change in fi value then arises from changes in b, the radius of the outer porous shell. In other words, the permeability of the porous shell is kept constant but its thickness (b - a) varies. The constant c( curves indicate that once /3 N lOo( (i.e. b N 10a) the composite sphere behaves as if it is a porous sphere of radius b. In other words, the presence of the solid core becomes insignificant. For large a(a > 2), the solid core becomes insignificant at B values of about 2a. Figure 4 clearly shows that except for very low a values, the composite sphere experiencs a drag fairly close to that of a porous sphere of radius b and irrespective of ~values, all composite spheres experience the same drag as a porous sphere when /? is large. This lower bound is, of course, given by eq. (33) in conjunction with eq. (32a). However, it was generated using the general expression for the drag force [eq. (21) with eq. (32a)]. It should be noted that the data presented in Fig 2(aE(d) were generated using double precision ar-

249

ithmetic (14 to 15 digits). Unreliable data would be generated if single precision arithmetic had been used.

EXPERIMENTAL

To test the theory described above, we chose to study the sedimentation of a solid sphere to which various flexible threads were attached. Although flexible, these threads all assumed a complete stretched configuration when attached to a sedimenting sphere. The reasons for choosing this system are two-fold: firstly by having a sufficient number of threads, this provides a convenient well characterized porous shell; secondly, the model has bearing on the hydrodynamics behavior of colloidal particles on to which polymers are anchored or adsorbed. The settling velocities of spheres with threads attached to their surface are determined and the drag force experienced by such spheres is then computed. The drag force is then related to that of a composite sphere. It will be shown that there is a good agreement between the experimental drag force and that of a composite sphere model. Apparatus

Experiments were carried out in a plexiglass tank having inner dimensions of 42 x 42 x 45 cm. Two side walls of the tank were fitted with high quality optical glass windows used to view the settling sphere and to photograph the sphere as it settles down the tank. Figure 5 shows a schematic view of the experimental set-up. A Linhoff camera with a stroboscopic illumination system was used. The stroboscopic lamp was triggered by a quartz timer. Precision of the timer was checked by taking photographs of a stopwatch and it was accurate to within about 1%. The calibrated time intervals between flashes were used in the calculation of the sphere velocity. Single frame multiple image technique was applied, i.e. the experiments were carried out in a dimmed room, the camera shutter was permanently open and every flash of the stroboscopic lamp resulted in a recording on the photograph_ of a subsequent position of the sphere. Photographs of a single frame multiple image exposures of a sphere with various number ofattached threads are shown in Fig. 6.

solid

sphere

of radius

08

04

Fig. 4. Variation of the drag ratio with p for various a values.

b

250

JACOB

H. MASLIYAH

et al.

was 1.60 and their diameter was 0.13a. The working fluid was silicone oil having a viscosity of0.99 Pa s and a density of 974 kg/m3 at 23°C. Procedure

J

A B C D E

tank stroboscopic lamp centering device vacuum raeas* valve pipette

F 0

cmnera rubber

hose

U

Fig. 5. Schematicview of experimentalset-up. A nylon sphere of radius 0.318 T 0.003 cm was used. The sphericity of the sphere was checked by measuring its diameter in different directions and it was better than 0.2%. The sphere density was 1116.6 kg/m’. Polyester threads of density 1060 kg/m3 were used to construct the composite sphere. Their average length

The nylon sphere and the threads were cleaned in methyl alcohol before and after gluing additional threads onto the sphere. Every separate thread was glued to the sphere surface using Colle Super Glue. A micro pipette capillary was used to apply the glue. The sphere with the attached thread(s) was conditioned in the silicone oil for a period of at least 24 h before settling measurements were made. The first six threads were affixed symmetrically on the sphere. At the start of each experiment, the sphere was held at the top of a pipette by applying suction at the other end of the pipette. At all times the sphere was below the silicone surface. By releasing the suction, the sphere begins its descent in the silicone oil. When the sphere is in line of view of the camera, the stroboscope flash light is triggered at pre-set intervals and the sphere position with respect to time is recorded on a poloroid photographic plate. At least five experimental runs were performed for each given number of attached threads.

Fig. 6. Photographs of settling sphere with threadsA, n = 1; B, n = 4; C, n = 5; D, n = 6; E, n = 7; F, n = to; G, a = 15; H. n = 21.

Creeping Aow over a composite sphere The position of the sphere was analyzed using a specially built optico-mechanical deviice which allowed accurate measurement to within 0.01 mm. The sphere velocity, U, was evaluated using the expression L(=

f%-AI

-‘t+Ar

(34)

2At

where z,_& and tt+Al are the sphere positions at times t - At and t + At, respectively, and At is the interval between two flashes. All measurements are made when the sphere has reached its terminal velocity. An estimate of the time required to reach steady-state for a bare sphere is given by Clift et al. (1978). Discussion of experimental results The settling velocity of the sphere without

the threads was measured to be 0.312 + 0.003 cm/s. The velocity calculated from Stokes Law is 0.3 18 cm/s_ The measured value is lower than the theoretical value by about 1.8%. This is attributed mainly to wall effects where the ratio of the container width to the sphere diameter is 66. The value of the settling velocity of the sphere with attached threads is a consequence of the balance existing between the buoyancy force on the “composite sphere” and the overall drag force. The settling velocity is of course affected by the buoyancy due to the attached threads. Consequently, mere observation of the settling velocity may not be sufficient to ascertain the effect of the attached threads. In this respect, it is preferable to examine the drag force experienced by a sphere held stationary with the fluid flowing over it. A force balance on a composite sphere gives s(v,(P,--Pr)+

UP,---p/l)

= 6n~uaK

(35)

where V, and V, are the volume of the threads and the sphere, respectively. P,, ps and p/ are the densities of thread, sphere and working fluid, respectively. u is the settling velocity of a composite sphere and K is a correction factor for the drag. The settling velocity, u,, of a bare sphere of radius a is given by 9 K(P,-P,)

= ~WQ%

(36)

combining eqs (35) and (36) gives

SK(P,---P/l

Kc?+

6xpua

.

(37)

F = 6npaU,K

for the composite sphere and F, = 6xpaU,

for the bare solid sphere of radius a. The above two equations lead to

Fa

K.

Combining eqs (37) and (38) leads to F <=u

%+Sv,~P,-PJ) 6xpua

(38)

’

(39)

The second term of eq. (37) and (39) is the correction the needs to be applied to correct for the buoyancy effect due solely to the threads. In this study the contribution of the second term is small and it is about 5 y0 for the highest number of threads. Its effect is smaller when the number of threads is low. The drag ratio F/F, can be evaluated from the physical data and measurements of the settling velocity II. A plot of the velocity ratio u/u, vs the number of threads is shown in Fig. 7. The sphere velocity was found to decrease with increasing number of the threads, n. However, the velocity changes are not a smooth function of n but take place in a discrete manner. These discrete changes are not only due to the addition of threads but also due to changes in the preferable settling orientation of the sphere as the number of threads is increased. The approximate positions of the threads as settling occurs are marked on the experimental data points of Fig. 7. The coordinates are shown in the insert of Fig. 7. The first six threads were located in a symmetrical pattern and their orientations were recorded. However, for n 2 7 it became difficult to distinguish the orientation of the threads. The first attached thread caused a relatively small change in u/u, (ofabout 4 %). Irrespective of the thread position at the time of the sphel’e release, the sphere settling position became such that the thread pointed vertically upward. The addition of a second thread caused a rapid decrease in the fall velocity and the threads took a stable orientation of (0, 1, 0) and (0, - 1, 0). This configuration was such that the dew was perpendicular to the threads which were located on opposite sides of the spheres. Addition of the third thread has little effect on the fall velocity where the third thread took up the same position as that of the sphere with a single thread. Significant decrease in the particle velocity was observed after the addition of the fourth thread. All the four threads were in a horizontal plane normal to the line of fall of the sphere. The average .thread concentration (volume fraction) is given by c = 3nla:/4(b3

Assuming now that the sphere is held stationary with a fluid flowing past at velocity U,, the drag forces will be given by

F -=

251

- a3)

(40)

where n is the number of threads, 1 is the thread length and a, is its radius. b and a are the outer composite sphere radius and bare sphere radius respectively. Here, b-u = 1. Making use of the above expression for the thread concentration and eq. (39), the drag ratio F/F, variation with thread concentration is plotted in Fig. 8. The experimental data clearly shows that the drag on the composite sphere increases with the thread concentration. The drag ratio F/F. should reach the limiting value of 2.6 ( = b/u) as the concentration approaches unity. In order to compare the experimental data with the

252

JACOB

I

I 5

0.2

H. MASLIYAH

10

ef al.

I

I

n

20

15

Fig. 7. Variation of u/u, with number of threads.

-

Sphere

COmpOSlte

Model

Experimental

Volume

Fraction,

c

Fig. 8. Variation of drag ratio for a sphere with threads.

theoretical analysis presented earlier, we need to know the permeability of the porous medium created by the threads. To this end one needs a relationship between the permeability and the thread concentration for a specified flow configuration. The flow over the threads can be considered to be mainly that of a flow normal to an array of circular cylinders. In fact, the streamlines of Fig. 2 within the porous shell support such an assertion. For flow normal to fiber mats and arrays of cylinders, the theoretical models of Happel (1959) and Neale and Masliyah (1973) are in excellent agreement with the experimental data as shown by Guzy et al. (1983). Using Happel’s model the permeability is given by k-

-g[lnc+(l-c2)/(1+c2)]

where c is the fiber volume fraction.

(41)

Making use of eq. (41) the drag ratio F/Fa is evaluated using the composite sphere model developed above. The agreement between the experimental and theoretical values of the drag ratio is good as is shown in Fig. 8. It should be noted that no adjustable parameters appear in the model. There are two sources of errors that need some clarification. First, a sphere with threads does not possess a uniform thread concentration in the radial direction and the expression of eq. (40) is an average quantity. Moreover the threads do not give a homogeneous porous media. The second source of error is that no compensation is made for the effect of the container walls on the settling velocity of the composite sphere. Presumably, the wall effect is slightly larger for the sphere with threads since the ratio of the composite sphere diameter to the container width is higher. However, the error involved should not be more than about 2-3 O/_. CONCLUSION

The drag experienced by a composite sphere was derived using the Stokes and the Brinkman flow equations. The drag force experienced by a settling sphere with attached threads was modelled using a composite sphere model. Such a composite sphere is a model for a colloidal particle on which polymers are adsorbed. In polymer adsorption one usually distinguishes between trains [sequences of segments adsorbed onto the particle, loops and tails (dangling ends)]. The dangling ends can extend a substantial distance into the medium, Fleer and Scheutjens (1982). From the results of this study it can be concluded that

Creeping flow over a composite sphere just a few dangling ends protuding from a colloidal particle can result in substantial changes in the effective hydrodynamic particle radius. Acknowledgement-One of the authors (J. H. Masliyah) is grateful to Natural Science and Engineering Council of Canadafor providing a Senior Industrial Fellowship while on

Study Leave at the Pulp and Paper Research Institute of Canada.

a

z C

F F, Fb 9 k K

1 n P r II % u, V Z

NOTATION raidus of solid sphere radius of attached thread radius of outer porous shell concentration (volume fraction) drag force experienced by a stationary composite sphere drag force experienced by a stationary solid sphere of radius a drag force experienced by a stationary solid sphere of radius b acceleration due to gravity porous shell permeability correction factor length of attached thread number of attached threads pressure radial coordinate settling velocity of a composite sphere settling velocity of a solid sphere of radius a approach fluid velocity volume fall direction of a sphere

Greek letters a dimensionless solid sphere radius, af fi

P A & 8 P r P

r

* Y

dimensionless outer shell radius, b/J% dimensionless porous shell thickness porosity angular coordinate fluid viscosity dimensionless radial coordinate, r / fi density shear stress stream function dimensionsless stream function, ti / k U w

Subscripts

L 90

f

s

t

radial direction angular direction value of velocity at 0 = x/2 fluid solid sphere thread

253

Superscript *

pertaining to porous medium

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