Creeping Flow of a Sphere in Tubes Filled with a Bingham Plastic Material

l~-lqcmmi~

Mechanics ELSEVIER J. Non-Newtonian Fluid Mech., 70 (1997) 59-77

Creeping motion of a sphere in tubes filled with a Bingham plastic material

J. Blackery , E. Mi t sou l i s *

Department of Chemical Engineering, University of Ottawa, Ottawa, Ont. K1N 6N5, Canada

Received 20 May 1996; received in revised form 24 October 1996

Abstract

Numerical simulations have been undertaken for the creeping flow of a Bingham plastic past a sphere contained in a cylindrical tube. Different diameter ratios have been studied ranging from 2:1 to 50:1. The Bingham constitutive equation is used with an appropriate modification proposed by Papanastasiou, which applies everywhere in the flow field in both yielded and practically unyielded regions. The emphasis is on determining the extent and shape of unyielded/yielded regions along with the drag coefficient for the whole range of Bingham numbers. The present results extend previous simulations for creeping flow of a sphere in an infinite medium and provide calculations of the Stokes drag coefficient in the case of wall effects. © 1997 Elsevier Science B.V.

Keywords: Yield stress; Bingham plastic: Flow past a sphere; Wall effects; Drag coefficient; Viscoplasticity

1. Introduction

An important class of non-Newtonian materials exhibits a yield stress, which must be exceeded before significant deformation can occur. A list of several materials exhibiting yield was given in a seminal paper by Bird et al. [1]. The models presented for such so-called viscoplastic materials included the Bingham, Herschel -Bulkley and Casson. Analytical solutions were provided for the Bingham plastic model in simple flow fields. Since then a renewed interest has developed among several researchers to study these materials in non-trivial flows both numerically [2-20] and experimentally [21-26],

The creeping motion of spheres in an infinite medium behaving as a Bingham plastic has been studied in a key paper by Beris et al. [5]. This paper provided the location of the yield surfaces as part of the solution, and the Stokes drag coefficient for the whole range of the dimensionless Bingham number. On the other hand, and within the viscoelastic context, the mot ion of a sphere

* Corresponding author.

0377-0257/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PIIS0377-0257(96)01 536-4

60 J. Blackery, E. Mitsoulis / J . Non-Newtonian Fluid Mech. 70 (1997) 59-77

through a viscoelastic fluid contained in a cylindrical tube has been proposed as one of the benchmark problems to be used for comparison of reliability and accuracy of different numerical techniques [27]. Many papers have dealt with this subject and for a variety of diameter ratios (see, e.g. [28-30]).

Within the viscoplastic context, early work focused on the problem of a sphere falling in an infinite medium [31,32,5]. Fig. 1 shows schematically the shapes and extents of the yielded/unyielded regions postulated by Ansley and Smith [31] and by Yoshioka et al. [32], and accurately determined via simulation by Beris et al. [5]. The experimental work by Atapattu et al. [25,26] has dealt extensively with wall effects and has provided new important information on viscoplastic flows around spheres. However, these authors compare their results against simulations valid for an infinite medium [5]. It is the purpose of the present paper to provide the corresponding results for the motion of a sphere through a viscoplastic medium contained in a cylindrical tube for different tube/sphere diameter ratios, and also calculate the corresponding drag coefficients for Bingham plastics.

(a)

0 ( (b) (ii)

Fig. 1. Schematic representation of the shape of the yielded/unyielded regions surrounding a sphere in creeping motion through an infinite Bingham medium: (a) Ansley and Smith [31], slip-line theory; (b) Yoshioka et al. [32], (i) maximum principle, (ii) minimum principle; (c) Beris et al. [5], finite element method (FEM).

J. Blackery, E. Mitsoulis /J. Non-Newtonian Fluid Mech. 70 (1997) 59-77 61

1T IIVIFIT V

O Re.

Fig. 2. Schematic representation of the system of a sphere falling in a tube filled with a viscoplastic medium. The shaded regions are unyielded and so are the small black regions forming polar caps around the stagnation points on the sphere.

2. Mathematical modelling

The problem of the creeping motion of a sphere falling inside a tube filled with a viscoplastic medium is schematically depicted in Fig. 2. The flow is governed by the usual conservation equations of mass and momentum. For an incompressible fluid under isothermal, creeping flow conditions we have

V ' ~ = 0 , (1)

0 = - V p + V ' ~ , (2)

where ~ is the velocity vector, ~ is the extra-stress tensor, and p is the scalar pressure. To model the stress-deformation behaviour of a Bingham plastic, the Bingham constitutive

equation has been proposed [1]. In simple shear flow it takes the form (see Fig. 3):

= ~y + lay for I~l > ~y, (3a)

= 0 for I~[ ~< ~y, (3b)

where ~ is the shear stress, p is the shear rate, ~y is the yield stress, and / t is a constant plastic viscosity. Note that when the shear stress ~ falls below zy, a solid structure is formed (unyielded). To avoid the discontinuity inherent in any viscoplastic model, Papanastasiou [6] proposed a modification in the equation by introducing a material parameter, which controls the exponential growth of stress. This way the equation is valid for both yielded and unyielded areas. Papanastasiou's modification when applied to the Bingham model becomes

62 J. Blackery, E. Mitsoulis / J . Non-Newtonian Fluid Mech. 70 (1997) 59-77

Z" ---- Zy[1 - - exp( - m~)] + # i (4)

where rn is the stress growth exponent (with units in seconds). As shown in Fig. 3, this equation mimics the ideal Bingham plastic (for m/> 100 s). The exponential modification was previously applied to Bingham plastics and used by Abdali et al. [13] to study entry and exit flows and determine the shape and extent of yielded/unyielded regions, extrudate swell and excess pressure losses through planar and axisymmetric geometries as a function of a dimensionless yield stress or Bingham number.

In full tensorial form the constitutive Eq. (4) can be written as

r = ~ 7, (5)

where r/ is the apparent viscosity given by

"(y r/=/.t + [-~ [1 - exp( - ml#l)], (6)

and is the magnitude of the rate-of-strain tensor ~ - -Vg+ Vg T, which is given by

I~I = 1II~ = ~=TfJ , (7)

where II~ is the second invariant of 7. To track down yielded/unyielded regions, we shall employ the criterion that the material flows (yields) only when the magnitude of the extra stress tensor exceeds the yield stress Ty i.e.

yielded

unyielded: ~ Ty. (8b)

3.0 i i i l i

r=l~y.Ty(1-e -m~) 2.5

\ 2 . 0 . . . . .

1 . 0

0.5

0 , 0 0 . 0 2 , 0 4 , 0 6 . 0 8 . 0 1 0 . 0 12.

SHEAR RATE, ~ (~) o

Fig. 3. Shear stress vs. shear rate according to the modified Bingham constitutive Eq. (4) for several values of the stress growth exponent m.

J. Blackery, E. Mitsoulis / J. Non-Newtonian Fluid Mech. 70 (1997) 59-77 63

The viscoplastic character of the flow around a sphere is assessed by a dimensionless yield stress (r*) or Bingham number (Bn) (Bn is used here instead of Bi used previously [1,5,13] to avoid confusion with the Biot number of heat transfer), defined respectively by (Beris et al. [5])

, 2ry 7rR2 ~'Y - - F ' (9)

Bn - 2zyR (10) IL V '

where R is the sphere radius, V is the fall velocity of the sphere, and F is the external force acting on the sphere. For a sphere in free fall

4 F = ~ zcR3(ps- p)g, (11)

where p is the fluid density, Ps is the density of the sphere, and g is the acceleration of gravity. The Stokes drag coefficient is defined as (Beris et al. [5])

F Cs - 6 zc/2------~" (12)

The dimensionless yield stress r* (Eq. (9)) can then be written in terms of the Bingham number and drag coefficient as

Bn r* = 6Cs' (13)

The gravity-yield number Y~ is defined as (Chhabra and Uhlherr [24])

7TR2Ty Ty* Y~= 8F =-3-" (14)

In all cases, the Newtonian fluid corresponds to ~* = Bn = ¥~ = 0. However, at the other extreme of an unyielded solid, Bn--, o0, while ry* reaches a dimensionless value of 0.143, beyond which the sphere will not fall in an infinite medium [5]. For this case and using Eq. (13), Cs = 1.17Bn.

3. Method of solution

The constitutive equation for the Bingham plastic must be solved together with the conservation Eqs. (1) and (2) and appropriate boundary conditions. Fig. 4 shows the solution domain and boundary conditions. Because of the creeping flow assumption (Re = 0), there is symmetry in both the r- and z-directions in a cylindrical coordinate system (r, z, 0). It is therefore necessary to consider only one quadrant of the flow domain, as was done previously by Beris et al. [5]. Note that this is not the case for flows with inertia or viscoelastic flows, where the flow field is not symmetric before and after the sphere. The usual handling of the sphere-in-a-tube problem is to assume the sphere stationary while the fluid and the cylinder walls are moving

64 J. Blackery, E. Mitsoulis /J . Non-Newtonian Fluid Mech. 70 (1997) 59-77

0 i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

V V

I l l I l l E

. . . . . . . . . . . . .

i i

Re rib

C A

k

Fig. 4. Flow domain and boundary conditions for the flow of a sphere in a tube filled with a viscoplastic medium. A cylindrical coordinate system is used (r, z, 0). Because of the creeping flow assumption there is symmetry in all directions, so only one quadrant need be used for the simulations (area enclosed by ABCDE).

with a constant velocity V. This eliminates the time dependence associated with the problem of a moving sphere. The boundary conditions are, therefore: symmetry along AB (Vr = 0), symmetry along DE (Vr = 0), constant velocity along the tube wall BC (vz = V, vr = 0), constant velocity along the outflow boundary CD (Vz = V, Vr = 0), no slip at the sphere surface along EA (Vz = Vr = 0), while the reference pressure is set to zero at one point (here point B). All lengths are scaled with R, all velocities with V, and all pressures and stresses with /~ V/R.

The numerical solution is obtained with the Finite Element Method (FEM) employing as primary variables the two velocities and pressure (u-v-p formulation) as explained in our previous papers [13,14]. Simulations are carried out for the whole range of Bingham numbers (0 ~< Bn < ~ ) by increasing the yield stress values from 0 to about 3 000 000, while the radius R, fall speed V, and plastic viscosity /~ have been set equal to unity. In order to come as close as possible to the ideal Bingham model, we have used in the modified apparent-viscosity Eq. (6) a value for the exponent m = 200 s, since higher values (m = 1000 s) did not affect the results.

A zero-order continuation scheme has been used to increase the Bingham number. The solution process starts from the Newtonian solution (ry = 0), which is used to obtain a first approximation. Iterations are performed until convergence for the current yield stress is achieved (usually when both the norm-of-the-error and the norm-of-the-residuals <0.001, although runs conducted with tolerance <0.01 gave virtually identical results). The new solution is then used as an initial estimate for the higher yield stress value to obtain a solution for a higher Bingham number. The number of iterations increased substantially as the yield stress was increased (departure from Newtonian fluid and approach to solid plastic).


4. Results and discussion

4. I. Newtonian results

We first applied our numerical scheme to the calculation of the creeping flow of a Bingham plastic past a sphere contained in a cylindrical tube with a diameter ratio of 2:1. Since this is a benchmark problem, accurate solutions exist in the literature for the Newtonian fluid (Bn = 0), given by a number of researchers [28-30,33,34].

With regard to Fig. 4 let L be the length downstream of the sphere centre and Rc = 2R the cylinder radius. For our calculations, we have chosen the ratio L/R to be 6, as done previously by other researchers [29]. The corresponding L/R ratios for the other cases are 6, 8, 10, 50 for the 4:1, 8:1, 10:1 and 50:1, respectively.

Using previous experience with finite element meshes for the 2:1 benchmark problem and for other diameter ratios, we have used the meshes shown in Fig. 5. The number of elements, nodes and the degrees of freedom (DOF) associated with each mesh for the different Rc/R ratios are given in Table 1.

Table 2 shows the Newtonian results for the drag coefficient Cs for the 5 diameter ratios studied. The values are compared with the Bohlin approximation (see e.g. [29]) given by

[ ' 0 10444( )+ 088 7( ) 3 // R x~ 10"]- 1

- 4 1 9 - -

The differences are always below 0.5% and compare very well with the numerical values given in the literature [28-30,33,34] for very dense grids.

4.2. Viscoplastic results--yielded/unyielded regions

The results for the Bingham plastic model as given by Eqs. (5) and (6) are presented in Figs. 6-10, which show the shape and extent of the yielded/unyielded regions (shaded) for diameter ratios 2:1, 4:1, 8:1, 10:1 and 50:1, respectively. For the 8:1, 10:1, and 50:1 ratios, the figures on the right-hand side are 4 x 4 windows showing more details for higher values of Bn. For the 50:1 ratio, the figures on the left-hand side are 10 x 10 windows showing a portion of the 50 x 50 flow domain. All figures show the progressive growth of the unyielded region as Bn or ry increases.

For the 2:1 case (Fig. 6), the fluid zone extends to the wall of the tube for all Bingham numbers shown, i.e. up to Bn = 544.6. The polar caps noted by previous researchers [5] become noticeable at Bn = 8.047 and increase slightly in size as Bn increases.

For the 4:1 case (Fig. 7), the fluid zone extends to the wall of the tube for the first three Bingham numbers (Bn = 0.108, Bn--0.747, Bn = 2.299), but then an envelope of fluid develops around the sphere. Again the polar caps are visible at Bn = 8.047 and higher.

For the 8:1 case (Fig. 8, note change of window in the right-hand side column), the fluid zone extends to the wall of the tube only for the first Bingham number (Bn = 0.108). As Bn increases,

66 J. Blackery, E. Mitsoulis / J. Non-Newtonian Fluid Mech. 70 (1997) 59-77

the region expands slightly in the radial direction (Bn = 0.747), then in the axial direction (Bn = 2.299), while at higher values the region is symmetric about both axes. Again the polar caps become noticeable at Bn = 8.047 and increase slightly in size as Bn increases.

2.0

1.0

0.0

8.0

7.0

6,0

5.0

~ . 4.0

3.0

2.0

1.0

0.0

50.0

40.0

30.0

20.0

10.0

0.0

0.0 1.0 2.0 3.0 4.0 5 .0

z / R (a)

l ' L

t N* 0,0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

(c) z / R

tt ' MI

IIJI~PPPl IUl~lHI mllHHII llll~tltH IIII~'.' 1 ~ 11

0.0 10.0 20.0 30.0 40.0 z / R

(e)

0.0

8.0

50.0

d)

4.0

3.0

2.0

1.0

0.0 0.0 1.0 2.0

Z

(b)

°°iiiij/ i .o. , 6 .0 ,

4.0 , !

2.0

--I t 1 0.0

0.0 2.0 (d)

i

! I ~ ~j

3.0 4.0 5.0 6.0 /R

4 .0 6.0 8.0 10.0

z / R 4.0

3.0

r,."

~ , 2.0

1.0

0 .0 0,0 1.0 2.0 3.0 4.0

(e) z / R (ii)

Fig. 5. Finite element meshes used in the simulations: (a) 2:1, (b) 4:1, (c) 8:1, (d) 10:1, (e) 50:1 (i) full view, (ii) blown-up view (see also Table 1 for more information).


Table 1 Summary of the finite element meshes used for the calculations

Rc/R Elements Nodes DOF

2:1 476 2001 4286 4:1 432 1823 3889 8:1 477 2007 4308

10:1 423 1785 3819 50:1 720 2997 6504

For each mesh the number of elements, computational nodes and u-v-p degrees of freedom (DOF) are given.

For the 10:l case (Fig. 9, note change of window in the right-hand side column), an interesting phenomenon occurs near the tube walls for the lowest Bn = 0.108, where the fluid is sheared next to the wall (yielded) and then again around the sphere, having an unyielded region in between, where the velocity profile is plug-like. As Bn increases, the fluid zone extends in the axial direction, but tends to become more symmetric with further Bn increases. The polar caps become visible at Bn = 59.59.

For the 50:1 case (Fig. 10, note change of window in the right-hand side column), at Bn = 0.108 the fluid region resembles that of Beris et al. [5] (see also Fig. l(c)), except that the region is rotated by 90 °. At Bn = 0.747, the slight indentations present on the side of the cylinder walls of the region for Bn = 0.108 are not present. As Bn increases, the region expands slightly in the radial direction, and at higher values the region is symmetric about both axes. Again the polar caps become visible at Bn = 8.047 and increase slightly in size as Bn increases.

We have further extended the computations to as high as Bn = 6 000 000, where r* = 0.141, but still less than the asymptotic value of 0.143 given by Beris et al. [5]. The results for these high Bn numbers are shown in Fig. 11. The envelope of fluid becomes smaller and the polar caps slightly spread more around the stagnation points, but not much else is changed except the tremendous increase in the drag coefficient (in the same order of magnitude as the Bn number). A complete set of results for the whole range of Bn numbers is given by Blackery [35].

The patterns of Fig. 10 are compared with those for the infinite medium [5] and for the same Bn in Fig. 12. It is clear that the regions are more or less the same in size and extent, except that the present simulations do not give the indentation in the direction of flow shown by Beris et al. [5] for all cases. It would appear that the disagreement is unexpected between the two sets of

Table 2 Comparison of drag coefficient results with Bohlin's approximation for Newtonian fluids

Rc/R Cs. N (FEM) CS.N (Bohlin's Eq. (15)) % difference

2:1 5.9389 5.9229 0.27 4:1 1.9779 1.9789 0.05 8:1 1.3529 1.3495 0.25

10:1 1.2619 1.2632 0.10 50:1 1.0471 1.0439 0.31


• ¢~ = 0 . 0 0 0 1 9 6 B n = O . O 0 7

6.0

5,0

4"0

3.0

2.0

1.0 0 ~.'0

-2.o ~-

-3,0 ~

-2.0 -1.0 "0 1"0 2°0 r / R

"¢~ = 0 .0745 B n = 8 . 0 4 7

6"0

B.O

4"0

&O

2.0

1.0

~..o

- I ' 0

_~'0 L ~11

-3.0

--4"0

-5"0

-6.0 -2.0 -1.0 ,0 1"0 2.0

r / R

• ~ = 0.00294

6,0

5.0

4,0

3.0

2.0

1.o 0 ~ . ' 0 N

- L o

-2.0

-4.11

-5.0

-5.0-2.0 -1"0 "0 1.0 r / R

x~ = 0.116

6.0

5.0

4.0

3"0

2,0

t.0 a: ~ . .0

-1.0

-2.0

-5.0

-4.0

-5"0

-11"0

-2.0 -1.0 .0 r / R

Bn = 0 .108

2,0

B n = 5 9 . 5 9

1 "0 2.0

x~ = 0 .0174 Bn = 0.747

6"0

5.0

4.0

3.0 IF, ,~--- ,~I 2.O

1.0

~''0" 0 -1,0

-2.0

l I -2.0 - 1.0 "0 1.0 2.0

r / R

x~, = 0 .134 B n = 544 .6

-2.0 -1"0 "0 1.0 2.0 r / R

Fig. 6. Progressive growth of the unyielded zone (shaded) for flow of Bingham plastics around a sphere contained in a tube with a 2:1 diameter ratio.

simulations. However, the simulations for an infinite medium have been carried out with a spherical coordinate system and a total absence of wall effects. The present simulations have been carried out with a cylindrical coordinate system and the walls present. The similarity (but not identity) of the two results implies that for the 50:1 diameter ratio, the walls are far enough away (but not exactly absent) from the sphere that the flow field behaves almost like an infinite medium.


4.3. Viscoplastic results--drag coefficient results

The dependence of the Stokes drag coefficient on Bingham number (Bn) for each diameter ratio is shown in Fig. 13. At low values of Bn, the curves are distinct and separated. As Bn increases further, the curves merge and are difficult to distinguish. It is also noted that the curve

e.o

L o

~.0

2.0

1.0

- I .o

-2.0

~o

: .0

I :

x; = 0.00809 Bn = 0.108

® i

- u

-s~-4,o -3.0 . 2 . o -1.o .o 1.0 2,o 3.o 4.0 , / R

~; = 0.0350 Bn = 0.?47

~; = 0.0885 tan = 8.047

• ~, = 0.121 Bn = 59.59

~ . : . a - I . 0 .0 I .o 2 . 0 ~ 4.o

~; = 0.134 Bn = 544.6

-"~'-4~ -3,0 -2 .0 - I .0 .O ~.0 2.0 ~ 0 4.0 -4.(I -3*0 -2.0 -1.0 .0 1.0 2*0 3.0 ,1~ , / R


70 J. Blackery, E. Mitsoulis /J . Non-Newtonian Fluid Mech. 70 (1997) 59 77

x i = 0 . 0 1 0 3 B n = 0 . 1 0 8 x ' = 0 . 0 8 9 1 Bn- -8 .047

4 . O -SO -4.0 -2*O .0 2.0 4-O SO SO -4.O -3.O -2.0 - 1 ~ .0 1.0 2.O 3.0 r / R , / ~

x~ = 0.0368 Bn = 0.747 x~ = 0.121 Bn = 59.59

=

- s o - s o -4.o -2*o .0 2*0 4.o s o s o -4.0 -3 ,0 -2.0 -1.0 .0 1.0 2.0 3.0 r / R

r / ~

• "c~=0.0611 B n = 2 . 2 9 9 x ~ = 0 . 1 3 5 B n = 5 4 4 . 6

-4.O -3.0 -2*O -I.o .o I.o r / R

r / m


for the 2:1 diameter ratio is substantially higher that the other curves, owing to the fact that the drag coefficient for the Newtonian case is much higher than that of the other ratios. A non-linear regression analysis was performed on the data in the range 0 ~< Bn < 1000 and gave the following relationship for the drag coefficient (continuous curves on the graphs):

Cs = CS,N + a(Bn) b (0 <~ Bn < 1000), (16)


where Cs,N is the Newtonian drag coefficient, and a and b are constants given in Table 3. The infinite case is based on the results by Beris et al. [5] for the same range. The dashed lines in the figure correspond to the two asymptotes for low and high Bn numbers, For low Bn numbers (Bn~O) the asymptote for Cs is given by the Bohlin approximation (Eq. (15)), i.e.

Cs = Cs,N (Bn~O), (17)

'~ = 0 . 0 1 0 5 B n = 0 . 1 0 8 B n -- 8 . 0 4 7

, / I

:i = 0.0896

u

l u l

s,a

- I i

- u

,/m

• ¢~ = 0 . 0 3 7 2 B n = 0 . 7 4 7 '~ = 0 . 1 2 1 B n = 5 9 . 5 9

, / n r /n

"~i = 0 . 0 6 1 5 B n -- 2 . 2 9 9 -c~ = 0 . 1 3 7 B n = 5 4 4 . 6

| 0 - t u - e . o . . 4 , o . - 4 e - , u ~ ~,o 4.0 u u t u

r im -4.o . . ~ ~ - I J ~ t,a ~ U 4.o

, I I



Y I E L D E D / U N Y I E L D E D R E G I O N S - 50:1

x~ = 0 . 0 1 1 6 Bn = 0 . 1 0 8 x~ = 0 . 0 9 0 8 Bn = 8 . 0 4 7

I

• / q r / i

.__ "~ = 0.0389 Bn = 0.747 ,., ~ = 0.1215 Bn = 59.59

0 m

, / R , / m

x~ = 0.0632 Bn = 2.299 ~; = 0.1373 Bn = 544.6

- I ~ - U - S O - s , * 0 - z a ~ l a 4~ **0 U Io~o

, / R

4*0

! . 0

.a

-1.0

-a.a

.4.0

, / m


and for high Bn numbers (Bn ~ ~ ) the asymptote is given by

Cs = 1.17(Bn) (Bn ~ oQ), (18)

based on Eq. (13) for z * - - 0 . 1 4 3 . The present results indicate that increasing the yield stress increases substantially the drag

coefficient in the system as the Bingham number increases. For Bn > 10, the curves are very close together, indicating that the drag is more or less independent o f the diameter ratio Rc/R.


The dependence of the Stokes drag coefficient on the dimensionless yield stress (r*) for each diameter ratio is shown in Fig. 14. This figure has trends which are similar to those of the previous figure. The results by Beris et al. [5] for the infinite case showed that the curve asymptotically approaches a line of constant dimensionless yield stress value equal to 0.143. The same observation is made in Fig. 14, implying that for all diameter ratios the sphere will only fall if the dimensionless yield stress is below 0.143 (r* < 0.143). This corresponds to a Y~ value of 0.048. It can also be noted that for all diameter ratios except the 2:1 case, the curves are very

* plotted. This indicates that for these ratios, the drag close together, for virtually all values of ~y * > 0.04. is more or less independent of the diameter ratio, especially for ry

: y = 0 .140 B n = 8 0 0 0

-4.O -3.O -2*O -1.0 .O 1.0 2.O 3.O 4.O z / R

$

:y = 0.141 Bn = 6,000,000

z / R

Fig. 11. Progressive growth of the unyielded zone (shaded) for flow of Bingham plastics around a sphere contained in a tube with a 50:1 diameter ratio for very high values of the Bn number.


n,-

N

6.0

4.0

2.0

.0

4.0

Present Results (Rc/R = 50) Bn = 0 .747

.0 2.0 4.0 6.0 r / R

Bn = 14.91

Results by Beris et al. (1985) (Rc/R = =)

Bn = 0 .747

0 It ,=(~' l

Bn = 14.91

&O

~- 2.0 q

1.0

.9 I.O 2.0 r / R

4.0

C01 COO;

C .qO01 i ....

Bn = 544 .6

&O

2.0 h

1.o

.0 .o LO

t / R

Bn = 544 .6

G [ K q

J J ~ q I Of OJOTi 2

Fig. 12. Comparison of the present results for a 50:1 diameter ratio with the results by Beris et al. [5] for the infinite medium, regarding the extent and shape of the yielded/unyielded regions for flow of Bingham plastics around a sphere.

5. Conclusions

Finite element simulations have been undertaken for creeping flows of Bingham plastics past a sphere contained in a cylindrical tube. The ideal Bingham constitutive equation has been modified as proposed by Papanastasiou [6] with an exponential growth term to make it valid in both the yielded and unyielded regions, thus eliminating the need for tracking the location of


o ~ 10 2

~3

~ 101 if)

10 3

I- i L " 8:1 | " 10:1 / 7 / • 50:1 /; ' I X Best Fit ~ '

i / ////

.- 10 o

10-3 10-2 10-1 100 10 ~ 102 103

Bingham number, Bn

Fig. 13. Drag coefficient vs. Bingham number (Bn) for Bingham plastics flowing around a sphere contained in a tube for different diameter ratios. The continuous curves are drawn from a non-linear regression analysis on the numerical simulation data according to Eq. (16) (see also Table 3 for details). The dashed lines correspond to the asymptotes for low Bn (horizontal lines) and high Bn (Cs= 1.17 Bn).

yield surfaces. The present results confirm earlier studies on Bingham plastics [5] regarding a dramatic increase of the drag coefficient as the dimensionless yield stress or Bingham number increases. Furthermore, the extent and shape of yielded/unyielded regions has been accurately captured using the criterion of the magnitude of the extra stress tensor exceeding the yield stress.

New results include the determination of the drag coefficient as a function of a dimensionless yield stress ry or Bingham number (Bn) for different tube/sphere diameter ratios. The drag coefficient goes to infinity as Bn--, ~ .

Table 3 Constants obtained from the best fit of the drag coefficient results for Bingham plastics (Eq. (16)) for 0 ~< Bn < 1000

Rc/R Cs, N a b

2:1 5.94 1.63 0.95 4:1 1.98 1.92 0.92 8:1 1.35 2.28 0.89

10:1 1.26 2.33 0.88 50:1 1.05 2.59 0.86 oc 1.00 2.93 0.83

The infinite case is based on results by Beris et al. [5] for the same range. Note that for very high Bn values (Bn> 1000), Cs is given by the asymptote 1.17Bn.

76 J. Blackery, E. Mitsoulis /J. Non-Newtonian Fluid Mech. 70 (1997) 59-77

C

£

103 0.143

102

101

10 0 ,, I .... I , ,~,I .... I .... I .... I .... I *

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

Yield stress, "~y*

* for Bingham plastics flowing around a sphere contained in Fig. 14. Drag coefficient vs. dimensionless yield stress Z-y a tube for different diameter ratios. The value of 3" = 0.143 corresponds to the infinite case [5] and represents the ultimate value of dimensionless yield stress beyond which the sphere will not fall.

Acknowledgements

Financial assistance from the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Ontario Centre for Materials Research (OCMR) is gratefully acknowledged.

References

[1] R.B. Bird, G.C. Dai and B.J. Yarusso, The rheology and flow of viscoplastic materials, Rev. Chem. Eng., 1 (1983) 1-70.

[2] G.G. Lipscomb and M.M. Denn, Flow of Bingham fluids in complex geometries, J. Non-Newtonian Fluid Mech., 14 (1984) 337-346.

[3] E.J. O'Donovan and R.I. Tanner, Numerical study of the Bingham squeeze film problem, J. Non-Newtonian Fluid Mech., 15 (1984) 75 83.

[4] D.K. Gartling and N. Phan-Thien, A numerical simulation of a plastic fluid in a parallel plate plastometer, J. Non-Newtonian Fluid Mech., 14 (1984) 347-360.

[5] A.N. Beris, J.A. Tsamopoulos, R.C. Armstrong and R.A. Brown, Creeping motion of a sphere through a Bingham plastic, J. Fluid Mech., 158 (1985) 219-244.

[6] T.C. Papanastasiou, Flow of materials with yield, J. Rheol., 31 (1987) 385-404. [7] P.S. Scott, F. Mirza and J. Vlachopoulos, Finite-element simulation of laminar viscoplastic flows with regions of

recirculation, J. Rheol., 32 (1988) 387-400. [8] C.R. Beverly and R.I. Tanner, Numerical analysis of extrudate swell in viscoelastic materials with yield stress, J.

Rheol., 33 (1989) 989-1009. [9] K.R.J. Ellwood, G.C. Georgiou, T.C. Papanastasiou and J.O. Wilkes, Laminar jets of Bingham-plastic liquids,

J, Rheol., 34 (1990) 787-812. [10] I.C. Walton and S.H. Bittleston, The axial flow of a Bingham plastic in a narrow eccentric annulus, J. Fluid

Mech., 222 (1991) 39-60.

J. Blacker)', E. Mitsoulis / J. Non-Newtonian Fluid Mech. 70 (1997) 59-77 77

[11] P. Szabo and O. Hassager, Flow of viscoplastic fluids in eccentric annular geometries, J. Non-Newtonian Fluid Mech., 45 (1992) 149-169.

[12] C.R. Beverly and R.I. Tanner, Numerical analysis of three-dimensional Bingham plastic flow, J. Non-Newtonian Fluid Mech., 42 (1992) 85-115.

[13] S.S. Abdali, E. Mitsoulis and N.C. Markatos, Entry and exit flows of Bingham fluids, J. Rheol., 36 (1992) 389-407.

[14] E. Mitsoulis, S.S. Abdali and N.C. Markatos, Flow simulation of Herschel-Bulkley fluids through extrusion dies, Can. J. Chem. Eng., 71 (1993) 147-160.

[15] A.I. Isayev and Y.H. Huang, Two-dimensional planar flow of a viscoelastic plastic medium, Rheol. Acta, 32 (1993) 181-191.

[16] T.V. Pham and E. Mitsoulis, Entry and exit flows of Casson fluids, Can. J. Chem. Eng., 72 (1994) 1080 1084. [17] H. Loest~ R. Lipp and E. Mitsoulis, Numerical flow simulation of viscoplastic slurries and design criteria for a

tape casting unit, J. Am. Ceram. Soc., 77 (1994) 254-262. [18] E.A. Toorman, An analytical solution for the velocity and shear rate distribution of non-ideal Bingham fluids in

concentric cylinder viscometers, Rheol. Acta, 33 (1994) 193-202. [19] O. Wuensch, Oscillating sedimentation of spheres in viscoplastic fluids, Rheol. Acta, 33 (1994) 292-302. [20] R.R. Huilgol and M.P. Panizza, On the determination of the plug flow region in Bingham fluids through the

application of variational inequalities, J. Non-Newtonian Fluid Mech., 58 (1995) 207-217. [21] N.O. Dzuy and D.V. Boger, Direct yield stress measurement with the vane method, J. Rheol., 29 (1985) 335-347. [22] A. Magnin and J.M. Piau, Shear rheometry of fluids with a yield stress, J. Non-Newtonian Fluid Mech., 24

(1987) 91 106. [23] A. Magnin and J.M. Piau, Cone-and-plate rheometry of yield stress fluids. Study of an aqueous gel, J.

Non-Newtonian Fluid Mech., 36 (1990) 85 108. [24] R.P. Chhabra and P.H.T. Uhlherr, Static equilibrium and motion of spheres in viscoplastic liquids, in N.P.

Cheremisinoff (Ed.), Encyclopedia of Fluid Mechanics, Vol. 7, Ch. 21, Gulf, Houston, TX, 1988, pp. 611 633. [25] D.D. Atapattu, R.P. Chhabra and P.H.T. Uhlherr, Wall effect for spheres falling at small Reynolds number in

a viscoplastic medium, J. Non-Newtonian Fluid Mech., 38 (1990) 3142. [26] D.D. Atapattu, R.P. Chhabra and P.H.T. Uhlherr, Creeping sphere motion in Herschel-Bulkley fluids: Flow field

and drag, J. Non-Newtonian Fluid Mech., 59 (1995) 245-265. [27] Preface in special issue, J. Non-Newtonian Fluid Mech., 29 (1988) 4 7. [28] O.G. Harlen, High-Deborah-number flow of a dilute polymer solution past a sphere falling along the axis of a

cylindrical tube, J. Non-Newtonian Fluid Mech., 37 (1990) 157-173. [29] R. Zheng, N. Phan-Thien and R,I. Tanner, The flow past a sphere in a cylindrical tube: Effects of inertia,

shear-thinning and elasticity, Rheol. Acta, 30 (1991) 499 510. [30] J.V. Satrape and M.J. Crochet, Numerical simulation of the motion of a sphere in a Boger fluid, J.

Non-Newtonian Fluid Mech., 55 (1994) 91--111. [31] R.W. Ansley and T.N. Smith, Motion of spherical particles in a Bingham plastic, AIChE J., 13 (1967)

1193-1196. [32] N. Yoshioka, K. Adachi and H. Ishimura, On creeping flow of a visco-plastic fluid past a sphere, Kagaku

Kogaku, 10 (1971) 1144-1152 (in Japanese); K. Adachi and N. Yoshioka, On creeping flow a visco-plastic fluid past a cylinder, Chem. Eng. Sci., 28 (1973) 215-226.

[33] M.J. Crochet and V. Legat, The consistent streamline-upwind/Petrov-Galerkin method for viscoelastic flow revisited, J. Non-Newtonian Fluid Mech., 42 (1992) 283 299.

[34] W.J. Lunsmann, L. Genieser, R.C. Armstrong and R.A. Brown, Finite element analysis of steady viscoelastic flow around a sphere in a tube: Calculations with constant viscosity models, J. Non-Newtonian Fluid Mech., 48 (1993) 63-99.

[35] J. Blackery, Viscoplastic flows around spheres, B.Sc. Thesis, Dept. Chem. Eng., University of Ottawa, Ottawa, Ontario, Canada, 1996.

Creeping Flow of a Sphere in Tubes Filled with a Bingham Plastic Material

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