Creeping Flow of a Sphere in Tubes Filled with Herschel-Bulkley Fluids

J. Non-Newtonian Fluid Mech., 72 (1997) 55–71

Creeping motion of a sphere in tubes filled withHerschel–Bulkley fluids

M. Beaulne, E. Mitsoulis *Department of Chemical Engineering, Uni6ersity of Ottawa, Ottawa, Ontario K1N 6N5, Canada

Received 1 November 1996; received in revised form 7 March 1997

Abstract

Previous numerical simulations for the flow of Bingham plastics past a sphere contained in cylindrical tubes ofdifferent diameter ratios are extended to Herschel–Bulkley fluids with the purpose of comparing them withexperiments. The emphasis is on determining the extent and shape of yielded/unyielded regions along with the dragcoefficient as a function of the pertinent dimensionless groups. Good overall agreement is obtained between thenumerical results and the experimental studies. © 1997 Elsevier Science B.V.

Keywords: Yield stress; Herschel-Bulkley fluid; Yielded/unyielded regions; Drag; Viscoplasticity

1. Introduction

A recent study [1] on Bingham plastics in flows around a sphere falling in tubes of differentdiameter ratios showed the extent and shape of yielded/unyielded regions and the calculation ofthe Stokes drag coefficient as a function of the Bingham number. The constitutive equation usedwas the continuous viscoplastic equation proposed by Papanastasiou [2], which reduces thedifficulty of solving for the location of the unyielded surface, and which holds for alldeformation rates. The numerical simulations showed that all previous postulates [3,4] andnumerical solutions [5] about the shape of the yielded/unyielded regions are correct under theright physical conditions, i.e., a combination of geometry and Bingham number.

The experimental work by Atapattu et al. [6,7] deals extensively with creeping sphere motionin Herschel–Bulkley fluids and provides new information on wall effects, drag behaviour andthe size of the yielded/unyielded regions. Here, the drag coefficient is also represented as afunction of a dynamic parameter, which includes the Bingham and Reynolds numbers. The

* Corresponding author. Fax: +1 613 5625172.

0377-0257/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved.PII S0377 -0257 (97 )00024 -4

M. Beaulne, E. Mitsoulis / J. Non-Newtonian Fluid Mech. 72 (1997) 55–7156

yielded/unyielded regions for different diameter ratios postulated by Atapattu et al. [6] areshown in Fig. 1.

It is the purpose of the present work to reexamine through numerical simulation these flowsand to compare the present numerical results with the experimental studies of Atapattu et al. [6].The numerical simulations are extended over a wide range of Bingham and Reynolds numbers,similarly focusing on wall effects, drag behaviour, and the extent and shape of the yielded/un-yielded regions.

2. Mathematical modeling

The problem at hand, i.e. a sphere of radius R falling with a terminal velocity V in a tube ofradius Rc filled with a viscoplastic fluid, has been described in [1] and shown here in Fig. 2. Theflow phenomena postulated by previous workers for viscoplastic flows around spheres includepolar caps appearing on the top and bottom of the sphere due to the stagnation points in theflow as predicted by Beris et al. [5]. Solid regions, or islands, similar to those proposed by Ansleyand Smith [3], may also appear on both sides of the sphere as will be shown by the presentnumerical simulations.

Fig. 1. Schematic representation of the shape of the yielded/unyielded regions surrounding a sphere in creepingmotion through a Herschel–Bulkley fluid as given by Atapattu et al. [6]: (a) RC/R=3.021, Bn*=2.526; (b)RC/R57.604, Bn*=2.526; (c) RC/R=9.635, Bn*=3.140.

M. Beaulne, E. Mitsoulis / J. Non-Newtonian Fluid Mech. 72 (1997) 55–71 57

Fig. 2. Schematic representation of the system of a sphere falling in a tube filled with a viscoplastic medium. Theshaded regions are unyielded and so are the small black regions forming polar caps around the stagnation points onthe sphere and the two islands on each side of the sphere.

The flow is governed by the usual conservation equations of mass and momentum for anincompressible fluid under isothermal conditions. The constitutive equation that relates thestress to the deformation is the Herschel–Bulkley equation. In simple shear flow, the Herschel–Bulkley equation takes the form (see Fig. 3)

t=ty+Kg; n for �t �\ty (1a)

g; =0 for �t �5ty (1b)

where t is the shear stress, g; is the shear rate, ty is the yield stress, K is the consistency indexand n is the power-law index. Note that when the shear stress t falls below ty, a solid structureis formed (unyielded). Also when the power-law index is unity and the consistency index isequivalent to the plastic viscosity, the Herschel–Bulkley model reduces to the Bingham model.

Applying Papanastasiou’s [2] modification to the Herschel–Bulkley model gives


t=ty[1−e−mg; ]+Kg; n (2)

where m is the stress growth exponent with units of time (s). Fig. 3 shows that for values ofm\1000 s, this modified equation mimics the ideal Herschel–Bulkley model. In addition, Eq.(2) approximates well the ideal Herschel-Bulkley fluid in the limiting cases of g; �� and of g; �0when the apparent viscosity h=t/g; �Kg; n−1 when h�Kg; n−1+mty. The exponential modifica-tion was previously applied to Herschel–Bulkley fluids and used by Mitsoulis et al. [8] to studythe non-isothermal flow through extrusion dies and to determine the shape and extent ofyielded/unyielded regions, extrudate swell, and the development of the temperature field. Thefull tensorial form of Eq. (2) is given in [8] and follows readily from the Bingham modification(see [1]).

3. Dimensionless parameters

The relative importance of inertia forces is assessed by the generalized Reynolds number Re*,defined for power-law fluids as [9]

Re*=V2−ndnr

K(3)

where d is the diameter of the sphere, r is the density of the fluid, and V is the terminal velocityof the sphere. For the numerical simulations performed in the present work, Re*�1 for mostcases, and the creeping flow approximation is valid.

For materials with a yield stress obeying the Herschel–Bulkley model, a generalized Binghamnumber Bn* is defined as

Fig. 3. Shear stress vs. shear rate according to the modified Herschel–Bulkley constitutive Eqs. (1a) and (1b) forseveral values of the stress growth exponent m for fluid S-10 (a Carbopol solution with K=23.89 Pa×sn, ty=46.47Pa, n=0.5).


Bn*=ty

K(V/d)n, (4)

where an apparent shear rate can be defined as g; =V/d.A corresponding equation for the Stokes drag coefficient can be written for Herschel–Bulkley

fluids following equivalent definitions for Re* for power-law fluids [9]:

C*s =F

6pKVnR2−n21−nX(n). (5)

Here, F is the external force acting on the sphere due to gravity, and X(n) is the drag correctionfactor which is a function of the power-law index, n. In the present work, the values for X(n)were taken from experimental measurements done by Gu and Tanner [9]. Ansley and Smith [3]also incorporate a general definition of the drag coefficient which does not neglect inertial effectsand is given as

CD=43

(rS−r) gdrV2 =

24C*SX(n)Re �

(6)

where rS is the sphere density and g is the acceleration due to gravity.A dimensionless yield stress is then defined in terms of the modified Bingham number and

drag coefficient as

t*y =Bn*

6C*SX(n)(7)

Chhabra and Uhlherr [10] also define the gravity-yield number given by

YG=ty

2gR(rs−r)=

Bn*18C*SX(n)

=t*y3

(8)

In all cases, the Newtonian fluid corresponds to Bn*=0, ty*=0, and YG=0.Ansley and Smith [3] reduced the number of dimensionless groups for Bingham plastics by

defining a dynamic parameter, Q, which is a function of the Reynolds and Bingham numbers.The equivalent generalized dynamic parameter Q* for Herschel–Bulkley fluids is given byAtapattu et al. [6] as

Q�=Re*

1+kBn*(9)

where k is postulated to be equal to 7p/24 (=0.9163) for Bingham plastics. Ansley and Smith[3] showed that the drag coefficient from Eq. (6) was a function of the dynamic parameter.

4. Method of solution

The continuous viscoplastic constitutive equation for Herschel–Bulkley fluids is solvedtogether with the conservation equations of mass and momentum, and appropriate boundaryconditions by the Finite Element Method (FEM) as explained in the previous papers [1,8]. Alllengths are scaled with R, all velocities with V, and all pressures and stresses with KVn/Rn.


Table 1Properties of the S-10, S-11 and S-12 test fluids (Carbopol solutions) used in the experimental studies by Atapattuet al. [6] and in the numerical simulations

r (kg−1 m3)Fluid K (Pa · sn) ty (Pa) n (−) X(n) (−)

1000S-10 23.89 46.47 1.420.501.391000S-11 0.5943.15 38.151.43S-12 9.14 5.32 0.48 1000

X(n) is the drag correction factor for power-law fluids [9]

Simulations are carried out for a wide range of Bingham numbers (05Bn*B�) bydecreasing the value of the velocity V, while the radius R, consistency index K, the yield stressty, and the power-law index n are held constant, representing the material properties of eachviscoplastic fluid. Three of the twelve fluids considered by Atapattu et al. [6] are simulated in thisstudy, namely the fluids designated as S-10, S-11, and S-12 (aqueous solutions of different gradesof Carbopol 940 and 941 resins supplied by Goodrich, USA). The values of the parameters inthe Herschel–Bulkley model for these fluids are given in Table 1, together with the dragcorrection factor X(n) determined by Gu and Tanner [9] for power-law fluids.

It was found during the numerical simulations that the results were very dependent on thevalues of the stress growth exponent, m, at large Bingham numbers, but were less affected forvery high values of m. The values of the stress growth exponent m used in this study vary foreach fluid and Bingham number so as to maintain the ratio of viscosities at low and high shearrates sufficiently large. The method for calculating the correct value of m to use for acorresponding Bingham number is by implementing the criterion that the ratio of viscosities atg; �0 and g; ��, given by the expression mg; (Bn*), be equal to 1000. By maintaining the valueof m sufficiently large (103BmB106), the results become independent of m, and better resultsfor the drag coefficient and the extent of the yielded/unyielded regions are achieved. Thecriterion of keeping the ratio of viscosities large is equivalent to that used by Tsamopoulos et al.[11] and similar to that used by O’Donovan and Tanner [12], and Beverly and Tanner [13] intheir biviscosity model for viscoplastic materials.

The values of the diameter ratios, RC/R, used in the simulations are those ratios used in theexperimental work undertaken by Atapattu et al. [6] (i.e., RC/R=3.021, 4.531, 5.469, 7.604 and9.635). The flow length L/R ratio was taken to be about the same as the RC/R ratio for all cases,except for the 3.021:1 and 4.531:1 ratios, where L/R was set equal to 6. The corresponding L/Rvalues for the other cases are 6,8 and 10 for the 5.469:1, 7.604:1 and 9.635:1 diameter ratios,respectively. The finite element meshes are similar in grid density to the ones used in theprevious work by Blackery and Mitsoulis [1], ranging from 470–637 quadrilateral Lagrangianelements, 2000–2700 nodes, and 4200–5800 degrees of freedom for the velocities–pressures.


5. Results and discussion

5.1. Newtonian results

We first applied our numerical scheme to the calculation of the creeping flow of a Newtonianfluid past a sphere to evaluate the meshes constructed for the given diameter ratios. Bohlin’sapproximation [1] was used to compare the values of the drag coefficient for a Newtonian fluid,and the results for the different RC/R ratios are given in Table 2. The differences between thecalculated values and those predicted by Bohlin’s approximation are always below 0.2%.

5.2. Viscoplastic results—yielded/unyielded regions

Numerical results on the extent and shape of the yielded/unyielded (shaded) regions with theHerschel–Bulkley model have been produced for all three fluids and five diameter ratios (9.635,7.604:1, 5.469:1, 4.531:1, 3.021:1). In the interest of brevity, we present here only the results forfluid S-10 and for the lowest and highest diameter ratios, in Figs. 4 and 5, respectively. All ofthe figures show the progressive growth of the unyielded region as Bn* and ty* increase, up tothe point where the region becomes completely plastic. It can also be seen that as Bn* increases,the solid regions on each side of the sphere (islands) increase in size as do the solid regionslocated around the front and rear stagnation points (polar caps).

For the 3.021:1 case (Fig. 4), the fluid region extends to the wall of the tube for low Binghamnumbers. The similarity of the results for Bn*=0.108 with the schematic representation of Fig.1(a) is remarkable. Solid regions around the front and rear stagnation points (caps) as well ason each side of the sphere (islands) first appear in these figures around Bn*=27.36. At anintermediate Bingham number (i.e., Bn*=197.5), the fluid zone disappears near the wall anddevelops around the sphere. As the Bingham number continues to increase, the fluid zonediminishes, while the solid island regions on each side of the sphere grow and at Bn*=368.4they have already reached the outer plastic boundary. The diminishing of the fluid regioncontinues as the Bingham number increases, and the solid polar caps grow and reach the plasticboundary around Bn*=1360. Also note the isolated fluid regions at the front and the rear ofthe sphere for Bn*=1360. For the higher Bingham numbers (i.e., Bn*=1768) all that remainsare crescent-shaped fluid regions on each side of the sphere until they too disappear, and acompletely plastic medium is present around the sphere.

Table 2Comparison of drag coefficient results with Bohlin’s approximation [1] for Newtonian fluids

RC/R CS,N (FEM) CS,N (Bohlin’s Appr’n) Per cent difference

1.27729.635 1.2757 0.127.604 1.3758 1.3737 0.15

1.59301.5955 0.165.4691.7937 1.7942 0.034.531

2.67332.6718 0.053.021


Fig. 4. Progressive growth of the unyielded zone (shaded) for the flow of Herschel–Bulkley fluid S-10 (see Table 1)around a sphere contained in a tube with a 3.021:1 diameter ratio.

For the 9.635:1 case (Fig. 5, note the change of window size (4×4) in the right-hand sidecolumn showing more details for the higher Bn* values), the fluid region does not extend to thewall of the tube for any Bingham number shown. The similarity of the results for Bn*=3.140with the schematic representation of Fig. 1(c) is remarkable. The solid polar caps and solidislands first appear around Bn*=59.59. The solid islands on each side of the sphere grow andmeet the outer plastic regions around Bn*=723.1. The solid polar caps grow as the Binghamnumber increases and join with the plastic region around Bn*=2990. Crescent-shaped fluidregions form for Bn*=3128, and as the Bingham number increases, the medium around thesphere is completely plastic.

The results for the intermediate diameter ratios follow similar patterns as described above andare not repeated here. The same is true for the other two fluids S-11 and S-12.

A comparison between the present simulation results and the experimental results by Atapattuet al. [6] for the shape and extent of the yielded/unyielded regions is now in order. In their effort


to provide comparisons with simulations for a Bingham plastic falling in an infinite medium,Atapattu et al. [6] have used not a generalized Bn*, but a Bingham number (their Bi ) based onan equivalent plastic viscosity m evaluated at an apparent shear rate of V/d. With the propermodifications to correspond to Bn*, Fig. 6 shows a comparison between our numerical resultsand the experiments by Atapattu et al. [6] for the corresponding fluids (fluid S-11 for the highest

Fig. 5. Progressive growth of the unyielded zone (shaded) for the flow of Herschel–Bulkley fluid S-10 (see Table 1)around a sphere contained in a tube with a 9.635:1 diameter ratio.


Fig. 6.

diameter ratio of 9.635 and fluid S-10 for the other ratios). As can be seen, the results in theouter yield surface agree well for the higher diameter ratios, especially along the axial symmetryline (z/R). However, the agreement of results for RC/R=3.021 is poor. It is interesting to notethat the yielded/unyielded region for RC/R=3.021 postulated by Atapattu et al. [6] at aBingham number of 2.526 is very similar to that of the present results at a Bingham number of0.108 (see Fig. 4). The disagreement between results is rather puzzling for the lowest RC/R ratio.To that effect, the present results for Bn*=0.108 are shown vs. the result by Atapattu et al. [6]


for Bn*=2.526, where the agreement is good. This suggests that perhaps a clerical error in theliterature may be attributed to the discrepancy of results for this particular geometry.

The use of a continuous viscoplastic model, such as the one used here, to determine the exactshape of yielded/unyielded regions can be criticized, since the model predicts deformation (albeitextremely small) for all values of the exponent m. A careful examination of the velocities andvelocity gradients in disputed regions, such as the islands and the polar caps, showed that therethe velocity gradients are extremely small (but not identically zero), and these regions are reallyapparently unyielded regions (AUR), as opposed to truly yielded regions that do exist whenevera plug velocity profile occurs. However, in the present simulations we do not differentiatebetween the two, and the separating line has been drawn as the contour of the magnitude of theextra stress tensor having a value equal to the yield stress, as in our previous publications [1,8].

Fig. 6. Comparison of the present results (left) with the results by Atapattu et al. [6] (right), regarding the extent andshape of the yielded/unyielded regions for flow of a Herschel–Bulkley fluid around a sphere. All results are for fluidS-10 except the first (RC/R=9.635), which is for fluid S-11.


Fig. 7. Comparison of the present results (solid lines) with the results by Atapattu et al. [6] (symbols), regarding theaxial velocities, 6z, along the equatorial line for different diameter ratios (i.e., RC/R=3.021, 4.531, 5.469 and 7.604).All results are for fluid S-10 and correspond to the conditions of Fig. 6.

5.3. Viscoplastic results—6elocity profiles

The results for the dimensionless axial velocities, 6z/V, along the equatorial line for differentdiameter ratios are shown in Fig. 7 for fluid S-10 and for the same conditions given in Fig. 6.Included in this figure are the experimental results of the velocity profiles by Atapattu et al. [6]for the same fluid. The present results, represented by the solid lines, agree well with those byAtapattu et al. [6] for the larger diameter ratios. However, the agreement of results for the3.021:1 diameter ratio is poor regarding the location of the maximum velocity (i.e., r/RC=1.7and r/RC=2.1 for the present results and the Atapattu et al. [6] results, respectively). Note thatthe same was true for the yielded/unyielded regions for this case as discussed above (see Fig. 6).

For each diameter ratio, the rise in the dimensionless axial velocities with radial distanceprecedes those of Atapattu et al. [6], however, the maxima of the dimensionless axial velocitiesagree well with the experimental results. Points to notice are the changing slope of the velocityprofile before and after the maxima, suggesting the absence of islands for these conditions,which was also borne out by the numerical simulations. However, the leveling-off of the velocityprofile after some distance suggests no velocity changes (plug profile) corresponding to practi-cally unyielded regions, which was also found in the numerical simulations (see Fig. 6).

5.4. Viscoplastic results—drag coefficient

The dependence of the drag coefficient (Eq. (6)) with respect to the generalized dynamicparameter is shown in Fig. 8, where the solid line represents the expression CD=24X(n)/Q*.


Also included in this figure are the experimental results by Atapattu et al. [6] shown as hollowcircular symbols, and the present results shown as solid square symbols. The present resultsincorporate the values of X(n) given experimentally by Gu and Tanner [9] (see Eq. (16)),whereas the results for the drag coefficient extend over 15 orders of magnitude in the dynamicparameter (10−165Q*50.89), which is roughly twice the range of the results from Atapattu etal. [6], which extend over seven orders of magnitude (10−85Q*50.3).

A non-linear regression analysis was performed on the data with respect to the dragcoefficient as a function of the dynamic parameter in the form of

CD=24X(n)(1+kBn*)

Re*. (10)

This form is used to evaluate the value of k, for which Ansley and Smith [3] postulated a valueof 7p/24 (=0.9163). The physical significance of k relates to the ratio of the drag term involvingyield stresses to the drag term involving viscous stresses [3]. The values of k found for eachdiameter ratio for the S-10 Herschel–Bulkley fluid are given in Table 3 in the following rangesof conditions: 1.43×10−125Re*50.00175; 0.7475Bn*5952.4. It can be seen from Table 3that the values of k for each diameter ratio are close to the one postulated by Ansley and Smith[3].

5.5. Viscoplastic results—Stokes drag coefficient

The numerical values of the Stokes drag coefficient (Eq. (5)) with corresponding values of thegeneralized Bingham and Reynolds numbers are shown in Table 4 for the case of 3.021:1

Fig. 8. Drag coefficient (defined by Eq. (6)) vs. the dynamic parameter for Herschel–Bulkley fluids flowing arounda sphere contained in a tube for different diameter ratios. The solid line represents CD=24X(n)/Q* (symbols: ,present results; �, results by Atapattu [6]).


Table 3Values of k in the expression of the dynamic parameter defined by Eq. (9) obtained from the regression analysis forthe S-10 Herschel–Bulkley fluid Eq. (10)

k (fluid S-10)RC/R

3.021 0.8924.531 0.8845.469 0.857

0.8217.6040.8779.635

The value of k=7p/24=0.9163 postulated by Ansley and Smith [3]

diameter ratio and the S-10 Herschel–Bulkley fluid. It is to be noted that Bn* and CS* are of thesame order of magnitude, while Re* decreases very rapidly from the value of about 1 for thelowest Bn* to many orders of magnitude, thus justifying the creeping flow approximation.

The dependence of the Stokes drag coefficient (Eq. (7)) on the dimensionless yield stress (ty*)is shown in Fig. 9 for different diameter ratios for the S-10 fluid simulations. Similar results areobtained for the other two fluids. It is noted that at low yield stress values, the curve for the3.021:1 diameter ratio is higher than the other curves, owing to the fact that the drag coefficientfor this geometry even for the Newtonian case is much higher than that of the other diameterratios. The results of Beris et al. [5] for the infinite case showed that the curve asymptoticallyapproaches a line of constant dimensionless yield stress value equal to 0.143, where the sphere

Table 4Values of the Stokes drag coefficient (CS*), generalized Bingham number (Bn*), and generalized Reynolds number(Re*) for the 3.021:1 diameter ratio for the S-10 Herschel–Bulkley fluid

Bn* CS*Re*

0.108 1.4290.97822.1012.956×10−30.747

2.299 3.6001.014×10−4

3.140 4.3833.980×10−5

7.5477.287×10−65.5302.365×10−6 8.7838.0473.718×10−7 14.7714.91

25.406.017×10−827.3659.59 52.525.824×10−9

167.0197.5 1.600×10−10

340.7 282.63.116×10−11

544.6 7.629×10−12 434.4748.5 568.72.939×10−12

952.4 690.21.426×10−12

7.971×10−13 801.211561360 4.897×10−13 904.2

10013.220×10−13156410912.230×10−131768


Fig. 9. Stokes drag coefficient vs. dimensionless yield stress ty* for the S-10 Herschel–Bulkley fluid flowing around asphere contained in a tube for different diameter ratios.

becomes motionless. Atapattu et al. [6] also discovered a critical dimensionless yield stress valuewhere the sphere becomes motionless at ty*:0.183. In reality, the latter number is anexperimental estimate of ‘no motion’, i.e. motion too slow to observe in a reasonable time of afew minutes or hours.

It can be argued that the present results up to ty*=0.14 conform with the previous numericalresults by Beris et al. [5] and Blackery and Mitsoulis [1], where the curves appear to beasymptotic about ty*=0.143. However, once the curves reach the asymptote line, they benddramatically in favour of the yield stress, and they tend to level off towards the value proposedby Atapattu et al. [6], near 0.183. This corresponds to a tendency towards total encapsulationof the sphere by the unyielded region and cessation of motion. It appears that the sudden bendin the curves is due to the two-viscosity model inherent even in the Papanastasiou modificationas the Bn* becomes very large. The regime ty*B0.143 represents ‘flow’ governed by the lowerviscosity, while the regime ty*\0.143 represents ‘creep’ governed by the higher viscosity. Thiscreep should continue indefinitely, and an accurate value of the critical dimensionless yield stresswhere the sphere becomes absolutely motionless is difficult to determine, since there is nocriterion for the motion of the sphere based on the shape of the yielded/unyielded regions.

The present results differ from those by Beris et al. [5] and Blackery and Mitsoulis [1], sincethere is a difference in how the simulations were carried out. In the previous simulations, theBingham number was increased by increasing the yield stress value but maintaining the velocityconstant. In the present simulations, the yield stress of the test fluid was used, and the Binghamnumber was increased by decreasing the velocity. This also gives an insight into why Beris et al.[5] and Blackery and Mitsoulis [1] were not able to attain the limit of cessation of sphere motion,since the boundary condition of having a constant velocity did not allow the sphere to become


motionless, within the context of the simulations. Physically, Beris et al. [5] and Blackery andMitsoulis [1] have solved for pulling a sphere through viscoplastic materials, whereas in thepresent work, the sphere speed is reduced so that at the limit, it ceases to move, and it is totallysurrounded by unyielded material. This may be closer to the experimental situation encounteredwith real fluids, such as the Carbopol solutions, where the existence of a true yield stress belowwhich there is absolutely no deformation, may be disputed.

6. Conclusions

Finite element simulations have been undertaken for creeping flows of Herschel–Bulkleyfluids past a sphere falling in cylindrical tubes. The ideal Herschel–Bulkley equation has beenmodified as proposed by Papanastasiou [2] with an exponential growth term to make it valid inboth yielded and unyielded regions, thus eliminating the need for tracking the location of theyield surfaces. The present results conform with the previous study by Atapattu et al. [6]regarding the dependence of the drag coefficient on the dynamic parameter proposed by Ansleyand Smith [3]. The extent and shape of the yielded/unyielded regions has been determined usingthe criterion of the magnitude of the extra stress tensor exceeding the yield stress.

New results include the determination of the drag coefficient, extending over a 15 orders-of-magnitude variation in the dynamic parameter and a four orders-of-magnitude variation in theBingham number. The Stokes drag coefficient was also determined as a function of a dimension-less yield stress ty*, up to the limit where the sphere appears to become motionless. Thetransitional shapes of the yielded/unyielded regions from a fluid state to a completely plastic statehave also been demonstrated. Furthermore, the inconsistency of results relating to the extentand shape of the yielded/unyielded regions postulated by different researchers can now beexplained, i.e., all postulates are correct under the right physical conditions, i.e., a combinationof geometry and Bingham number.

Acknowledgements

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

References

[1] J. Blackery and E. Mitsoulis, Creeping Motion of a Sphere in Tubes Filled with a Bingham Plastic Material, J.Non-Newtonian Fluid Mech., (1997) in press.

[2] T.C. Papanastasiou, Flow of Materials with Yield, J. Rheol. 31 (1987) 385–404.[3] R.W. Ansley, T.N. Smith, Motion of Spherical Particles in a Bingham Plastic, AIChE J. 13 (1967) 1193–1196.[4] N. Yoshioka, K. Adachi, H. Ishimura, On Creeping Flow of a Visco-plastic Fluid past a Sphere, Kagaku

Kogaku, 10 (1971) 1144–1152 (in Japanese). See also K. Adachi and N. Yoshioka, On Creeping Flow of aVisco-plastic Fluid past a Cylinder, Chem. Eng. Sci. 28 (1973) 215–226.


[5] A.N. Beris, J.A. Tsamopoulos, R.C. Armstrong, R.A. Brown, Creeping Motion of a Sphere through a BinghamPlastic, J. Fluid Mech. 158 (1985) 219–244.

[6] D.D. Atapattu, R.P. Chhabra, P.H.T. Uhlherr, Creeping Sphere Motion in Herschel–Bulkley Fluids: Flow Fieldand Drag, J. Non-Newtonian Fluid Mech. 59 (1995) 245–265.

[7] D.D. Atapattu, R.P. Chhabra, P.H.T. Uhlherr, Wall Effect for Spheres Falling at Small Reynolds Number in aViscoplastic Medium, J. Non-Newtonian Fluid Mech. 38 (1990) 31–42.

[8] E. Mitsoulis, S.S. Abdali, N.C. Markatos, Flow Simulation of Herschel–Bulkley Fluids through Extrusion Dies,Can. J. Chem. Eng. 71 (1993) 147–160.

[9] D. Gu, R.I. Tanner, The Drag on a Sphere in a Power-Law Fluid, J. Non-Newtonian Fluid Mech. 17 (1985)1–12.

[10] 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, Chapter 21, Gulf Publishing, Houston, TX, 1988,pp. 611–633.

[11] J.A. Tsamopoulos, M.F. Chen, A.V. Borkar, On the Spin Coating of Viscoplastic Fluids, Rheol. Acta 35 (1996)597–615.

[12] E.J. O’Donovan, R.I. Tanner, Numerical Study of the Bingham Squeeze Film Problem, J. Non-Newtonian FluidMech. 15 (1984) 75–83.

[13] C.R. Beverly, R.I. Tanner, Numerical Analysis of Three-Dimensional Bingham Plastic Flow, J. Non-NewtonianFluid Mech. 42 (1992) 85–115.

.

Creeping Flow of a Sphere in Tubes Filled with Herschel-Bulkley Fluids

Documents