Journal of Non-Newtonian Fluid Mechanics 203 (2014) 51–53

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Journal of Non-Newtonian Fluid Mechanics

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Short Communication

Mechanism of drag increase on spheres in viscoelastic cross-shear flows

0377-0257/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.jnnfm.2013.10.007

⇑ Corresponding author. Tel.: +61 2 9351 7153; fax: +61 2 9351 7060.E-mail address: roger.tanner@sydney.edu.au (R.I. Tanner).

Roger I. Tanner a,⇑, Kostas D. Housiadas b, Fuzhong Qi a

a School of Aerospace, Mechanical and Mechatronic Engineering, University of Sydney, Sydney, NSW 2006, Australiab Department of Mathematics, University of the Aegean, Karlovassi, Samos 83200, Greece

a r t i c l e i n f o

Article history:Received 29 July 2013Received in revised form 28 October 2013Accepted 29 October 2013Available online 6 November 2013

Keywords:ViscoelasticitySphereSettling speedNormal stressesSuspension

a b s t r a c t

A sphere falling along the vorticity axis of a shear flow in a viscoelastic fluid is known to experience anincrease in drag from the value in a quiescent fluid (van den Brule and Gheissary, 1993). Two recentpapers have analysed the problem and this paper seeks to provide a simpler explanation of the phenom-enon by considering deformation of the streamline pattern and the consequent generation of lift on thesphere by normal stresses.

� 2013 Elsevier B.V. All rights reserved.

4 3

1. Introduction

In 1993 van den Brule and Gheissary [1] found by experimentthat a sphere settling in a viscoelastic fluid could be retarded byimposing a cross-shear flow. Here we consider a shear imposedin the x–y plane, where the velocity is given by v ¼ _cyi; _c is theshear rate, and the sphere moves along the z axis or vorticity direc-tion with speed U. The phenomenon is of intrinsic rheologicalinterest and is also relevant to the settling of proppants in oilrecovery technology, but little was done in analysing the flow until2012 [2] and 2013 [3]. In [2] the authors used a perturbation tech-nique to solve the problem for small Deborah (De) numbers, whilstin [3] a computational approach was used. In the former paper [2]it was hinted that the extra tension due to the normal stress differ-ence (N1) was possibly the cause of the extra drag on the sphere,while in [3] the authors essentially said that the direct stress inthe direction of motion was increased due to a break of symmetryin the stress pattern, thus slowing the sphere. The present paperseeks to provide a simple explanation based on observables to ac-count for the effect. In addition we will also briefly mention theproblems of walls and neighbouring particles in a concentratedswarm of spheres.

2. Force balance and dimensional analysis

Consider a sphere, radius R, falling under the effect of gravityalong the z-axis in a viscoelastic medium. The net force (m� � g)

along the axis due to gravity is 3 pR gðqs � qÞ, where g is the accel-eration of gravity, and qs and q are the sphere and fluid mass den-sities, respectively. The speed along the z-axis (U) depends also onthe viscosity (g) and relaxation time (k) of the fluid, the width ofthe channel (2L), the cross-shear rate ð _cÞ, the size of the sphere(R), and, for multiple spheres, the volume fraction (/), so

Us

U¼ f k _c;

UkR;/;

LR;q

URg

� �; ð1Þ

where Us is the Stokes speed:

Us ¼29

R2

ggðqs � qÞ: ð2Þ

We shall at first ignore the effect of walls, and suppose the fluidis unbounded. Also the Reynolds number qUR/g is supposed negli-gible and to begin only a single sphere is considered, so / = 0. Ananalytical result is given in [2] for an Oldroyd-B model with a ratioof polymer viscosity/total viscosity denoted by 1 � b(b = 0 for anupper convected Maxwell (UCM) model; and b = 1 returns to theNewtonian case) as

Us

U¼ 1þ ð1� bÞy1 _c2k2 � ð1� bÞy0De2; ð3Þ

where y0 ¼ 25825;025þ

1�b175 ; y1 ¼ 81;005

84;084� ð1� bÞ 1;185;0251;513;512 and the Deborah

number De ¼ UkR (Note in Ref. [3] the characteristic number h used

instead of De is Uk/2R.).Table 1 gives the values of y0 and y1 for various b-values; recall

b = 0 is the upper convected Maxwell (UCM) limit, b = 1 is the New-tonian case.

Table 1Coefficients y0 and y1.

b y0 y1 (1 � b)y0 (1 � b)y1

0 0.0160 0.1804 0.0160 0.18040.2 0.0149 0.3370 0.0119 0.26960.4 0.0137 0.4936 0.0082 0.29620.6 0.0126 0.6502 0.0050 0.26010.7 0.0120 0.7285 0.0036 0.21860.8 0.0115 0.8068 0.0023 0.16140.9 0.0109 0.8851 0.0011 0.08851.0 0.0103 0.9634 0 0

Table 2Us/U for b = 0.7 as a function of Weissenberg number Wið¼ k _cÞ.

_c Wi Eq. (4a) Eq. (4b) Calculation [3]

0.0 0.0 1.0 1.0 1.02.0 0.6 1.079 1.063 1.063.33 1.0 1.219 1.100 1.124.0 1.2 1.315 1.068 1.185.0 1.5 1.492 0.889 –

ig. 1. (a) Shear flow in the x–y plane disturbed by a sphere falling with speed Ulong the z-direction. (b) Force balance on the sphere. For the unbounded case Lust be a multiple of R, and d � Uk, where k is a characteristic relaxation time.

52 R.I. Tanner et al. / Journal of Non-Newtonian Fluid Mechanics 203 (2014) 51–53

In the experiments of van den Brule and Gheissary [1] the valueof b was about 0.69 for the fluid B, which is the case of most inter-est here as its viscosity was nearly constant (5.94 Pa s); the relax-ation time was about 0.284 s [3]. Hence for the fluid B [1] we find(taking b = 0.7) from Eq. (3):

Us

U¼ 1þ 0:2186ðk _cÞ2 � 0:0036De2: ð4aÞ

Including higher order corrections (the details for the derivationof the higher order terms will be given elsewhere) results in:

Us

U¼ 1þ 0:2186ðk _cÞ2 � 0:119ðk _cÞ4 � 0:0036De2 þ 0:0041De4

þ 0:025ðk _cDeÞ2: ð4bÞ

For De << 1, Dek _c << 1 the last term in Eq. (4a), and the lastthree terms in Eq.(4b) are negligible and can be ignored. It wasshown (Fig. 7 in [3]) that up to k _cð�WiÞ of about 0.6 that Eq.(4a) is in rough agreement with the drag computed in Ref. [3],although the computed drag was somewhat smaller, even atk _c ¼ 0:6, than that predicted by Eq. (4a) [Table 2]. Note howeverthat when higher order corrections are included, as shown in Eq.(4b), the disagreement between the analytical (Eq. (4b)) and calcu-lated [3] results becomes smaller. From Fig. 3a of Ref. [3] one findsthat the drag on a sphere settling in a tube, as computed by theauthors, is also less than that found by Yang and Khomami [4]for the same problem.

We now propose a mechanism for the increased drag.

3. Mechanism of drag enhancement

We assume that N2 = 0 in the Boger fluid. This leaves, in the simpleshearing flow, a normal stress system rxx, ryy and rzz in the x, y and zdirections respectively. Since N2 = ryy� rzz is assumed zero, there is astate of tension of amount N1 = rxx� ryy acting along the streamlines,relative to the pressure in the y and z directions. This tension is as-sumed to be essentially unchanged by the motion of the sphere.

In Fig. 1a we sketch the streamline pattern along which a ten-sion N1 is exerted, and Fig. 1b shows a force balance on the sphere.The streamlines are assumed to be bent down by a small amount d,due to the sphere motion.

The net downward force F on the sphere is, from Fig. 1b,

F ¼ mg � 2AdL

N1; ð5Þ

Fam

where we assume that the normal stress acts over on area A, whichis a multiple of the projected area pR2. Now because there is nolength scale in the x-direction, we assume L is a multiple of R, andhence Eq. (5) becomes

F ¼ 43pR3ðqs � qÞg � 3CdN1pR; ð6Þ

where 3C is a constant. The depression of the streamline d is as-sumed to be proportional to the sphere speed (U) multiplied by arelaxation time k. Hence

F ¼ 43pR3ðqs � qÞg � 3C

kUR

N1pR2: ð7Þ

If we assume that De ¼ kUR is small, then the speed U of the

sphere in the unbounded fluid is given by F/6pgR, so we find

U ¼ Us �C

2gDeN1R; or

Us

U¼ 1þ C

2gkN1: ð8Þ

In the Oldroyd-B model case, N1 ¼ 2gkð1� bÞ _c2, so Eq. (8)becomes

Us

U¼ 1þ Cð1� bÞk2 _c2: ð9Þ

This agrees exactly with Eq. (3) above when the small De2 termis ignored, provided we take C = y1 � O(1). If a finite wall distance Lis taken into account, this will introduce a further factor into Eq. (8)and (9). In the experiments [1] the value of R/L was up to 0.2, whichwill increase the sphere drag considerably, see [5]. The model istherefore consistent with the analysis [2,3], and gives a simpleexplanation of extra drag induced by the cross-shearing.

(s-1

)

100101

N1,G

'(P

a)

1

10

100

1000

N1 (Padhy)

N1 (O-B)

N1 (Exp.)

G' (Padhy)

G' (O-B)

G' (Exp.)

(N1)ex

Fig. 2. The models used for N1 and G0: s: N1 [3]; D: N1, Oldroyd-B [2]; r: N1,Experiment, fluid B [1]; d: G0 [3], N: G0 Oldroyd-B [2]; .: G0 Experiment, fluid B [1];h: N1, Power-law fit, Eq. (10). Clearly G0 is not closely modelled, but N1 is reasonablyapproximated.

R.I. Tanner et al. / Journal of Non-Newtonian Fluid Mechanics 203 (2014) 51–53 53

4. Comparison with experiments

Eq. (9) ignores the increase in drag which is due to wall effects[5,3], and from Fig. 1 of Ref. [1] it is clear that although the viscosityis nearly constant, the N1 versus _c curve is not a square law as ispredicted by the Oldroyd-B model. Experimental values of N1 werenot given for _c < 10 s�1, even though this is the region of the great-est interest. For _c > 10 s�1, from Fig. 1 of Ref. [1], we find,approximately

N1 / 1:79ð _cÞ1:35: ð10Þ

Hence the single-mode models used in [2,3] are of unknownaccuracy in the region _c < 10 s�1, Fig. 2 shows the various fittingsfor N1 and G0; the experimental data are from [1], and the Padhyet al. [3] data are shown, as is the Oldroyd-B fit [2]. The power-law of Eq. (10) is also plotted, and the dashed line follows thislaw. Clearly the G0 fits to the experimental data are poor, but theN1 fits, including Eq. (10), are plausible in the range _c < 10 s�1.

We can replace the ðk _cÞ2 factor in Eq. (4a) by kN1=2gð1� bÞ totry and avoid the small-Weissenberg number limitation of theperturbation analysis, and use the power-law N1 of Eq. (10). Thenwe find (b = 0.7, k = 0.35 s and g = 6 Pa s)

Us

U¼ 1þ 0:1093

kN1

gð1� bÞ : ð11Þ

For Wi = 1.23, this yields Us/U = 1.24, slightly below the Old-royd-B result which gives 1.33. Hence the mechanism seems tobe confirmed, but it is sensitive to the value of N1 used.

We can also try and use Eq. (3) on fluid C of Ref. [1], which isshear-thinning. Fitting the parameters is difficult. Letting b = 0.7,k = 0.9 s and g = 12 Pa s, at x = 1 s�1 we find G0 = 1.8 Pa, matchingthe measured G0 of 1.8 Pa. Then using Eq. (3), ignoring the (De)2

term, and setting b = 0.7, we find Us=U � 1þ 0:177 _c2. At _c ¼ 3, thisgives Us/U = 2.6, whereas the measured value is 2.2. Clearly moredata would be welcome.

5. Conclusion

The simple model appears to be realistic, and it can be used togive quantitative results. It does not include the effect of channelwalls, although it is clear from [5] that O(10%) changes in dragare to be expected from this cause. For a concentrated suspension,it is known that N1 increases with / [6], and so does the viscosity.Both of these effects will give increased retardation of the sphere.Hence, assuming d is unchanged, one can expect a considerable de-crease in settling speed with concentration, as in the experimentsdescribed in [3].

Acknowledgement

We are grateful to the Australian Research Council for financialsupport via Grant DP110103414.

References

[1] B.H.A.A. van den Brule, G. Gheissary, Effects of fluid elasticity on the static anddynamic settling of a spherical particle, J. Non-Newton. Fluid Mech. 49 (1993)123–132.

[2] K.D. Housiadas, R.I. Tanner, The drag of a freely sedimenting sphere in a shearedweakly viscoelastic fluid, J. Non-Newton. Fluid Mech. 183 (2012) 52–56.

[3] S. Padhy, E.S.G. Shagfeh, G. Iaccarino, J.F. Morris, N. Tonmukayakal, Simulationof a sphere sedimenting in a viscoelastic fluid with cross shear flow, J. Non-Newton. Fluid Mech. 197 (2013) 48–60.

[4] B. Yang, B. Khomami, Simulations of sedimentation of a sphere in a viscoelasticfluid using molecular based constitutive models, J. Non-Newton. Fluid Mech. 82(1999) 429–452.

[5] J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics, M. NijhoffPublishers, The Hague, 1983. p. 327.

[6] R.I. Tanner, F. Qi, K.D. Housiadas, A differential model for the rheologicalproperties of concentrated suspensions with weakly viscoelastic matrices,Rheol. Acta 49 (2010) 169–176.