Laminar flow of power-law fluids past a rotating cylinder

J. Non-Newtonian Fluid Mech. 165 (2010) 1442–1461

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Laminar flow of power-law fluids past a rotating cylinder Saroj K. Panda, R.P. Chhabra ∗ Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208016, India

a r t i c l e

i n f o

Article history: Received 9 May 2010 Received in revised form 17 July 2010 Accepted 20 July 2010

Keywords: Rotating cylinder Power-law fluid Drag Lift Reynolds number

a b s t r a c t In this work, the continuity and momentum equations have been solved numerically to investigate the flow of power-law fluids over a rotating cylinder. In particular, consideration has been given to the prediction of drag and lift coefficients as functions of the pertinent governing dimensionless parameters, namely, power-law index (1 ≥ n ≥ 0.2), dimensionless rotational velocity (0 ≤ ˛ ≤ 6) and the Reynolds number (0.1 ≤ Re ≤ 40). Over the range of Reynolds number, the flow is known to be steady. Detailed streamline and vorticity contours adjacent to the rotating cylinder and surface pressure profiles provide further insights into the nature of flow. Finally, the paper is concluded by comparing the present numerical results with the scant experimental data on velocity profiles in the vicinity of a rotating cylinder available in the literature. The correspondence is seen to be excellent for Newtonian and inelastic fluids. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Flow past a cylinder constitutes a classical problem in the domain of fluid mechanics. The interest in this flow configuration stems from both fundamental and pragmatic considerations. For instance, a circular cylinder denotes the simplest bluff body shape which is free from geometrical singularities and it has indeed provided valuable insights into the underlying physics of the flow including the wake phenomena, vortex shedding, drag and lift characteristics. On the other hand, the flow past a cylinder also represents idealization of several industrially important applications such as the flow in tubular and pin-type heat exchangers, filtration screens, membrane based separation modules, etc. Additional examples are found in the use of obstacles of various shapes to form weld-lines in polymer processing applications, in the resin transfer process of producing fiber-reinforced composites, and as a model for lungs. The imposition of rotation is used to delay and/or suppress the propensity for vortex shedding thereby extending the steady flow regime. Indeed, the flow over a cylinder is influenced by a large number of parameters such as the nature of the far flow field (uniform, or shear, or extensional), confined or unconfined cylinder, type of fluid (compressible, or incompressible, or nonNewtonian), stationary or rotating cylinder, etc. For the simplest case of the incompressible uniform flow of Newtonian fluids over an unconfined stationary cylinder, the flow undergoes several transitions with a gradual increase in the value of the Reynolds number. Thus, for instance, the flow remains attached to the surface of the

∗ Corresponding author. Tel.: +91 512 259 7393; fax: +91 512 259 0104. E-mail address: [email protected] (R.P. Chhabra). 0377-0257/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2010.07.006

cylinder up to about the Reynolds number value of 4–5. As the value of the Reynolds number is gradually incremented, the occurrence of an adverse pressure gradient at some point on the surface of the cylinder results in the separation of flow from the surface thereby leading to the formation of the so-called wake region. This region is characterized by the loss of fore and aft symmetry of the flow field, albeit the flow is still steady and two-dimensional. The attached twin-vortices grow in size with further increase in the value of the Reynolds number and the flow continues to be symmetric about mid-plane. At about Re = 46–47, the wake becomes asymmetric and vortices begin to break off alternately from the upper and lower halves of the cylinder respectively thereby resulting in the so-called laminar vortex shedding regime. Under these conditions, the flow is still two-dimensional, but no longer steady and one must seek solutions to the time-dependent equations. This flow regime occurs up to about the Reynolds number of about 165, at which the wake itself becomes turbulent and early signatures of 3D flow also begin to manifest. Evidently, the underlying changes in the kinematics of the flow associated with each flow regime also impact the macroscopic flow characteristics like the force coefficients (drag and lift), Strouhal number and Nusselt number, etc., especially in the way these scale with the Reynolds number. Due to concentrated research efforts expended in elucidating the aforementioned as well as many other associated aspects, a wealth of information has accrued in the literature as far as the flow of Newtonian fluids past a stationary cylinder is concerned and this body of information has been reviewed in several excellent sources [1–5]. Similarly the flow imposed over a rotating cylinder is not only another fundamental flow of the bluff body family, but the rotation of cylinder is also used to increase lift, to stabilize flow and/or to control boundary layers on aerofoils. For Newtonian fluids, this flow con-

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Nomenclature CD CL CDN CLN CDP CDF CLP CLF D De FD FL H I2 L m n Ni p R Re Ux Uy U0 x* y*

Drag coefficient Lift coefficient Total drag coefficient in a Newtonian fluid Total lift coefficient in a Newtonian fluid Pressure component of drag coefficient Frictional component of drag coefficient Pressure component of lift coefficient Frictional component of lift coefficient Diameter of the cylinder (m) Deborah number Drag force per unit length of the cylinder (N/m) Lift force per unit length of the cylinder (N/m) Height (and width) of the square domain (m) Second invariant of the rate of deformation tensor (s−2 ) Length of the cylinder (m) Power-law consistency index (Pa sn ) Power-law index Number of points on the surface of the cylinder Pressure (Pa) Radius of the cylinder (m) Reynolds number x-Component of velocity (m/s) y-Component of velocity (m/s) Uniform velocity of the fluid at the inlet (m/s) Stream wise co-ordinate, x* = x/R Transverse co-ordinate, y* = y/R

Greek letters ˛ Rotational velocity ˝ Angular velocity of the cylinder (rad/s)  Density of fluid (kg/m3 )  ij Extra stress tensor (Pa)  Viscosity (Pa s) εij Component of the rate of strain tensor (s−1 )  Angular displacement from the front stagnation point  Relaxation time (s)

figuration has also been explored, though the resulting literature is neither as extensive nor as coherent as that for a stationary cylinder. Much of the pertinent literature has been reviewed in [1–6]. It is thus fair to say that an adequate body of information is available on the flow of Newtonian fluids past a stationary and a rotating cylinder. On the other hand, it is readily acknowledged that many fluids of industrial significance exhibit non-Newtonian flow characteristics including shear-dependent viscosity (shear-thinning), yield stress, visco-elasticity, etc. Typical examples include polymer melts and solutions, foams, emulsions and suspensions, etc. [7]. The flow of non-Newtonian fluids over a rotating cylinder denotes an idealization of some engineering applications encountered in coating operations, rotary drum filters used for non-Newtonian slurries, roller bearing applications, in oil drilling operations, in mixing vessels with novel impeller designs, etc. In spite of their fundamental and pragmatic significance, very little is known about the fluid mechanical aspects of the non-Newtonian fluid flow over a rotating cylinder and this work endeavours to fill this gap in the current literature. However, in order to facilitate the presentation of the new results obtained in this study, it is instructive to briefly review the body of knowledge available on the flow of non-Newtonian (especially power-law fluids) past a sta-

Fig. 1. (a) Schematic representation of the physical model, (b) computational domain, and (c) close up view of the grid in the vicinity of cylinder.

tionary cylinder and that of Newtonian fluids past a rotating cylinder. 2. Previous work As noted earlier, since a succinct summary of the pertinent literature on the flow of Newtonian fluids over a rotating cylinder is available in numerous references, e.g., see [6,8], only the salient features are recounted here. For the simplest case of an unconfined cylinder, the flow is now governed by two dynamic parameters, namely, Reynolds number and dimensionless rotational velocity, ˛, of the cylinder. While many studies are available at moderate to high Reynolds numbers at which the flow field is three-dimensional (e.g., see [9–12]), since the present work is concerned with the steady flow regime, the ensuing discussion is limited to the steady flow regime. At Re = 46–47, the flow becomes unsteady even for a

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Table 1 Selection of optimum domain. H/D Re = 0.1, n = 1, ˛ = 6 120 220 240 Re = 0.1, n = 0.2, ˛ = 6 120 220 240 Re = 5, n = 1, ˛ = 6 100 140 160

CDP

CDF

CD

CLP

CLF

CL

31.404 29.290 29.078

32.805 30.600 30.371

64.209 59.890 59.450

−13.230 −12.304 −12.218

−8.469 −7.860 −7.806

−21.698 −20.164 −20.024

132.154 132.050 132.049

16.382 16.370 16.371

148.536 148.42 148.42

−22.580 −22.427 −22.427

0.141 0.155 0.154

−22.438 −22.271 −22.273

−3.390 −3.389 −3.389

3.150 3.149 3.149

−0.240 −0.239 −0.240

−20.955 −20.953 −20.953

−4.575 −4.574 −4.574

−25.530 −25.527 −25.527

5.089 5.088 5.089

0.185 0.185 0.185

5.274 5.273 5.274

−7.893 −7.892 −7.892

−0.043 −0.043 −0.043

−7.936 −7.935 −7.935

Re = 5, n = 0.2, ˛ = 6 100 140 160

stationary cylinder, i.e., ˛ = 0. Broadly speaking, small values of the rotational velocity ˛ do not seem to influence this transition. However, once the value of ˛ exceeds a critical value (˛1 ) for Re > 46–47, the tendency for vortex shedding is somewhat suppressed [13]. However, subsequent more detailed studies [6,8] also point to the occurrence of two more transitions. At high Reynolds numbers and for ˛ > ˛2 , the flow again becomes unsteady and finally, again the flow reverts to a steady flow regime at ˛ > ˛3 at Re = 200, albeit the second transition ˛2 ≤ ˛ ≤ ˛3 occurs over a rather narrow range of conditions. As expected, for a fixed value of the Reynolds number, drag and lift coefficients decrease with the increasing value of ˛ [14] and so does the rate of heat transfer [6,15] in the range 5 ≤ Re ≤ 166 and ˛ ≤ 4. In contrast to the aforementioned studies based on the solution of the complete governing equations, Kendoush [16] has employed the boundary layer approximation to predict the value of Nusselt number for a rotating cylinder. All in all, the flow is known to be steady up to Re ≤ 40 and ˛ ≤ 6 over a rotating cylinder [6]. In contrast, the literature on the flow of power-law fluids over a stationary cylinder is not only of recent vintage, but is also much less extensive than that for Newtonian fluids. Most of these studies have been reviewed recently [17,18]. Broadly, reliable results are now available on the momentum transfer characteristics for a nonrotating cylinder in unconfined power-law fluids in the steady flow regime [19–23] for both shear-thinning and shear-thickening fluids. At low Reynolds numbers, shear-thinning behaviour enhances drag above its Newtonian value and as expected, shear-thickening

Table 2 Details of grids used for grid independence study. Grid

Ni

ı/D

H/D = 220 G1 G2 G3

100 200 400

0.005 0.005 0.005

117,600 151,100 218,100

H/D = 100 G1 G2 G3

100 200 400

0.005 0.005 0.005

86,496 151,100 178,896

Ncells

has the opposite effect. However, it is appropriate to add here that whether the drag is reduced or augmented with respect to the Newtonian value is also linked to the choice of the characteristic viscosity used to define the Reynolds number [24]. Furthermore, the role of power-law index gradually diminishes with the increasing value of the Reynolds number. Similarly, heat transfer is facilitated in shear-thinning fluids both in forced and mixed convection regimes [25–27]. Qualitatively, similar trends are observed for a cylinder confined symmetrically in between two parallel walls [28,29]. Finally, there has been only one set of study on the flow and heat transfer phenomena from a cylinder submerged in powerlaw fluids in the laminar vortex shedding regime [17,18]. For the sake of completeness, it is worthwhile to add here that scant results are also available for square [30,31] and elliptic cylinders [32,33]

Table 3 Effect of grid details on the results at Re = 0.1 (H/D = 220) and Re = 5 (H/D = 100). Grid Re = 0.1, n = 1, ˛ = 6 G1 G2 G3 Re = 0.1, n = 0.2, ˛ = 6 G1 G2 G3 Re = 5, n = 1, ˛ = 6 G1 G2 G3 Re = 5, n = 0.2, ˛ = 6 G1 G2 G3

CDP

CDF

CD

CLP

CLF

CL

29.318 29.290 29.289

30.618 30.600 30.586

59.936 59.890 59.875

−12.37 −12.304 −12.285

−7.907 −7.86 −7.845

−20.276 −20.164 −20.13

132.130 132.050 132.048

16.380 16.370 16.366

148.510 148.420 148.414

−22.491 −22.427 −22.416

0.153 0.155 0.156

−22.338 −22.271 −22.261

−3.380 −3.391 −3.392

3.141 3.150 3.151

−0.232 −0.240 −0.241

−20.920 −20.959 −20.960

−4.560 −4.575 −4.577

−25.480 −25.531 −25.537

5.280 5.275 5.266

−7.892 −7.893 −7.893

−0.046 −0.044 −0.044

−7.930 −7.936 −7.937

5.095 5.090 5.082

0.1850 0.1855 0.1858

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Table 4 Comparison between the present and literature results for a rotating cylinder in Newtonian fluids. Source Present Stojkovic et al. [8] Badr et al. [15]

Re

CDP

5

˛=1 1.9531 – – ˛=6 – –

Present Stojkovic et al. [8] Present Paramane and Sharma [6] Stojkovic et al. [8] Badr et al. [15]

20

Present Paramane and Sharma [6] Stojkovic et al. [8] Present Paramane and Sharma [6]

40

Present Paramane and Sharma [6]

CDF

˛=1 1.01 1 – – ˛=6 −2.734 −2.791 – ˛=1 0.7774 0.775 ˛=6 −2.0841 −1.9

−0.6076 – 0.55

CL −2.922 2.9022 2.84

– –

– –

0.8312 0.8368 – –

1.8412 1.8368 – 1.91

−2.387 −2.3634 2.424 2.44

−0.3614 −0.357 0.3501 0.35

−2.7483 −2.7204 2.7741 2.79

2.4227 2.4467 –

−0.3112 −0.3443 –

−28.86 −28.8764 28.86

−2.9412 −2.9314 2.9461

−31.8003 −31.8078 31.806

0.5458 0.54

1.3233 1.315

−2.3345 −2.3405

−0.2645 −0.2608

−2.599 −2.6013

−32.894 −32.9103

−2.2621 −2.248

−35.1562 −35.1583

−0.018 0.217

2.0661 2.117

˛=0 CDP

CDF

CD

69.5039 72.3863

270.5431 267.5328

Present Sivakumar et al. [41]

0.1

201.0391 195.1465

Present Sivakumar et al. [41]

1

20.0084 19.6615

7.0551 7.2447

27.0636 26.9062

Present Sivakumar et al. [41]

10

2.4251 2.4388

0.7318 0.7178

3.1570 3.1566

40

0.9915 0.9954

0.1450 0.1443

1.1365 1.1397

Present Sivakumar et al. [41]

−2.315 – 2.29

CLF

3.8574 3.8013 3.81

Table 5 Comparison between the present and literature values for a stationary cylinder in shear-thinning fluids (n = 0.2). Re

CLP

1.9043 – –

submerged in confined and unconfined power-law fluids, most of which are based on the assumption of the steady flow regime except for the recent results reported in [34] on the flow over a square cylinder in the laminar vortex shedding regime. On the other hand, a vast literature exists on the flow of visco-elastic fluids past a non-rotating circular cylinder [24]; however, most of it relates to the creeping (zero Reynolds number) flow regime and endeavours to elucidate the role of fluid visco-elasticity on the fluid mechanical aspects in the absence of shear-thinning behaviour. While most experimental studies report significant changes in the detailed flow field which are attributed to visco-elasticity, but the numerical simulations predict only minor changes. At low Reynolds numbers, visco-elasticity reduces the drag on a cylinder below its Newtonian value and it increases the drag at high Reynolds numbers. Therefore, the literature is inundated with conflicting reports regarding the role of elasticity in this case. As far as known to us, there have been only two studies on the flow of non-Newtonian fluids past a rotating cylinder. Townsend [35] numerically studied the uniform flow of an Oldroyd model fluid past a rotating cylinder over the range of conditions as ˛ ≤ 5 and 10 ≤ Re ≤ 60. His results suggest that the visco-elasticity tends to increase both the drag and lift experienced by a rotating cylinder whereas the shear-thinning behaviour tends to reduce their values. On the other hand, Christiansen [36,37] has reported detailed experimental data on the local velocity field for

Source

CD

−20.9557 –

−4.575 –

−25.5308 25.4764

a cylinder rotating in Newtonian and polymer solutions of varying levels of shear-thinning and visco-elasticity. It is thus safe to conclude that very little is known about the flow of non-Newtonian fluids past a rotating cylinder. Admittedly, complex fluids exhibit a range of rheological features such as shearthinning, visco-elasticity and yield stress, it seems reasonable to begin with the simplest type of non-Newtonian behaviour, namely, shear-thinning viscosity and the level of complexity can gradually be built up to incorporate other aspects of real fluid behaviour in a systematic manner. This work is thus concerned with the flow of power-law fluids past a rotating cylinder over the range of conditions Re ≤ 40, ˛ ≤ 6 and 0.2 ≤ n ≤ 1 over which the flow is expected to be steady and two-dimensional. In particular, extensive numerical results are presented elucidating the effect of rheological and dynamic parameters on the detailed kinematics of flow and on drag and lift coefficients. Finally, the present predictions are compared with the experimental results available in the literature [36,37]. 3. Problem statement and governing equations Let us consider the incompressible steady flow of a powerlaw fluid (with uniform velocity U0 ) past a cylinder of radius R or diameter D (infinitely long in z-direction) rotating with an angular velocity of ˝ in the anti-clockwise direction, as shown schematically in Fig. 1(a). Since it is not possible to simulate numerically truly an unconfined flow, it is customary to introduce an artificial domain in the form of a box as shown schematically. Following the previous studies [6], the cylinder is placed at the center of a square of size H, as shown in Fig. 1(b). The value of H is chosen in such a manner that it does not unduly influence the flow field and at the same time it necessitates only modest computational resources. The flow phenomenon is described by the continuity and momentum equations written in their compact forms as follows: • Continuity:

∇ .V = 0

(1)

• Momentum: 

DV = −∇ p + ∇ ·  + g Dt

(2)

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Fig. 2. Streamline contours near the cylinder for n = 0.2 for ˛ = 1 and 6.

where , V, p,  and g, respectively, are the fluid density, velocity vector, isotropic pressure, extra stress tensor and gravity. For power-law fluids, the components of the extra stress tensor  ij are related to the rate of deformation tensor as

where the rate of deformation tensor is given by εij =

1 2



∂Uj ∂Ui + ∂xj ∂xi



(4)

And the scalar viscosity function, , for a power-law fluid is given by ij = 2εij

(3)

 = m(2I2 )(n−1)/2

(5)

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Fig. 3. Streamline contours near the cylinder for n = 0.6 for ˛ = 1 and 6.

where n is the so-called power-law index. Evidently, n = 1 corresponds to the Newtonian fluid behaviour and n < 1 denotes shear-thinning behaviour. It is customary to introduce dimensionless variables. In this work, the free stream velocity, U0 , and diameter of cylinder, D, have been used as scaling variables. Thus, the pressure has been scaled using U02 , stress components using m(U0 /D)n , time with D/U0 , etc.

The physically realistic boundary conditions for this flow are written as follows: (1) At the inlet plane: It is the uniform flow in x-direction, i.e., Ux = U0 ,

Uy = 0

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Fig. 4. Streamline contours near the cylinder for n = 1 for ˛ = 1 and 6.

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Fig. 5. Vorticity contours near the cylinder for n = 0.2 for ˛ = 1 and 6.

(2) The top and bottom walls are assumed to be slip boundaries so that there is no dissipation at these walls. The corresponding mathematical description is given by:

(3) On the surface of the solid cylinder: The standard no-slip boundary condition is used, i.e., Ux = −˛ sin 

∂Ux = 0; ∂y

Uy = 0

and

Uy = −˛ cos 

(4) At the exit plane: The default outflow boundary condition option in FLUENT (a zero diffusion flux for all flow variables) was used

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Fig. 6. Vorticity contours near the cylinder for n = 0.6 for ˛ = 1 and 6.

in this work. This choice implies that the conditions of the outflow plane are extrapolated from within the domain and as such have negligible influence on the upstream flow conditions. The extrapolation procedure used by FLUENT updates the outflow velocity and the pressure in a manner that is consistent with the fully developed flow assumption, when there is no area change at the outflow boundary. However, the gradients in the cross-

stream direction may still exist at the outflow boundary. Also, the use of this condition obviates the need to prescribe a boundary condition for pressure. This is similar to the homogeneous Neumann condition, that is, ∂Ux =0 ∂x

and

∂Uy =0 ∂x

S.K. Panda, R.P. Chhabra / J. Non-Newtonian Fluid Mech. 165 (2010) 1442–1461

Fig. 7. Vorticity contours near the cylinder for n = 1 for ˛ = 1 and 6.

The numerical solution of the governing Eqs. (1) and (2) together with the aforementioned boundary conditions maps the flow domain in terms of Ux , Uy and p which, in turn, can be post processed to evaluate the derived quantities such as stream function, vorticity, drag and lift coefficients as functions of the pertinent governing parameters, namely, the Reynolds number (Re), power-law index (n) and the non-dimensional rotational velocity of the cylinder (˛). At this juncture, it is appropriate to introduce the pertinent definitions of some of these parameters as follows:

• Reynolds number (Re)

Re =

U02−n Dn m

• Rotational velocity (˛)

˛=

R˝ U0

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Fig. 8. Representative pressure profiles on the surface of the rotating cylinder.

• Drag coefficient (CD )

CD =

2FD U02 D

where FD is the drag force in the direction of flow exerted on the cylinder per unit length. It is also customary to split the total drag force into two components arising from the shear and pressure forces thereby giving rise to the corresponding drag coefficient components as CDF and CDP respectively.

S.K. Panda, R.P. Chhabra / J. Non-Newtonian Fluid Mech. 165 (2010) 1442–1461

Fig. 9. Dependence of CD , CDP , CL on Reynolds number for ˛ = 1 and 6.

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Fig. 10. Variation of CD , CDP , CL with ˛ for Re = 0.1 and 40.

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Table 6 Numerical values of drag and lift coefficients as functions of ˛, Re and n. CD ˛=0

Re

0.1 1 10 40

n = 0.8

n = 0.6

n = 0.4

n = 0.2

n=1

n = 0.8

n = 0.6

n = 0.4

n = 0.2

61.1275 10.5663 2.7983 1.5136

100.8522 13.2468 2.8609 1.4378

164.2520 17.4697 2.9348 1.3455

231.3182 23.3118 3.0268 1.2393

270.5431 27.0635 3.1570 1.1365

60.9866 10.1578 2.1340 0.8453

89.9372 11.7208 2.2173 0.8517

133.9509 14.2309 2.3359 0.8592

176.8087 17.8904 2.5245 0.8798

199.819 20.2951 2.6436 0.9115

˛=4

Re

0.1 1 10 40

˛=2

n=1

˛=6

n=1

n = 0.8

n = 0.6

n = 0.4

n = 0.2

n=1

n = 0.8

n = 0.6

n = 0.4

n = 0.2

60.5715 8.9340 0.5891 −0.1440

84.4540 10.8012 1.3992 0.3397

119.1578 12.9498 2.2384 0.7686

151.0860 15.7119 2.9974 1.0976

169.8143 17.3883 3.3288 1.3608

59.8808 6.9582 −0.7747 −0.0180

80.6602 10.0769 1.5118 0.6089

108.9539 12.5536 2.8136 0.9679

133.0632 15.0304 3.3974 1.2528

148.426 16.6408 3.5269 1.3297

CL ˛=0

Re

0.1 1 10 40 Re

n=1

n = 0.8

n = 0.6

n = 0.4

n = 0.2

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

˛=4 n=1

0.1 1 10 40

˛=2

−13.4117 −12.6919 −14.5583 −16.1329

n = 0.8

n = 0.6

n = 0.4

n = 0.2

−3.0726 −4.0993 −5.3654 −5.3832

−2.6343 −3.0404 −4.8587 −5.0545

−3.9018 −3.8614 −4.5096 −4.7375

−7.2012 −6.6750 −4.5950 −4.4573

˛=6 n = 0.8

n = 0.6

n = 0.4

n = 0.2

n=1

n = 0.8

n = 0.6

n = 0.4

n = 0.2

−6.3120 −8.3219 −12.7017 −15.2928

−5.6369 −6.2170 −10.7257 −14.1065

151.086 −7.6299 −8.6285 −12.4331

−14.1945 −11.9027 −7.5009 −10.7610

−20.1638 −20.2742 −28.6372 −35.1562

−9.6371 −12.855 −22.633 −25.557

−8.9071 −9.371 −15.137 −19.400

−12.8874 −10.935 −10.080 −15.069

−22.269 −15.432 −7.9919 −12.260

• Lift coefficient (CL ) CL =

n=1 −6.6968 −6.1190 −5.9608 −5.7205

2FL U02 D

where FL is the lift force acting in the y-direction on the cylinder per unit length. Here also, it is not uncommon to split the lift force into two components due to pressure and shear as CLP and CLF respectively. The scaling of the governing equations and the boundary conditions suggests the integral flow parameters, CD and CL to be functions of the Reynolds number, power-law index and ˛. This functional relationship is explored and developed in this work. 4. Numerical solution method Since detailed descriptions of the numerical solution procedure are available elsewhere [17,18,28], only the salient features are recapitulated here. In this study, the field equations have been solved using FLUENT (version 6.2.26). The structured quadrilateral cells of uniform and non-uniform grid spacing were generated using the commercial grid tool GAMBIT (version 2.3.16). The twodimensional, laminar, segregated solver was used to solve the incompressible flow on the collocated grid arrangement. Both steady and unsteady solvers have been used in this study. While the major thrust of this work is on the steady flow regime, limited time-dependent simulations for extreme values of the governing parameters such as Re = 40, n = 0.2 and ˛ = 6, etc. were also conducted to ascertain that the flow regime indeed was steady over the range of conditions covered in this study. The second order upwind scheme has been used to discretize the convective terms in the momentum equations. The semi-implicit method

for the pressure linked equations (SIMPLE) scheme was used for solving the pressure–velocity coupling. The constant density and non-Newtonian power-law viscosity modules were used to input the physical properties of the flow. However, the input values of the physical properties , m, n and kinematic parameters like D, U0 , ˝, etc. are of no consequence as the final results are reported in a dimensionless form. FLUENT solves the system of algebraic equations using the Gauss–Siedel (G–S) point-by-point iterative method in conjunction with the algebraic multi-grid (AMG) method solver. The use of the AMG scheme greatly reduces the number of iterations (thereby accelerating convergence) and thus economizing the CPU time required to obtain a converged solution, particularly when the model contains a large number of control volumes. Relative convergence criteria of 10−8 for the residuals of the continuity and x- and y-component of the momentum equations were prescribed in this work. Furthermore, a simulation was deemed to have converged when the values of the global parameters had stabilized to at least four significant digits. 5. Choice of numerical parameters Undoubtedly, the reliability and accuracy of the numerical results is strongly influenced by the choice of domain size (H), grid characteristics (number of cells on the surface of the cylinder, grid spacing, stretching, etc.) and to some extent by the convergence criterion, etc. In this work, the values of these parameters have been selected after extensive exploration. The values of (H/D) ranging from 80 to 240 have been employed here to arrive at an optimal value of this parameter. Intuitively, it appears that a larger domain is required at low Reynolds numbers than that at high Reynolds numbers. This is so simply due to the slow spatial decay of the velocity field at low Reynolds numbers. Therefore, the domain independence study has been carried out at the lowest value of Re = 0.1

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Fig. 11. Effect of power-law index on the normalized values of CD and CL with ˛ for Re = 0.1 and 40.

used in this study. After a detailed examination of the results (drag and lift coefficients), suffice it to say here that, the value of (H/D) used in this work is 220 at low Reynolds numbers (Re < 5), whereas (H/D) = 100 was used for Re ≥ 5 for the domain effects to be negligible as can be seen in Table 1. These findings are consistent with that reported by others, e.g., see [6] over this range of Reynolds numbers. Having fixed the value of (H/D), an optimal grid should meet two conflicting requirements: namely, it should be sufficiently fine to resolve the thin boundary layers and the steep gradients near the rotating cylinder, and of course without being prohibitively computation intensive. For this purpose, the relative performance of three grids has been studied in detail to arrive at the choice of an optimal grid. Each grid was characterized in terms of the number of points (Ni ) on the surface of the cylinder and the value of (ı/D) near the cylinder, as summarized in Table 2. A typical grid is shown in

Fig. 1(c). For the same combinations of the values of Re, n and ˛ as that used in the domain independence study, Table 3 summarizes the relative performance of the three grids in terms of the resulting values of the individual components of drag and lift coefficients. A detailed examination of these results reveals that very little is gained in terms of accuracy of the present results as one moves from G2 to G3 whereas the CPU time required for G3 is many folds larger than that needed for G2 to satisfy the same criterion of convergence. Thus, grid G2 denotes a good compromise between the accuracy and computational efforts and hence all results reported herein are based on the use of grid G2. Finally the adequacy of the values of (H/D) and grid G2 is further demonstrated in the next section by benchmarking the present results against the literature values for a few limiting cases like for a rotating cylinder in Newtonian media, and for a non-rotating cylinder in power-law fluids.

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Fig. 12. Comparison between the predicted and experimental velocity profiles for 86% aqueous glycerin solution. Re = 0.411, n = 1: (a) ˛ = 0, (b) ˛ = 1.18, and (c) ˛ = 1.18.

6. Results and discussion As noted earlier, the present flow is governed by five dimensionless parameters, namely, CD , CL , Re, n and ˛ and the principal aim of the present study is to develop this functional relationship. However, prior to embarking upon the presentation of new results, it is useful to demonstrate the adequacy of the numerical computations which will also help ascertain the accuracy of the new results reported herein. 6.1. Validation of results Since detailed comparisons are available elsewhere for the flow of Newtonian fluids past a stationary cylinder [22,23], suffice it to say here that the present results are within ∼1–2% of the literature values. Next, Table 4 presents a comparison for a rotating cylinder in Newtonian fluids; both the values of drag and lift coefficients are included here for ˛ = 1 and 6. Broadly speaking, the present results are within 1–2% of that reported by Stojkovic et al. [8] and Paramane and Sharma [6] for Re ≥ 1,

though the present values are seen to deviate by up to 5–6% at Re = 0.1 from that reported in [8]. Furthermore, at ˛ = 6 and Re = 40, while the present values of CDP and CDF are in excellent agreement with that reported in Ref. [6], but the total drag values differ by an order of magnitude. This is so due to two reasons: both CDP and CDF are of similar magnitudes but of opposite sign. Secondly, the values of CDP and CDF corresponding to Ref. [6] have been read off their figures. Thus, even a small error incurred in extracting these values gets accentuated in this case. While this may seem like a large difference, it needs to be emphasized here that the differences of this magnitude are not at all uncommon in such numerical studies due to underlying variations in different studies arising from the choices of solution methodologies, size and shape of domain, convergence criterion, etc. [38]. Similarly, Table 5 reports a representative comparison between the present and literature values for a stationary cylinder in shear-thinning fluids. While more extensive comparisons are reported elsewhere [23], it is fair to say that the present results are seen to be in excellent agreement with the literature values in Table 5.

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Fig. 13. Comparison between the predicted and experimental velocity profile for 1% cellosize hydroxyethyl cellulose (HEC) solution. Re = 0.079, n = 0.43: (a) ˛ = 0, (b) ˛ = 1.706, and (c) ˛ = 1.706.

Aside from the aforementioned benchmark comparisons, the flow of Newtonian and power-law fluids was also studied in the standard lid driven square cavity. The present results for the centerline velocities were found to be within ±2% of the corresponding Newtonian results [39] and within ±2.5% for power-law fluids as reported by Neofytou [40]. These comparisons lend further support to the credibility of the numerical solution methodology employed herein. Based on the above-noted comparisons, the present results for a cylinder rotating in power-law fluids are believed to be reliable to within ±2%. 6.2. Detailed kinematics Figs. 2–4 show representative results elucidating the influence of Re, ˛ and n on the streamlines in the vicinity of the cylinder. For the range of Reynolds number encompassed here, the flow is known to be steady for all values of the power-law index at ˛ = 0, i.e., for a

non-rotating cylinder [41]. As expected, the flow field is symmetric about the mid-plane, and at low Reynolds numbers (Re ≤ 1), it also exhibits fore and aft symmetry. This is so due to the fact that the viscous forces far outweigh the inertial forces under these conditions. However, as the Reynolds number is gradually increased, the flow detaches itself from the surface of the cylinder which leads to the formation of the wake region in the rear of the cylinder. This marks the formation of a pair of standing vortices which grow in size with the increasing Reynolds number up to a critical value of the Reynolds number. For Newtonian fluids, the first signature of flow separation is seen at about Re = 4–5 whereas this transition is delayed in shear-thinning fluids [41]. For instance, it occurs at about Re  11–12 in a highly shear-thinning fluid (n = 0.3) for a non-rotating cylinder [41]. A detailed examination of Figs. 2–4 reveals the following overall trends: As soon as the rotation of the cylinder is superimposed on the uniform flow, the flow becomes asymmetric as can clearly be seen even at Re = 0.1, albeit there is no

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indication of flow separation at this stage. In fact, a distinct saddle point is seen to form in Newtonian and mildly shear-thinning fluids (n = 0.6) which disappears in highly shear-thinning fluids (such as at n = 0.2). This is simply due to the rapid decay of the flow field in highly shear-thinning fluids and it is almost tantamount to a cylinder rotating in a cavity formed by low-viscosity fluid which itself is surrounded by a high viscosity fluid, akin to the rotation of an impeller in a highly shear-thinning and/or viscoplastic fluid [42]. Owing to the weak advection, this effect is particularly striking at Re = 0.1, where no saddle point is seen in the streamlines plot. The viscous effects diminish with the increasing Reynolds number, or rotational velocity (due to enhanced levels of deformation), or a combination of both. Thus, as the value of the Reynolds number is progressively increased, one can see the formation of a streamline enclosing the cylinder at lower and lower values of ˛. For instance, while this phenomenon occurs at ˛ = 6 for a weak uniform flow such as that at Re = 1 for n = 0.2, it was found to occur at ˛ = 4 for Re = 10 and n = 0.2. Similarly, the saddle point is seen to shift a little in the anti-clockwise direction with the increasing Reynolds number, or decreasing value of power-law index. For Newtonian fluids, at vanishingly small values of the Reynolds numbers (Re → 0), while the effect of rotation and uniform flow is easily delineated by superimposing the two individual solutions due to the linearity of the Navier–Stokes equations. This is clearly not possible for powerlaw fluids due to the non-linearity of the viscous term even when the non-linear inertial terms are altogether neglected. As Reynolds number is gradually increased, the inertial effects progressively become important. While for ˛ = 0, the flow field is found to be symmetric about the mid-plane, but with the increasing value of ˛, the symmetry is lost and the upper vortex detaches from the cylinder (e.g., see Fig. 3 for Re = 10). Similarly, the lower vortex also dislodges itself and eventually disappears with increasing ˛. Furthermore, the streamlines near the cylinder are drawn closely around it until a ring of trapped fluid is formed which rotates along with the cylinder. This behaviour is seen for all values of the power-law index considered herein (1 ≥ n ≥ 0.2), except for the fact that the size of the so-called separatrix [6] is seen to be smaller in shear-thinning fluids than that in Newtonian fluids otherwise under identical conditions. Thus, in summary broadly, qualitatively similar streamline patterns are seen in power-law shear-thinning and Newtonian fluids in the steady flow regime. Typical value of the stream function ( * = /U0 D) range from ∼10−8 m2 /s at the bottom of the box (y = −H/2) to 4000 m2 /s at top of the box (y = H/2) depending upon the values of n, Re, ˛. Naturally, lower the value of Re, smaller is the value of * near the cylinder. Representative vorticity contours are shown in Figs. 5–7 for the same combinations of Re, n and ˛ which also reinforce the trends seen in streamline patterns. For instance, at low Reynolds numbers, the vorticity contours become increasingly asymmetric as the value of ˛ is increased. Indeed, larger is the value of Reynolds number, lower is the value of ˛ needed to break the symmetry. The crowding of the iso-vorticity contours underneath the cylinder show the intensity of the vorticity distribution in this region. With the increasing Reynolds number, the vortices are seen to be transported in the main direction of flow. The surface vorticity is naturally maximum on the surface of the cylinder (ω* = ω/U0 D) and it ranges from ∼1 to ∼220 at Re = 0.1 as the value of n is decreased from n = 1 to n = 0.2. Of course, it decays to very small values of the order of 10−12 to 10−7 far away from the cylinder. Fig. 8 shows representative surface pressure profiles for a range of combinations of Re, ˛ and n. Obviously, since the flow is not symmetric about the x- or y-axis, pressure profiles are shown over the entire surface (0 ≤  ≤ 360). At low Reynolds numbers, e.g., at Re = 0.1, the surface pressure in the front of the cylinder progressively increases with the increasing degree of shear-thinning (decreasing values of n), albeit the effect is most dramatic at the

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front stagnation point. In the rear of the cylinder, the influence of power-law index is not monotonic on the surface pressure. With the superimposition of rotation, the pressure is no longer maximum at the front stagnation point and it occurs at about  = 20–30◦ . At low Reynolds numbers, as the value of ˛ is gradually increased, the surface pressure profile exhibits increasing asymmetry. Similarly, for a fixed value of ˛, as the Reynolds number is gradually increased, the surface pressure drops progressively, and the effect of n also weakens up to about ˛ = 3. However, as expected, for ˛ = 6, the pressure is negative over most of the surface of the cylinder (e.g., see plot for Re = 40 and ˛ = 6), where as it undergoes a sign change at lower Reynolds numbers. The complex dependence seen in Fig. 8 obviously stems from different scaling of the viscous and inertial forces on velocity and power-law index. How these changes at the microscopic level influence the values of macroscopic quantities like drag and lift coefficient will be seen in the next section, as one would expect the pressure drag to be negative under certain circumstances. 6.3. Drag and lift coefficients Figs. 9 and 10 show the influence of ˛, Re and n on pressure drag coefficient, total drag coefficient and lift coefficient. At low Reynolds numbers, both total drag coefficient and its pressure component are seen to decrease with the increasing value of rotational velocity which is consistent with the results of Stojkovic et al. [8] and Paramane and Sharma [6] for Newtonian fluids (n = 1). While at low Reynolds numbers, the rotation of the cylinder augments both the total drag and its pressure component, but it is evident that the pressure drag contributes to the total drag significantly under these conditions. On the other hand, at high Reynolds numbers, Re = 40, the dependence of CD and CDP on ˛ is more complex. For Newtonian fluids, the total drag decreases with the increasing value of ˛ up to about ˛  5 beyond which it shows an increase. It is interesting to see that under certain conditions both CD and CDP can be negative which is consistent with the pressure profiles seen in the previous sections due to a very small contribution of friction component. Shear-thinning fluids are also seen to exhibit qualitatively similar behaviour, except for the fact that while in some cases the value of CDP can become slightly negative but the total drag always remains positive thereby suggesting that friction probably contributes in greater proportion here than that in the case of Newtonian fluids. Indeed, smaller is the value of power-law index (highly shear-thinning fluid), higher is the total drag coefficient. Fig. 10 shows these results in a slightly different form where it is clearly seen that the effect of power-law index on both CDP and CD gradually weakens with the increasing Reynolds number. For instance, for ˛ = 1, the curves for different values of n collapse onto a single curve at about Re ∼ 1–2, whereas at ˛ = 6 this behaviour occurs at about Re  40. The lift coefficient is seen to be negative (i.e. acting in the downward direction) at all values of the Reynolds number considered in this work and it decreases almost linearly with the increasing rotational velocity. At low Reynolds numbers, it goes through a maximum value at about n  0.6, as can be seen clearly in Fig. 9. At low Reynolds numbers, the rate of change also increases with the increasing degree of shear-thinning behaviour, i.e., decreasing value of n. On the other hand, at Re = 40, the effect of n is rather insignificant up to about ˛ = 2–3 and beyond this value, the lift coefficient increases with the decreasing value of n. While these results are in line with that of Stojkovic et al. [8] for Newtonian fluids, but these are at variance from that of Townsend [35] who predicted the lift to increase with ˛ to a maximum value, but then it falls steeply. This is an unusual result which has been attributed to the inadequacy of their computations [8]. A representative summary of the present numerical values of drag and lift coefficients as func-

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tions of Re, n and ˛ is given in Table 6. Intuitively, one would expect the product of drag coefficient and Reynolds number to approach a limiting value depending upon the values of n and ˛ as the value of the Reynolds number diminishes, but unfortunately the lowest value of Re = 0.1 used here is not sufficiently small for this limiting behaviour to be observed in the present case. Finally, Fig. 11 shows the influence of the power-law index, Reynolds number, and ˛ on the normalized values of CD , CDP and CL to delineate the role of power-law index in an unambiguous fashion. As noted earlier, at low Reynolds numbers, the total drag on a cylinder can be as much as 3–4 times of that in a Newtonian fluid otherwise under identical conditions; however, this effect diminishes with the increasing value of ˛. On the other hand, this effect is seen to be even more dramatic at Re = 40. In the extreme case, not only CD can undergo a change of sign depending upon the values of ˛ and n, but their ratio too can be as high as −70. In contrast, the lift coefficient is seen to display a more regular dependence on n and ˛, both at high and low Reynolds numbers, though it peaks at about n  0.6 at low Reynolds numbers. 7. Comparison with experiments As noted earlier, while no experimental results are available on drag and lift coefficients for a cylinder rotating in Newtonian and in power-law fluids in the steady flow regime, Christiansen [36,37] reported detailed velocity measurements for a rotating cylinder in a Newtonian glycerol solution and in two polymer solutions. Two test cylinders 102 mm long, but of two different diameters, namely, 1.59 mm and 4.76 mm were used. The (L/D) ratios are ∼22 and ∼65, which are modest for the end effects to be negligible. The two polymer solutions used in their study exhibited viscoelastic behaviour, with relaxation times of 0.041 s (HEC solution) and 8.6 s (Separan solution). The value of power-law index for both solutions is identical, as n = 0.43. The values of ˛ varied from 0 to 2.29 and of the Reynolds number are in the range 0.03 to ∼0.5. He reported the values of the x- and y-component of the velocity along (x = 0) and (y = 0) lines both upstream and downstream from the cylinder. Fig. 12 shows a comparison between the present predictions and the experimental data for their Newtonian fluid at Re = 0.411 for a stationary cylinder (˛ = 0) and a rotating cylinder (˛ = 1.18). An excellent agreement is seen to exist between the present numerical results and the experimental results. Fig. 13 shows a similar comparison for the weakly elastic HEC solution wherein the correspondence is though less good, but may still be regarded as satisfactory thereby suggesting this solution to be virtually inelastic over this range of conditions. The corresponding value of Deborah number, defined as De = U0 /D, is 0.32 which is rather low and therefore, visco-elastic effects are expected to be negligible here. On the other hand, the value of Deborah number ranges from 48 to 150 for the Separan solution and therefore, it is not appropriate to compare these results with the present predictions. Overall, the experimental verification seen in Figs. 12 and 13 lends further credibility to the accuracy and reliability of the results presented herein. 8. Concluding remarks In this work, the flow of shear-thinning power-law fluids past a rotating cylinder in the 2D laminar steady flow regime has been investigated numerically. The range of conditions encompassed include power-law index (1 ≥ n ≥ 0.2), Reynolds number (0.1 ≤ Re ≤ 40) and rotational velocity (0 ≤ ˛ ≤ 6). The flow is visualized in terms of streamline contours, iso-vorticity profiles and surface pressure profiles for representative combinations of the values of Re, n and ˛. The lift is found to be negative under most

conditions studied here, whereas the total drag force is usually positive but can be negative when the friction makes a little contribution and the pressure drag coefficient itself is negative such as that at high values of ˛ and Re. The power-law index exerts a much stronger influence on drag and lift at low Reynolds numbers than that at high Reynolds numbers. Indeed, shear-thinning behaviour can augment drag values by up to a factor of ∼4 at low values of Re and ˛, albeit the extent of increase decreases rapidly with the increasing values of ˛. Finally, the present numerical predictions of velocity at x = 0 and y = 0 planes are shown to be in fair agreement with the scant experimental results available in the literature for Newtonian and relatively inelastic polymer solutions.

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