1-s2.0-S0377025710002028-main

L

SD

a

ARRA

KRPDLR

1

dsiwvicrtfitttmssbficNcasT

0d

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

Contents lists available at ScienceDirect

Journal of Non-Newtonian Fluid Mechanics

journa l homepage: www.e lsev ier .com/ locate / jnnfm

aminar flow of power-law fluids past a rotating cylinder

aroj K. Panda, R.P. Chhabra ∗

epartment of Chemical Engineering, Indian Institute of Technology, Kanpur 208016, India

r t i c l e i n f o

rticle history:eceived 9 May 2010eceived in revised form 17 July 2010ccepted 20 July 2010

a b s t r a c t

In this work, the continuity and momentum equations have been solved numerically to investigate theflow of power-law fluids over a rotating cylinder. In particular, consideration has been given to theprediction 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

eywords:otating cylinderower-law fluidrag

number (0.1 ≤ Re ≤ 40). Over the range of Reynolds number, the flow is known to be steady. Detailedstreamline and vorticity contours adjacent to the rotating cylinder and surface pressure profiles providefurther insights into the nature of flow. Finally, the paper is concluded by comparing the present numericalresults with the scant experimental data on velocity profiles in the vicinity of a rotating cylinder availablein the literature. The correspondence is seen to be excellent for Newtonian and inelastic fluids.

ifteynolds number

. Introduction

Flow past a cylinder constitutes a classical problem in theomain of fluid mechanics. The interest in this flow configurationtems from both fundamental and pragmatic considerations. Fornstance, a circular cylinder denotes the simplest bluff body shape

hich is free from geometrical singularities and it has indeed pro-ided valuable insights into the underlying physics of the flowncluding the wake phenomena, vortex shedding, drag and liftharacteristics. On the other hand, the flow past a cylinder alsoepresents idealization of several industrially important applica-ions such as the flow in tubular and pin-type heat exchangers,ltration screens, membrane based separation modules, etc. Addi-ional examples are found in the use of obstacles of various shapeso form weld-lines in polymer processing applications, in the resinransfer process of producing fiber-reinforced composites, and as a

odel for lungs. The imposition of rotation is used to delay and/oruppress the propensity for vortex shedding thereby extending theteady flow regime. Indeed, the flow over a cylinder is influencedy a large number of parameters such as the nature of the far floweld (uniform, or shear, or extensional), confined or unconfinedylinder, type of fluid (compressible, or incompressible, or non-ewtonian), stationary or rotating cylinder, etc. For the simplest
ase of the incompressible uniform flow of Newtonian fluids overn unconfined stationary cylinder, the flow undergoes several tran-itions with a gradual increase in the value of the Reynolds number.hus, 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).

377-0257/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.jnnfm.2010.07.006

© 2010 Elsevier B.V. All rights reserved.

cylinder up to about the Reynolds number value of 4–5. As the valueof the Reynolds number is gradually incremented, the occurrenceof an adverse pressure gradient at some point on the surface of thecylinder results in the separation of flow from the surface therebyleading to the formation of the so-called wake region. This region ischaracterized by the loss of fore and aft symmetry of the flow field,albeit the flow is still steady and two-dimensional. The attachedtwin-vortices grow in size with further increase in the value of theReynolds number and the flow continues to be symmetric aboutmid-plane. At about Re = 46–47, the wake becomes asymmetricand vortices begin to break off alternately from the upper andlower halves of the cylinder respectively thereby resulting in theso-called laminar vortex shedding regime. Under these conditions,the flow is still two-dimensional, but no longer steady and one mustseek solutions to the time-dependent equations. This flow regimeoccurs up to about the Reynolds number of about 165, at which thewake itself becomes turbulent and early signatures of 3D flow alsobegin to manifest. Evidently, the underlying changes in the kine-matics of the flow associated with each flow regime also impact themacroscopic flow characteristics like the force coefficients (dragand lift), Strouhal number and Nusselt number, etc., especially inthe way these scale with the Reynolds number. Due to concen-trated research efforts expended in elucidating the aforementionedas well as many other associated aspects, a wealth of informationhas accrued in the literature as far as the flow of Newtonian fluidspast a stationary cylinder is concerned and this body of information
has been reviewed in several excellent sources [1–5]. Similarly theflow imposed over a rotating cylinder is not only another funda-mental flow of the bluff body family, but the rotation of cylinderis also used to increase lift, to stabilize flow and/or to controlboundary layers on aerofoils. For Newtonian fluids, this flow con-
dx.doi.org/10.1016/j.jnnfm.2010.07.006

http://www.sciencedirect.com/science/journal/03770257

http://www.elsevier.com/locate/jnnfm

mailto:[email protected]

dx.doi.org/10.1016/j.jnnfm.2010.07.006

S.K. Panda, R.P. Chhabra / J. Non-Newtonian Fluid Mech. 165 (2010) 1442–1461 1443

Nomenclature

CD Drag coefficientCL Lift coefficientCDN Total drag coefficient in a Newtonian fluidCLN Total lift coefficient in a Newtonian fluidCDP Pressure component of drag coefficientCDF Frictional component of drag coefficientCLP Pressure component of lift coefficientCLF Frictional component of lift coefficientD Diameter of the cylinder (m)De Deborah numberFD Drag force per unit length of the cylinder (N/m)FL Lift force per unit length of the cylinder (N/m)H Height (and width) of the square domain (m)I2 Second invariant of the rate of deformation tensor

(s−2)L Length of the cylinder (m)m Power-law consistency index (Pa sn)n Power-law indexNi Number of points on the surface of the cylinderp Pressure (Pa)R Radius of the cylinder (m)Re Reynolds numberUx x-Component of velocity (m/s)Uy y-Component of velocity (m/s)U0 Uniform velocity of the fluid at the inlet (m/s)x* Stream wise co-ordinate, x* = x/Ry* 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

findioc

iivsniorvdtogptfl

point� Relaxation time (s)

guration has also been explored, though the resulting literature iseither as extensive nor as coherent as that for a stationary cylin-er. Much of the pertinent literature has been reviewed in [1–6]. It

s thus fair to say that an adequate body of information is availablen the flow of Newtonian fluids past a stationary and a rotatingylinder.

On the other hand, it is readily acknowledged that many fluids ofndustrial significance exhibit non-Newtonian flow characteristicsncluding shear-dependent viscosity (shear-thinning), yield stress,isco-elasticity, etc. Typical examples include polymer melts andolutions, foams, emulsions and suspensions, etc. [7]. The flow ofon-Newtonian fluids over a rotating cylinder denotes an ideal-

zation of some engineering applications encountered in coatingperations, rotary drum filters used for non-Newtonian slurries,oller bearing applications, in oil drilling operations, in mixingessels with novel impeller designs, etc. In spite of their fun-amental and pragmatic significance, very little is known abouthe fluid mechanical aspects of the non-Newtonian fluid flow
ver a rotating cylinder and this work endeavours to fill thisap in the current literature. However, in order to facilitate theresentation of the new results obtained in this study, it is instruc-ive to briefly review the body of knowledge available on theow of non-Newtonian (especially power-law fluids) past a sta-
Fig. 1. (a) Schematic representation of the physical model, (b) computationaldomain, and (c) close up view of the grid in the vicinity of cylinder.

tionary cylinder and that of Newtonian fluids past a rotatingcylinder.

2. Previous work

As noted earlier, since a succinct summary of the pertinent lit-erature on the flow of Newtonian fluids over a rotating cylinderis available in numerous references, e.g., see [6,8], only the salientfeatures are recounted here. For the simplest case of an unconfinedcylinder, 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 tohigh Reynolds numbers at which the flow field is three-dimensional(e.g., see [9–12]), since the present work is concerned with thesteady flow regime, the ensuing discussion is limited to the steadyflow regime. At Re = 46–47, the flow becomes unsteady even for a

1444 S.K. Panda, R.P. Chhabra / J. Non-Newtonian Fluid Mech. 165 (2010) 1442–1461

Table 1Selection of optimum domain.

H/D CDP CDF CD CLP CLF CL

Re = 0.1, n = 1, ˛= 6120 31.404 32.805 64.209 −13.230 −8.469 −21.698220 29.290 30.600 59.890 −12.304 −7.860 −20.164240 29.078 30.371 59.450 −12.218 −7.806 −20.024

Re = 0.1, n = 0.2, ˛= 6120 132.154 16.382 148.536 −22.580 0.141 −22.438220 132.050 16.370 148.42 −22.427 0.155 −22.271240 132.049 16.371 148.42 −22.427 0.154 −22.273

Re = 5, n = 1, ˛= 6100 −3.390 3.150 −0.240 −20.955 −4.575 −25.530140 −3.389 3.149 −0.239 −20.953 −4.574 −25.527160 −3.389 3.149 −0.240 −20.953 −4.574 −25.527

5.274 −7.893 −0.043 −7.9365.273 −7.892 −0.043 −7.9355.274 −7.892 −0.043 −7.935

sretHofflscdaaseNt

alhnrrid

Table 2Details of grids used for grid independence study.

Grid Ni ı/D Ncells

H/D = 220G1 100 0.005 117,600G2 200 0.005 151,100G3 400 0.005 218,100

H/D = 100

TE

Re = 5, n = 0.2, ˛= 6100 5.089 0.185140 5.088 0.185160 5.089 0.185

tationary cylinder, i.e., ˛= 0. Broadly speaking, small values of theotational velocity ˛ do not seem to influence this transition. How-ver, once the value of˛ exceeds a critical value (˛1) for Re > 46–47,he tendency for vortex shedding is somewhat suppressed [13].owever, subsequent more detailed studies [6,8] also point to theccurrence of two more transitions. At high Reynolds numbers andor ˛>˛2, the flow again becomes unsteady and finally, again theow reverts to a steady flow regime at ˛>˛3 at Re = 200, albeit theecond transition ˛2 ≤˛≤˛3 occurs over a rather narrow range ofonditions. As expected, for a fixed value of the Reynolds number,rag and lift coefficients decrease with the increasing value of˛ [14]nd so does the rate of heat transfer [6,15] in the range 5 ≤ Re ≤ 166nd ˛≤ 4. In contrast to the aforementioned studies based on theolution of the complete governing equations, Kendoush [16] hasmployed the boundary layer approximation to predict the value ofusselt number for a rotating cylinder. All in all, the flow is known

o 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

stationary cylinder is not only of recent vintage, but is also muchess extensive than that for Newtonian fluids. Most of these studiesave been reviewed recently [17,18]. Broadly, reliable results are
ow available on the momentum transfer characteristics for a non-otating cylinder in unconfined power-law fluids in the steady flowegime [19–23] for both shear-thinning and shear-thickening flu-ds. At low Reynolds numbers, shear-thinning behaviour enhancesrag above its Newtonian value and as expected, shear-thickening
able 3ffect of grid details on the results at Re = 0.1 (H/D = 220) and Re = 5 (H/D = 100).

Grid CDP CDF CD

Re = 0.1, n = 1, ˛= 6G1 29.318 30.618 59.936G2 29.290 30.600 59.890G3 29.289 30.586 59.875

Re = 0.1, n = 0.2, ˛= 6G1 132.130 16.380 148.510G2 132.050 16.370 148.420G3 132.048 16.366 148.414

Re = 5, n = 1, ˛= 6G1 −3.380 3.141 −0.232G2 −3.391 3.150 −0.240G3 −3.392 3.151 −0.241

Re = 5, n = 0.2, ˛= 6G1 5.095 0.1850 5.280G2 5.090 0.1855 5.275G3 5.082 0.1858 5.266

G1 100 0.005 86,496G2 200 0.005 151,100G3 400 0.005 178,896

has the opposite effect. However, it is appropriate to add here thatwhether the drag is reduced or augmented with respect to theNewtonian value is also linked to the choice of the characteristic vis-cosity used to define the Reynolds number [24]. Furthermore, therole of power-law index gradually diminishes with the increasingvalue of the Reynolds number. Similarly, heat transfer is facili-tated in shear-thinning fluids both in forced and mixed convectionregimes [25–27]. Qualitatively, similar trends are observed for acylinder confined symmetrically in between two parallel walls
[28,29]. Finally, there has been only one set of study on the flowand heat transfer phenomena from a cylinder submerged in power-law fluids in the laminar vortex shedding regime [17,18]. For thesake of completeness, it is worthwhile to add here that scant resultsare also available for square [30,31] and elliptic cylinders [32,33]
CLP CLF CL

−12.37 −7.907 −20.276−12.304 −7.86 −20.164−12.285 −7.845 −20.13

−22.491 0.153 −22.338−22.427 0.155 −22.271−22.416 0.156 −22.261

−20.920 −4.560 −25.480−20.959 −4.575 −25.531−20.960 −4.577 −25.537

−7.892 −0.046 −7.930−7.893 −0.044 −7.936−7.893 −0.044 −7.937


Table 4Comparison between the present and literature results for a rotating cylinder in Newtonian fluids.

Source Re CDP CDF CD CLP CLF CL

5

˛= 1Present 1.9531 1.9043 3.8574 −2.315 −0.6076 −2.922Stojkovic et al. [8] – – 3.8013 – – 2.9022Badr et al. [15] – – 3.81 2.29 0.55 2.84

˛= 6Present – – – −20.9557 −4.575 −25.5308Stojkovic et al. [8] – – – – – 25.4764

20

˛= 1Present 1.01 0.8312 1.8412 −2.387 −0.3614 −2.7483Paramane and Sharma [6] 1 0.8368 1.8368 −2.3634 −0.357 −2.7204Stojkovic et al. [8] – – – 2.424 0.3501 2.7741Badr et al. [15] – – 1.91 2.44 0.35 2.79

˛= 6Present −2.734 2.4227 −0.3112 −28.86 −2.9412 −31.8003Paramane and Sharma [6] −2.791 2.4467 −0.3443 −28.8764 −2.9314 −31.8078Stojkovic et al. [8] – – – 28.86 2.9461 31.806

40

˛= 1Present 0.7774 0.5458 1.3233 −2.3345 −0.2645 −2.599Paramane and Sharma [6] 0.775 0.54 1.315 −2.3405 −0.2608 −2.6013

˛= 6

soeshattcefiuvvtroioovbtr

TCs

Present −2.0841 2.0661Paramane and Sharma [6] −1.9 2.117

ubmerged in confined and unconfined power-law fluids, mostf which are based on the assumption of the steady flow regimexcept for the recent results reported in [34] on the flow over aquare cylinder in the laminar vortex shedding regime. On the otherand, a vast literature exists on the flow of visco-elastic fluids pastnon-rotating circular cylinder [24]; however, most of it relates to

he creeping (zero Reynolds number) flow regime and endeavourso elucidate the role of fluid visco-elasticity on the fluid mechani-al aspects in the absence of shear-thinning behaviour. While mostxperimental studies report significant changes in the detailed floweld which are attributed to visco-elasticity, but the numerical sim-lations predict only minor changes. At low Reynolds numbers,isco-elasticity reduces the drag on a cylinder below its Newtonianalue and it increases the drag at high Reynolds numbers. Therefore,he literature is inundated with conflicting reports regarding theole of elasticity in this case. As far as known to us, there have beennly two studies on the flow of non-Newtonian fluids past a rotat-ng cylinder. Townsend [35] numerically studied the uniform flowf an Oldroyd model fluid past a rotating cylinder over the rangef conditions as ˛≤ 5 and 10 ≤ Re ≤ 60. His results suggest that the
isco-elasticity tends to increase both the drag and lift experiencedy a rotating cylinder whereas the shear-thinning behaviour tendso reduce their values. On the other hand, Christiansen [36,37] haseported detailed experimental data on the local velocity field for
able 5omparison between the present and literature values for a stationary cylinder inhear-thinning fluids (n = 0.2).

Source Re ˛= 0

CDP CDF CD

Present0.1

201.0391 69.5039 270.5431Sivakumar et al. [41] 195.1465 72.3863 267.5328

Present1

20.0084 7.0551 27.0636Sivakumar et al. [41] 19.6615 7.2447 26.9062

Present10

2.4251 0.7318 3.1570Sivakumar et al. [41] 2.4388 0.7178 3.1566

Present40

0.9915 0.1450 1.1365Sivakumar et al. [41] 0.9954 0.1443 1.1397

−0.018 −32.894 −2.2621 −35.15620.217 −32.9103 −2.248 −35.1583

a cylinder rotating in Newtonian and polymer solutions of varyinglevels of shear-thinning and visco-elasticity.

It is thus safe to conclude that very little is known about theflow of non-Newtonian fluids past a rotating cylinder. Admittedly,complex fluids exhibit a range of rheological features such as shear-thinning, visco-elasticity and yield stress, it seems reasonable tobegin with the simplest type of non-Newtonian behaviour, namely,shear-thinning viscosity and the level of complexity can graduallybe built up to incorporate other aspects of real fluid behaviour ina systematic manner. This work is thus concerned with the flow ofpower-law fluids past a rotating cylinder over the range of condi-tions Re ≤ 40, ˛≤ 6 and 0.2 ≤ n ≤ 1 over which the flow is expectedto be steady and two-dimensional. In particular, extensive numer-ical results are presented elucidating the effect of rheological anddynamic parameters on the detailed kinematics of flow and on dragand lift coefficients. Finally, the present predictions are comparedwith the experimental results available in the literature [36,37].

3. Problem statement and governing equations

Let us consider the incompressible steady flow of a power-law fluid (with uniform velocity U0) past a cylinder of radius R ordiameter D (infinitely long in z-direction) rotating with an angularvelocity of ˝ in the anti-clockwise direction, as shown schemati-cally in Fig. 1(a). Since it is not possible to simulate numerically trulyan unconfined flow, it is customary to introduce an artificial domainin the form of a box as shown schematically. Following the previousstudies [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 thatit does not unduly influence the flow field and at the same time itnecessitates only modest computational resources. The flow phe-nomenon is described by the continuity and momentum equationswritten in their compact forms as follows:

• Continuity:

∇.V = 0 (1)

• Momentum:

�DV

Dt= −∇p+ ∇ · � + �g (2)


the cy

wv

�

�

Fig. 2. Streamline contours near

here �, V, p, � and g, respectively, are the fluid density, velocityector, 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
ij = 2�εij (3)

linder for n = 0.2 for ˛= 1 and 6.

where the rate of deformation tensor is given by

εij = 12

(∂Ui∂x

+ ∂Uj∂x

)(4)

j i

And the scalar viscosity function, �, for a power-law fluid is givenby

� = m(2I2)(n−1)/2 (5)


the cy

wrs

wbu

Fig. 3. Streamline contours near

here n is the so-called power-law index. Evidently, n = 1 cor-esponds to the Newtonian fluid behaviour and n < 1 denotes
hear-thinning behaviour.
It is customary to introduce dimensionless variables. In thisork, the free stream velocity, U0, and diameter of cylinder, D, have

een used as scaling variables. Thus, the pressure has been scaledsing �U2

0 , stress components using m(U0/D)n, time with D/U0, etc.


The physically realistic boundary conditions for this flow arewritten as follows:

(1) At the inlet plane: It is the uniform flow in x-direction, i.e.,

Ux = U0, Uy = 0


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


he cy

(

Fig. 5. Vorticity contours near t

2) The top and bottom walls are assumed to be slip boundaries sothat there is no dissipation at these walls. The corresponding
mathematical description is given by:
∂Ux∂y

= 0; Uy = 0


(3) On the surface of the solid cylinder: The standard no-slip bound-ary condition is used, i.e.,

Ux = −˛ sin � and Uy = −˛ cos �

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


he cyl
Fig. 6. Vorticity contours near t
in this work. This choice implies that the conditions of the out-flow plane are extrapolated from within the domain and as such
have negligible influence on the upstream flow conditions. Theextrapolation procedure used by FLUENT updates the outflowvelocity and the pressure in a manner that is consistent with thefully developed flow assumption, when there is no area changeat the outflow boundary. However, the gradients in the cross-
inder for n = 0.6 for ˛= 1 and 6.

stream direction may still exist at the outflow boundary. Also,the use of this condition obviates the need to prescribe a bound-
ary condition for pressure. This is similar to the homogeneousNeumann condition, that is,
∂Ux∂x

= 0 and∂Uy∂x

= 0


the cy

tflpftpodf

Fig. 7. Vorticity contours near

The numerical solution of the governing Eqs. (1) and (2)ogether with the aforementioned boundary conditions maps theow domain in terms of Ux, Uy and p which, in turn, can beost processed to evaluate the derived quantities such as streamunction, vorticity, drag and lift coefficients as functions of the per-
inent governing parameters, namely, the Reynolds number (Re),ower-law index (n) and the non-dimensional rotational velocityf the cylinder (˛). At this juncture, it is appropriate to intro-uce the pertinent definitions of some of these parameters asollows:
linder for n = 1 for ˛= 1 and 6.

• Reynolds number (Re)

Re = �U2−n0 Dn

m

• Rotational velocity (˛)

˛ = R˝

U0


les on

•

Fig. 8. Representative pressure profi

Drag coefficient (CD)

CD = 2FD

�U20D

the surface of the rotating cylinder.

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


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


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


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

CD

Re ˛= 0 ˛= 2

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

0.1 61.1275 100.8522 164.2520 231.3182 270.5431 60.9866 89.9372 133.9509 176.8087 199.8191 10.5663 13.2468 17.4697 23.3118 27.0635 10.1578 11.7208 14.2309 17.8904 20.2951

10 2.7983 2.8609 2.9348 3.0268 3.1570 2.1340 2.2173 2.3359 2.5245 2.643640 1.5136 1.4378 1.3455 1.2393 1.1365 0.8453 0.8517 0.8592 0.8798 0.9115

Re ˛= 4 ˛= 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

0.1 60.5715 84.4540 119.1578 151.0860 169.8143 59.8808 80.6602 108.9539 133.0632 148.4261 8.9340 10.8012 12.9498 15.7119 17.3883 6.9582 10.0769 12.5536 15.0304 16.6408

10 0.5891 1.3992 2.2384 2.9974 3.3288 −0.7747 1.5118 2.8136 3.3974 3.526940 −0.1440 0.3397 0.7686 1.0976 1.3608 −0.0180 0.6089 0.9679 1.2528 1.3297

CL

Re ˛= 0 ˛= 2

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

0.1 0 0 0 0 0 −6.6968 −3.0726 −2.6343 −3.9018 −7.20121 0 0 0 0 0 −6.1190 −4.0993 −3.0404 −3.8614 −6.6750

10 0 0 0 0 0 −5.9608 −5.3654 −4.8587 −4.5096 −4.595040 0 0 0 0 0 −5.7205 −5.3832 −5.0545 −4.7375 −4.4573

Re ˛= 4 ˛= 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

45270910

•

dff

4

arseudistticoot

0.1 −13.4117 −6.3120 −5.6369 151.086 −14.191 −12.6919 −8.3219 −6.2170 −7.6299 −11.90

10 −14.5583 −12.7017 −10.7257 −8.6285 −7.5040 −16.1329 −15.2928 −14.1065 −12.4331 −10.76

Lift coefficient (CL)

CL = 2FL

�U20D

where FL is the lift force acting in the y-direction on the cylinderper unit length. Here also, it is not uncommon to split the liftforce into two components due to pressure and shear as CLP andCLF respectively.

The scaling of the governing equations and the boundary con-itions suggests the integral flow parameters, CD and CL to beunctions of the Reynolds number, power-law index and ˛. Thisunctional relationship is explored and developed in this work.

. Numerical solution method

Since detailed descriptions of the numerical solution procedurere available elsewhere [17,18,28], only the salient features areecapitulated here. In this study, the field equations have beenolved using FLUENT (version 6.2.26). The structured quadrilat-ral cells of uniform and non-uniform grid spacing were generatedsing the commercial grid tool GAMBIT (version 2.3.16). The two-imensional, laminar, segregated solver was used to solve the

ncompressible flow on the collocated grid arrangement. Bothteady and unsteady solvers have been used in this study. Whilehe major thrust of this work is on the steady flow regime, limitedime-dependent simulations for extreme values of the govern-
ng parameters such as Re = 40, n = 0.2 and ˛= 6, etc. were alsoonducted to ascertain that the flow regime indeed was steadyver the range of conditions covered in this study. The secondrder upwind scheme has been used to discretize the convectiveerms in the momentum equations. The semi-implicit method
−20.1638 −9.6371 −8.9071 −12.8874 −22.269−20.2742 −12.855 −9.371 −10.935 −15.432−28.6372 −22.633 −15.137 −10.080 −7.9919−35.1562 −25.557 −19.400 −15.069 −12.260

for the pressure linked equations (SIMPLE) scheme was used forsolving the pressure–velocity coupling. The constant density andnon-Newtonian power-law viscosity modules were used to inputthe physical properties of the flow. However, the input values ofthe physical properties �, m, n and kinematic parameters like D,U0,˝, etc. are of no consequence as the final results are reported ina dimensionless form. FLUENT solves the system of algebraic equa-tions using the Gauss–Siedel (G–S) point-by-point iterative methodin 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 CPUtime required to obtain a converged solution, particularly when themodel contains a large number of control volumes. Relative conver-gence criteria of 10−8 for the residuals of the continuity and x- andy-component of the momentum equations were prescribed in thiswork. Furthermore, a simulation was deemed to have convergedwhen the values of the global parameters had stabilized to at leastfour significant digits.

5. Choice of numerical parameters

Undoubtedly, the reliability and accuracy of the numericalresults is strongly influenced by the choice of domain size (H), gridcharacteristics (number of cells on the surface of the cylinder, gridspacing, stretching, etc.) and to some extent by the convergence cri-terion, etc. In this work, the values of these parameters have beenselected after extensive exploration. The values of (H/D) rangingfrom 80 to 240 have been employed here to arrive at an optimal
value of this parameter. Intuitively, it appears that a larger domainis required at low Reynolds numbers than that at high Reynoldsnumbers. This is so simply due to the slow spatial decay of thevelocity field at low Reynolds numbers. Therefore, the domain inde-pendence study has been carried out at the lowest value of Re = 0.1


alized

uau(gtn

crrptopt

Fig. 11. Effect of power-law index on the norm

sed in this study. After a detailed examination of the results (dragnd lift coefficients), suffice it to say here that, the value of (H/D)sed in this work is 220 at low Reynolds numbers (Re < 5), whereasH/D) = 100 was used for Re ≥ 5 for the domain effects to be negli-ible as can be seen in Table 1. These findings are consistent withhat reported by others, e.g., see [6] over this range of Reynoldsumbers.

Having fixed the value of (H/D), an optimal grid should meet twoonflicting requirements: namely, it should be sufficiently fine toesolve the thin boundary layers and the steep gradients near theotating cylinder, and of course without being prohibitively com-
utation intensive. For this purpose, the relative performance ofhree grids has been studied in detail to arrive at the choice of anptimal grid. Each grid was characterized in terms of the number ofoints (Ni) on the surface of the cylinder and the value of (ı/D) nearhe cylinder, as summarized in Table 2. A typical grid is shown in
values of CD and CL with ˛ for Re = 0.1 and 40.

Fig. 1(c). For the same combinations of the values of Re, n and ˛ asthat used in the domain independence study, Table 3 summarizesthe relative performance of the three grids in terms of the resultingvalues of the individual components of drag and lift coefficients.A detailed examination of these results reveals that very little isgained in terms of accuracy of the present results as one movesfrom G2 to G3 whereas the CPU time required for G3 is many foldslarger than that needed for G2 to satisfy the same criterion of con-vergence. Thus, grid G2 denotes a good compromise between theaccuracy and computational efforts and hence all results reportedherein are based on the use of grid G2.

Finally the adequacy of the values of (H/D) and grid G2 is fur-ther demonstrated in the next section by benchmarking the presentresults against the literature values for a few limiting cases likefor a rotating cylinder in Newtonian media, and for a non-rotatingcylinder in power-law fluids.


F 6% aq

6

saHitr

6

flfitrasb

ig. 12. Comparison between the predicted and experimental velocity profiles for 8

. Results and discussion

As noted earlier, the present flow is governed by five dimen-ionless parameters, namely, CD, CL, Re, n and ˛ and the principalim of the present study is to develop this functional relationship.owever, prior to embarking upon the presentation of new results,

t is useful to demonstrate the adequacy of the numerical computa-ions which will also help ascertain the accuracy of the new resultseported herein.

.1. Validation of results

Since detailed comparisons are available elsewhere for theow of Newtonian fluids past a stationary cylinder [22,23], suf-ce it to say here that the present results are within ∼1–2% of
he literature values. Next, Table 4 presents a comparison for aotating cylinder in Newtonian fluids; both the values of dragnd lift coefficients are included here for ˛= 1 and 6. Broadlypeaking, the present results are within 1–2% of that reportedy Stojkovic et al. [8] and Paramane and Sharma [6] for Re ≥ 1,
ueous glycerin solution. Re = 0.411, n = 1: (a) ˛= 0, (b) ˛= 1.18, and (c) ˛= 1.18.

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 andRe = 40, while the present values of CDP and CDF are in excel-lent agreement with that reported in Ref. [6], but the total dragvalues differ by an order of magnitude. This is so due to tworeasons: both CDP and CDF are of similar magnitudes but of oppo-site sign. Secondly, the values of CDP and CDF corresponding toRef. [6] have been read off their figures. Thus, even a small errorincurred in extracting these values gets accentuated in this case.While this may seem like a large difference, it needs to be empha-sized here that the differences of this magnitude are not at alluncommon in such numerical studies due to underlying vari-ations in different studies arising from the choices of solutionmethodologies, 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 cylin-der in shear-thinning fluids. While more extensive comparisonsare reported elsewhere [23], it is fair to say that the present resultsare seen to be in excellent agreement with the literature values inTable 5.


F cellosa

flstNrthft

6

ott

ig. 13. Comparison between the predicted and experimental velocity profile for 1%nd (c) ˛= 1.706.

Aside from the aforementioned benchmark comparisons, theow of Newtonian and power-law fluids was also studied in thetandard lid driven square cavity. The present results for the cen-erline velocities were found to be within ±2% of the correspondingewtonian results [39] and within ±2.5% for power-law fluids as

eported by Neofytou [40]. These comparisons lend further supporto the credibility of the numerical solution methodology employederein. Based on the above-noted comparisons, the present results

or a cylinder rotating in power-law fluids are believed to be reliableo within ±2%.

.2. Detailed kinematics

Figs. 2–4 show representative results elucidating the influencef Re, ˛ and n on the streamlines in the vicinity of the cylinder. Forhe range of Reynolds number encompassed here, the flow is knowno be steady for all values of the power-law index at ˛= 0, i.e., for a

ize hydroxyethyl cellulose (HEC) solution. Re = 0.079, n = 0.43: (a)˛= 0, (b)˛= 1.706,

non-rotating cylinder [41]. As expected, the flow field is symmetricabout the mid-plane, and at low Reynolds numbers (Re ≤ 1), it alsoexhibits fore and aft symmetry. This is so due to the fact that theviscous forces far outweigh the inertial forces under these condi-tions. However, as the Reynolds number is gradually increased, theflow detaches itself from the surface of the cylinder which leadsto the formation of the wake region in the rear of the cylinder.This marks the formation of a pair of standing vortices which growin size with the increasing Reynolds number up to a critical valueof the Reynolds number. For Newtonian fluids, the first signatureof flow separation is seen at about Re = 4–5 whereas this transi-tion 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) fora non-rotating cylinder [41]. A detailed examination of Figs. 2–4reveals the following overall trends: As soon as the rotation of thecylinder is superimposed on the uniform flow, the flow becomesasymmetric as can clearly be seen even at Re = 0.1, albeit there is no

tonian

ip(ahdiiORvrcpewaaadieiNltnbs˛caFaaptiusastdR

ssbtnTdWtnifo

ostag(

S.K. Panda, R.P. Chhabra / J. Non-New

ndication of flow separation at this stage. In fact, a distinct saddleoint is seen to form in Newtonian and mildly shear-thinning fluidsn = 0.6) which disappears in highly shear-thinning fluids (such ast n = 0.2). This is simply due to the rapid decay of the flow field inighly shear-thinning fluids and it is almost tantamount to a cylin-er rotating in a cavity formed by low-viscosity fluid which itself

s surrounded by a high viscosity fluid, akin to the rotation of anmpeller in a highly shear-thinning and/or viscoplastic fluid [42].wing to the weak advection, this effect is particularly striking ate = 0.1, where no saddle point is seen in the streamlines plot. Theiscous effects diminish with the increasing Reynolds number, orotational velocity (due to enhanced levels of deformation), or aombination of both. Thus, as the value of the Reynolds number isrogressively increased, one can see the formation of a streamlinenclosing the cylinder at lower and lower values of ˛. For instance,hile this phenomenon occurs at˛= 6 for a weak uniform flow such

s that at Re = 1 for n = 0.2, it was found to occur at ˛= 4 for Re = 10nd n = 0.2. Similarly, the saddle point is seen to shift a little in thenti-clockwise direction with the increasing Reynolds number, orecreasing value of power-law index. For Newtonian fluids, at van-

shingly small values of the Reynolds numbers (Re → 0), while theffect of rotation and uniform flow is easily delineated by super-mposing the two individual solutions due to the linearity of theavier–Stokes equations. This is clearly not possible for power-

aw fluids due to the non-linearity of the viscous term even whenhe non-linear inertial terms are altogether neglected. As Reynoldsumber is gradually increased, the inertial effects progressivelyecome important. While for ˛= 0, the flow field is found to beymmetric about the mid-plane, but with the increasing value of, the symmetry is lost and the upper vortex detaches from theylinder (e.g., see Fig. 3 for Re = 10). Similarly, the lower vortexlso dislodges itself and eventually disappears with increasing ˛.urthermore, the streamlines near the cylinder are drawn closelyround it until a ring of trapped fluid is formed which rotateslong with the cylinder. This behaviour is seen for all values of theower-law index considered herein (1 ≥ n ≥ 0.2), except for the facthat the size of the so-called separatrix [6] is seen to be smallern shear-thinning fluids than that in Newtonian fluids otherwisender identical conditions. Thus, in summary broadly, qualitativelyimilar streamline patterns are seen in power-law shear-thinningnd Newtonian fluids in the steady flow regime. Typical value of thetream function ( * = /U0D) range from ∼10−8 m2/s at the bot-om of the box (y = −H/2) to 4000 m2/s at top of the box (y = H/2)epending upon the values of n, Re, ˛. Naturally, lower the value ofe, smaller is the value of * near the cylinder.

Representative vorticity contours are shown in Figs. 5–7 for theame combinations of Re, n and ˛ which also reinforce the trendseen in streamline patterns. For instance, at low Reynolds num-ers, the vorticity contours become increasingly asymmetric ashe value of ˛ is increased. Indeed, larger is the value of Reynoldsumber, lower is the value of ˛ needed to break the symmetry.he crowding of the iso-vorticity contours underneath the cylin-er show the intensity of the vorticity distribution in this region.ith the increasing Reynolds number, the vortices are seen to be

ransported in the main direction of flow. The surface vorticity isaturally maximum on the surface of the cylinder (ω* =ω/U0D) and

t ranges from ∼1 to ∼220 at Re = 0.1 as the value of n is decreasedrom n = 1 to n = 0.2. Of course, it decays to very small values of therder of 10−12 to 10−7 far away from the cylinder.

Fig. 8 shows representative surface pressure profiles for a rangef combinations of Re, ˛ and n. Obviously, since the flow is not
ymmetric about the x- or y-axis, pressure profiles are shown overhe entire surface (0 ≤ �≤ 360). At low Reynolds numbers, e.g.,t Re = 0.1, the surface pressure in the front of the cylinder pro-ressively increases with the increasing degree of shear-thinningdecreasing values of n), albeit the effect is most dramatic at the
Fluid Mech. 165 (2010) 1442–1461 1459

front stagnation point. In the rear of the cylinder, the influence ofpower-law index is not monotonic on the surface pressure. Withthe superimposition of rotation, the pressure is no longer maximumat the front stagnation point and it occurs at about � = 20–30◦. Atlow Reynolds numbers, as the value of ˛ is gradually increased,the surface pressure profile exhibits increasing asymmetry. Simi-larly, for a fixed value of ˛, as the Reynolds number is graduallyincreased, the surface pressure drops progressively, and the effectof 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 signchange at lower Reynolds numbers. The complex dependence seenin Fig. 8 obviously stems from different scaling of the viscous andinertial forces on velocity and power-law index. How these changesat the microscopic level influence the values of macroscopic quan-tities like drag and lift coefficient will be seen in the next section,as one would expect the pressure drag to be negative under certaincircumstances.

6.3. Drag and lift coefficients

Figs. 9 and 10 show the influence of ˛, Re and n on pressuredrag coefficient, total drag coefficient and lift coefficient. At lowReynolds numbers, both total drag coefficient and its pressure com-ponent are seen to decrease with the increasing value of rotationalvelocity which is consistent with the results of Stojkovic et al. [8]and Paramane and Sharma [6] for Newtonian fluids (n = 1). Whileat low Reynolds numbers, the rotation of the cylinder augmentsboth the total drag and its pressure component, but it is evidentthat the pressure drag contributes to the total drag significantlyunder these conditions. On the other hand, at high Reynolds num-bers, Re = 40, the dependence of CD and CDP on ˛ is more complex.For Newtonian fluids, the total drag decreases with the increasingvalue of ˛ up to about ˛� 5 beyond which it shows an increase.It is interesting to see that under certain conditions both CD andCDP can be negative which is consistent with the pressure profilesseen in the previous sections due to a very small contribution offriction component. Shear-thinning fluids are also seen to exhibitqualitatively similar behaviour, except for the fact that while insome cases the value of CDP can become slightly negative but thetotal drag always remains positive thereby suggesting that frictionprobably contributes in greater proportion here than that in thecase of Newtonian fluids. Indeed, smaller is the value of power-lawindex (highly shear-thinning fluid), higher is the total drag coeffi-cient. Fig. 10 shows these results in a slightly different form whereit is clearly seen that the effect of power-law index on both CDP andCD gradually weakens with the increasing Reynolds number. Forinstance, for ˛= 1, the curves for different values of n collapse ontoa single curve at about Re ∼ 1–2, whereas at ˛= 6 this behaviouroccurs at about Re � 40.

The lift coefficient is seen to be negative (i.e. acting in the down-ward direction) at all values of the Reynolds number consideredin this work and it decreases almost linearly with the increasingrotational velocity. At low Reynolds numbers, it goes through amaximum value at about n � 0.6, as can be seen clearly in Fig. 9.At low Reynolds numbers, the rate of change also increases withthe increasing degree of shear-thinning behaviour, i.e., decreasingvalue of n. On the other hand, at Re = 40, the effect of n is ratherinsignificant up to about˛= 2–3 and beyond this value, the lift coef-ficient increases with the decreasing value of n. While these resultsare in line with that of Stojkovic et al. [8] for Newtonian fluids, but
these are at variance from that of Townsend [35] who predictedthe lift to increase with ˛ to a maximum value, but then it fallssteeply. This is an unusual result which has been attributed to theinadequacy of their computations [8]. A representative summaryof the present numerical values of drag and lift coefficients as func-

1 tonian

ttaovb

Rticoiso˛

dt

7

diriTnaTeas2Hifpadtswrtvrnrntlp

8

abp(asv

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

460 S.K. Panda, R.P. Chhabra / J. Non-New

ions of Re, n and˛ is given in Table 6. Intuitively, one would expecthe product of drag coefficient and Reynolds number to approachlimiting value depending upon the values of n and ˛ as the valuef the Reynolds number diminishes, but unfortunately the lowestalue of Re = 0.1 used here is not sufficiently small for this limitingehaviour to be observed in the present case.

Finally, Fig. 11 shows the influence of the power-law index,eynolds number, and ˛ on the normalized values of CD, CDP and CLo delineate the role of power-law index in an unambiguous fash-on. As noted earlier, at low Reynolds numbers, the total drag on aylinder can be as much as 3–4 times of that in a Newtonian fluidtherwise under identical conditions; however, this effect dimin-shes with the increasing value of˛. On the other hand, this effect iseen to be even more dramatic at Re = 40. In the extreme case, notnly CD can undergo a change of sign depending upon the values ofand n, but their ratio too can be as high as −70.In contrast, the lift coefficient is seen to display a more regular

ependence on n and ˛, both at high and low Reynolds numbers,hough it peaks at about n � 0.6 at low Reynolds numbers.

. Comparison with experiments

As noted earlier, while no experimental results are available onrag and lift coefficients for a cylinder rotating in Newtonian and

n power-law fluids in the steady flow regime, Christiansen [36,37]eported detailed velocity measurements for a rotating cylindern a Newtonian glycerol solution and in two polymer solutions.wo test cylinders 102 mm long, but of two different diameters,amely, 1.59 mm and 4.76 mm were used. The (L/D) ratios are ∼22nd ∼65, which are modest for the end effects to be negligible.he two polymer solutions used in their study exhibited visco-lastic behaviour, with relaxation times of 0.041 s (HEC solution)nd 8.6 s (Separan solution). The value of power-law index for botholutions is identical, as n = 0.43. The values of ˛ varied from 0 to.29 and of the Reynolds number are in the range 0.03 to ∼0.5.e reported the values of the x- and y-component of the veloc-

ty along (x = 0) and (y = 0) lines both upstream and downstreamrom the cylinder. Fig. 12 shows a comparison between the presentredictions and the experimental data for their Newtonian fluidt Re = 0.411 for a stationary cylinder (˛= 0) and a rotating cylin-er (˛= 1.18). An excellent agreement is seen to exist betweenhe present numerical results and the experimental results. Fig. 13hows a similar comparison for the weakly elastic HEC solutionherein the correspondence is though less good, but may still be

egarded as satisfactory thereby suggesting this solution to be vir-ually inelastic over this range of conditions. The correspondingalue of Deborah number, defined as De =�U0/D, is 0.32 which isather low and therefore, visco-elastic effects are expected to beegligible here. On the other hand, the value of Deborah numberanges from 48 to 150 for the Separan solution and therefore, it isot appropriate to compare these results with the present predic-ions. Overall, the experimental verification seen in Figs. 12 and 13ends further credibility to the accuracy and reliability of the resultsresented herein.

. Concluding remarks

In this work, the flow of shear-thinning power-law fluids pastrotating cylinder in the 2D laminar steady flow regime has

een investigated numerically. The range of conditions encom-
assed include power-law index (1 ≥ n ≥ 0.2), Reynolds number0.1 ≤ Re ≤ 40) and rotational velocity (0 ≤˛≤ 6). The flow is visu-lized in terms of streamline contours, iso-vorticity profiles andurface pressure profiles for representative combinations of thealues of Re, n and ˛. The lift is found to be negative under most
[

[

Fluid Mech. 165 (2010) 1442–1461

conditions studied here, whereas the total drag force is usuallypositive but can be negative when the friction makes a little con-tribution and the pressure drag coefficient itself is negative such asthat at high values of ˛ and Re. The power-law index exerts a muchstronger influence on drag and lift at low Reynolds numbers thanthat at high Reynolds numbers. Indeed, shear-thinning behaviourcan augment drag values by up to a factor of ∼4 at low values ofRe and ˛, albeit the extent of increase decreases rapidly with theincreasing values of ˛. Finally, the present numerical predictions ofvelocity at x = 0 and y = 0 planes are shown to be in fair agreementwith the scant experimental results available in the literature forNewtonian and relatively inelastic polymer solutions.

References

[1] M.M. Zdravkovich, Flow Around Circular Cylinders, vol. 1: Fundamentals,Oxford University Press, New York, 1997.

[2] M.M. Zdravkovich, Flow Around Circular Cylinders, vol. 2: Applications, OxfordUniversity Press, New York, 2003.

[3] C.H.K. Williamson, Vortex dynamics in the cylinder wake, Annu. Rev. FluidMech. 28 (1996) 477–539.

[4] M. Coutanceau, J.R. Defaye, Circular cylinder wake configuration: a flow visu-alization survey, Appl. Mech. Rev. 44 (1991) 255–305.

[5] B.M. Sumer, J. Fredsoe, Hydrodynamics Around Cylindrical Structures, WorldScientific, Singapore, 2006.

[6] S.B. Paramane, A. Sharma, Numerical investigation of heat and fluid flow acrossa rotating circular cylinder maintained at constant temperature in 2-D laminarflow regime, Int. J. Heat Mass Transfer 52 (2009) 3205–3216.

[7] R.P. Chhabra, J.F. Richardson, Non-Newtonian Flow and Applied Rheology, 2nded., Butterworth-Heinemann, Oxford, 2008.

[8] D. Stojkovic, M. Breuer, F. Durst, Effect of high rotation rates on the laminarflow around a circular cylinder, Phys. Fluids 14 (2002) 3160–3178.

[9] Y.-M. Chen, Y.-R. Ou, A.J. Pearlstein, Development of the wake behind a circularcylinder impulsively started into rotary and rectilinear motion, J. Fluid Mech.253 (1993) 449–484.

10] M. Coutanceau, C. Menard, Influence of rotation on the near-wake develop-ment behind an impulsively started circular cylinder, J. Fluid Mech. 158 (1985)399–446.

11] H.M. Badr, S.C.R. Dennis, Time-dependent viscous flow past an impulsivelystarted rotating and translating circular cylinder, J. Fluid Mech. 158 (1985)447–488.

12] M.-H. Chou, Numerical study of vortex shedding from a rotating cylinderimmersed in a uniform flow field, Int. J. Num. Methods Fluids 32 (2000)545–567.

13] G. Hu, D. Sun, X. Yin, B. Tong, Hopf bifurcation in wakes behind a rotating andtranslating circular cylinder, Phys. Fluids 8 (1996) 1972–1974.

14] S. Kang, H. Choi, S. Lee, Laminar flow past a rotating circular cylinder, Phys.Fluids 11 (1999) 3312–3321.

15] H.M. Badr, S.C.R. Dennis, P.J.S. Young, Steady and unsteady flow past a rotatingcircular cylinder at low Reynolds numbers, Comp. Fluids 4 (1989) 579–609.

16] A.A. Kendoush, An approximate solution of the convection heat transfer froman isothermal rotating cylinder, Int. J. Heat Fluid Flow 17 (1996) 439–441.

17] V.K. Patnana, R.P. Bharti, R.P. Chhabra, Two dimensional unsteady flow ofpower-law fluid over a cylinder, Chem. Eng. Sci. 64 (2009) 2978–2999.

18] V.K. Patnana, R.P. Bharti, R.P. Chhabra, Two dimensional unsteady forced con-vection heat transfer in power-law fluids from a cylinder, Int. J. Heat MassTransfer 53 (2010) 4152–4167.

19] R.I. Tanner, Stokes paradox for power-law flow around a cylinder, J. Non-Newt.Fluid Mech. 50 (1993) 217–224.

20] M.J. Whitney, G.J. Rodin, Force–velocity relationship for rigid bodies translatingthrough unbounded shear-thinning power-law fluids, Int. J. Non-Linear Mech.36 (2001) 947–953.

21] R.P. Chhabra, A.A. Soares, J.M. Ferreira, Steady non-Newtonian flow past a cir-cular cylinder: a numerical study, Acta Mech. 172 (2004) 1–16.

22] A.A. Soares, J.M. Ferreira, R.P. Chhabra, Flow and forced convection heat transferin cross flow of non-Newtonian fluids over a circular cylinder, Ind. Eng. Chem.Res. 44 (2005) 5815–5827.

23] R.P. Bharti, R.P. Chhabra, V. Eswaran, Steady flow of power-law fluids across acircular cylinder, Can. J. Chem. Eng. 84 (2006) 406–421.

24] R.P. Chhabra, Bubbles, Drops and Particles in Non-Newtonian Fluids, 2nd ed.,CRC Press, Boca Raton, FL, 2006.

25] R.P. Bharti, R.P. Chhabra, V. Eswaran, Steady forced convection heat transferfrom a heated circular cylinder to power-law fluids, Int. J. Heat Mass Transfer50 (2007) 977–990.

26] A.T. Srinivas, R.P. Bharti, R.P. Chhabra, Mixed convection heat transfer from acylinder in power-law fluids: effect of aiding buoyancy, Ind. Eng. Chem. Res. 48(2009) 9735–9754.

27] A.A. Soares, J. Anacleto, L. Caramelo, J.M. Ferreira, R.P. Chhabra, Mixed convec-tion from a circular cylinder to power-law fluids, Ind. Eng. Chem. Res. 48 (2009)8219–8231.

tonian

[

[

[

[

[

[

[

[

[

[[

[

[

S.K. Panda, R.P. Chhabra / J. Non-New

28] R.P. Bharti, R.P. Chhabra, V. Eswaran, Two-dimensional steady Poiseuille flowof power-law fluids across a circular cylinder in a plane confined channel: walleffect and drag coefficient, Ind. Eng. Chem. Res. 46 (2007) 3820–3840.

29] R.P. Bharti, R.P. Chhabra, V. Eswaran, Effect of blockage on heat transfer from acylinder to power-law liquids, Chem. Eng. Sci. 62 (2007) 4729–4741.

30] A.K. Dhiman, R.P. Chhabra, V. Eswaran, Flow and heat transfer across a confinedsquare cylinder in the steady flow regime: effect of Peclet number, Int. J. HeatMass Transfer 48 (2005) 4598–4614.

31] A.K. Dhiman, R.P. Chhabra, V. Eswaran, Steady flow of power-law fluids acrossa square cylinder, Chem. Eng. Res. Des. 84 (2006) 300–310.

32] P. Sivakumar, R.P. Bharti, R.P. Chhabra, Steady flow of power-law fluids acrossan unconfined elliptical cylinder, Chem. Eng. Sci. 62 (2007) 1682–1702.

33] R.P. Bharti, P. Sivakumar, R.P. Chhabra, Forced convection heat transfer froman elliptic cylinder to power-law fluids, Int. J. Heat Mass Transfer 51 (2008)1838–1853.

34] A.K. Sahu, R.P. Chhabra, V. Eswaran, Two dimensional laminar flow of a power-law fluid across a square cylinder, J. Non-Newtonian Fluid Mech. 165 (2010)752–763.

[

[

Fluid Mech. 165 (2010) 1442–1461 1461

35] P. Townsend, A numerical simulation of Newtonian and visco-elastic flowpast stationary and rotating cylinders, J. Non-Newtonian Fluid Mech. 6 (1980)219–243.

36] R.L. Christiansen, A laser-doppler anemometry study of visco-elastic flow pasta rotating cylinder, Ind. Eng. Chem. Fundam. 24 (1985) 403–408.

37] R.L. Christiansen, PhD Thesis, University of Wisconsin, Madison, WI, 1980.38] P.J. Roache, Perspective:, A method for uniform reporting of grid refinement

studies, J. Fluids Eng. 116 (1994) 405–413.39] U. Ghia, K.N. Ghia, C.T. Shin, High-Re solutions for incompressible flow using

the Navier–Stokes equations and a multigrid method, J. Comp. Phys. 48 (1982)387–411.

40] P. Neofytou, A 3rd order upwind finite volume method for generalized Newto-
nian fluid flows, Adv. Eng. Softw. 36 (2005) 664–680.
41] P. Sivakumar, R.P. Bharti, R.P. Chhabra, Effect of power law index on criticalparameters for power law fluid flow across an unconfined circular cylinder,Chem. Eng. Sci. 61 (2006) 6035–6046.

42] R.P. Chhabra, Fluid mechanics and heat transfer with non-Newtonian liquidsin mechanically agitated vessels, Adv. Heat Transfer 37 (2003) 77–178.

Documents

1-s2.0-S0377025710002028-main