Steady non–Newtonian flow pasta circular cylinder: a numerical study

R. P. Chhabra, Kanpur, India, and A. A. Soares and J. M. Ferreira, Vila Real, Portugal

Received May 19, 2004; revised June 7, 2004Published online: August 30, 2004 � Springer-Verlag 2004

Summary. The steady and incompressible flow of non–Newtonian fluids past a circular cylinder is

investigated for power law indices n between 0.2 and 1.4, blockage ratios of 0.037, 0.082 and 0.164, and the

Reynolds numbers Re of 1, 20 and 40, using a stream function/vorticity formulation. The governing field

equations have been solved by using a second-order accurate finite difference method to determine the drag

coefficient, wake length, separation angle and flow patterns, and to investigate their dependence on power

law index, blockage ratio and Reynolds number. The results reported here provide fundamental know-

ledge on the dependence of engineering flow parameters on blockage ratio and power law index, and

further show that the effects on stream line and iso-vorticity patterns which result from an increase in the

blockage ratio are similar to those which result from a decrease in the power law index.

1 Introduction

The flow of fluids past a circular cylinder represents a classical problem in fluid mechanics and

thus has received considerable attention in the literature. Thus, over the years, the flow of a

Newtonian fluid around circular cylinders has attracted a great deal of interest from the

experimental, analytical, and numerical point of view, e.g. [1]–[5] and references therein, and

indeed excellent reviews are available (e.g., [6]–[8]). An examination of these survey articles

shows that an adequate body of knowledge (especially on the prediction of gross engineering

parameters such as drag, wake characteristics, etc.) is now available on many aspects relating to

the flow of Newtonian fluids past a long circular cylinder under most conditions of interest,

albeit there are still some unresolved issues related to the detailed structure of the flow field. It is

readily acknowledged that many materials encountered in industrial practice show non–

Newtonian behavior [9]. Typical examples include polymer melts and their solutions, multi-

phase mixtures (suspensions, emulsions, foams) and soap solutions, etc. A knowledge of the

hydrodynamic forces experienced by submerged objects like spheres and cylinders is needed in

connection with the design of support structures and piers exposed to non–Newtonian muds, in

the use of wires and thin cylinders as measurement probes and sensors in non–Newtonian flows

and in the design of slurry pipelines where large particles are conveyed in a non–Newtonian

vehicle. Additional examples are found in polymer processing operations such as the use of

submerged surfaces to form weld lines. In addition to these potential applications, there is an

intrinsic theoretical interest in elucidating the role of non–Newtonian characteristics on the

structure of the flow field and on the drag for such highly idealized shapes like spheres and

cylinders. In spite of such overwhelming importance and frequent occurrence of non–Newtonian

fluids, there has been very little work reported on the non–Newtonian flow past a cylinder, even

Acta Mechanica 172, 1–16 (2004)

DOI 10.1007/s00707-004-0154-6

Acta MechanicaPrinted in Austria

for the simplest and possibly also the most common type of non–Newtonian behavior, namely,

shear thinning and shear thickening behavior. This type of fluid behavior is frequently modelled

using the simple two constant power law model. This work sets out to investigate the steady

two-dimensional flow of power law liquids past a circular cylinder. It is, however, instructive to

briefly summarize the previous scant results available in the literature prior to undertaking the

formulation of the present problem.

From a theoretical point of view there have been only very limited studies reported in the

literature. Thus, Tanner [10] demonstrated that in the limit of the zero Reynolds number there

is no Stokes paradox for shear thinning fluids whereas it was still relevant for shear thickening

fluids. This is clearly due to the fact that the effective shear rate in the fluid decreases rapidly

away from the cylinder. For shear thinning fluids, this implies a progressive increase in the

viscous forces whereas the viscous forces diminish for a shear thickening fluid. Subsequently a

similar inference was also drawn by others [11]. Tanner [10] was able to obtain approximate

analytical results for the creeping flow of power law fluids past a circular cylinder for shear

thinning fluids. He also supplemented these results with numerical predictions in the range

n ¼ 0:4��0:9; the correspondence was found to be reasonable. Subsequently, these results

have been extended to even smaller values of the power law index up to n ¼ 0:2 [12]. However,

all these results only relate to the zero Reynolds number and shear thinning fluid behavior.

D’Alessio and Pascal [13] numerically investigated the steady power-law flow around a cylinder

at Reynolds numbers Re ¼ 5, 20 and 40 using a first-order accurate difference method for a

fixed blockage ratio b = 0.037, where b is defined as the ratio between the cylinder diameter

and the distance H from the external boundary to the cylinder surface (Fig. 1). They investi-

gated the dependence of critical Reynolds number, wake length, separation angle and drag

coefficient on the power-law index. They reported that as the value of the Reynolds number was

progressively increased there was a decreasing degree of convergence which restricted the range

of power law indices for which a numerical solution was possible. Thus, for instance, for

Re ¼ 20, they were able to obtain fully convergent results only for weakly non–Newtonian fluid

behavior. Furthermore, their expression for the calculation of drag appears to be in error due to

an inadvertent omission of a factor of 2. Fortunately, this does not influence their other results

on wake characteristics, flow patterns, etc. However, it is also important to determine the

dependence of the aforementioned parameters on the distance from the cylinder surface to the

external numerical boundary (the blockage effect), since an increase in this distance approxi-

mates the conditions of flow in an infinite extent of fluid or, equivalently, decreases the wall

y

r = e

xcylinder

fieldequations

upstream downstream

Htransformed field equations

downstream

upstream

cylin

der

0 q

q

pe

ee•a b

Fig. 1. The real (x; y) plane and the computational (e; h) plane

2 R. P. Chhabra et al.

effects. Although the effect of blockage on flow parameters and/or stream line patterns is well

documented for Newtonian fluids (e.g., [14]–[16]) as well as for non–Newtonian viscoelastic

fluids, both numerically and experimentally in the creeping flow region (e.g., [17]–[18]), a

corresponding investigation for non–Newtonian viscoinelastic power law fluids is lacking. The

aim of the present study is to extend the work of D’Alessio and Pascal [13] using a more

accurate second-order finite difference method, more refined computational meshes and a

greater power law index range in order to investigate the effect of blockage on drag coefficient,

wake length, separation angle, and flow patterns (stream line and iso-vorticity contours) over

wide ranges of conditions.

2 Basic theory

From a theoretical viewpoint, the steady incompressible flow of a power law liquid past a long

circular cylinder is described by the continuity and momentum equations. The oncoming steady

incompressible flow is in the x-direction normal to the axis of the cylinder. Due to the infinite

extent of the cylinder axis along the z-direction, the flow is two-dimensional, i.e., no flow

variable depends upon the z-coordinate and mz ¼ 0. The equation of continuity and the r and hcomponents of the equations of motion in cylindrical coordinates [19] can be expressed in terms

of the polar coordinates (e; h) with e ¼ ln r=að Þ, giving the equation of continuity

1

ee

@

@eee @w@h

� �� @

@hwþ @w

@e

� �¼ 0; ð1Þ

the e-component

@w@h

@2w@e@h

� @w@eþ w

� �@2w

@h2þ @w@eþ w

� �¼ � 1

2

@p

@e� 2n

Ree�e @

@eeesrrð Þ þ @srh

@e� shh

� �; ð2:1Þ

and the h-component

� @w@h

@2w@e2þ @w@e

� �þ @w

@eþ w

� �@2w@e@h

¼ � 1

2

@p

@h� 2n

Ree�2e @

@ee2esrh� �

þ @shh

@h

� �; ð2:2Þ

where the scaled dimensionless stream function w, vorticity x and pressure p are related to their

dimensional counterparts as eewUa, e�exU=a and pU2q�

2, respectively. U is the flow velocity,

a the cylinder radius and q the fluid density. The dimensionless components sij of the extra

stress tensor are related to their dimensional counterpart through K Ua

� �nsij, and the Reynolds

number Re is defined as

Re ¼ q 2að ÞnU2�n

K; ð3Þ

where K denotes the power law consistency index and n the power law index. For a shear

thinning fluid n < 1, and n > 1 describes shear thickening behavior.

The constitutive equation for a power law fluid is written as

sij ¼ �geij; ð4Þ

where g is the dimensionless viscosity and eij are the dimensionless components of the rate-of-

deformation tensor (e.g., [19]).

The equation for the power law dimensionless viscosity is

g ¼ In�1

2

2 ; ð5Þ

Steady non–Newtonian flow past a circular cylinder 3

and I2 is the dimensionless second invariant of the rate-of-deformation tensor given as

I2 ¼ e�2e J2 þ 4@2w@h@e

� �2" #

; ð6Þ

with

J ¼ w� @2w@e2þ @

2w

@h2: ð7Þ

The vorticity in its scaled form is given as

@2w@e2þ @

2w

@h2þ 2

@w@eþ wþ x ¼ 0: ð8Þ

Eliminating the pressure in Eqs. (2) by the method of cross-differentiation and introducing the

vorticity x with some rearrangement leads to

g@2x@e2þ @

2x

@h2

� �þ 2k

@x@eþ 2l

@x@hþ cx ¼ F; ð9Þ

where

k ¼ @g@e� g� Re ee

2nþ1

@w@h

; ð10Þ

l ¼ @g@hþ Re ee

2nþ1

@w@eþ w

� �; ð11Þ

c ¼ �2@g@eþ gþ Re ee

2n

@w@h

; ð12Þ

F ¼ J@2g

@h2� @

2g@e2þ 2

@g@e

� �� 4

@2w@h @e

@g@h� @2g@h @e

� �: ð13Þ

Due to the two-dimensional nature of the problem (xy-plane) and since the oncoming flow is in

the x-direction, we only need to consider the region y � 0 and x2 þ y2 � 1. Thus, the corre-

sponding region in the (e; h)-plane is defined by e � 0 and 0 � h � p (Fig.1).

The boundary conditions are expressed as follows:

On the cylinder surface, the usual no-slip condition is applied, i.e.,

@w@e¼ @w@h¼ 0 for e ¼ 0: ð14:1Þ

Equation (14.1) together with Eq. (8) gives

w ¼ 0 and x ¼ � @2w@e2

: ð14:2Þ

Along the x-direction,

w ¼ x ¼ 0 for h ¼ 0; p: ð14:3Þ

Since both the stream function and vorticity equations are of elliptic nature, it is necessary to

establish boundary conditions for w and x at the external boundary, i.e., at a large distance

(r1) from the cylinder. Imai [20] has given asymptotic formulas for the stream function and

vorticity which are generally applied as external boundary conditions for both Newtonian (e.g.,

[3], [4], [14] ) and power law [13] fluids. In polar coordinates, the first terms of w and x at the

external boundary are obtained inserting the viscosity equation (5) into Imai’s [20] equations,

giving the two following external boundary conditions (both in scaled and dimensionless form):

4 R. P. Chhabra et al.

w � sin hð Þ þ Cd

2e�e h

p� erf Qð Þ

� �; ð15Þ

x � �Cd Re I1�n

2

2

2nþ1ffiffiffipp Qe�Q2

; ð16Þ

where Cd is the drag coefficient,

Q ¼ ee2

ffiffiffiffiffiffiRe

2n

rI

1�n4

2 sinh2

� �; ð17Þ

and erf(Q) is the standard error function. It is appropriate to mention here that for n ¼ 1,

Eqs. (16) and (17) reduce to the Newtonian situation (e.g., [3], [4], [14]). Although Eqs. (16) and

(17) differ from those of D’Alessio and Pascal [13] by numerical factors of 0.5 and 1.4,

respectively, their corresponding equations appear to be in error, as they do not reduce to the

expected Newtonian limiting behavior for n ¼ 1.

The exponential scaling for the stream function and vorticity is appropriate since the stream

function is exponentially large far from the cylinder, and the vorticity is exponentially small

everywhere except in the region of the wake [13]. For the external boundary conditions used in

the present study, the scaling procedure keeps the values of w between zero and 1 and xbetween zero and )4, and thus suppresses numerical instabilities.

Once the values ofx,w and g are known in the flowdomain, the total drag coefficient is given by

Cd ¼ 2nþ1

Re

Zp

0

�In�1

2

2

@ �x@e� �x

� �þ n� 1

2

@�I2

@e�I

n�32

2 �x

� �e¼0

sin hð Þdh; ð18Þ

where �x is the dimensionless vorticity in real space ( �x ¼ e�ex), and �I2 is the dimensionless

second invariant of the rate-of-deformation tensor in real space which is obtained through the

insertion of the dimensionless stream function in real space, w ¼ e�e �w, into Eq. (6). Finally, it is

also appropriate to add here that the factor of 2 in the denominator of the rightmost term in

Eq. (18) is missing from the corresponding equation in [13].

3 Numerical solution method

The numerical solutions obtained for the computational domain shown are in Fig. 1b. For an

(N+1) · (M+1) computational mesh, the spacings in the e- and h-directions are e1=N and

p=M, respectively. The governing stream function and vorticity equations (8) and (9) are

rewritten as finite difference equations using the central difference of second-order accuracy. A

second-order upwind differencing technique [21] is used to solve Eq. (9) with one-sided dif-

ference approximations to the first derivatives of x. Numerically, this technique yields a tri-

diagonal matrix which is diagonally dominant and, therefore, unconditionally stable. The

second order upwind differencing technique was used in the present study in preference to the

central difference approximation, because preliminary tests showed that the latter approxi-

mation resulted in numerical instability, consistent with observations by other authors (e.g.,

[22]). The steady-state solutions for the governing equations, namely stream function (Eq. (8)),

vorticity (Eq. (9)) and power law viscosity (Eq. (5)), are obtained using the Gauss-Seidel

relaxation iterative method [23]. To obtain consistent approximations for all variables, for each

iteration a sweep is made through all mesh points and an updated value of the drag coefficient is

determined by numerical integration of Eq. (18) on the cylinder surface using Simpson’s rule.

Steady non–Newtonian flow past a circular cylinder 5

To obtain convergence, it was also necessary to apply an under-relaxation factor between 0.5

and 1 to the vorticity variable. The iteration procedure was repeated until the variation of Cd

per iteration was less than a preset value of 10)10. The preliminary values of w and x to

initialise the computations were obtained at every mesh point using the following equations

which are valid for ideal flow:

0.2 0.4 0.6 0.8 1.0 1.2 1.48

10

12

14

16

18

20

22

b = 0.164b = 0.082

b = 0.037

n

n

n0.2 0.4 0.6 0.8 1.0 1.2 1.4

2.2

2.0

1.8

1.6

1.4

b = 0.164

b = 0.082

b = 0.037

a

c

b

1.8

1.6

1.4

1.2

1.0

0.8

0.60.2 0.4 0.6 0.8 1.0 1.2 1.4

b = 0.164b = 0.082b = 0.037

Cd

Cd

Cd

Fig. 2a. Drag coefficient (Cd) as a function of power law index (n) and blockage ratio (b) at Re ¼ 1 for

which the critical power law index nc � 0:8; bDrag coefficient (Cd) as a function of power law index (n)and blockage ratio (b) at Re ¼ 20 for which the critical power law index nc � 0.42 ± 0.08; c Drag

coefficient (Cd) as a function of power law index (n) and blockage ratio (b) at Re ¼ 40 for which thecritical power law index nc � 0.40 ± 0.05

0.2 0.4 0.6 0.8 1.0 1.2 1.434

36

38

40

42

44

46

48

s

b = 0.082

b = 0.037

s

0.2 0.4 0.6 0.8 1.0 1.2 1.452

54

56

a b

b = 0.164b = 0.082 b = 0.037

b = 0.164

n n

qq

Fig. 3a. Separation angle (hs) as a function of the power law index (n) and blockage ratio (b) for Re ¼ 20;b Separation angle (hs) as a function of the power law index (n) and blockage ratio (b) for Re ¼ 40

6 R. P. Chhabra et al.

w ¼ 1� e�2e� �

sin hð Þ ð19:1Þ

and

x ¼ 0: ð19:2Þ

The preliminary values thus obtained were then used to determine the values of w and x at

every point for a Newtonian fluid, which in turn were used as the initial guesses for non–

Newtonian flow. This procedure ensured accelerated convergence of the numerical solution.

For all computational meshes used in the present study the radial step size e was 0.02 and the

angular step size for h was 1.8� for Re ¼ 20, 40, except for Re ¼ 1 where both these step sizes

were doubled. Additional tests carried out by us for different step sizes showed that the above

step sizes were adequate because they were found to provide sufficient numerical resolution.

The numerical solutions for each Reynolds number and power law index were obtained using

three rectangular computational meshes with different blockage ratios. The mesh sizes used for

Re ¼ 20, 40 were 200 · 100, 160 · 100 and 125 · 100, but for Re ¼ 1 the sizes used were

100 · 50, 80 · 50 and 62 · 50. In all cases the outer boundaries were positioned at e ¼ 4, 3.2

and 2.5, corresponding respectively to asymptotic boundary conditions at distances 54.6, 24.5

and 12.2 radii away from the cylinder. The corresponding blockage ratios, b, were approxi-

mately 0.037, 0.082 and 0.164.

The aforementioned numerical method was tested for Newtonian flow using values of Re in

the range 1 to 200, and was in excellent agreement with others authors [1]–[4], [16]. However,

since at Reynolds numbers greater than 40 real flow becomes unsteady (e.g., [6], [24]), there is

no practical interest in simulating steady flow outside this range and, therefore, only values of

Re up to 40 were used. Moreover, no values of n below 0.6 with b ¼ 0:164, below 0.2 with

b ¼ 0:037 or 0.082 for Re ¼ 1, or below 0.3 for Re ¼ 20, 40, were used in the present study due

to the numerical instability which was found to occur for lower values of n and higher values of

b, possibly due to the increase in magnitude of the vorticity on the cylinder surface as n

decreases. Such numerical instability produces divergent or oscillatory results.

Flow parameters such as drag coefficient Cd, separation angle hs, and wake length L were

computed for values of n up to 1.4 and Re ¼ 1, 20 and 40, and the corresponding behavior is

described below. In our work the results for non–Newtonian flow patterns are presented for

Reynolds numbers up to 40. At Re ¼ 40 the flow separated from the cylinder and formed a pair

of standing vortices, but was not yet unsteady (Fig. 8). Unfortunately there are no experimental

0.2 0.4 0.6 0.8 1.0 1.2 1.4

3.0

2.5

2.0

1.5

1.0

b = 0.164b = 0.082b = 0.037

0.2 0.4 0.6 0.8 1.0 1.2 1.4

6.0

5.5

5.0

4.5

4.0

3.5

3.0

b = 0.164 b = 0.082 b = 0.037

a b

L L

n n

Fig. 4a. Wake length (L) as a function of the power law index (n) and blockage ratio (b) for Re ¼ 20;

b Wake length (L) as a function of the power law index (n) and blockage ratio (b) for Re ¼ 40

Steady non–Newtonian flow past a circular cylinder 7

0.01

-1

-1

-1

2 -1 0

0.8

0.20.01

0.3 0.4

0.8

0.20.01

0.3 0.4

0.01

0.6 0.4 0.3

0.8

0.2

0.20.8

0.6 0.40.3

a

b

c

d

Fig. 5. Stream lines (top) and iso-vorticity lines (bottom) for Re ¼ 1 (nc � 0.8) and blockage ratio0.037: a n ¼ 1:4, b n ¼ 1:0; c n ¼ 0:6, and d n ¼ 0:2; flow from left to right

8 R. P. Chhabra et al.

results for power law fluid flow across a solid cylinder to validate the numerical predictions of

the present study, which are described below.

4 Results and discussion

The results showed that for each of the Re ¼ 1, 20 and 40, the variation of the drag coefficient

with the power law index was nonlinear, consistent with Tanner [10], Ferreira and Chhabra

-1

-1

-1

0.8

0.20.01

0.40.3

0.8

0.20.01

0.40.3

0.01

0.8

0.2

0.60.4 0.

3a

b

c

Fig. 6. Stream lines (top) and iso-vorticity lines (bottom) for Re ¼ 1 (nc � 0.8) and blockage ratio

0.164: a n ¼ 1:4, b n ¼ 1:0; and c n ¼ 0:6; flow from left to right

Steady non–Newtonian flow past a circular cylinder 9

[25], and Whitney and Rodin [12]. For Re ¼ 1, the drag coefficient peaked at n ¼ 0:4 (Fig. 2a),

consistent with Tanner [10] and with the results for the flow of power law fluids around a sphere

[26]. Such a peak was, however, not observed by Whitney and Rodin [10]. At higher Reynolds

numbers (Re ¼ 20, 40), no such peak was observed and there was an overall increase of the

drag coefficient with the power law index (Fig. 2b and c). It was also found that there was a

critical power law index n ¼ nc for which the drag coefficient became almost independent of

the blockage ratio, and that this value receded from nc ¼ 0:8 to nc ¼ 0:40 ± 0.05 as the

Reynolds number increased from Re ¼ 1 to Re ¼ 40 (Fig. 2). Moreover, for values of n > nc,

an increase of the blockage ratio always resulted in an increase of the drag coefficient (Fig. 2).

The present results show a nonlinear variation of the drag coefficient with power law index are

different from those of D’Alessio and Pascal [13] due to their omission of a factor of 2 from the

expression for the drag coefficient. However, a detailed comparison between our results and

those of D’Alessio and Pascal [13] showed that this omission did not significantly affect their

findings concerning the variation of streamline patterns with the power law index at a fixed

blockage ratio (b ¼ 0:037). Furthermore, the results showed that both the separation angle and

the wake length were strongly dependent on the power law index for Re = 20 and 40, and both

these parameters reached their minimum values around n � 0.4 to 0.6 (Figs. 3 and 4). For

power-law index values greater than 0.9, an increase in the blockage ratio resulted in a decrease

-1

-1

-2 -1

0.8

0.20.01

0.50.7

0.8

0.20.01

0.7 0.5

0.8

0.20.01

0.7 0.5

a

b

c

Fig. 7. Stream lines (top)and iso-vorticity lines (bot-

tom) for Re ¼ 20 (nc�0.42±0.08) and blockage

ratio 0.037: a n ¼ 1:4,b n ¼ 1:0, and c n ¼ 0:3,negative iso-vorticity lines(arrowed); flow from left to

right

10 R. P. Chhabra et al.

in the separation angle (Fig. 3). The aforementioned observation is also consistent, for n ¼ 1,

with the results obtained by Anagnastopoulos and Iliadis [15] at Re ¼ 106 using blockage ratios

of 0.05, 0.15 and 0.25. Furthermore, for larger values of the power law index, the wake length

became more dependent on the blockage ratio and was a decreasing function of this parameter

(Fig. 4). The results also showed that the wake length became more dependent on the blockage

ratio as the Reynolds number increased from 20 to 40 (Fig. 4), an observation which holds for

all values of n, and for n ¼ 1 is consistent with Anagnostopoulos and Iliadis [15]. A similar

variation in drag coefficient, separation angle and wake length for n ¼ 1 is observed in Table 1

of Takami and Keller [14] for Re between 1 and 20, except for their separation angle at

Re ¼ 20, which remains independent of blockage ratio.

A detailed analysis of the flow patterns at Re ¼ 1 showed that for blockage ratios of

b ¼ 0:037 and b ¼ 0:082 a decrease in the power law index always resulted in a shift of the

streamlines towards the direction of the oncoming flow (Fig. 5). However, for higher blockage

ratios (e.g., b ¼ 0:164) this effect was only observed for 1.4> n > 1 (Fig. 6a, b), whereas for

n < 1 a decrease in the power law index resulted in a shift of the streamlines away from the

direction of oncoming flow (Fig. 6b, c). Moreover, all the aforementioned effects on streamline

patterns which resulted from a decrease in the power law index were found to be similar to

those which resulted from an increase in the blockage ratio (Figs. 5 and 6). In contrast, at

-2 -1

-1

-2 -1

0.8

0.2

0.01-0.015

-0.03

0.50.7

0.8

0.20.01-0.015-0.03

0.50.7

0.8

0.20.01-0.015

0.7 0.5

a

b

c

Fig. 8. Stream lines (top)and iso-vorticity lines (bot-

tom) for Re ¼ 40 (nc �0.40±0.05) and blockage

ratio 0.037: a n ¼ 1:4, b

n ¼ 1:0, and c n ¼ 0:3,negative iso-vorticity lines(arrowed); flow from left to

right

Steady non–Newtonian flow past a circular cylinder 11

-2 -1

-1

-1

0.8

0.20.01

-0.015-0.03

0.7 0.5

0.8

0.20.01-0.015-0.03

0.70.5

0.8

0.2

0.7 0.5

0.01-0.015

a

b

c

Fig. 9. Stream lines (top)

and iso-vorticity lines(bottom) for Re ¼ 40

(nc� 0.40±0.05) and block-age ratio 0.164: a n ¼ 1:4,b n ¼ 1:0, and c n ¼ 0:3,negative iso-vorticity lines

(arrowed); flow from left toright

0 45 90 135 1800

2

4

6

8

10

(degrees)

n = 0.2n = 0.4n = 0.7n = 1.0n = 1.4

0

q

w

Fig. 10. Vorticity distribution (x0) over the surface ofthe cylinder as a function of power law index n at

Re ¼ 1 and blockage ratio 0.037

12 R. P. Chhabra et al.

higher Reynolds numbers (e.g., Re ¼ 20 and 40) a decrease in the power law index always

resulted in a shift of the streamlines towards the direction of oncoming flow in the downstream

region as well as in downstream convection of the iso-vorticity lines (Figs. 7–9). Moreover, all

the aforementioned effects on streamline and iso-vorticity patterns which resulted from a de-

crease in the power law index were found to be similar to those which resulted from an increase

in blockage ratio (Figs. 8 and 9). For low values of n (n < 0:5) our results showed the

appearance of negative iso-vorticity lines in the upstream region (Figs. 7c, 8c and 9c), which

became more pronounced as n decreased.

0 45 90 135 180

0

1

2

3

4

= 0.164= 0.082= 0.037

ω0

(degrees)

bbb

q

Fig. 11. Vorticity distribution (x0) over the surface of

the cylinder for Re ¼ 40 (nc � 0.40±0.05) andn ¼ 1:4 at three different blockage ratios (b)

0 45 90 135 1800.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0

(degrees)

w

q

b = 0.164b = 0.082b = 0.037

Fig. 12. Vorticity distribution (x0) over the surface

of the cylinder for Re ¼ 1 (nc � 0.8) and n ¼ 1:4 atthree different blockage ratios (b)

Steady non–Newtonian flow past a circular cylinder 13

The present results also seem to suggest that for n > 0:4 the critical Reynolds number

increased when n decreased, an observation which is consistent with D’Alessio and Pascal [13].

A study of the vorticity contours around the cylinder surface showed that a decrease in the

power law index resulted in an overall increase in vorticity (Fig. 10), consistent with D’Alessio

and Pascal [13] who used a constant blockage ratio of 0.037. An increase in blockage ratio also

resulted in an overall increase in vorticity (Figs. 11 and 12), except for Re ¼ 1 and values of

n < nc where this trend was reversed (Fig. 13). Furthermore, for n ¼ 1, our observation of an

increase in vorticity with blockage ratio is consistent with Anagnostopoulos and Iliadis [15].

Our study also showed that an increase in Re resulted in downstream convection of the iso-

vorticity lines (Figs. 7 and 8), thus extending this well known result to power law fluids. Finally,

suffice it to say here that it can readily be shown that since the second invariant of the rate-of-

deformation tensor, I2, is equal to the square of the vorticity on the surface of the cylinder, the

vorticity profiles shown in Figs. 10–13 also correspond to the variation of the shear rate on the

surface of the cylinder. Obviously, these in turn can be combined with the power law model to

deduce the corresponding shear stress profiles on the surface of the cylinder.

5 Conclusions

The steady non–Newtonian flow across a cylinder was found to be dependent on blockage

ratio, power law index and Reynolds number. However, there was a critical power law index nc

for which the drag coefficient became independent of the blockage ratio, and the value of nc

receded as the Reynolds number increased. A larger blockage ratio caused an increase in drag

as well as a decrease in separation angle for n > 0:9. For n > 0:6, an increase in the power law

index resulted in increased wake length and separation angles, and in greater dependence of

wake length on blockage ratio. However, an increase in blockage ratio or a decrease in the

power law index were both found to have similar effects on stream line and iso-vorticity

patterns, and increased the vorticity on the cylinder surface as well as its dependence on the

0 45 90 135 1800.0

0.5

1.0

1.5

2.0

q (degrees)

0w

b = 0.164b = 0.082b = 0.037

Fig. 13. Vorticity distribution (x0) over the surface of

the cylinder for Re ¼ 1 (nc � 0.8) and n ¼ 0:6 at threedifferent blockage ratios (b)

14 R. P. Chhabra et al.

blockage ratio for larger Reynolds numbers (Re ¼ 20, 40). The results pertaining to the effect of

blockage ratio on non–Newtonian flow provide useful clues for future research on wall effects

on cylinder motion in power law fluids.

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Authors’ addresses: R. P. Chhabra, Department of Chemical Engineering, Indian Institute ofTechnology, Kanpur, 208016, India (E-mail: chhabra@iitk.ac.in); A. A. Soares and J. M. Ferreira,

Departamento de Fısica, Universidade de Tras-os-Montes e Alto Douro, Apartado 1013, 5000-911Vila Real, Portugal

16 R. P. Chhabra et al.: Steady non–Newtonian flow past a circular cylinder