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Steady flow of Bingham plastic fluids past anelliptical cylinder

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Impact Factor: 1.94 DOI: 10.1016/j.jnnfm.2013.09.006

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Swati Patel

Indian Institute of Technology Kanpur

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Indian Institute of Technology Kanpur

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Retrieved on: 15 August 2015
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Steady flow of Bingham plastic fluids past an elliptical cylinder

S.A. Patel, 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 24 July 2013

Received in revised form16 September 2013

Accepted 19 September 2013Available online 26 September 2013

Keywords:

Elliptical cylinder

Bingham plastic fluid

Reynolds number

Bingham number

Yielded/unyielded zones

Drag coefficient

a b s t r a c t

In the present work, the flow of Bingham plastic fluids past an elliptical cylinder has been investigated

numerically elucidating the effect of yield stress and fluid inertia on the momentum transfer character-

istics at finite Reynolds numbers for a 100-fold variation in the aspect ratio. The governing differentialequations have been solved over wide ranges of Reynolds number (0.01 6 Re6 40) and Bingham number

(0.016 Bn6 100) in the laminar flow regime employing the finite element method. Furthermore, the

effect of the aspect ratio (E) of the elliptical cylinder on the detailed flow characteristics has been studied

by varying it from E= 0.1 to E= 10 thereby spanning varying levels of streamlining of the submerged

object. In particular, new extensive results on streamline contours, shape and size of yielded/unyielded

regions, shear rate profiles, surface pressure distribution and drag coefficient as functions of the Reynolds

number, Bingham number and aspect ratio are presented and discussed. The functional dependence of

the individual and total drag coefficients on the governing dimensionless parameters, aspect ratio, Rey-

nolds number and Bingham number, is explored. The present results reveal a significant influence of the

shape of the cylinder, i.e., aspect ratio on the detailed flow patterns and the overall hydrodynamic flow

behavior of elliptical cylinders.

2013 Elsevier B.V. All rights reserved.

1. Introduction

Owing to the wide occurrence of viscoplastic fluid behavior in

suspensions, foams and multiphase systems encountered in scores

of industrial settings including food, pharmaceutical, personal-care

product sectors, polymer composites, geological applications, etc.,

there has been a renewed interest in studying their fluid mechan-

ical behavior in various configurations[13]. One of the main dis-

tinguishing aspects of viscoplastic fluids is the fact that the flow

domain is spanned by the so-called yielded (fluid-like) and unyiel-

ded (solid-like) regions depending upon the prevailing stress levels

vis-a-vis the value of the fluid yield stress. From a theoretical/

numerical standpoint, not only this aspect itself poses enormous

challenges in resolving such regions but such dual nature of the

flow field also has a deleterious effect on the degree of mixingand convective transport of heat and mass, for diffusion is the chief

mode of heat and mass transfer operating in the unyielded regions.

Thus, the current interest in studying the behavior of such media in

complex geometries stems from both pragmatic and fundamental

considerations. Consequently, over the past fifty years or so, signif-

icant advances have been made in the behavior of viscoplastic flu-

ids in internal flows[1,3], porous media flows[4], mixing vessels

[5], etc., though the fluid mechanical aspects have been studied

much more thoroughly than the corresponding heat and mass

transfer phenomena. In contrast, the currently available body ofknowledge on the so-called external or boundary layer type flows

is very limited indeed[6]. The bulk of the available literature re-

lates to the prediction of drag and stability of spherical particles

settling in such fluids in the creeping flow regime, e.g., see

[712] or on interactions between them[13]. Detailed discussion

and cross-comparisons between various numerical and/or experi-

mental studies have been presented elsewhere [14,15]. Suffice it

to add here that based on a combination of the experimental and

numerical studies, reliable results are now available on the wall ef-

fects, drag coefficient and the size/shape of the yielded regions for

spherical particles undergoing steady translation in viscoplastic

fluids in the creeping flow regime. These comparisons clearly re-

veal that the predictions and experiments for drag on a single

sphere are in reasonable agreement in the creeping flow regime.Indeed, the effect of finite Reynolds numbers (up to 100) on drag

and heat transfer characteristics of a heated sphere in Bingham

plastic and Herschel Bulkley fluids has been reported only very re-

cently[14,15]. Broadly, while the fluid yield stress acts to stabilize

the flow field by postponing the flow detachment to higher values

of the Reynolds number than that in Newtonian fluids, it obviously

increasingly restricts the size of the yielded fluid-like regions close

to the surface of the sphere where the stress level exceeds the fluid

yield stress. On the other hand, with the increasing Reynolds

number, the fluid-like domains tend to expand spatially thereby

facilitating convective transport [14,15]. In contrast, much less

attention has been accorded to the other two-dimensional shapes

0377-0257/$ - see front matter 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.jnnfm.2013.09.006

Corresponding author. Tel.: +91 512 2597393; fax: +91 512 2590104.

E-mail address:[email protected](R.P. Chhabra).

Journal of Non-Newtonian Fluid Mechanics 202 (2013) 3253

Contents lists available at ScienceDirect

Journal of Non-Newtonian Fluid Mechanics

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such as circular cylinders [1622]and square bars [23,24]. Whilethe currently available results for a circular cylinder are restricted

to the creeping flow only (zero Reynolds number), limited results

for a square cylinder are available at finite Reynolds numbers up

toRe = 40[24]. Indeed, not only these studies reveal the existence

of different types of yielded/unyielded domains, but their shapes

and sizes are also modulated by the shape of the object as well

as by the values of the governing parameters. The simplest devia-

tion from a circular cylinder is an elliptical shape which not only

allows the varying levels of streamlining simply by varying its as-

pect ratio but it is also free from geometric singularities such as a

square cylinder. Therefore, this work is concerned with the two-

dimensional flow of Bingham plastic fluids past an elliptical cylin-

der oriented with its long axis transverse to the flow. At the outset,

it is instructive to briefly recount the available results on the flowof Newtonian fluids past elliptical cylinders and the analogous re-

sults for viscoplastic fluids which, in turn, facilitate the presenta-

tion and discussion of the new results obtained in this work.

1.1. Previous work

The flow past elliptical cylinders denotes a classical problem

in the realm of fluid mechanics and transport phenomena and

has been studied widely over the past 100 years or so for New-

tonian fluids. Early attempts at studying the flow of Newtonian

fluids past elliptical cylinders are invariably based on the use

of the Oseens linearized form of the NavierStokes equations

to obviate the so-called Stokes paradox. This approach is exem-

plified by the works of Tomotika and Aoi [25], Imai [26] andHasimoto [27]. Subsequent results [28,29] based on the numeri-

cal solutions of the complete NavierStokes equations revealedthe results obtained in [2527] to be grossly inadequate for

Re>2 for unconfined flow conditions. Since the first numerical

study of Epstein and Masliyah [28], numerous numerical studies

pertaining to the steady flow regime [29], elucidating the influ-

ence of incidence [30], etc. have been reported in the literature

which are mutually consistent as far as the values of the drag,

recirculation length, etc. are concerned. Depending upon the val-

ues of the Reynolds number and aspect ratio, the flow past a cyl-

inder exhibits a variety of flow regimes, akin to that seen for a

circular cylinder. Thus, for instance, Faruquee et al. [31] have

extensively studied the influence of aspect ratio on the wake

characteristics at a fixed Reynolds number of 40. At Re= 40,

the critical aspect ratio was reported to be 0.34 for the onset

of flow separation. Subsequently, Stack and Bravo [32]presentedthe critical Reynolds number denoting the onset of flow separa-

tion for aspect ratios ranging from 0 (plate normal to flow) to 1

(circular cylinder) by solving the complete NavierStokes equa-

tions. As the value of Ebecomes increasingly larger than unity,

the degree of streamlining increases and the flow remains at-

tached to the surface of the cylinder up to much larger values

of the Reynolds number than the oft reported value of Re= 56

for a circular cylinder. The effect of confinement on the vortex

shedding characteristics of an elliptical cylinder has been inves-

tigated using the lattice Boltzmann method recently [33]. At the

other extreme, the high Reynolds number limit has also been ap-

proached by employing the standard integral boundary layer

analysis for the prediction of skin friction and Nusselt number

for an elliptical cylinder [34]. More detailed reviews of the per-tinent studies are available elsewhere [3537].

Nomenclature

a semi-axis of the elliptical cylinder along the direction offlow, m

b semi-axis of the elliptical cylinder normal to the direc-tion of flow, m

Bn Bingham number so2blB

V1 , dimensionlessBnc critical Bingham number denoting the disappearance of

flow separation, dimensionlessCD drag coefficient, dimensionlessCD;1 limiting plastic drag coefficient, dimensionlessCDF frictional drag coefficient, dimensionlessCDP pressure drag coefficient, dimensionlessCp pressure coefficient, dimensionlessCp modified pressure coefficient, Eq.(16), dimensionlessCpo pressure coefficient at the front stagnation point,

dimensionlessD diameter of circular cylinder, mD1 diameter of the computational domain, mE aspect ratio of the elliptical cylinder, (=a/b), dimension-

less

FD drag force per unit length of the cylinder, N m1

FDF frictional component of drag force per unit length of thecylinder, N m1

FDP pressure component of drag force per unit length of thecylinder, N m1

lR length of the static rigid zone (Zr2) from the center ofthe cylinder, m

lw distance from the center of the cylinder to the point ofreattachment of the near closed streamline along thex-axis, m

L length of the cylinder in thez-direction, m

LR length of the unyielded rigid static zone (Zr2) lRa

2a

,

dimensionlessLw recirculation length

lwa2a

, dimensionless

m regularization parameter, dimensionlessn power-law flow behavior index, dimensionlessns unit vector normal to the surface of cylinder, dimen-

sionlessnx, ny x- and y-components of the unit vector normal to the

surface of cylinder, dimensionlessp pressure, dimensionlessps local pressure on the surface of cylinder, Pap1 reference pressure far away from the cylinder, Pa

Re Reynolds number qV12blB

, dimensionless

Re modified Reynolds number, Eq. (17), dimensionlessS surface area of the cylinder, m2

V velocity vector, dimensionlessV1 free stream velocity, m s

1

Greek symbols

_c rate of strain tensor, dimensionlesslB plastic viscosity, Pa sly yielding viscosity, Pa sq density of the fluid, kg m3

h angular position on the surface of the cylinder measuredfrom the front stagnation point,

s extra stress tensor, dimensionlesss0 yield stress,Pa

Subscriptsi, j, x, y Cartesian coordinates

S.A. Patel, R.P. Chhabra / Journal of Non-Newtonian Fluid Mechanics 202 (2013) 3253 33
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In contrast, as far as known to us, within the framework of the

generalized Newtonian fluids, there have been only three studies

dealing with the flow of power- law fluids past elliptical cylinders

[3537]. Sivakumar and co-workers[35,36]reported extensive re-

sults on the momentum and forced convection heat transfer char-

acteristics in the steady flow regime (Re6 40) for shear-thinning

and shear- thickening fluids. Both the drag and Nusselt number

values were found to be enhanced in shear-thinning fluids andthese were suppressed in shear-thickening media with reference

to that in Newtonian fluids otherwise under identical conditions.

However, these results are based on a priori assumption of the

steady flow regime to prevail over the range of conditions spanned

in their study. Indeed, the limits of the steady flow regime for ellip-

tical cylinders of various aspect ratios have been delineated only

recently[37]. Based on these findings, some of the results reported

by Sivakumar and co-workers [35,36]might be less reliable than

initially thought. Also, as expected, for blunt shapes (E< 1) flow

separation occurs at lower values of the Reynolds number than

that for a circular (E= 1) cylinder and the critical Reynolds number

increases with the increasing value ofE. This finding is consistent

with that of Faruquee et al.[31].

Even less is known about the flow of viscoplastic fluids past

elliptical cylinders. Putz and Frigaard[38]presented very limited

results for a two-dimensional planar flow over an elliptical cylin-

der using the standard TaylorHood finite element method in the

creeping flow regime. Similarly, in an attempt to mimic the behav-

ior of an artificial lung, Zierenberg et al. [39]have considered the

pulsatile flow of Casson model fluid (blood) over a circular cylin-

der. While they have considered three values of the Reynolds num-

ber (5, 10 and 40), the yield stress values are extremely small

(corresponding to blood) and therefore very small deviations from

the corresponding Newtonian kinematics are predicted in their

study. From the aforementioned discussion, it is thus fair to con-

clude that there is only scant information available on the flow

of viscoplastic fluids over an elliptical cylinder. For a given value

of the aspect ratio (E), it is expected that with the increasing Rey-

nolds number, the fluid-like yielded domains must grow in size,but this tendency is countered to some extent by the fluid yield

stress. Intuitively therefore, it appears that for a given Reynolds

number and aspect ratio, there must be a critical value of the Bing-

ham number above which the flow remains attached due to the

equilibrium between the yield stress and viscous forces on one

hand and the inertial forces on the other. Conversely, for a given as-

pect ratio of the cylinder and Bingham number, it is expected that

the flow would remain attached to the surface of the cylinder up to

higher Reynolds numbers than that in Newtonian fluids. This work

endeavors to fill this gap in the literature.

In particular, the main objective of the present work is to solve

the field equations (continuity and momentum) numerically for

the flow of Bingham plastic fluids past an elliptical cylinder eluci-

dating the effect of fluid yield stress and inertia on the fluidmechanical aspects in the range of conditions as: Reynolds number

0.016 Re 6 40, Bingham number 0.01 6 Bn6 100 and aspect ratio

0.16 E6 10. This work also reports the limiting values of the Bing-

ham number above which the flow does not detach itself from the

surface of the elliptical cylinder. The present results are compared

with the previous studies wherever possible.

2. Problem statement and formulation

The flow of an incompressible Bingham fluid with uniform

velocityV1over a long elliptical cylinder of aspect ratio E=a/bori-

ented transverse to the direction of flow is considered here, as

shown schematically inFig. 1a. Since the cylinder is infinitely longin the z-direction, the flow is considered to be two-dimensional,

i.e., Vz= 0 and @@z

0. The unconfined flow condition is reached

here by enclosing the elliptical cylinder in a hypothetical concen-

tric cylindrical envelope of fluid of diameterD1as shown schemat-

ically inFig. 1b. The diameter of the outer circular boundaryD1 is

taken to be sufficiently large to minimize the boundary effects.

While no information exists about the flow regimes in Bingham

plastic fluids for elliptical cylinders, by analogy with the transitions

observed in Newtonian fluids[37,40], the flow is expected to besteady and symmetric about the mid plane (y= 0) over the range

of conditions spanned here and therefore the computations have

been carried out only in half-domain (yP 0) to economize on

the computational effort.

For 2-D, incompressible and steady flow, the continuity and

momentum equations in their dimensionless forms are given by:

Continuity:

rV 0 1

Momentum:

VrV rp 1

Rer : s 2

For a Bingham plastic fluid, the deviatoric part of the stress ten-

sors is given by

_c 0 if jsj 6 Bn 3

s 1 Bn

j _cj

_c if jsj> Bn 4

wherej _cj ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

12

tr _c2q

is the magnitude of rate of deformation tensor

and jsj ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

12

trs2q

is the magnitude of deviatoric stress tensor. In

these equations, the two dimensionless parameters are the familiar

Reynolds number (Re) and Bingham number (Bn) which are defined

a little later in Eqs. (9) and (10).

The rate-of-strain tensor _cis given by

_c rVrVT 5

There have been several approaches developed to obviate the

discontinuity inherent in the Bingham constitutive equation[41].

However, the two such approaches have gained wide acceptance,

namely that of Papanastasiou[42]and bi-viscosity[43]in the liter-

ature. While primarily the former is used in this work, limited re-

sults were also obtained with the latter to corroborate the

reliability of our results. Papanastasiou[42]modified the classical

Bingham model by introducing an exponential term for the stress

growth. The proposed BinghamPapanastasiou model which trans-

forms the solid regions to a viscous one of high viscosity is given

by:

s 1 Bn1 expmj _cj

j _cj

_c 6

where m, the regularization parameter, controls the exponential

growth of the stress. Evidently, in the limit ofm ?1, this model

coincides with the Bingham model. Similarly, the bi-viscosity model

approach[43]postulates:

slylB

_c for jsj 6 Bn 7

s Bn _c Bn

ly=lB

! for jsj> Bn 8

The relative merits and demerits of different regularization

methods and cross-comparisons between their predictions basedon different regularization techniques for specific geometries like

34 S.A. Patel, R.P. Chhabra/ Journal of Non-Newtonian Fluid Mechanics 202 (2013) 3253
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the creeping flow over a sphere are available elsewhere [41,44].

Potential difficulties in locating the yield surfaces through such

regularization methods have also been discussed in Ref.[44].

In order to complete the problem statement, the following

boundary conditions have been used in this work.

The front-half of the fluid envelope (of diameter D 1) is desig-

nated as the inlet and at this surface, a uniform flow in thex-direc-tion is prescribed, i.e.,Vx= 1 and Vy= 0.

The rear-half of the surrounding fluid envelope is designated as

the outlet and here the disturbance to the flow field caused by the

elliptical cylinder is assumed to have subsided and thus, zero-dif-

fusion flux condition for the both velocity components, i.e.,@Vx@x

0 and @Vy

@x 0 is used here on this plane.

On the surface of the cylinder: The standard no-slip boundary

condition, i.e.,Vx=Vy= 0 is used.

Over the range of conditions spanned here, the flow is expected

to be symmetric about y= 0 plane and therefore, the symmetry

conditions are implemented here, i.e., @Vx@y

0 and Vy= 0.

The preceding governing equations and the boundary condi-

tions have been rendered dimensionless by using V1 and 2b as

the characteristic velocity and length scales respectively. These,in turn, can be used to obtain the corresponding scales as lB

V12b

,

qV21 and 2b

V1

for the stress components, pressure and regulariza-

tion parameter respectively. Naturally, one could have chosen 2a

instead of 2bas the characteristic linear scale, but since the aspect

ratioEis dimensionless on its own, one can convert these results

from one format (based on the choice of 2b) to another (based on

the choice of 2a). Evidently, in this case, the momentum character-

istics are governed by the following three dimensionlessparameters:

Bingham number: This represents the ratio of the yield stress

to viscous forces, i.e.,

Bn solB

V12b

9Reynolds number: This denotes the ratio of the inertial to vis-

cous forces, i.e.,

Re qV21lB

V12b

10Of course, the aspect ratio,E=a/b, which describes the shape of

the cylinder cross-section, is the third dimensionless parameter.The preceding definitions of the Reynolds and Bingham numbers

x

E = 10

E = 1

y

x

No slip- wall

symmetry

Out flowUniform velocity

D

E = 0.1

(b)

(a)

E = 0.1

y

E = 10

V

E = 1

Cylinder

a

b

Fig. 1. Schematics of the flow past an elliptical cylinder: (a) physical model (b) computational domain.

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are based on the assumption that the characteristic shear rate is of

the order of (V1/2b) and the effective viscous stress is given simply

aslB(V1/2b) thereby disregarding the influence of the fluid yieldstress. However, the inclusion of the yield stress in estimating

the representative viscosity will only rescale the Reynolds number

by incorporating the effect of the Bingham number, as seen in Eq.

(17)here and elsewhere[14,24].

It is customary to present the detailed kinematics of the flow interms of the streamlines in the vicinity of the cylinder and the dis-

tribution of pressure coefficient along the surface of the cylinder.

The overall gross behavior is denoted in terms of the recirculation

length, individual and total drag coefficients. In the case of visco-

plastic fluids, the size and shape of the yielded zones also depend

on the values of three parameters, namely, Re, Bn and E. Some of

these characteristics are defined here.

Drag coefficient (CD): This is a measure of the net hydrody-

namic force exerted by the fluid on the immersed cylinder along

the direction of flow. The drag coefficient is made up of two com-

ponents, namely, friction drag (CDF) due to the shearing forces and

form drag (CDP) due to the normal forces acting on the cylinder.

These are defined as follows and are essentially evaluated by the

surface integrals as shown below:

CD CDF CDP 11

CDF FDF

1

2qV212b

2

Re

Zs

sxxnx sxynydS 12

wherenx and ny are the components of the unit normal vector, ns,

normal to the surface of the cylinder given as

ns x=a2ex y=b

2eyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

xa2

2 y

b2

2r nxexnyey 13

The form drag is defined and evaluated as follows

CDP FDP1

2qV212b

Zs

CpnxdS 14

In Eq.(14), Cpis the dimensionless pressure coefficient defined

as the ratio of the static to dynamic pressure on the surface of the

cylinder, i.e.,

Cp psp1

1

2qV21

15

In Eq.(15),psis the local pressure at a point which varies along

the surface of the cylinder and p1 is the reference pressure far

away from the cylinder.

Further insights into the nature of this flow can be gained by

rescaling the pressure coefficient (ratio of the static pressure to

yield stress) on the surface of the cylinder. The modified pressurecoefficient is defined as:

Cp CpRe 16

where

Re Re

1 Bn 17

Recirculation (or wake) length (Lw): It is the dimensionless

distance measured from the rear of the cylinder to the point of

reattachment for the near closed streamlineVx=Vy= 0 on the lineof symmetry (y= 0).

Lw lwa

2a 18

wherelwis the distance from the center of the cylinder to the point

of reattachment for the near closed streamline as shown schemati-

cally inFig. 2a. In the context of Newtonian fluids, this is a direct

measure of the wake length. However, in the present situation, as

will be seen in Sections5.2 and 5.3,there is an unyielded zone at-

tached in the rear of the cylinder which is engulfed in the recircu-

lating region. Therefore, it is not uncommon to introduce another

characteristic parameter to describe the length of this static zone.

Length of the unyielded rigid zone (LR): It is the dimensionless

length of the static rigid zone Zr2 measured from the rear of thecylinder.

LR lRa

2a 19

where lR is the length of static zone downstream of the cylinder

measured from the center of the cylinder (Fig 2b).

Finally, the scaling considerations suggest that the detailed

kinematics and macroscopic momentum characteristics in the

present case are influenced to varying extents by three dimension-

less groups or combinations thereof, namely, Reynolds number

(Re), Bingham number (Bn) and the aspect ratio of the cylinder

(E). This work endeavors to understand and develop this

relationship.

3. Numerical methodology

The preceding partial differential equations subject to the afore-

mentioned boundary conditions have been solved here numeri-

cally using the finite element method based solver, COMSOL

Multiphysics (Version 4.2a). The computational domain was

meshed using a non-uniform grid structure created by the built-

in meshing function employing quadrilateral and triangular ele-

ments. Owing to the expected steep gradients close to the surface

of the cylinder and near the interface demarcating the yielded and

unyielded regions, a fine mesh was used in these regions which

was progressively made coarse to economize on the required com-

putational effort. The resulting system of equations is solved using

the steady, 2-D, laminar flow module with parallel direct linear sol-ver (PARDISO). The deviatoric part of the stress tensor in the

Fig. 2. Schematic representation of (a) recirculating wake and (b) static zone characteristics.

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7/23

momentum equation is approximated using the Papanastasiou

modified Bingham model. Based on our previous studies

[14,15,23,24], the relative convergence criterion of 105 for the

equations of continuity and momentum is used in this work. With-

in the framework of this criterion, the drag values had also stabi-

lized at least up to four significant digits. Besides, the results

obtained using more stringent convergence criterion were virtually

indistinguishable from the present results. Appropriate values ofthe fluid characteristics like density (q), yield-stress (so), plasticviscosity (lB) and the geometric parameters were specified toachieve the desired values of the three dimensionless parameters,

namely, Re,BnandE. However, these specific values of the physical

properties of the fluid are of no particular significance because the

final results are presented here in their nondimensional form.

Finally, the yield-surfaces denoting the boundaries between the

yielded and unyielded regions were located by manually refining

the computational mesh in this region and by comparing the mag-

nitude of the dimensionless extra stress tensor with the fluid yield

stress (Bingham number) within a tolerance of 106107 at each

point. Further reduction in the tolerance criterion did not produce

any noticeable changes in the shape or size of the unyielded

regions.

4. Choice of numerical parameters

Undoubtedly, the accuracy and reliability of the numerical re-

sults is strongly influenced by the choice of numerical parameters,

namely, domain size, quality of grid, convergence criterion, the va-

lue of the regularization parameter,mand of the yielding viscosity,

ly. Much has been written about these aspects elsewhere[14,23,24], and therefore only the salient points are recapitulated

here. Bearing in mind the fact that the velocity field decays slowly,

at low Reynolds numbers and/or Bingham numbers, several values

of (D1/2b) were used in this study to choose its optimum value

without a significant loss of accuracy and keeping the required

computational effort at a reasonable level (Table 1a). An inspection

of Table 1a suggests that the domain sizes ofD1/(2b) = 300, 520,

900 and 800 are believed to be adequate for E= 0.1, 0.2, 0.5 and

1 respectively while for E= 2, 5 and 10, the domain size of D1/

(2b) = 1000, 2500 and 5000 respectively are seen to be adequate

for the ranges of conditions of interest here: 0.016 Re6 40 and

0.01 6 Bn6 100.

Having fixed the domain size, a grid independence study hasbeen carried out using three non- uniform grids (G1,G2 and G3)

with the increasing level of refinement for a range of values of

the aspect ratio, E at Re= 40 and Bn = 100. The grid used in the

present work is divided into two subregions. The first zone in the

vicinity of cylinder where the mesh is highly concentrated consists

of triangular elements, as shown inFig. 3for extreme values of the

aspect ratio E= 0.1 and E= 10. Otherwise non-uniform quadrilat-

eral elements were employed for the remaining intermediate val-

ues of the aspect ratio. The second sub-region away from the

cylinder employed non-uniform quadrilateral elements with man-

ual refining near the yield surfaces. Fig. 3shows the schematics of

the grid structures used for aspect ratioE= 0.1 and 10 with their

close-up view near the cylinder expanded for the three grids G1,

G2 and G3 tested in this work. These results for grid tests are sum-

marized inTable 1b which clearly show that the values of the drag

coefficient change negligibly as one goes from grid G2 to G3. Fur-

thermore,Fig. 4shows the influence of grids on the velocity profile

in thex- andy-directions for the extreme values of the aspect ratio

E= 0.1 and E= 10 at Re= 40 and Bn= 100. Detailed examination of

the results shown inTable 1b andFig. 4reveals that grid G2 is ade-

quate to resolve the thin boundary layers under the extreme con-

ditions considered herein. Indeed, the comparison shown inFig. 4

constitutes a more stringent test of the grid effects than the results

shown inTable 1b. Thus, on both counts, G2 seems to be adequate

in the present study.

A reliable prediction of the unyielded zones also depends

strongly on an appropriate choice of the regularization parameter,

Table 1

Choice of computational parameters: (a) Domain effects. (b) Grid effects.

(a) Domain independence test (b) Grid independence test

Re= 0.01 Re= 40

E Domain size Bn =0.01 E Grid Elementsb Bn= 0.01 Bn= 100

D1/(2b) CD CDP CD CDP CD CDP

0.1 260 833.54 737.43 0.1a G1 26,878 1.6600 1.5735 61.333 58.879

300 834.01 737.47 G2 35,722 1.6596 1.5771 61.278 59.508

350 834.07 737.40 G3 40,322 1.6596 1.5773 61.267 59.567

0.2 360 836.30 686.88 0.2 G1 12,000 1.6355 1.4994 62.231 58.591

520 836.61 688.49 G2 23,640 1.6313 1.4768 61.749 58.342

700 836.70 690.14 G3 26,000 1.6303 1.4720 61.672 58.088

0.5 400 848.31 622.01 0.5 G1 12,000 1.5631 1.2335 63.031 52.856900 848.16 573.45 G2 23,640 1.5626 1.2367 62.835 53.537

1300 848.34 578.92 G3 26,000 1.5624 1.2376 62.815 53.708

1 500 874.29 441.23 1 G1 12,000 1.5078 0.98117 66.247 49.281

800 875.50 447.36 G2 23,640 1.5076 0.98593 65.989 49.704

1800 874.64 446.33 G3 28,000 1.5076 0.98663 65.970 49.819

2 600 929.57 317.62 2 G1 12,000 1.4940 0.7237 74.920 47.333

1000 929.46 313.20 G2 23,640 1.4945 0.7185 74.286 46.000

2000 929.39 313.90 G3 28,000 1.4943 0.7186 74.260 46.006

5 1500 1076.7 183.72 5 G1 13,200 1.6515 0.4515 108.09 46.269

2500 1077.5 186.00 G2 18,200 1.6513 0.4489 107.36 45.545

5000 1077.8 187.68 G3 21,000 1.6513 0.4485 107.20 45.387

10 3000 1270.2 122.24 10a G1 7527 1.95041 0.2964 164.40 43.862

5000 1270.4 127.91 G2 16,332 1.95506 0.2802 161.53 43.700

8000 1270.3 127.50 G3 20,609 1.95338 0.2808 161.51 43.332

a

Free triangular grid in the vicinity of cylinder.b Refer to half-domain (yP 0).

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8/23

m. Naturally, its unduly small values do not correctly capture the

behavior of the fluid whereas its excessively high values lead to a

very stiff coefficient matrix thereby leading to oscillations in the

predicted contours of the unyielded zones and convergence prob-

lems [44,45]. In this study, the values of m ranging from 104 to

107 have been examined for 0.1 6 E6 10 at Bn = 10 and Bn = 100.

Fig. 5shows the influence of this parameter on the predictions of

the yielded/unyielded zones for extreme values of the aspect ratio,

i.e., E= 0.1 and E= 10. Evidently, the results change very little formP 107. Based on these observations, the results reported herein

are based on the value ofm = 107. Similarly, in the case of the bi-

viscosity model, one needs to examine the effect of the value of

the yielding viscosity (ly) on the accuracy of the solution.Table 2summarizes the results showing the influence of this parameter on

the values of the pressure and drag coefficients at the minimum

and maximum values of the Bingham numbers used in this work.

Evidently, the value of (ly/lB) = 105 is seen to be satisfactory over

the range of conditions spanned in the present work. Finally, the

adequacy of these choices is demonstrated in the next section bypresenting a few benchmark comparisons.

E

=1

0

E

=0.1

G3G2G1

G3G2G1

Fig. 3. Schematic representation of non-uniform computational grid structure with their expanded view near the cylinder.

Fig. 4. Influence of grid size on the variation of velocity profiles in x- and y-directions at Re = 40 andBn = 100.

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5. Results and discussion

In this study, the governing differential equations of flow havebeen solved numerically to examine the effects of aspect ratio

(0.16 E6 10) on the fluid flow around an elliptical cylinder over

wide ranges of dimensionless parameters as: 0.016 Re6 40 and

0.01 6 Bn6 100. In particular, the present results endeavor to elu-cidate the role of these parameters on streamline patterns, yielded/

Fig. 5. Influence of the regularization parameter,m, on the location of unyielded zones at Re= 40 (a)E= 0.1 (b)E= 10.

Table 2

Influence of the yielding viscosity ly, on the total drag (CD) and pressure drag (CDP) coefficients.

lylB

CD CDP

Re = 0.01 Re = 40 Re = 0.01 Re = 40

Bn = 0.01 Bn = 100 Bn = 0.01 Bn = 100 Bn = 0.01 Bn = 100 Bn = 0.01 Bn = 100

E = 0.1

104 835.47 242,882 1.6633 61.525 732.17 234,144 1.5559 59.149

105 835.49 242,903 1.6644 61.531 732.16 234,164 1.5570 59.156

106 835.50 242,906 1.6652 61.532 732.16 234,167 1.5578 59.156

E = 10

104 1269.4 646,055 1.9555 161.53 98.131 174,693 0.2802 43.700

105 1269.4 646,056 1.9555 161.53 98.131 174,693 0.2802 43.700

106 1269.4 646,056 1.9555 161.53 98.131 174,693 0.2802 43.700

Table 3

Comparison of drag coefficients (CD) for elliptical cylinders (E= 0.2 and E= 5) in Newtonian fluids.

Re Dennis and Young[30] DAlessio and Dennis[29] Sivakumar et al.[35] Present

E = 0.2

0.01 404.53 409.97

0.1 54.247 54.748

1 9.806 9.839

5 3.854 3.862 3.790 3.782

20 2.119 2.140 2.065 2.062

40 1.876 1.621 1.618

E = 5

0.1 67.109 66.586

1 8.096 8.222 8.110 8.014

5 2.712 2.7361 2.665

10 1.765 1.848 1.768 1.730

20 1.169 1.228 1.168 1.147

40 0.789 0.794 0.786 0.774

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10/23

unyielded zones, flow kinematics and drag coefficient. At the out-

set, it is, however, important to validate the solution methodologyused in this study by comparing the present results with the liter-

ature values for a few limiting cases, as this will help ascertain the

level of the accuracy of the new results for Bingham plastic fluids

for elliptical cylinders presented herein.

5.1. Validation of results

Initially, a few results were obtained for the flow of Bingham

plastic fluids in a lid-driven square cavity and the resulting values

of the centerline velocities at the horizontal and vertical positions

of the vortex center were found to be within 2% of the literature

values for Bingham plastic fluids[9,46]and the results of center-

line velocities for Newtonian fluids show deviations around 3%

with those reported by Neofytou [47]. Next, as reliable results

are now available for the flow of Newtonian fluids (Bn= 0) past

elliptical cylinders over the range of conditions spanned here,Ta-

ble 3shows a comparison between the present and literature val-

ues culled from a few sources employing different numerical

solution schemes, domains, etc. With the exception of one datapoint of Dennis and Young[30], the present values are within 3

4% of the previous results [29,30,35]. Furthermore, Table 4 com-

pares the present values of the pressure coefficient at the front

stagnation point and drag coefficient for the Newtonian fluids

[48]. Barring the results forE= 5, the other values are seen to be

within 4% of each other. While no prior results are available for

an elliptical cylinder with E= 0.1, these are expected to be very

close to that for a plane surface oriented normal to the oncoming

fluid stream. Table 5 and Fig. 6 show comparisons between the

present results forE= 0.1 and that for a plate with the literature re-

sults culled from several sources [4952]. The close correspon-

dence seen inTable 5and inFig. 6is particularly instructive and

lends credibility to the present solution methodology. Finally,

Table 6 shows a comparison between the present and literaturevalues for elliptical and circular cylinders in terms of the limiting

Table 4

Comparison of front stagnation point pressure coefficient Cpo(h= 0) and drag coefficient of elliptical cylinders in Newtonian fluids.

E Re Cpo CD

Masliyah and Epstein[48] Present Masliyah and Epstein[48] Present

0.2 1 10.810 9.839

5 1.634 1.619 3.942 3.782

15 1.212 1.226 2.586 2.309

40 1.049 1.088 1.814 1.618

2 0.5 18.820 17.835

5 2.047 2.004 4.298 4.225

15 1.468 1.417 2.379 2.370

40 1.200 1.176

5 5 2.649 2.408 5.019 5.037

10 2.037 1.839 3.490 3.417

20 1.656 1.481 2.424 2.350

40 1.436 1.262 1.771 1.637

Table 5

Comparison of drag coefficient between the present results for E= 0.1 and that of a

vertical flat plate in Newtonian fluids.

Re Dennis et al.[51] Present

0.5 15.08 15.991 9.66 9.95

5 3.75 3.81

10 2.75 2.76

20 2.09 2.10

30 1.82 1.82

40 1.68 1.66

Fig. 6. Comparison of drag coefficient values for a vertical flat plate and an elliptical

cylinder (E= 0.1).

Table 6

Validation of the present results (Bn= 105) for elliptical cylinders in the fully plastic

limit.

Ref. CD;1

E= 0.5 E= 1 E= 2 E= 10

Randolph and Houlsby[53] 11.94

Mitsoulis[16] 11.7

Tokpavi et al. [19] 11.94

Putz and Frigaard[38] 13.1 11.94 11.56 11.35

Present 13.205 11.939 11.581 11.331

Table 7

Comparison of the present and literature values of drag coefficient at finite Reynolds

numbers for a circular cylinder (E= 1).

CDRe

Re

Bn= 0.2 Re

Bn= 1

Mossaz et al.[20] Present Mossaz et al.[20] Present

0.0083 25.628 24.720 0.005 59.279 59.239

0.0833 25.723 24.717 0.05 59.281 59.317

0.8333 26.678 25.291 0.5 59.478 60.103

4.1667 30.921 30.879 2.5 62.920 63.593

8.3333 36.225 37.429 5 68.522 67.955

16.6667 46.834 48.732 10 78.912 76.680

33.3333 68.050 67.826 20 97.052 94.130

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11/23

drag values (Bn?1) while Table 7 compares the values of the

drag coefficient at finite Reynolds numbers for a circular cylinder

in a Bingham plastic fluid. Once again, an excellent match is seen

to exist in these tables. Similar extensive comparisons for the drag

Fig. 7. Representative streamline profiles for an elliptical cylinder (a) E= 0.1 (b)E= 0.2 (c)E= 1 (d)E= 5 (e)E= 10.

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12/23

of a sphere in Bingham plastic and HerschelBulkley fluids have

been recently presented elsewhere[14,15] and therefore, these

are not repeated here. Based on the foregoing extensive compari-

sons, the new results for elliptical cylinders reported herein are be-

lieved to be reliable to within 23%.

5.2. Streamlines contours and recirculation length

Representative streamline profiles close to the surface of an

elliptical cylinder (E= 0.1, 1 and 10) are shown in Fig. 7for a range

of values of the Reynolds number and Bingham number. At low

Reynolds numbers, the fluid inertia is small and therefore a fluidelement is able to negotiate the body contour without incurring

any loss of kinetic energy and thus the flow remains attached tothe surface of the cylinder. Similarly, the yield stress of the fluid

also tends to delay the onset of flow detachment from the surface

of the cylinder. This is ascribed to the fact that away from the sub-

merged cylinder, the material is by and large unyielded which acts

as a virtual wall and it is thus tantamount to that the flow occurs in

a confined geometry. This, in turn, tends to suppress the propensity

for flow separation, in line with the available results in Newtonian

fluids. Thus, while the tendency for flow separation increases with

the increasing Reynolds number, it is suppressed with the increas-

ing Bingham number for a given shape, i.e., value ofE. Naturally

both these mechanisms are modulated by the shape of the object.

Thus, for instance, atE= 0.1 which behaves like a plane surface ori-

ented normal to flow, due to sudden changes in the flow direction,

flow separation is likely to occur at low Reynolds numbers; thecritical value beingRe= 0.08 for Newtonian fluids [32]. Thus, for in-

stance, atRe = 1, there is a visible separated region in the form of

twin counter rotating vortices atBn = 0.01 which seems to disap-

pear completely at BnP 0.1. Intuitively, it appears that higher

the Reynolds number, larger would be the value of the Bingham

number needed to prevent the flow separation. This observation

is clearly borne out by the results shown in Fig. 7 irrespective of

the value of E6 1. However, for E> 1, flow separation occurs at

much larger Reynolds numbers even in Newtonian and power-

law fluids[37]and with the introduction of yield stress, this trend

is likely to continue even up to higher Reynolds numbers, as can be

seen inFig. 7and inTable 8. These trends are qualitatively consis-

tent with that reported for a circular cylinder[20] and a sphere

[14].Table 8summarizes the functional dependence of the recircu-lation lengthLwon the Reynolds number and Bingham number for

Table 8

Effect of Reynolds number and Bingham number on the recirculation length.

Bn Re Lw

E = 0.1 E = 0.2 E = 0.5 E = 1 E = 2 E = 5 E = 10

0.01 1 0.8915

5 5.2663 2.0811 0.3125

10 10.127 4.3815 1.1269 0.2106

20 21.151 9.4296 2.7842 0.8683 0.1271

40 47.325 21.201 6.3974 2.1867 0.5706

0.1 1 0.2940

5 4.2747 1.5548 0.1121

10 8.477 3.5784 0.8010 0.0619

20 17.786 7.8883 2.2418 0.6057

40 39.876 17.877 5.3753 1.7784 0.3886

1 5 1.4640

10 3.6493 1.1840 0.0552

20 8.1874 3.1989 0.5397

40 18.254 7.8966 1.9409 0.2825

5 10 0.5789

20 2.0011 0.4853

40 5.1902 1.7259 0.1212

10 20 0.5175

40 2.2307

Table 9

Values of critical Bingham number for elliptical cylinders.

Re E= 0.1 E = 0.2 E =0.5 E =1 E = 2

Bn(wake) Bnc (no wake) Bn(wake) Bnc (no wake) Bn(wake) Bnc (no wake) Bn (wake) Bnc (no wake) Bn(wake) Bnc (no wake)

1 0.5 0.75

5 4.5 4.75 1.75 2 0.2 0.3

10 7.75 8 4.25 4.5 1.25 1.5 0.2 0.25

20 13.5 13.75 8.25 8.5 3 3.25 0.8 0.85 0.075 0.08

30 18.75 19 11.75 12 4.75 5 1.4 1.45 0.30 0.35

40 24 24.24 15.5 15.75 6.50 6.75 2 2.25 0.45 0.50

Table 10

Comparison of recirculation length Lw for Bingham plastic fluid flow past circular

cylinder.

Lw

Mossaz et al. [20] Present %error

Bn Re= 20

0.08 0.6310 0.6567 4.08

0.19 0.3793 0.4016 5.88

Re= 40

0.08 1.8574 1.7954 3.34

0.18 1.5079 1.4857 1.47

0.28 1.2422 1.1931 3.95

0.39 1.0064 0.9809 2.54

0.59 0.5980 0.6195 3.60

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a range of values ofE. For a fixed value of the Bingham number and

aspect ratio, the recirculation length shows a positive dependence

on the Reynolds number which is in line with the behavior seen in

Newtonian and power-law fluids. On the other hand, for a fixed

Reynolds number and aspect ratio, the recirculation length de-

creases with the increasing Bingham number. The decreasing wake

size and disappearance of the standing vortices is also expected

with the increasing value of the aspect ratio (E) due to the increas-

ing degree of streamlining of the bluff body. Thus for instance, no

flow separation is observed for the range of Bingham and Reynolds

numbers considered in this study forE> 2 which is also in line with

the previous results [37]. Similarly, no flow separation was ob-

served atRe 6 0.1 for the ranges of Bingham number, Bnand aspect

ratio,Eembraced in this study.Table 9summarizes the critical val-

ues of the Bingham number (within 0.13) as a function of the as-

pect ratio and Reynolds number above which the flow remains

attached to the surface of the submerged body. It is worthwhile

to add here the values of the critical Bingham number (for a fixed

Reynolds number) reported here (denoting the cessation of the

flow separation) are complimentary to the critical values of the

Reynolds number, for a fixed Bingham number, reported by Moss-

az et al.[20]which denote the onset of the formation of the recir-

culating regions in the rear of the cylinder. Therefore, while it is not

possible to contrast these two results, however,Table 10contrasts

the present values of the recirculation length, Lw, with that of

Mossaz et al.[20]in the limit ofE= 1 andn= 1, and the two values

are seen to be in good agreement.

Finally, attention is drawn to the fact that in one case corre-

sponding toRe = 10,Bn = 0.1 and E= 1, there is a second recircula-

tion region, smaller than the primary wake present, while no wake

was observed at Bn = 0.2 and only one recirculating region was

seen atBn = 0.080.09. Therefore, it is likely that this point is justtoo close to the critical point corresponding to the suppression of

the wake formation atBn= 0.2. It is likely that the primary recircu-

lating region splits into smaller regions before disappearing alto-

gether. On the other hand, the presence of the second

recirculation region is not a numerical artifact because this case

was repeated at least with two different meshes and for a few val-

ues of the Bingham number in the vicinity of Bn= 0.1. No more

explanation can be given at this stage for this effect.

5.3. Delineation of yielded/unyielded zones

One of the distinct features of the flow of viscoplastic media is

the simultaneous existence of the fluid-like (yielded) and solid-like

(unyielded) regions, both in the vicinity of the submerged objectand far away from it, as have been reported for a sphere, circular

cylinder and a square bar. Similarly, in the present case, three dis-

tinct unyielded zones are observed, shown schematically inFig. 8,

where the unyielded zones are shaded while the unshaded regions

represent the deforming fluid zones; however, these differ in shape

and size depending upon the aspect ratio (especially Zr1 and Zr2)

of the cylinder from that seen for a circular cylinder (E= 1). These

are briefly described below:

Two triangular shaped unyielded zones (Zr1 and Zr2) attached

to the front and rear of the cylinder at the stagnation points

which are static in nature. The triangular shape of these zones

observed in this study was also reported by Mossaz et al.[21].

Two symmetric rigid cores (Zr3), equidistant from the cylinderon the either side about the horizontal axis of symmetry. These

are dynamic in nature, i.e., these are undergoing a rigid body-

like rotation without deformation of the fluid.

A rigid envelope enclosing the fluid zone, far away from the cyl-

inder referred to here as Zr4. This is also dynamic in nature in so

far that it is moving as a solid plug with a uniform velocityV1,

without deforming.

The existence of the above-mentioned rigid zones has been also

confirmed by comparing the location of the yielded/unyielded

Zr1

Zr3

Zr2

Zr4

Zr3

Bn = 100 Re = 40

Fig. 8. Schematic representation of the rigid zones around a circular cylinder (E= 1)

(flow is from left to right).

Fig. 9. Comparison of unyielded zones of (a) Tokpavi et al. [19](creeping flow) with that of (b) present work (Re= 0.01) for Bingham plastic fluid.

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regions for a circular cylinder (E= 1) at low values of the Reynolds

number (Re= 0.01) in the present study with that of Tokpavi et al.

[19]for the creeping flow regime at Bn = 10 and Bn = 100 (Fig. 9).

Notwithstanding the inherently different values of the Reynolds

number in the two cases and the numerical solution methodolo-

gies, the two predictions are seen to be qualitatively similar.

Naturally, the size of each of these unyielded segments will

vary not only with the kinematic parameters (Re and Bn), butalso with the aspect ratio of the cylinder. For the extreme values

of E= 0.1 and E= 10 considered here, the shape of the cylinder

corresponds to a vertical (E? 0) or to a horizontal (E?1) flat

plate. The static zones Zr1 and Zr2 are observed to be the largest

corresponding to E= 0.1 (Fig. 10a). These regions shrink gradu-

ally as the aspect ratio increases and there is no evidence of

the formation of these static zones for E> 1 (Fig. 11b). On the

other hand, the size of zone Zr3 is observed to be the largest

for the extreme geometry given by E= 10 (Fig. 11b) due to the

increased extent of streamlining of the cylinder. Their size de-

creases progressively, as the body shape becomes increasingly

blunt, due to the enhanced levels of deformation and it vanishes

altogether for aspect ratio, E< 0.5, as shown inFig. 10. Similarly,

the kinematic parameters, Reynolds number and Bingham num-

ber, also exert significant influence on the size of these zones.

With the increasing Reynolds number, the size of zone Zr3 de-

creases for a given value of the aspect ratio at low Bingham

numbers, while at high values of Bn, this effect is not so signif-

icant, as can be seen clearly in Fig. 11. The size of the static zone

Zr1 (in the front side of cylinder) decreases as the Reynolds

number increases at low Bingham numbers while Zr2 (formedin the rear of the cylinder) increases and this is discussed more

later. However, the role of Reynolds number is somewhat coun-

tered by the increasing Bingham number in suppressing these

regions. Finally, irrespective of the value of the aspect ratio,

the far away rigid fluid envelope Zr4, surrounding the fluid zone

increases in size as the value ofBn increases, attaining a limiting

behavior corresponding to the fully plastic limit reaching at a

limiting value of Bingham number. Included in these figures

are also the predictions of the bi-viscosity model (with

ly/lB= 105) where the two results are seen to be in very good

agreement thereby suggesting that it is possible to use either

of these approaches with suitably chosen values ofm or ly. Thisfinding is also in line with our previous studies [14,23,24].

Fig. 10. Unyielded fluid zones (shaded): (a) E= 0.1 (b)E= 0.5 (dashed lines represent bi-viscosity model predictions) (flow is from left to right).

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Before leaving this section, it is worthwhile to analyze the func-

tional relationship between the size of static zone Zr2 on one hand

and the Reynolds number and Bingham number on the other.

Fig. 12 shows the representative results for 0.16 E6 1. These

trends are seen to be qualitatively similar to that for a circular cyl-

inder[20]. However, forEP 2, this zone was not observed due tothe streamlining of the cylinder shape.

5.4. Flow kinematics

Figs. 13 and 14show the variation of thex-component of veloc-

ity,Vx, along the positivex-axis andy-axis at the extreme values of

the Reynolds number,Re= 0.01 and Re= 40 for a range of values of

the Bingham number and for representative values of the aspect

ratio. An inspection of the velocity profiles along the y-axis for

E= 10 (Fig. 13) shows that there are four different segments of

curve in the case of high Bingham numbers. These segments are

characterized as:

III: Rapid change in velocity Vx where fluid experiences rela-tively a high rate of deformation.

Fig. 11. Unyielded fluid zones (shaded): (a) E= 1 (b)E= 10 (dashed lines represent bi-viscosity model predictions) (flow is from left to right).

Fig. 12. Dependence of the size of the static rigid zone Zr2 on the Reynolds numberand Bingham number.

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IIIII: Solid body rotation, representing unyielded zone Zr3.

IIIIV: Corresponds to a flow region with very high strain rate.

IVV: Corresponds to a dynamic zone Zr4 moving with a con-

stant velocity without shearing.

As the aspect ratio of the elliptic cylinder decreases, the

size of zone Zr3 shrinks and ultimately it vanishes. So only

the segments III, IIIIV and IVV are observed for aspect ratioE6 0.5 which suggest altogether the disappearance of the zone

Zr3 for this configuration of elliptical cylinders as shown in

Fig. 14. On the other hand, for E= 0.1, an examination of the

velocity profile along the x-direction shows three different

regions irrespective of the value of the Bingham number, Bn,

spanned here (Fig. 14). These segments are characterized as

follows:

III: Static (Vx= 0), corresponds to the rigid zone (static zoneZr2) adhering to the surface of the cylinder.

Fig. 13. Velocity profile along (i) y = 0, x > 0 (ii) x = 0, y > 0 for E= 10 andE= 1.

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IIIII: Velocity changes from 0 toVxcorresponding to the fluid-

like zone between the rigid envelope Zr4 and static zone Zr2.

IIIIV: Constant velocityVx= 1, corresponding to the translation

of the rigid envelope Zr4.

As the aspect ratio of the cylinder increases, the static zone Zr2

decreases in size (Fig. 14) and disappears above aspect ratioE= 1

as shown inFig. 13, hence one only observes the segments IIIII

and IIIIV in this case.

Fig. 15shows the profiles of the second invariant of the strain

rate tensor at the equator and on the vertical axis of the symmetry

atRe= 5 for a range of Bingham numbers and for the extreme val-

ues of aspect ratio (E= 0.1 and E =10). For an elliptical cylinder

with E= 0.1 shown in Fig. 15a, for very small values of Bingham

Fig. 14. Velocity profile along (i) y = 0, x > 0 (ii) x = 0, y> 0 for E= 0.5 andE= 0.1.



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number (Bn= 0.01 and Bn= 0.1), i.e., small deviations from the

Newtonian fluid behavior, two peaks (atx= 2.5 and x= 5) are pres-

ent at Re= 5. Under these conditions, the yield stress effects are

rather weak and the fluid behaves nearly like a Newtonian fluid

and there is a well formed wake in the rear of the plate which

probably does not extend up to the top edge of the cylinder. Hence,

the two peaks probably correspond to the sharp turning of thestreamlines at the two points along the wake contour. With the

increasing Bingham number (at a fixed Reynolds number), as the

unyielded zone Zr2 appears and grows which behaves like a so-

lid-region thereby extending the body contour in the downstream

direction. This, in turn, leads to a gradual turning of the streamlines

and hence, the first minor peak disappears altogether. Thus, there

is only one maximum in the shear rate plot inx-direction located in

the fluid zone between Zr2 and Zr4 for BnP 1. While for aspectratio E= 10, Fig. 15b, the presence of one peak in the x-direction

Fig. 15. Shear rate magnitude profiles at the equator (y= 0) and on the vertical axis (x= 0) at Re= 5: (a)E= 0.1 and (b)E= 10.

Fig. 16. Influence of the regularization parameter, m on the velocity profiles inx- and y-directions at Re = 5 andBn= 100.

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confirms the fluid zone between the cylinder and Zr4. On the other

hand, in the case of an elliptical cylinder with E= 10 (shown in

Fig. 15b) there are two zones of high shear rate in the positive y-

direction which manifest in the form of two peaks of the velocity

profile in the y-direction. As aspect ratio approachesE= 0.1, only

one peak located in the fluid zone between cylinder and Zr4 is ob-

served (Fig. 15a). It is, however, appropriate to mention that the

shear rate is scaled here using (V1/2b) as the characteristic shear

rate. The only other possibility is to employ (V1/2a) as the charac-

teristic shear rate. These two values are, however, inter-related via

the value of the aspect ratio,E. Both these choices approximate the

shear rate in an average sense, as actual shear rate could be signif-

icantly higher than this value in some parts of the flow domain.However, since the values of the regularization parameter (m) have

been varied here by 23 orders of magnitude accompanied by a

very little change in the detailed velocity profile (shown in

Fig. 16) and/or in the value of drag coefficients clearly demon-

strates the robustness of the values ofm used here. This, as such,

lends further credibility to the reliability of the present results.

Figs. 17 and 18show the pressure variation along the surface of

an elliptical cylinder for a range of values of the aspect ratios span-

ning the range 0.16 E6 10 and Bingham number 0.01 6 Bn6 100

atRe= 10 andRe= 40 in terms of the modified pressure coefficient,

Cp. Evidently, the aspect ratio is seen to have a strong influence on

the pressure coefficient distribution along the surface of the cylin-

der, similar to the case of Newtonian fluids. These results confirm

that as the aspect ratio increases, the pressure decrease becomessharper in the front part of the cylinder. For each configuration

Fig. 17. Variation of the modified pressure coefficient along the surface of cylinder for (i) E= 0.1 (ii)E = 0.5 (iii)E = 1.



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of the elliptical cylinder, it is clear from these figures that the mag-

nitude of the pressure on the surface of the cylinder shows a posi-

tive dependence on the both Reynolds number and Bingham

number.

5.5. Drag coefficients

The drag coefficient is a gross parameter which describes the

macroscopic fluid mechanical behavior and it consists of two com-

ponents, i.e., viscous drag due to shear stress and form drag (CDP)

due to the pressure field, as defined in Eqs.(12) and (14).Fig. 19

shows the dependence of the total (CD) and pressure (CDP) drag

coefficients on the Reynolds number and Bingham number for arange of values of the aspect ratio considered in this study. Both

drag coefficients exhibit the classical inverse dependence on the

Reynolds number while positive dependence on the Bingham

number irrespective of the shape of the cylinder. The relative con-

tributions of the friction and form drag depend upon the shape of

the cylinder, as can be clearly seen inFig. 20.For E6 1, the ellipti-

cal cylinder acts more like a bluff body and thus the total drag is

dominated by the form drag drawing little contribution from the

viscous drag. As the aspect ratioEincreases above unity, the object

becomes more streamlined where the total drag is dominated by

the viscous drag.Fig. 20also reveals that the ratioCDP/CDFbecomes

independent of the Reynolds number above the value of Bingham

number 50 for a given value of the aspect ratio while the total

drag coefficient increases with the increasing Bingham number(Fig. 19). It is desirable to correlate the present numerical results

Fig. 18. Variation of the modified pressure coefficient along the surface of cylinder for (i) E= 2 (ii)E = 5 (iii)E = 10.

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using regression which will facilitate the interpolation of the

present results for the intermediate values of the parameters.

The present numerical values of the total (CD) and pressure (CDP)

drag coefficients for elliptical cylinders have been correlated over

the range of conditions (0.016 Re6 40, 0.016 Bn6 100 and0.16 E6 10) as follows:

CD I1m11 m2Re

f

Re 1 m3Bn

a m4Bnb 20

CDP I2 k1

Rek1 k3Bn

a k4Bnb 21

Table 11 summarizes the values of the empirically fitted

constants in Eqs. (20) and (21) for the total and pressure drag

coefficients respectively which exhibit additional dependence on

the aspect ratioE. Eqs.(20) and (21)reproduce the present numer-

ical data (343 data points) for Reynolds number (16 Re6 40) and

aspect ratio 0.16 E6 10 with an average deviation(davg) < 6% ex-cept at the lowest Bingham number values of 0.01 and 0.1 where the

maximum deviations are of the order of33% as shown inTable 11.

On the other hand, at low Reynolds numbers (0.016 Re< 1), Eqs.

(20) and (21) approximate the present numerical results with an

average error of less than 1% which rises to a maximum of 3.43%

over the range of values of E spanned here. Further statisticalexamination of the results showed that the deviations between

the numerical and predicted, using Eqs. (20) and (21), values in-

creased with the increasing values of the Reynolds number and as-

pect ratio and with the decreasing values of the Bingham number.

Therefore, there is a degree of self-cancelation of errors to some ex-

tent depending upon the combination of values of the parameters.

Unfortunately, Eqs.(20) and (21)do not seem to approach the ex-

pected Newtonian values asBn? 0. While the reasons for this are

not immediately obvious, similar difficulties in reconciling the

numerical and experimental results in Bingham plastic media have

been observed for spheres[14,15]and for square cylinders[23,24].

Before leaving this section, it is worthwhile to compare the

present predictions with the numerical predictions of drag for a

plate oriented normal to the direction of flow available in the liter-ature [54]. In terms of the present notations, their results

Fig. 19. Dependence of drag coefficient (CD) and pressure drag coefficient (CDP) on Reynolds number and Bingham number.

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correspond to E= 0.15 but the two ends of the plate were cham-

fered at an angle of 30. Strictly speaking, therefore, it is neither

possible nor justified to make a comparison with their results.

However, limited results were obtained in the present study for

an elliptic cylinder with E= 0.15 for this purpose, and these are

plotted inFig. 21together with the results of Savreux et al. [54].

Notwithstanding the differences in the two geometries, the agree-

ment is seen to be good inFig. 21; the two values differ from each

other at most by 7.5%. All in all, the present results seem to be con-

sistent with the previously available results for a circular cylinder

and a flat plate oriented normal to the direction of flow.

6. Conclusions

In this work, extensive numerical results are reported for the

steady flow of Bingham plastic fluids past an elliptical cylinder in

an infinite medium over the range of conditions as: 0.01 6 Re6 40,

0.016 Bn6 100, and 0.16 E6 10. The extreme values of the as-

pect ratio,E, correspond to the limiting cases of a plane surface ori-

ented normal and parallel to the direction of flow respectively.

Detailed results on the streamline contours, yielded/unyielded

zones, wake characteristics and drag coefficients are presented to

delineate the influence of the inertial and yield stress forces on

the velocity and shear rate distribution in the close proximity of

the cylinder. Broadly speaking, while the increasing Reynolds num-

ber tends to eliminate the unyielded zones due to the increased

fluid inertia, this tendency is strongly suppressed by the increasing

Bingham number due to the stronger yield-stress effects. In addi-

tion, the flow remains attached to the surface of the submerged

body up to higher Reynolds numbers in viscoplastic fluids than

that in Newtonian fluids. Indeed, for fixed values of the Reynolds

number and aspect ratio, there exists a limiting Bingham number

beyond which the flow does not detach itself from the surface of

Fig. 20. Influence of the Reynolds number, Bingham number and aspect ratio on the

drag ratio (CDP/CDF).

Table 11

Values of fitted constants in Eqs. (20) and (21).

CD CDP

E 0.1 0.2 0.5 1 2 5 10 E 0.1 0.2 0.5 1 2 5 10

0.016 Re < 1a 0.016 Re < 1b

I1 0.370 0.411 0.382 0.407 0.501 0.534 0.863 I2 0.434 0.439 0.318 0.247 0.198 0.169 0.148

m1 3.910 3.973 4.051 4.351 4.412 4.698 4.834 k1 3.497 3.049 2.434 1.924 1.457 0.617 1.800

m3 5.348 5.286 7.388 7.347 6.155 8.352 12.427 k3 5.841 6.616 7.735 9.284 6.979 10.087 0.056

m4 7.425 7.359 5.363 5.289 8.127 10.123 12.950 k4 7.468 7.809 7.786 7.739 11.932 2 8.56 1 9.54 0

a 1.013 1.013 0.424 0.440 0.997 0.998 0.997 a 1.013 1.014 1.012 1.008 0.414 0.222 0.50b 0.418 0.421 1.010 1.006 0.444 0.460 0.468 b 0.414 0.401 0.401 0.409 1.003 1.005 1.004

davg 0.08 0.09 0.10 0.13 0.12 0.13 0.15 davg 0.07 0.08 0.07 0.11 0.16 0.72 0.94

dmax 0.48 0.43 0.45 0.54 0.43 1.00 1.69 dmax 0.42 0.59 0.60 0.77 0.87 3.29 3.43

16 Re 6 40 1 6 Re6 40

I1 1.315 1.282 1.148 1.103 0.968 0.947 1.070 I2 1.234 1.172 0.973 0.774 0.560 0.350 0.240

m1 6.711 6.963 8.670 7.854 3.034 3.816 4.265 k1 6.040 5.591 4.536 3.601 2.785 1.023 1.800

m2 0.003 0.002 0.152 0.001 1.887 1.805 1.972 k3 3.141 3.362 3.767 4.684 3.286 17.181 9.524

m3 3.964 3.858 3.746 3.812 2.722 3.193 4.189 k4 3.943 3.842 3.888 3.736 5.947 4.318 0.093

m4 2.888 2.756 2.667 2.603 3.857 4.256 4.776 k 1.002 1.002 1.002 1.001 1.001 1.001 1.001

f 0.412 0.450 0.009 0.681 0.002 0.002 0.001 a 1.023 1.023 0.494 1.016 0.520 1.006 1.004a 0.497 0.508 0.516 0.534 1.014 1.016 1.012 b 0.495 0.489 1.021 0.503 1.009 0.294 0.271b 1.023 1.025 1.023 1.022 0.544 0.565 0.571 davg 2.70 2.51 2.87 2.97 3.64 4.54 3.91

davg 2.58 2.06 3.83 4.11 4.55 5.40 5.57 dmax 18.57 17.56 17.12 18.16 20.34 24.58 21.47

dmax 18.63 18.95 21.80 22.98 27.76 31.78 33.37

d: Percent relative r.m.s. deviation from the numerical data (Total data points = 49 7 = 343).a

m2= 0.b k= 1.

Fig. 21. Comparison of the present drag coefficient results for E= 0.15 (hollow

symbols with solid lines) and for a normal flat plate [54](filled symbols with

dashed lines).

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the cylinder. Similarly, for highly streamlined shapes (E> 1), the

flow does not detach even at Re= 40, the maximum value of the

Reynolds number used in this study. The numerical drag values ob-

tained in this work have been correlated using empirical expres-

sions thereby enabling their interpolation for the intermediate

values of the governing parameters. The present results are consis-

tent with the previous studies in the limits ofE= 1 (circular cylin-

der) and corresponding to the flow transverse to a plane surface(E= 0.15). Finally, this work also demonstrates that it is possible

to use either the bi-viscous or the exponential regularization meth-

od to predict the location of the yield surfaces with comparable

levels of precision.

Acknowledgement

We gratefully acknowledge the detailed and constructive com-

ments made by the two anonymous reviewers.

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