Effect of confinement on forced convection from a heated sphere in Bingham plastic … · 2015. 5....

© 2015 The Korean Society of Rheology and Springer 75

Korea-Australia Rheology Journal, 27(2), 75-94 (May 2015)DOI: 10.1007/s13367-015-0009-9

www.springer.com/13367

pISSN 1226-119X eISSN 2093-7660

Effect of confinement on forced convection from a heated sphere in Bingham plastic fluids

Pradipta K. Das1, Anoop K. Gupta

1, Neelkanth Nirmalkar

1,2 and Raj P. Chhabra

1,*1Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208016, India

2Department of Chemical Engineering, University of Birmingham, Birmingham, B15 2TT, United Kingdom

(Received December 4, 2014; final revision received March 30, 2015; accepted March 31, 2015)

In this work, the momentum and heat transfer characteristics of a heated sphere in tubes filled with Binghamplastic fluids have been studied. The governing differential equations (continuity, momentum and thermalenergy) have been solved numerically over wide ranges of conditions as: Reynolds number, 1 ≤ Re ≤ 100;Prandtl number, 1 ≤ Pr ≤ 100; Bingham number, 0 ≤ Bn ≤ 100 and blockage ratio,0 ≤ λ ≤ 0.5 where λ isdefined as the ratio of the sphere to tube diameter. Over this range of conditions, the flow is expected tobe axisymmetric and steady. The detailed flow and temperature fields in the vicinity of the surface of thesphere are examined in terms of the streamline and isotherm contours respectively. Further insights aredeveloped in terms of the distribution of the local Nusselt number along the surface of the sphere togetherwith their average values in terms of mean Nusselt number. Finally, the wall effects on drag are present onlywhen the fluid-like region intersects with the boundary wall. However, heat transfer is always influencedby the wall effects. Also, the flow domain is mapped in terms of the yielded- (fluid-like) and unyielded(solid-like) sub-regions. The fluid inertia tends to promote yielding whereas the yield stress counters it. Fur-thermore, the introduction of even a small degree of yield stress imparts stability to the flow and therefore,the flow remains attached to the surface of the sphere up to much higher values of the Reynolds numberthan that in Newtonian fluids. The paper is concluded by developing predictive correlations for drag andNusselt number.

Keywords: sphere, Bingham number, drag, Nusselt number, wall effects

1. Introduction

Over the past 50 years or so, considerable research

effort has been expended in studying the settling behaviour

of an isolated sphere in yield-stress fluids (Chhabra, 2006).

Current interest in this flow configuration stems both from

theoretical and pragmatic considerations. From a funda-

mental standpoint, this is a classical problem in the

domain of transport phenomena dating back to Stokes

(Stokes, 1851) more than 150 years ago. So there is an

intrinsic interest to explore the influence of rheological

characteristics on the settling behaviour of a single sphere

in inelastic (Chhabra, 2006), visco-elastic (McKinley,

2002) and visco-plastic type fluids (Chhabra, 2006). The

fact that this configuration occupied the centre stage for

almost 10-15 years in the rheological community testifies

to its intrinsic significance and its appropriateness for the

purpose of bench-marking the efficacy of numerical solu-

tion methodologies (Chhabra, 2006; Walters and Tanner,

1992). On the other hand, the flow over a sphere also

denotes an idealization of many industrially important

applications. For instance, most of the processed food-

stuffs, pharmaceutical and personal-care products tend to

be in the form of suspensions and the particles must

remain in suspension to ensure their satisfactory end use.

Thus, the necessity to estimate the settling velocity of a

given particle in a fluid of known rheological properties

frequently arises in process engineering calculations. This

information is also used in the design of slurry pipelines,

solid-liquid mixing equipment, heating/cooling of such

fluids (Chhabra and Richardson, 2008; Suresh and Kan-

nan, 2011, 2012). Furthermore, it is readily conceded that

the confining walls exert a significant influence on the

detailed kinematics (velocity and temperature fields) as

well as on the global characteristics (drag coefficient and

Nusselt number) of a sphere. These aspects have been

explored extensively both in Newtonian (Clift et al., 1978)

and in power-law fluids up to moderate Reynolds numbers

(Suresh and Kannan, 2011, 2012; Song et al., 2009, 2010,

2011, 2012; Missirlis et al., 2001). Suffice it to say here

that the numerical predictions of wall effects on the set-

tling velocity of a sphere are consistent with the available

experimental results for Newtonian and power-law fluids,

at least in the creeping flow region (Chhabra, 2006).

However, this problem is accentuated in yield stress flu-

ids on two counts: firstly, the fluid-like (yielded) regions

are of finite size depending upon the values of the Reyn-

olds and Bingham numbers. Therefore, if such a region

does not extend up to the confining wall, the settling

#This paper is based on work presented at the 6th Pacific RimConference on Rheology, held in the University of Melbourne,Australia from 20th to 25th July 2014.*Corresponding author; E-mail: [email protected]

P.K. Das, Anoop K. Gupta, N. Nirmalkar and R.P. Chhabra

76 Korea-Australia Rheology J., 27(2), 2015

sphere should not experience any retardation effect due to

the walls. This is in line with the preliminary experimental

and numerical results available in the literature (Atapattu

et al., 1990; Blackery and Mitsoulis, 1997). Secondly, from

a heat transfer standpoint, convection is limited to the

yielded regions only and conduction is the sole heat trans-

fer mechanism in the solid-like or unyielded portions of

the fluid. In steady state situations, thus conduction could

be the overall rate limiting step which directly depends

upon the extent of yielded regions formed in the flow

domain. This work is concerned with the settling and

forced convection heat transfer aspects of a sphere falling

at the axis of a cylindrical tube filled with Bingham plastic

fluids up to intermediate Reynolds numbers. It is, how-

ever, desirable to present a terse review of the prior devel-

opments in this field in order to facilitate the presentation

and discussion of the new results obtained in this work.

2. Previous Works

Since an exhaustive review of the pertinent literature up

to 2006 is available in Chhabra (2006), only the key points

and the recent studies are reviewed here. A cursory

inspection of the available literature shows that the bulk of

the research effort in this field has been directed at the

fluid mechanical aspects. In particular, while the numeri-

cal studies, e.g., see (Blackery and Mitsoulis, 1997; Beaulne

and Mitsoulis, 1997; Putz and Frigaard, 2010; Prashant

and Derksen, 2011; Nirmalkar et al., 2013a, 2013b) have

concentrated on the prediction of yield surfaces, drag

behaviour and the attainment of the fully plastic limit, the

corresponding experimental studies are mainly limited to

the overall drag behaviour with the notable exception of

Atapattu et al. (1990) who studied the wall effects and

performed flow visualization studies to estimate the extent

of the fluid-like regions (Atapattu et al., 1990, 1995). Suf-

fice it to say here that reliable predictions of the drag for

an unconfined sphere are now available in both Bingham

and Herschel-Bulkley type visco-plastic fluids up to about

Re = 100 and these are in fair agreement with the currently

available experimental results (Blackery and Mitsoulis,

1997; Beaulne and Mitsoulis, 1997; Putz and Frigaard,

2010; Prashant and Derksen, 2011; Nirmalkar et al., 2013a,

2013b). Similarly, the scant experimental results on the

size of fluid-like cavities encapsulating the sphere are also

consistent with the numerical predictions, at least in the

creeping flow regime (Blackery and Mitsoulis, 1997; Nir-

malkar et al., 2013b), albeit the unyielded material adher-

ing to the sphere surface at its stagnation points have not

been observed in the flow visualization studies of Atapattu

et al. (1990, 1995). Other recent studies (Deglo de Besses

et al., 2004; Ferroir et al., 2004; Gumulya et al., 2011,

2014) have addressed the issues concerning the role of slip

at the sphere surface (Deglo de Besses et al., 2004), of

time-dependent fluid behaviour (Ferroir et al., 2004;

Gumulya et al., 2014) and development of generalized

settling velocity correlations (Gumulya et al., 2011), etc.

As far as known to us, there has been only one study per-

taining to the combined effects of fluid inertia (moderate

values of the Reynolds number) and confinement (λ =

0.25). Yu and Wachs (2007) employed the fictitious domain

method to study the drag behaviour of one and two-sphere

systems settling at the axis of a cylindrical tube filled with

a Bingham plastic fluid. In the creeping flow limit, their

limited results for λ = 0.25 are in line with that of Black-

ery and Mitsoulis (1997). Overall, their results of a single

sphere extend up to Reynolds number of 400 (based on

the inertial velocity scale). They also remarked that while

their method yields reliable predictions of drag and yield

surfaces, it is not effective in locating the condition of

motion/no motion of the sphere.

In contrast, very little information is available on heat

transfer from variously shaped objects submerged in yield-

stress fluids in general and from a confined sphere in par-

ticular (Chhabra, 2006; Chhabra and Richardson, 2008).

Indeed, the forced convection from an isolated isothermal

sphere in Bingham plastic and Herschel-Bulkley model

fluids has been studied only very recently (Nirmalkar et

al., 2013a, 2013b). Nirmalkar et al. (2013a) solved the

governing differential equations using the Papanastasiou's

exponential regularization method (Papanastasiou, 1987)

to circumvent the inherently discontinuous form of the

Bingham and Herschel-Bulkley model fluids. Extensive

results on the isotherm contours, yield-surfaces, drag coef-

ficient and Nusselt number were presented and correlated

over wide ranges of the Reynolds number (Re ≤ 100),

Bingham number (Bn ≤ 104 in Nirmalkar et al., 2013a and

Bn ≤ 10 and 0.2 ≤ n ≤ 1 in Nirmalkar et al., 2013b) and

Prandtl number (Pr ≤ 100). Overall, the rate of heat

transfer was shown to bear a positive dependence on the

Reynolds, Prandtl and Bingham numbers. The analogous

problem of free convection has been studied recently

(Nirmalkar et al., 2014a). In this case, the rate of heat

transfer (Nusselt number) decreases from its maximum

value in Newtonian fluids (at small Bingham numbers) to

the limiting value corresponding to the conduction limit,

i.e., Nu = 2 (at large Bingham numbers). In addition to

this, scant results are also available for isolated spheroids

(Gupta and Chhabra, 2014) in the forced convection regime

over the range of conditions as: Re ≤ 100; Pr ≤ 100; Bn ≤

100 and aspect ratio of the spheroids in the range from 0.2

to 5 including the limiting case of a sphere. In this case,

certain shapes were shown to facilitate heat transfer while

others impeded it. On the other hand, there has been quite

a bit of interest in studying heat transfer from two-dimen-

sional bluff bodies in such fluids in the forced-, mixed-

and free-convection regimes, e.g., for a circular cylinder

(Nirmalkar and Chhabra, 2014; Nirmalkar et al., 2014b),


Korea-Australia Rheology J., 27(2), 2015 77

elliptical cylinder (Patel and Chhabra, 2014, 2015) and a

square bar (Nirmalkar et al., 2013c), etc. While these stud-

ies are not of direct interest in the present context, these

are mentioned here for the sake of completeness.

From the foregoing discussion, it is thus abundantly

clear that very little information is available on the drag

and heat transfer aspects of a sphere falling at the axis of

a cylindrical tube filled with a Bingham fluid. In partic-

ular, in this work the governing partial differential equa-

tions (momentum and thermal energy) have been solved

numerically over the following ranges of conditions: sphere

Reynolds number (1 ≤ Re ≤ 100), Bingham number (0 ≤

Bn ≤ 100), sphere-to-tube diameter ratio (0 ≤ λ ≤ 0.5) and

Prandtl number (1 ≤ Pr ≤ 100). Over the range of Reyn-

olds numbers spanned here, the flow is known to be

steady and axisymmetric in Newtonian fluids (Wham et

al., 1996; Johnson and Patel, 1999) and it is expected to

be so for Bingham plastic fluids also due to the augmen-

tation of viscous effects by the fluid yield-stress which

impart stability to the flow.

3. Problem Formulation

Consider a heated sphere of diameter d falling along the

axis of a long cylindrical tube (no end effects) of diameter

D filled with an incompressible Bingham plastic fluid. In

this work, the case of the falling sphere in a stationary

Bingham fluid is mimicked by considering the cylinder

and the fluid to move in the upward direction at a constant

velocity over a stationary sphere as shown schemat-

ically in Fig. 1. The surface of the sphere is maintained at

a constant temperature Tw (> ) and the wall of the tube

is assumed to be adiabatic. The confinement or the block-

age ratio, λ, is simply defined by the ratio of the diameter

of sphere to that of the tube (λ = d/D). The thermo-phys-

ical properties of the fluid (density, ρ ; thermal conductiv-

ity, k; heat capacity, C; yield stress, τ0; plastic viscosity,

μB) are assumed to be temperature-independent and the

viscous dissipation term in the energy equation is assumed

to be negligible. The first of these two assumptions

restricts the applicability of the present results to situations

where the maximum temperature difference present in the

system is small and it is thus justified to

evaluate the physical properties of the fluid at the mean

film temperature, i.e., . The validity of the sec-

ond assumption, namely negligible viscous dissipation,

hinges on the value of the Brinkman number being small.

In this work, the maximum value of ΔT = 5 K is used

which yields the maximum value of the Brinkman num-

ber, and hence both

these simplifying assumptions are justified here. Over the

range of conditions considered here, the flow is assumed

to be steady, laminar and axisymmetric, as noted earlier.

The governing equations (continuity, momentum and

energy equations) are written in their dimensionless forms

as follows:

Continuity:

, (1)

Momentum equation:

, (2)

, (3)

Thermal energy equation:

. (4)

For incompressible fluids, the deviatoric stress tensor is

written as follows:

. (5)

The deviatoric part of the stress tensor τ is given by the

Bingham plastic constitutive relation which can be written

in its tensorial form as follows (Macosko, 1994):

U∞

T∞

ΔT = Tw T∞–

Tw T∞+( )/2

Br = μBU∞2/k Tw T∞–( ) = 0.002



, if , (6)

, if . (7)

Clearly, in the present form, Eqs. (6) and (7) are discon-

tinuous, non-differentiable and hence cannot be imple-

mented directly in a numerical solution scheme. Consequently,

over the years, a few regularization methods (Glowinski

and Wachs, 2011) have evolved which convert this abrupt

transition into a gradual one; the so-called exponential

method due to Papanastasiou (1987) has gained wide

acceptance in the literature (Glowinski and Wachs, 2011;

Balmforth et al., 2014) and hence will be used here also.

Within this framework, the Bingham plastic model is

rewritten as:

(8)

where m is a regularization parameter. Clearly, in the limit

of , Eq. (8) reduces exactly to Eq. (6). Thus, suf-

ficiently large values of m would result in accurate pre-

dictions of the flow and heat transfer characteristics.

Similarly, another regularization scheme which has gained

wide acceptance is the so-called bi-viscous fluid model

(O'Donovan and Tanner, 1984). In this approach, the

solid-like behaviour for the stress levels below the fluid

yield stress is approximated by treating the substance as

highly viscous (yielding viscosity ). The idealized

Bingham fluid is thus approximated as:

for , (9a)

for . (9b)

In the present work, while the exponential regularization

method, Eq. (8) was used to obtain most of the results,

limited simulations were also carried out by using the

bi-viscous fluid approach, Eq. (9), to contrast the two

predictions. Detailed discussions of the relative merits

and demerits of these two approaches as well as that of

the other regularization techniques are available in the lit-

erature (Glowinski and Wachs, 2011; Balmforth et al.,

2014).

The problem statement is completed by identifying the

physically realistic boundary conditions for the configu-

ration studied herein. On the surface of the sphere, the

usual no-slip (i.e., ) and the condition of con-

stant temperature, = 1 are used. At the inlet of the tube,

a uniform velocity in the z-direction and uniform tempera-

ture of the fluid are specified (i.e., ).

A zero diffusion flux condition for all dependent variables

is prescribed at the outlet (i.e., where ,

or ξ). This condition is consistent with the fully-devel-

oped flow assumption and is similar to the homogeneous

Neumann condition. Similarly, the conditions of no-slip

and adiabatic nature are prescribed on the tube wall, i.e.,

. (10)

The aforementioned equations are rendered dimension-

less using d, and as the characteristic length,

velocity and viscosity scales, respectively. The tempera-

ture is non-dimensionalized as . Evi-

dently, the velocity and temperature fields in the present

case are expected to be functions of the four dimension-

less groups, namely, Bingham number (Bn), Reynolds

number (Re) and Prandtl number (Pr) or combinations

thereof. Of course, the blockage ratio, λ, is the fourth

dimensionless parameter. For a Bingham plastic fluid,

these are defined as follows:

Bingham number:

, (11)

Reynolds number:

, (12)

Prandtl number:

. (13)

However, it is possible to use a slightly different scaling

(Nirmalkar et al., 2013a, 2013b, 2013c, for instance) of

the effective fluid viscosity being given by

which incorporates the effect of the fluid yield stress. Such

renormalization, however, modifies the preceding defini-

tions of the Reynolds and Prandtl numbers only by a fac-

tor of (1 + Bn) as:

and . (14)

This approach thus not only eliminates the Bingham num-

ber from the set of dimensionless parameters, but it has

also proved to be effective in consolidating both experi-

mental and numerical results for spheres (Ansley and

Smith, 1967; Atapattu et al., 1990, 1995; Nirmalkar et al.,

2013a, 2013b, 2013c, etc.). While Re, Bn and Pr coordi-

nates have been used in analyzing the streamline, isotherm

and yield surface results, the modified coordinates Re* and

Pr* have been used for the purpose of developing predic-

tive correlations for drag and Nusselt number.

τ = 1Bn

IIγ·

----------+⎝ ⎠⎛ ⎞ γ· IIτ Bn

2>

γ· = 0 IIτ Bn2≤

τ = 1Bn 1 exp– m II

γ·–( )[ ]

IIγ·

--------------------------------------------------+⎝ ⎠⎜ ⎟⎛ ⎞

γ·

m ∞→

μy >> μB

τ = μyμB-------⎝ ⎠⎛ ⎞ γ· IIτ Bn

2≤

τ = 1 + Bn

IIγ·

---------- 11

μy / μB---------------–⎝ ⎠

⎛ ⎞ γ· IIτ Bn2>

Ur = Uz = 0

ξ

Ur = 0, Uz = 1 & ξ = 0

∂ϕ/∂z = 0 ϕ = UrUz

Ur = 0, Uz = 1 & ∂ξ∂r------ = 0

U∞

μB

ξ = T ′ T∞

–( )/ Tw T∞–( )

Bn = τ0d

μBU∞-------------

Re = ρdU

∞

μB-------------

Pr = C μB

k-----------

μB + τ0d/U∞( )

Re* =

Re

1 Bn+-------------- Pr

* = Pr 1 Bn+( )



Finally, the numerical solution of the preceding govern-

ing equations subject to these boundary conditions maps

the flow domain in terms of the primitive variables (u-v-

p-T). The resulting velocity and temperature fields, in

turn, are post-processed to obtain the derived results like

streamline and isotherm contours, size and shape of the

yielded/unyielded regions, i.e., location of yield surfaces,

drag coefficients, the local Nusselt number distribution

over the surface of the sphere and the average Nusselt

number as functions of the relevant dimensionless groups,

as described in detail in our recent studies (Nirmalkar et

al., 2013a, 2013b).

4. Numerical Methodology and Choice of Parameters

As a detailed description of the solution methodology is

available in some of our recent studies (Nirmalkar et al.,

2013a, b), only the salient aspects are recapitulated here.

The governing equations subject to the aforementioned

boundary conditions have been solved numerically using

the finite element based solver COMSOL Multiphysics®

(Version 4.2a) with the linear direct solver (PARDISO).

The direct solver uses LU matrix factorization to solve the

system of linear algebraic equations in an efficient manner

thereby reducing the number of iterations to attain the

desired level of convergence. COMSOL Multiphysics®

has been used for both meshing the computational domain

as well as to map the flow domain in terms of the prim-

itive variables to obtained the steady state solutions. Due

to the boundary layers developed over the surface of the

sphere, the velocity and temperature gradients are expected

to be steep near the solid surface and near the yield sur-

faces due to the yielding/unyielding transition. Hence, a

relatively fine mesh is used in both these regions. A quad-

rilateral grid with non-uniform spacing has been used here

to mesh the computational domain. An axisymmetric, sta-

tionary model was used together with laminar flow and

heat transfer in fluids modules. The fluid viscosity was

estimated using either the Papanastasiou regularized Bing-

ham model, Eq. (8) or the bi-viscous model, Eq. (9), and

it was input via a user defined function. A relative toler-

ance of 10−6 is used as convergence criteria for the con-

tinuity, momentum and energy equations.

The influence of the numerical aspects, namely, the size

of the computational box (values of Lu and Ld), computa-

tional mesh (number of cells, grid spacing, etc.), value of

the regularization parameter (m or μy/μB) on the resulting

velocity and temperature fields and the global character-

istics need not be overemphasized here. Owing to the slow

spatial decay of the velocity and temperature fields at low

Reynolds and Prandtl numbers, domain tests have there-

fore been performed at small values of Re and Pr. Based

on a detailed examination of the present results for the

extreme values of λ, i.e., λ = 0 and λ = 0.5, the value of Lu= Ld = 50d was seen to be adequate over the range of con-

ditions spanned here and this is also in line with the values

used by others (Song et al., 2012; Wham et al., 1996) in

the context of Newtonian and power-law fluids. Similarly,

in order to arrive at an optimal computational mesh, three

numerical grids G1, G2 and G3 (detailed in Table 1) were

created and their influence on the results is also included

in Table 1 at the maximum values of the Reynolds (Re =

100) and Prandtl (Pr = 100) numbers for the extreme val-

ues of λ = 0 and λ = 0.5 and of Bn = 0 (Newtonian) and

Bn = 100. An inspection of this table suggests G2 to be

adequate to resolve satisfactorily thin boundary layers

under these conditions. In order to add weight to this

assertion, Fig. 2 shows the effect of grid on the detailed

kinematics in terms of the surface pressure and local Nus-

selt number distribution on the surface of the sphere. On

this count also, grid G2 is seen to be satisfactory. The grid

specifics include the smallest cell size of ~0.008d for λ =

0 and ~0.0065d for λ = 0.5 thereby leading to the number

of control volumes on the surface of the sphere (half), Np= 200 and 240 for λ = 0 and 0.5 respectively. The grid was

progressively made coarse by using a stretch ratio of

1.023.

Next, we turn our attention to the selection of a suitable

value of the regularization parameter, m. A summary of

these results is shown in Table 2 where it is clearly seen

that m = 104 is sufficient to obtain the drag and Nusselt

number values which are free from such numerical arti-

facts. Similarly, Table 3 shows a comparison of the gross

parameters obtained using the exponential regularization

with m = 104 and the bi-viscous model with μy/μB = 104

and once again, the two predictions are seen to be in a

Table 1. Grid independence study (Re = 100, Pr =100, m = 104).

λ Grid Np Elements δ/dBn = 0 Bn = 100

CD CDP Nu CD CDP Nu

0

G1 160 36600 0.0098 1.089 0.512 34.289 32.247 22.193 61.165

G2 200 42600 0.0079 1.088 0.517 34.063 32.245 22.351 60.197

G3 240 48600 0.0065 1.088 0.521 33.915 32.245 22.513 59.502

0.5

G1 200 20000 0.0079 2.324 1.317 39.990 33.039 22.788 59.393

G2 240 27600 0.0065 2.324 1.319 39.793 33.017 22.803 58.897

G3 280 35000 0.0056 2.324 1.319 39.673 33.004 22.816 58.589



near perfect agreement. The yield surfaces delineating the

yielded- and unyielded sub-regions were resolved using

the von Mises criterion with the relative tolerance of 10−6.

Thus, in summary, the numerical results reported in this

work are based on the following parameters: Lu = Ld = 50;

grid G2 and m = μy/μB = 104. Additional support for these

choices is provided in the next section by way of present-

ing a few benchmark comparisons.

5. Results and Discussion

In this work, extensive new results are obtained over

wide ranges of dimensionless parameters as: 1 ≤ Re ≤ 100,

Fig. 2. Effect of grid resolution on the variation of pressure coefficient and local Nusselt number over the surface of the sphere at

Re = 100 and Pr = 100.

Table 2. Effect of the value of the regularization parameter, m on

the drag coefficient and Nusselt number at Re = 100 and Pr = 100.

λ mBn = 0.1 Bn = 100

CD CDP Nu CD CDP Nu

0

103

1.1254 0.5376 34.008 32.225 22.348 60.091

104 1.1284 0.5394 34.031 32.245 22.351 60.197

105 1.1306 0.5405 34.053 32.249 22.351 60.215

0.5

103 2.3548 1.3358 39.767 33.015 22.811 58.889

104

2.3552 1.3360 39.769 33.017 22.803 58.897

105 2.3552 1.3361 39.769 33.018 22.796 58.898

Table 3. Comparison between the Papanastasiou and bi-viscosity

models in terms of drag coefficient and Nusselt number values at

Bn = 100 and Pr = 100 (m = 104, μy/μB = 104).

λ RePapanastasiou model Bi-viscosity model

CD CDP Nu CD CDP Nu

0

1 3209.7 2223.2 10.242 3209.8 2223.2 10.236

5 641.95 444.65 18.964 641.62 446.7 18.934

10 320.99 222.33 24.731 320.82 223.36 24.670

50 64.276 44.527 45.761 64.234 44.735 45.465

100 32.245 22.351 60.197 32.231 22.456 59.746

0.5

1 3277.1 1989.4 10.537 3276.9 1977.6 10.537

5 655.44 397.89 19.477 654.99 396.45 19.454

10 327.74 198.97 25.312 327.51 198.25 25.272

50 65.679 39.975 45.786 65.637 39.834 45.627

100 33.017 20.224 58.897 33.015 20.106 58.898



1 ≤ Pr ≤ 100, 0 ≤ Bn ≤ 100 and 0 ≤ λ ≤ 0.5. The detailed

flow and heat transfer characteristics of the isothermal

sphere are analyzed in terms of the streamline and iso-

therm contours, yield surfaces separating the yielded and

unyielded domains and the variation of the local Nusselt

number on the surface of the sphere. At the next level, the

individual and total drag coefficient, and the average Nus-

selt number are employed to characterize the overall gross

behaviour of the sphere in Bingham plastic fluids. How-

ever, before undertaking the detailed presentation of the

new results, it is desirable to establish the precision and

reliability of the solution methodology used. This objec-

tive is accomplished here by way of a few benchmark

comparisons.

5.1. Comparison with previous resultsReliable results are now available for an isolated sphere

in Bingham plastic fluids all the way from the creeping

regime (Blackery and Mitsoulis, 1997; Beris et al., 1985)

to finite Reynolds numbers (Nirmalkar et al., 2013a; Gupta

and Chhabra, 2014). Fig. 3 shows a comparison with the

results of Blackery and Mitsoulis (1997) for a confined

sphere in a tube in the zero Reynolds number limit; need-

less to add here that the analogous results for an uncon-

fined sphere are in excellent agreement (within ± 1.5%)

with the numerical values of Beris et al. (1985) and within

± 10% of the experimental results of Ansley and Smith

(1967) (these are not shown here for the sake of brevity).

Similarly, in order to compare the present results at finite

Reynolds number for an unconfined sphere, numerical

simulations have been performed here for λ = 0.01 after

due grid independence tests. Table 4 shows a typical com-

parison between three sets of results for an unconfined

sphere. Overall the agreement is seen to be good, except

in a few cases where the present values are about 2%

overestimated. While assessing the comparison shown in

Table 4, it must be borne in mind that both in Nirmalkar

et al. (2013a) and in the present case, a tubular compu-

tational domain has been used with the tube diameter

being 60d and 100d respectively whereas Gupta and

Chhabra (2014) employed a spherical domain of diameter

100d. Notwithstanding these differences as well as that

stemming from the differences in grids, value of m, etc.,

these values of the drag coefficient and Nusselt number

are seen to be quite robust and reliable to within 2%. Next,

Table 4. Numerical results on the drag and Nusselt number for an isolated sphere in Bingham plastic fluids (Pr = 100).

Bn ReNirmalkar et al. (2013a) Gupta and Chhabra (2014) Present

CD CDP Nu CD CDP Nu CD CDP Nu

0

1 27.333 9.1481 5.7721 27.375 9.0313 5.7375 27.352 9.2498 5.8086

10 4.2991 1.5223 12.602 4.3116 1.5101 12.584 4.3098 1.5474 12.751

50 1.5788 0.6538 24.289 1.5772 0.6561 24.167 1.5768 0.6694 24.376

100 1.096 0.5119 33.551 1.0887 0.5082 33.778 1.0884 0.5166 34.063

1

1 96.101 41.222 7.3441 96.784 41.267 7.2517 95.983 41.535 7.3883

10 10.023 4.3486 15.068 10.069 4.3405 14.990 9.9944 4.3748 15.167

50 2.5093 1.1649 26.387 2.5196 1.1652 26.309 2.5015 1.1762 26.641

100 1.5034 0.7556 33.814 1.5027 0.7556 32.955 1.5089 0.7597 34.761

10

1 433.55 248.92 8.8422 433.56 248.24 8.8214 433.86 250.36 8.9192

10 43.509 24.903 19.229 43.501 24.802 19.187 43.494 25.112 19.337

50 9.0212 5.1998 32.989 9.0172 5.1866 32.957 9.0152 5.2551 33.751

100 4.7297 2.7686 43.032 4.7321 2.7613 42.454 4.7301 2.7993 43.499

100

1 3208.1 2222.1 10.179 3207.1 2213.5 10.177 3209.7 2223.2 10.242

10 320.49 222.14 24.545 320.44 221.66 24.438 320.99 222.33 24.731

50 64.182 44.298 45.856 64.171 44.198 44.991 64.276 44.528 45.761

100 32.184 22.035 58.942 32.197 21.992 58.476 32.245 22.351 60.197

Fig. 3. Comparison of the present values of the Stokes drag

coefficient (symbols) with that of Blackery and Mitsoulis (1997)

(solid lines) for a confined sphere in Bingham fluids.



the present results for a confined sphere in Bingham plas-

tic fluids were compared with that of Yu and Wachs

(2007) in Fig. 4 for λ = 0.25 corresponding to the so-

called inertial Reynolds number of 100. An excellent

agreement is seen to exist here also. Similarly, the present

values of the average Nusselt number for a unconfined

and confined sphere respectively in Newtonian fluids for

scores of values of the Prandtl number were compared and

these were found to be in excellent agreement with that of

Song et al. (2009), Dhole et al. (2006) and Feng and

Michaelides (2000). Based on the preceding comparisons

coupled with our past experience, the new results reported

herein are therefore considered to be reliable to within 2%

or so.

5.2. Streamlines and isotherm contoursTypically, streamline and isotherm contours provide a

visual representation of the flow and temperature fields in

terms of “dead zones” and/or hot and cold spots which

may be relevant from a view point of mixing and/or in the

processing of temperature-sensitive materials. Fig. 5 shows

representative results for a range of combinations of the

values of the Reynolds number, Bingham number, Prandtl

number and two extreme blockage ratios. In Newtonian

Fig. 4. Comparison of the present values with that of Yu and

Wachs (2007) for Bingham plastic fluids at λ = 0.25. Definition

of ReI is same as that used by Yu and Wachs (2007).

Fig. 5(a). (Color online) Representative streamline (right half) and isotherm (left half) contours at λ = 0: (a) Re = 1, (b) Re = 100 (flow

is from bottom to top).



fluids (Bn = 0), the flow remains attached to the surface of

the sphere at Re = 1, but there is a well-developed recir-

culation region (wake) at Re = 100, for the critical Reyn-

olds number is known to be ~22-23 (Clift et al., 1978). In

line with the prior studies in Newtonian fluids, the con-

fining walls not only delay the wake formation but also

the resulting wakes are shortened. The present results in

Newtonian fluids are in quantitative agreement with the

previous studies in this field (Clift et al., 1978; Song et al.,

2009; Wham et al., 1996). As expected, all else being

equal, the introduction of yield stress has even more dra-

matic effect both on the propensity of wake formation as

well as on its size. Thus, wake formation is deferred to

even higher Reynolds number in visco-plastic fluids than

the oft quoted value of Re = 22-24 in unconfined Newto-

nian fluids. Qualitatively, this trend persists for all values

of λ. From another vantage point, for given values of Re

and λ, there exists a critical Bingham number beyond

which no wake is formed. This behaviour has also been

reported for circular and elliptical cylinders (Nirmalkar

and Chhabra, 2014; Patel and Chhabra, 2013). Thus, the

effects of blockage and yield stress go hand in hand as far

as the formation and size of wake are concerned but this

tendency is promoted by the increasing Reynolds number.

Now turning our attention to the corresponding isotherm

contours, it is readily seen that in the absence of yield

stress effects (i.e., Bn = 0), the isotherm contours are

almost concentric circles at low Peclet numbers (Re = 1,

Pr = 1) for an unconfined sphere which shows the dom-

inance of conduction. Due to the imposition of the wall,

the isotherms are seen to be somewhat extended in the

downstream direction even at low Peclet numbers and this

effect gets accentuated with the increasing blockage. Also,

the thinning of the thermal boundary layer is evident with

the increasing values of the Reynolds number or Prandtl

number or both. With the increasing blockage, further

sharpening of the temperature gradient occurs thereby

leading to some enhancement in heat transfer. Similarly,

with the introduction of yield stress effects (increasing

Bingham number), further sharpening of the temperature

gradients occurs because the fluid-like region is confined

to a thin layer in the vicinity of the heated sphere. There-

Fig. 5(b). (Color online) Representative streamline (right half) and isotherm (left half) contours at λ = 0.5: (a) Re = 1, (b) Re = 100

(flow is from bottom to top).



fore, the temperature distribution (hence the local Nusselt

number) is now determined by the relative importance of

convective (in yielded-regions) and conductive (in unyielded-

regions) transports. Such thinning of the thermal boundary

layer is also evident in Fig. 5b for a confined sphere. Suf-

fice it to say here that it is fair to postulate that for fixed

values of λ and Bn, the rate of heat transfer is expected to

bear a positive dependence on the Reynolds and Prandtl

numbers. On the other hand, for fixed values of Re and Pr,

the Nusselt number shows a rather intricate dependence

on the blockage ration (λ) and Bingham number (Bn).

This is so partly due to the simultaneous coexistence of

the yielded- and unyielded sub-domains in the flow region.

Hence the morphology of these regions is studied in the

next section.

5.3. Structure of yielded and unyielded regionsIntuitively, it appears that while the Bingham number

tends to suppress the extent of yielded regions, this effect

is countered by the increasing Reynolds number. Thus, the

morphology of the flow domain in terms of the yielded-

and unyielded-regions is determined by the relative mag-

nitudes of the yield stress (Bn) and inertial (Re) forces.

This balance is seen to vary with the values of Re and Bn

in Fig. 6a for λ = 0, i.e., an unconfined sphere. Broadly,

one can discern two unyielded regions: polar cap in the

rear of the sphere and faraway large body of fluid moving

en masse like a plug without shearing. At low Reynolds

numbers (e.g., Re = 1), the fluid-like zones are seen to

extend up to about four times the sphere radius at Bn = 5.

On the other hand, the yielded region is elongated in the

flow direction at high Reynolds number (e.g., at Re = 100)

due to strong advection. Also, the size of the polar cap

adhering at the rear stagnation point increases with the

Reynolds number but it decreases with the Bingham num-

ber. In contrast, for a confined sphere with moderate

blockage ratio of λ = 0.2 (Fig. 6b), the results are quali-

tatively similar except for the fact that the physical walls

and the fluid-like cavity do not intersect each other,

though the two are sufficiently close to each other at Re

Fig. 6(a). Structure of fluid-like and solid-like regions at λ = 0


Fig. 6(b). Structure of fluid-like and solid-like regions at λ = 0.2




= 100 and Bn = 5. Strictly speaking, the falling sphere

does not see the walls so to say and thus there should be

no wall effects on the falling velocity or the drag coeffi-

cient, in line with the experimental results available in the

literature (Atapattu et al., 1990). However, any small dif-

ferences present can safely be ascribed to the numerical

artifacts and due to the approximate nature of the regu-

larization approaches used here which replace the unyielded

parts by a highly viscous material. However at λ = 0.5

(Fig. 6c), this is no longer true as the physical boundary is

within the yielded region. One can also note the increasing

extent of the fluid-like zone in the axial direction, espe-

cially in the downstream side as well as the enlarged polar

cap at Re = 100 and Bn = 5. Furthermore, there is also a

small unyielded material adhering to the sphere at the

front stagnation point, but it is much smaller than that at

the rear stagnation point. In order to demonstrate that the

yield surfaces shown in Fig. 6 are reliable, Fig. 7 contrasts

the predictions of the two regularization methods, i.e., Eq.

(8) and (9). While the two results for λ = 0.5 are in near

perfect agreement, these differ slightly at the front and

rear stagnation points for an unconfined sphere. Further

increase in the values of m (or μy/μB) did not alter the

results, and also these minor differences exert virtually no

influence on the drag and Nusselt number values which

are determined solely by the velocity and temperature gra-

dients on the surface of the sphere.

5.4. Drag coefficientDue to the prevailing shearing and normal stress com-

ponents on the surface, the sphere experiences a net

hydrodynamic drag made up of two components, frictional

and form or pressure drag. Dimensional considerations

suggest the drag coefficient and its components to be

functions of the Reynolds number, Bingham number and

blockage ratio. Fig. 8a shows this functional relationship

for the total drag coefficient for a range of conditions.

Included in this figure (open symbols) are also the results

for an unconfined sphere as the base case. A quick inspec-

tion of these results reveals the following key trends: for

fixed values of the Bingham number and blockage ratio,

the drag coefficient exhibits the classic inverse depen-

dence on the Reynolds number which, however, weakens

with the increasing Reynolds number. On the other hand,

as postulated earlier, the falling sphere “sees” the bound-

ing walls only if the yielded regions extend up to the phys-

ical wall. As seen in Fig. 8a, for λ = 0.1 and λ = 0.2, this

is not the case and consequently, the drag coefficient val-

ues are identical to the corresponding values for an uncon-

fined sphere; the minor differences if any can safely be

ascribed to the numerics. On the other hand, at λ = 0.4 and

λ = 0.5, wall effects are evident up to a critical value of

the Bingham number which increases with the both Re

and λ. Thus, for instance, at λ = 0.4, the confining walls

are seen to augment the drag above the unconfined value

below 50, i.e., Bn < ~50. Beyond this value of the Bing-

ham number, the fluid solidifies before reaching the con-

fining walls. Furthermore, the wall effects are also seen to

diminish with the increasing Reynolds number which is

qualitatively consistent with the previous studies pertain-

ing to Newtonian and power-law fluids (Chhabra, 2006;

Chhabra and Richardson, 2008). All else being equal, the

drag coefficient bears a positive dependence on the block-

age ratio which is generally ascribed to the sharpening of

the velocity gradients on the surface of the sphere arising

from the upward flow of the fluid displaced by the sphere

and/or due to the additional energy dissipation on the

walls. However, this effect progressively weakens with the

increasing Reynolds number due to the diminishing role

Fig. 6(c). Structure of fluid-like and solid-like regions at λ = 0.5




of viscous forces. In order to delineate the role of yield

stress in an unambiguous manner, these results were

replotted (but not shown here) in the form of a normalized

drag coefficient (with respect to the corresponding value

at Bn = 0) as a function of Re, Bn and λ. In the limit of

, the normalized drag coefficient was seen toBn 0→

Fig. 7. Comparison between the predictions of Papanastasiou bi-viscosity (dashed lines) regularisation models: (a) λ = 0, (b) λ = 0.5


Fig. 8(a). Dependence of drag coefficient on the Bingham number, Reynolds number and blockage ratio (λ). Unfilled symbols show

the results for λ = 0.



approach the value of one, at least for λ = 0.4 and λ = 0.5.

In all other cases, this ratio was always greater than unity,

as the yield stress effects enhance the drag on the sphere

as do the confining walls. However, the extent of drag

enhancement shows a weak inverse dependence on the

Reynolds number. Some additional insights can be gained

by examining the relative contributions of the frictional

and form drags. Fig. 8b shows the representative results

for a range of combinations of conditions on the ratio CDF/

CDP. For fixed values of λ and Bn, this ratio is seen to

decrease with the increasing Reynolds number, similar to

that seen in Newtonian fluids (Clift et al., 1978). This can

safely be attributed to the decreasing role of viscous forces

with the increasing Reynolds number. Similarly, for fixed

values of λ and Re, this ratio decreases with the increasing

Bingham number, eventually leveling off at about CDF/CDP= 0.44-0.45 regardless of the values of λ and Re. This sug-

gests that the pressure drag increases faster than the fric-

tion drag with the increasing Reynolds number. For fixed

values of Bn and Re, this ratio also decreases with the

increasing value of λ. However, neither the exact limiting

value of CDF/CDP seen in Fig. 8b nor the reasons for such

a limiting value in the vicinity of 0.44-0.45 are immedi-

ately obvious.

Finally, Fig. 9 consolidates the present entire data set of

drag results in terms of the modified Reynolds number

(Re*) and the drag is seen to increase with the increasing

value of λ. In order to enhance the practical utility of these

results, the following best fits were obtained using the

non-linear regression approach:

(15)

where the values of the constants are: a = 0.32, b = 0.31,

c = 2.8 for and a = 0.75, b = 0.19 and c = 0.05 for

. The resulting average errors are of the order

of 14% which rise to a maximum of ~28% for about 400

data points. Naturally, the positive exponent of (1 + λ)

reflects the positive influence of confinement on drag. The

functional form of Eq. (15) is similar to that of the familiar

Schiller-Naumann equation for Newtonian fluids (Clift et

al., 1978). Other forms were attempted to improve the

degree of fit, but these proved to be unsuccessful without

increasing the number of fitted constants.

5.5. Distribution of local Nusselt numberFigures 10a-10d show representative results for a range

of combinations of conditions in terms of the values of Re,

λ, Bn and Pr. An inspection of these figures reveals the

following key trends. Firstly, irrespective of the values of

CD = 24

Re*

-------- 1 a Re*b

+( ) 1 λ+( )c

Bn 1≤1 Bn< 100≤

Fig. 8(b). (Color online) Dependence of the friction to pressure drag coefficient ratio (CDF/CDP) on the Bingham number, Reynolds num-

ber and blockage ratio (λ).



λ and Bn, there is very little variation in the value of the

local Nusselt number over the surface of the sphere at low

Peclet numbers due to weak advection. Conversely, heat

transfer increases with both Reynolds and Prandtl num-

bers due to the gradual thinning of the thermal boundary

layer. Needless to say that at low Reynolds numbers and

in Newtonian fluids (Bn = 0) such as Re = 1, Figs. 10a and

10c, there is no flow separation and hence the local Nus-

selt number decreases from its maximum value at the

front stagnation point all the way up to the rear stagnation

point. In contrast, at Re = 100 (Figs. 10b and 10d) there is

a well formed wake region and the Nusselt number

decreases along the surface up to the flow detachment

point (θ ~ 126-127o), but beyond this point ~ ,

the Nusselt number increases a little bit in Newtonian flu-

ids due to the enhanced fluid circulation; some further

augmentation in heat transfer is evident as the blockage

ratio increases. This is simply due to the acceleration of

the fluid in the annular region. The introduction of the

127 θ 180≤ ≤

Fig. 9. (Color online) Dependence of drag coefficient on the

modified Reynolds number and blockage ratio (λ).

Fig. 10(a). Distribution of the local Nusselt number over the surface of the sphere at Re = 1 and λ = 0.



yield stress (increasing Bingham number) brings about

qualitative and quantitative modifications to the Nusselt

number profiles along the surface of the sphere. Firstly,

the Nusselt number is no longer maximum at the front

stagnation point. The maximum value occurs increasingly

downstream along the surface of the sphere with the

increasing Bingham number. Also, the peak values are

seen to bear a positive dependence on Bn. Also, as seen

previously in Fig. 5, the flow remains attached to the

sphere surface up to much higher Reynolds numbers in

Bingham plastic fluids. The local Nusselt number contin-

uously decreases from its peak value (at θ ~ 10o to 40o) up

to the rear stagnation point. At high Reynolds numbers

(Figs. 10b and 10d), the Nusselt number plots exhibit

another distinct feature by displaying two local peaks at

θ ≤ ~45o and at θ ~ 90o. These peaks increasingly become

higher with the increasing blockage, e.g., see Figs. 10c

and 10d for λ = 0.5 in contrast to the behaviour at λ = 0.2

seen in Figs. 10a and 10b. These peak values increase due

to the diminishing size of the yielded regions in the lateral

direction and the fluid must experience appreciable accel-

eration in accord with the continuity equation. On the

other hand, the local Nusselt number is determined by the

local temperature gradient which, in turn, is linked inti-

mately with the formation of the unyielded region near or

at the stagnation points on the surface of the sphere.

Therefore, under some conditions of Re, Pr and Bn, the

maximum enhancement on account of confinement in heat

transfer occurs in Newtonian fluids and yield stress effects

tend to suppress this phenomenon. Therefore, the local

value of the Nusselt number is governed by an intricate

interplay between λ, Re and Pr on one side and Bn on the

other side; the former tend to augment the rate of heat

transfer which is somewhat offset by the Bingham number.

Indeed, this complexity also manifests itself even in terms

of the surface average Nusselt number, as is seen in the

next section.

5.6. Average Nusselt numberIt is readily conceded that it is the value of the surface

Fig. 10(b). Distribution of the local Nusselt number over the surface of the sphere at Re = 100 and λ = 0.



averaged Nusselt number which is frequently needed in

process design calculations. The scaling arguments sug-

gest the average Nusselt number to be a function of the

four dimensionless groups, namely, Re, Pr, Bn and λ. Fig.

11 shows this functional relationship for the extreme val-

ues of the Prandtl number for scores of values of Re, Bn

and λ. At low Reynolds numbers and/or Prandtl numbers,

the average Nusselt number shows a positive dependence

on the Bingham number which sharpens with the increas-

ing Reynolds number and/or with the blockage ratio. The

positive dependence on the Reynolds number can readily

be explained via the classical boundary layer consider-

ations. Similarly, the positive role of λ and Bn is ascribed

to the sharpening of the temperature gradient due to the

reduction in area available for the flow of fluid. These

trends are further accentuated at Pr = 100, Fig. 11b. Thus,

for instance, the yield stress can augment the value of the

Nusselt number by up to 100% for an unconfined sphere

and it drops to about 50-60% at λ = 0.5. Finally, the pres-

ent values can be adequately consolidated by using the

familiar Colburn factor, jH, defined as follows:

(16)

where a = 1.76 for and a = 2.11 for .

Both equations combined reproduce nearly 1600 data

points with an average error of 13-14% which rises to a

maximum of ~40%. This implies that the effect of λ on

heat transfer is well within 40%. This aspect was further

explored by refitting the data for individual values of λ

and Table 5 summarizes the resulting values of a along

with the average and maximum percentage deviations for

each value of λ investigated here. Evidently, the resulting

average and maximum deviations are slightly lower than

those when a single value of the constant is used in Eq.

(16). Naturally, this marginal improvement has come

about at the expense of extra disposable parameters. How-

ever, Eq. (16) does retain the widely accepted scaling of

jH = Nu

Re*

Pr*1/3×

-------------------------- = a

Re*2/3

-------------

Bn 1≤ 1 Bn< 100≤

Fig. 10(c). Distribution of the local Nusselt number over the surface of the sphere at Re = 1 and λ = 0.5.



and of .

6. Conclusions

In this work, the wall effects on the drag and Nusselt

number for an isothermal sphere falling axially in a cyl-

inder filled with Bingham fluids have been studied numer-

ically over the range of conditions as: 1 ≤ Re ≤ 100, 1 ≤

Pr ≤ 100, 0 ≤ Bn ≤ 100 and 0 ≤ λ ≤ 0.5. Detailed struc-

tures of the flow and temperature fields are studied in

terms of the streamline and isotherm contours in the close

proximity of the sphere and the location of yield surfaces.

While the fluid inertia (Reynolds number) fosters the

growth of fluid-like regions, this effect is somewhat coun-

tered by the yield stress effects. The signature of wall

effects in drag are only seen if the fluid-like region extends

up to the confining wall. This effect is clearly seen in

terms of the drag coefficient behaviour. However, even

under these conditions, heat transfer is influenced by the

confining walls via the thermal resistance to conduction

through unyielded material. The imposition of walls, how-

ever, sharpens the velocity and temperature gradients on

the surface of the sphere both of which enhance the values

of the drag and Nusselt number by varying amounts. On

the other hand, while the yield stress effects are seen to

enhance the value of the Nusselt number on the surface of

the sphere, but some of this advantage is lost by virtue of

the fact that the flow remains attached to surface and thus,

the local Nusselt number shows no recovery in the rear of

the sphere. The Nusselt number is influenced in decreas-

ing order by the values of the Reynolds number, Bingham

number, Prandtl number and blockage ratio. Finally, the

present numerical data of drag and Nusselt number are

consolidated in terms of the modified Reynolds and Prandtl

numbers thereby enabling their prediction in a new appli-

cation. The widely used scaling of the Nusselt number

with the Prandtl and Reynolds numbers is observed here

also.

Nu ~ Pr*1/3

jH ~ Re* 2/3–

Fig. 10(d). Distribution of the local Nusselt number over the surface of the sphere at Re = 100 and λ = 0.5.



Fig. 11. (Color online) Dependence of the average Nusselt number on Bingham number and Reynolds number at (a) Pr = 1 and (b)

Pr = 100.



Nomenclatures

Bn : Bingham number (−)C : Specific heat of fluid (J/kg·K)

CD : Drag coefficient (−)

CDF : Friction drag coefficient (−)

CDP : Pressure or form drag coefficient (−)

Cp : Pressure coefficient (−)

Cs : Stokes drag coefficient (−)

d : Diameter of sphere (m)

D : Diameter of the cylindrical tube (m)

FD : Drag force (N)

FDF : Friction drag force (N)

FDP : Pressure drag force (N)

Fs : Stokes drag force (N)

h : Local heat transfe coefficient (W/m2·K)

k : Thermal conductivity of fluid (W/m·K)

Ld : Downstream length (m)

Lu : Upstream length (m)

m : Regularization parameter (−)Np : Number of control volumes on the surface of

sphere (−) Nuθ : Local Nusselt number (−)

Nu : Average Nusselt number (−)

P : Pressure (−)

p0 : Reference pressure (Pa)

ps : Pressure at the surface of the sphere (Pa)

Pr : Prandtl number (−)

Re : Reynolds number (−)

r : Radial coordinate (m) T' : Temperature of the fluid (K)

: Temperature of fluid at the inlet (K)

Tw : Temperature on the surface of the sphere (K)

Ur : r-component of the velocity (−)

Uz : z-component of the velocity (−)

: Uniform inlet velocity (m/s)

Greek symbols: Rate of strain tensor (−)

δ : Minimum spacing between grid points (m)

η : Apparent viscosity of the fluid (Pa·s)

θ : Position on the surface of the sphere (deg) λ : sphere-to-tube diameter or blockage ratio (−)

(≡d/D)

μB : Plastic viscosity (Pa·s)

μy : Yielding viscosity (Pa·s)

ξ : Fluid temperature (−)

: Second invariant of extra stress tensor (−)

: Second invariant of strain rate tensor (−)

ρ : Density of fluid (kg/m3)

τ : Extra stress tensor (−)

τ0 : Fluid yield stress (Pa)

ω : Surface vorticity (−)

Subscriptsr, z : Cylindrical coordinates

w : Sphere surface condition

References

Ansley, R.W. and T.N. Smith, 1967, Motion of spherical particles

in a Bingham plastic, AIChE J. 13, 1193-1196.

Atapattu, D.D., R.P. Chhabra, and P.H.T. Uhlherr, 1990, Wall

effects for spheres falling at small Reynolds number in a vis-

coplastic medium, J. Non-Newton. Fluid Mech. 38, 31-42.

Atapattu, D. D., R.P. Chhabra, and P.H.T. Uhlherr, 1995, Creep-

ing sphere motion in Herschel-Bulkley fluids: flow field and

drag, J. Non-Newton. Fluid Mech. 59, 245-265.

Balmforth, N.J., I.A. Frigaard, and G. Ovarlez, 2014, Yielding to

stress: Recent developments in viscoplastic fluid mechanics,

Ann. Rev. Fluid Mech. 46, 121-146.

Beaulne, M. and E. Mitsoulis, 1997, Creeping motion of a sphere

in tubes filled with Herschel–Bulkley fluids, J. Non-Newton.

Fluid Mech. 72, 55-71.

Beris, A.N., J.A. Tsamopoulos, R.C. Armstrong, and R.A. Brown,

1985, Creeping motion of a sphere through a Bingham plastic,

J. Fluid Mech. 158, 219-244.

Blackery, J. and E. Mitsoulis, 1997, Creeping motion of a sphere

in tubes filled with a Bingham plastic material, J. Non-Newton.

Fluid Mech. 70, 59-77.

Chhabra, R.P., 2006, Bubbles, Drops and Particles in Non-New-

tonian Fluids, 2nd

ed., CRC Press, Boca Raton.

Chhabra, R.P. and J.F. Richardson, 2008, Non-Newtonian Fluid

and Applied Rheology: Engineering Applications, 2nd ed., But-

terworth-Heinemann, Oxford.

Clift, R., J.R. Grace, and M.E. Weber, 1978, Bubbles, Drops and

Particles, Academic Press, New York.

Deglo de Besses, B., A. Magnin, and P. Jay, 2004, Sphere drag

in a viscoplastic fluid, AIChE J. 50, 2627-2629.

Dhole, S.D., R.P. Chhabra, and V. Eswaran, 2006, Forced con-

8≡ FD/ρU∞2πd

2( ) 8≡ FDF/ρU∞

2πd

2( )

8≡ FDP/ρU∞2πd

2( )( 2≡ ps p0–( ) /ρU∞

2)

Fs≡ / 3πμBdU∞( )( )

T∞

U∞

γ·

T ′ T

∞–

Tw T∞–----------------≡⎝ ⎠

⎛ ⎞

IIτII

γ·

Table 5. Values of constant a used in Eq. (16).

Range λ a % Error

Averager Maximum

0 ≤ Bn ≤ 1

0 1.73 12.3 29.3

0.1 1.74 12.6 29.4

0.2 1.73 11.4 28.2

0.3 1.72 7.6 21.4

0.4 1.73 4.5 11.6

0.5 1.89 3.6 9.6

1 < Bn ≤ 100

0 2.17 9.1 22.2

0.1 2.16 9.1 21.2

0.2 2.14 8.1 22.9

0.3 2.07 7.5 23.7

0.4 2 6.5 25.6

0.5 2.13 6.5 22.0



vection heat transfer from a sphere to non-Newtonian power-

law fluids, AIChE J. 52, 3658-3667.

Feng, Z.-G. and E.E. Michaelides, 2000, A numerical study on

the transient heat transfer from a sphere at high Reynolds and

Peclet numbers, Int. J. Heat Mass Tran. 43, 219-229.

Ferroir, T., H.T. Huynh, X. Chateau, and P. Coussot, 2004, Motion

of a solid object through a pasty (thixotropic fluid), Phys. Flu-

ids 16, 594-601.

Glowinski, R. and A. Wachs, 2011, On the numerical simulation

of viscoplastic fluid flow, In: Glowinski, R. and J. Xu, eds.,

Handbook of Numerical Analysis, Elsevier, Amsterdam, 483-

717.

Gumulya, M.M., R.R. Horsley, K.C. Wilson, and V. Pareek,

2011, A new fluid model for particles settling in a viscoplastic

fluid, Chem. Eng. Sci. 66, 729-739.

Gumulya, M.M., R.R. Horsley, and V. Pareek, 2014, Numerical

simulation of the settling behaviour of particles in thixotropic

fluids, Phys. Fluids 26, 023102.

Gupta, A.K. and R.P. Chhabra, 2014, Spheroids in viscoplastic

fluids: drag and heat transfer, Ind. Eng. Chem. Res. 53, 18943-

18965.

Johnson, T.A. and V.C. Patel, 1999, Flow past a sphere up to a

Reynolds number 300, J. Fluid Mech. 378, 19-70.

Macosko, C.W., 1994, Rheology: Principles, Measurements and

Applications, Wiley-VCH, New York.

McKinley, G.H., 2002, Steady and transient motion of spherical

particles in viscoelastic liquids, In: Chhabra, R.P. and D. De

Kee, eds., Transport Processes in Bubbles, Drops and Parti-

cles, Taylor & Francis, New York, 338-375.

Missirlis, K.A., D. Assimacopoulos, E. Mitsoulis, and R.P.

Chhabra, 2001, Wall effects for motion of spheres in power-

law fluids, J. Non-Newton. Fluid Mech. 96, 459-471.

Nirmalkar, N., R.P. Chhabra, and R.J. Poole, 2013a, Numerical

predictions of momentum and heat transfer characteristics from

a heated sphere in yield-stress fluids, Ind. Eng. Chem. Res. 52,

6848-6861.

Nirmalkar, N., R.P. Chhabra, and R.J. Poole, 2013b, Effect of

shear-thinning behaviour on heat transfer from a heated sphere

in yield-stress fluids, Ind. Eng. Chem. Res. 52, 13490-13504.

Nirmalkar, N., R.P. Chhabra, and R.J. Poole, 2013c, Laminar

forced convection heat transfer from a heated square cylinder

in a Bingham plastic fluid, Int. J. Heat Mass Tran. 56, 625-

639.

Nirmalkar, N., A.K. Gupta, and R.P. Chhabra, 2014a, Natural

convection from a heated sphere in Bingham plastic fluids, Ind.

Eng. Chem. Res. 53, 17818-17832.

Nirmalkar, N., A. Bose, and R.P. Chhabra, 2014b, Free convec-

tion from a heated circular cylinder in Bingham plastic fluids,

Int. J. Therm. Sci. 83, 33-44.

Nirmalkar, N. and R.P. Chhabra, 2014, Momentum and heat

transfer from a heated circular cylinder in Bingham plastic flu-

ids, Int. J. Heat Mass Tran. 70, 564-577.

O’Donovan, E.J. and R.I. Tanner, 1984, Numerical study of the

Bingham squeeze film problem, J. Non-Newton. Fluid Mech.

15, 75-83.

Papanastasiou, T.C., 1987, Flow of materials with yield, J. Rheol.

31, 385-404.

Patel, S.A. and R.P. Chhabra, 2013, Steady flow of Bingham

plastic fluids past an elliptical cylinder, J. Non-Newton. Fluid

Mech. 202, 32-53.

Patel, S.A. and R.P. Chhabra, 2014, Heat transfer in Bingham

plastic fluids from a heated elliptical cylinder, Int. J. Heat Mass

Tran. 73, 671-692.

Patel, S.A. and R.P. Chhabra, 2015, Laminar free convection in

Bingham plastic fluids from a heated elliptical cylinder, J.

Thermophys. Heat Tran. doi:10.2514/1.T4578.

Prashant and J.J. Derksen, 2011, Direct simulations of spherical

particle motion in Bingham liquids, Comput. Chem. Eng. 35,

1200-1214.

Putz, A. and I.A. Frigaard, 2010, Creeping flow around particles

in a Bingham fluid, J. Non-Newton. Fluid Mech. 165, 263-280.

Song, D., R.K. Gupta, and R.P. Chhabra, 2009, Wall effects on a

sphere falling in power-law fluids in cylindrical tubes, Ind.

Eng. Chem. Res. 48, 5845-5856.

Song, D., R.K. Gupta, and R.P. Chhabra, 2010, Effect of block-

age on heat transfer from a sphere in power-law fluids, Ind.

Eng. Chem. Res. 49, 3849-3861.

Song, D., R.K. Gupta, and R.P. Chhabra, 2011, Drag on a sphere

in Poiseuille flow of shear-thinning power-law fluids, Ind. Eng.

Chem. Res. 50, 13105-13115.

Song, D., R.K. Gupta, and R.P. Chhabra, 2012, Heat transfer to

a sphere in tube flow of power-law liquids, Int. J. Heat Mass

Tran., 55, 2110-2121.

Stokes, G.G., 1851, On the effect of the internal friction of fluids

on the motion of pendulums, Trans. Cambridge Phil. Soc. 9, 8-

106.

Suresh, K. and A. Kannan, 2011, Effect of particle diameter and

position on hydrodynamics around a confined sphere, Ind.

Eng. Chem. Res. 50, 13137-13160.

Suresh, K. and A. Kannan, 2012, Effect of particle blockage and

eccentricity in location on the non-Newtonian fluid hydrody-

namics around a sphere, Ind. Eng. Chem. Res. 51, 14867-14883.

Walters, K. and R.I. Tanner, 1992, The motion of a sphere

through an elastic fluid, In: Chhabra, R.P. and D. De Kee, eds.,

Transport Processes in Bubbles, Drops, and Particles, Hemi-

sphere, New York, 73-86.

Wham, R.M., O.A. Basaran, and C.H. Byers, 1996, Wall effects

on flow past solid spheres at finite Reynolds number, Ind. Eng.

Chem. Res. 35, 864-874.

Yu, Z. and A. Wachs, 2007, A fictitious domain method for

dynamic simulation of particle sedimentation in Bingham flu-

ids, J. Non-Newton. Fluid Mech. 145, 78-91.

Effect of confinement on forced convection from a heated sphere in Bingham plastic … · 2015. 5....

Documents

Transcript of Effect of confinement on forced convection from a heated sphere in Bingham plastic … · 2015. 5....