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© 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),
Effect of confinement on forced convection from a heated sphere in Bingham plastic fluids
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
P.K. Das, Anoop K. Gupta, N. Nirmalkar and R.P. Chhabra
78 Korea-Australia Rheology J., 27(2), 2015
, 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+( )
Effect of confinement on forced convection from a heated sphere in Bingham plastic fluids
Korea-Australia Rheology J., 27(2), 2015 79
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
P.K. Das, Anoop K. Gupta, N. Nirmalkar and R.P. Chhabra
80 Korea-Australia Rheology J., 27(2), 2015
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
Effect of confinement on forced convection from a heated sphere in Bingham plastic fluids
Korea-Australia Rheology J., 27(2), 2015 81
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.
P.K. Das, Anoop K. Gupta, N. Nirmalkar and R.P. Chhabra
82 Korea-Australia Rheology J., 27(2), 2015
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).
Effect of confinement on forced convection from a heated sphere in Bingham plastic fluids
Korea-Australia Rheology J., 27(2), 2015 83
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).
P.K. Das, Anoop K. Gupta, N. Nirmalkar and R.P. Chhabra
84 Korea-Australia Rheology J., 27(2), 2015
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
(flow is from bottom to top).
Fig. 6(b). Structure of fluid-like and solid-like regions at λ = 0.2
(flow is from bottom to top).
Effect of confinement on forced convection from a heated sphere in Bingham plastic fluids
Korea-Australia Rheology J., 27(2), 2015 85
= 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
(flow is from bottom to top).
P.K. Das, Anoop K. Gupta, N. Nirmalkar and R.P. Chhabra
86 Korea-Australia Rheology J., 27(2), 2015
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
(flow is from bottom to top).
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.
Effect of confinement on forced convection from a heated sphere in Bingham plastic fluids
Korea-Australia Rheology J., 27(2), 2015 87
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).
P.K. Das, Anoop K. Gupta, N. Nirmalkar and R.P. Chhabra
88 Korea-Australia Rheology J., 27(2), 2015
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.
Effect of confinement on forced convection from a heated sphere in Bingham plastic fluids
Korea-Australia Rheology J., 27(2), 2015 89
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.
P.K. Das, Anoop K. Gupta, N. Nirmalkar and R.P. Chhabra
90 Korea-Australia Rheology J., 27(2), 2015
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.
Effect of confinement on forced convection from a heated sphere in Bingham plastic fluids
Korea-Australia Rheology J., 27(2), 2015 91
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.
P.K. Das, Anoop K. Gupta, N. Nirmalkar and R.P. Chhabra
92 Korea-Australia Rheology J., 27(2), 2015
Fig. 11. (Color online) Dependence of the average Nusselt number on Bingham number and Reynolds number at (a) Pr = 1 and (b)
Pr = 100.
Effect of confinement on forced convection from a heated sphere in Bingham plastic fluids
Korea-Australia Rheology J., 27(2), 2015 93
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
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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
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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
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