Effect of confinement on forced convection from a heated sphere in Bingham plastic fluidsPradipta K. Das

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DOI: 10.1007/s13367-015-0009-9

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

Pradipta K. Das¹, Anoop K. Gupta¹, 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 Bingham plastic fluids have been studied. The governing differential equations (continuity, momentum and thermal energy) 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 is defined as the ratio of the sphere to tube diameter. Over this range of conditions, the flow is expected to be axisymmetric and steady. The detailed flow and temperature fields in the vicinity of the surface of the sphere are examined in terms of the streamline and isotherm contours respectively. Further insights are developed in terms of the distribution of the local Nusselt number along the surface of the sphere together with their average values in terms of mean Nusselt number. Finally, the wall effects on drag are present only when the fluid-like region intersects with the boundary wall. However, heat transfer is always influenced by 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 number than that in Newtonian fluids. The paper is concluded by developing predictive correlations for drag and Nusselt 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 solution 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 settling 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 fluids 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 Rim Conference on Rheology, held in the University of Melbourne, Australia from 20^th to 25^th July 2014.

*Corresponding author; E-mail: chhabra@iitk.ac.in

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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 transfer 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, however, desirable to present a terse review of the prior developments 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 numerical 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 adhering 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 particular (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 coefficient 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-dimensional 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),

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elliptical cylinder (Patel and Chhabra, 2014, 2015) and a square bar (Nirmalkar et al., 2013c), etc. While these studies 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 particular, in this work the governing partial differential equations (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 augmentation 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 T_w (> ) and the wall of the tube is assumed to be adiabatic. The confinement or the blockage ratio, λ, is simply defined by the ratio of the diameter of sphere to that of the tube (λ = d/D). The thermo-physical properties of the fluid (density, ρ ; thermal conductivity, 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 second 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 = T_w–T_∞

T_w+T_∞

( )/2

Br = μBU_∞²/k T( _w–T_∞) = 0.002<<<1

1 ---r∂ rU( _r)

--- + ∂r ∂U_z --- = 0∂z

U_r∂U_r

--- + U∂r _z∂U_r --- = −∂∂z P

--- + ∂r 1 Re--- 1

---r∂ rτ( _rr) ---∂r ∂τ_zr

---∂z

⎝ + ⎠

⎛ ⎞

U_r∂U_z

--- + U∂r _z∂U_z --- = ∂z −∂p

--- + ∂z 1 Re--- 1

---r∂ rτ( _rz) ---∂r ∂τ_zz

---∂z

⎝ + ⎠

⎛ ⎞

U_r∂ξ --- + U∂r _z∂ξ

--- = ∂z 1 Re Pr× --- 1

--- ∂r

∂r--- r∂ξ ---∂r

⎝ ⎠

⎛ ⎞ + ∂²ξ

∂z² ---

⎝ ⎠

⎛ ⎞

τ = ηγ·

Fig. 1. (Color online) Schematics of the flow geometry and computational domain.

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, if , (6)

, if . (7)

Clearly, in the present form, Eqs. (6) and (7) are discontinuous, 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, sufficiently large values of m would result in accurate predictions 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 literature (Glowinski and Wachs, 2011; Balmforth et al., 2014).

The problem statement is completed by identifying the physically realistic boundary conditions for the configuration studied herein. On the surface of the sphere, the usual no-slip (i.e., ) and the condition of constant temperature, = 1 are used. At the inlet of the tube, a uniform velocity in the z-direction and uniform temperature 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-developed 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 dimensionless using d, and as the characteristic length, velocity and viscosity scales, respectively. The temperature is non-dimensionalized as . Evi- dently, the velocity and temperature fields in the present case are expected to be functions of the four dimensionless 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 factor of (1 + Bn) as:

and . (14)

This approach thus not only eliminates the Bingham number from the set of dimensionless parameters, but it has also proved to be effective in consolidating both experimental 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 coordinates 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 predictive correlations for drag and Nusselt number.

τ = 1 Bn II_γ·

---

⎝ + ⎠

⎛ ⎞ γ· II_τ>Bn² γ· = 0 II_τ≤Bn²

τ = 1 Bn 1 exp[ – (–m II_γ·)] II_γ·

---

⎝ + ⎠

⎜ ⎟

⎛ ⎞

γ·

m→∞

μ_y >> μ_B

τ = μ_y μ_B ---

⎝ ⎠

⎛ ⎞γ· II_τ≤Bn² τ = 1 + Bn

II_γ·

--- 1 1 μ_y/μ_B ---

⎝ – ⎠

⎛ ⎞ γ· II_τ>Bn²

U_r = U_z = 0 ξ

U_r = 0, U_z = 1 & ξ = 0

∂ϕ/∂z = 0 ϕ = U_r U_z

U_r = 0, U_z = 1 & ∂ξ --- = 0∂r

U_∞ μ_B

ξ = T′ T( – ∞)/ T( w–T_∞)

Bn = τ₀d μ_BU_∞ ---

Re = ρdU_∞ μ_B ---

Pr = Cμ_B ---k

μ_B + τ₀d/U_∞

( )

Re^* = Re 1 Bn+

--- Pr^* = Pr 1 Bn( + )

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Finally, the numerical solution of the preceding governing 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 primitive 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 surfaces 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, stationary 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 tolerance of 10⁻⁶ is used as convergence criteria for the continuity, momentum and energy equations.

The influence of the numerical aspects, namely, the size of the computational box (values of L_u and L_d), computational 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 characteristics 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 therefore 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

= L_d = 50d was seen to be adequate over the range of conditions 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 values 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), N_p

= 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 artifacts. 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/d Bn = 0 Bn = 100

C_D C_DP Nu C_D C_DP 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

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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: L_u= L_d = 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 m Bn = 0.1 Bn = 100

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 Re Papanastasiou model Bi-viscosity model

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

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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 isotherm 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 results

Reliable 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; needless to add here that the analogous results for an unconfined 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 comparison 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 computational 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 Re Nirmalkar et al. (2013a) Gupta and Chhabra (2014) Present

C_D C_DP Nu C_D C_DP Nu C_D C_DP 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.

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the present results for a confined sphere in Bingham plastic 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 contours

Typically, 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 Re_I 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).

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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 confining 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).

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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 regions

Intuitively, 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 number. In contrast, for a confined sphere with moderate blockage ratio of λ = 0.2 (Fig. 6b), the results are qualitatively 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).

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= 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 coefficient, in line with the experimental results available in the literature (Atapattu et al., 1990). However, any small differences present can safely be ascribed to the numerical artifacts and due to the approximate nature of the regularization 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 gradients on the surface of the sphere.

5.4. Drag coefficient

Due to the prevailing shearing and normal stress components 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 inspection 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 dependence 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 physical wall. As seen in Fig. 8a, for λ = 0.1 and λ = 0.2, this is not the case and consequently, the drag coefficient values are identical to the corresponding values for an unconfined 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 confining 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 blockage 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).

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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 to Bn→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.

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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 C_DF/ C_DP. 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 C_DF/C_DP

= 0.44-0.45 regardless of the values of λ and Re. This suggests that the pressure drag increases faster than the friction 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 C_DF/C_DP 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 number

Figures 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

C_D = 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 (C_DF/C_DP) on the Bingham number, Reynolds number and blockage ratio (λ).

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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 numbers 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 ~ 126-127^o), but beyond this point ~ , the Nusselt number increases a little bit in Newtonian fluids 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.

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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 ~ 10ô to 40ô) 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 ≤ ~45ô and at θ ~ 90ô. 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 acceleration 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 number

It 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.

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averaged Nusselt number which is frequently needed in process design calculations. The scaling arguments suggest 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 values 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 increasing Reynolds number and/or with the blockage ratio. The positive dependence on the Reynolds number can readily be explained via the classical boundary layer considerations. 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, j_H, 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

j_H = 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.

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and of . 6. Conclusions

In this work, the wall effects on the drag and Nusselt number for an isothermal sphere falling axially in a cylinder filled with Bingham fluids have been studied numerically 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 countered 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, however, 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 decreasing 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 j_H ~ 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.

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Fig. 11. (Color online) Dependence of the average Nusselt number on Bingham number and Reynolds number at (a) Pr = 1 and (b) Pr = 100.

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Nomenclatures

Bn : Bingham number (−)

C : Specific heat of fluid (J/kg·K) C_D : Drag coefficient (−)

C_DF : Friction drag coefficient (−)

C_DP : Pressure or form drag coefficient (−) C_p : Pressure coefficient (−)

C_s : Stokes drag coefficient (−) d : Diameter of sphere (m)

D : Diameter of the cylindrical tube (m) F_D : Drag force (N)

F_DF : Friction drag force (N) F_DP : Pressure drag force (N) F_s : Stokes drag force (N)

h : Local heat transfe coefficient (W/m²·K) k : Thermal conductivity of fluid (W/m·K) L_d : Downstream length (m)

L_u : Upstream length (m) m : Regularization parameter (−)

N_p : Number of control volumes on the surface of sphere (−)

Nu_θ : Local Nusselt number (−) Nu : Average Nusselt number (−) P : Pressure (−)

p₀ : Reference pressure (Pa)

p_s : 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)

T_w : Temperature on the surface of the sphere (K) U_r : r-component of the velocity (−)

U_z : 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 (−) Subscripts

r, 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 viscoplastic 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, 2^nd ed., CRC Press, Boca Raton.

Chhabra, R.P. and J.F. Richardson, 2008, Non-Newtonian Fluid and Applied Rheology: Engineering Applications, 2^nd 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∞2πd²

( )

8≡ F_DF/ρU_∞²πd²

( )

8≡ FDP/ρU∞2πd²

( )

( 2≡ p( _s–p₀) /ρU_∞²) F≡ / 3πμ_s ( _BdU_∞)

( )

T_∞

U_∞

γ·

T′ T– ∞

T_w–T_∞ ---

⎝≡ ⎠

⎛ ⎞

II_τ II_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

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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 convection 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 blockage 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 hydrodynamics 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 fluids, J. Non-Newton. Fluid Mech. 145, 78-91.