Effect of buoyancy-assisted flow on convection from an isothermal spheroid in power-law fluids

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Korea-Australia Rheology Journal, 28(2), 87-110 (May 2016) DOI: 10.1007/s13367-016-0009-4

www.springer.com/13367

Effect of buoyancy-assisted flow on convection from an isothermal spheroid in power-law fluids

Anoop K. Gupta and Rajendra Prasad Chhabra*

Department of Chemical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India (Received November 16, 2015; final revision received March 10, 2016; accepted March 12, 2016) In this work, the coupled momentum and energy equations have been solved to elucidate the effect of aiding-buoyancy on the laminar mixed-convection from a spheroidal particle in power-law media over wide ranges of the pertinent parameters: Richardson number, 0≤Ri≤5; Reynolds number, 1≤Re≤100; Prandtl number, 1≤Pr≤100; power-law index, 0.3≤n≤1.8, and aspect ratio, 0.2≤e≤5 for the case of constant thermo-physical properties. New results for the velocity and temperature fields are discussed in terms of the streamline and isotherm contours, surface pressure and vorticity contours, drag coefficient, local and surface averaged Nusselt number. The effect of particle shape on the flow is seen to be more pronounced in the case of oblates (e < 1) than that for prolates (e > 1). The propensity for wake formation reduces with the rising values of power-law index, Richardson number and slenderness of the body shape (e > 1). Also, the drag coefficient is seen to increase with the Richardson number and power-law index. All else being equal, the Nusselt number shows a positive dependence on the Richardson number and Reynolds number and an inverse dependence on the power-law index and aspect ratio of the spheroid. Limited results were also obtained by considering the exponential temperature dependence of the power-law consistency index. This factor can increase the values of the average Nusselt number by up to ~10-12% with reference to the corresponding values for the case of the constant thermo-physical properties under otherwise identical conditions. Finally, the present values of the Nusselt number have been consolidated in the form of Colburn j-factor as a function of the modified Reynolds and Prandtl numbers for each value of the aspect ratio (e).

The effect of the temperature dependent viscosity is included in this correlation in terms of a multiplication factor.

Keywords: spheroid, oblate, prolate, power-law fluids, Richardson number, Nusselt number

1. Introduction

From a theoretical standpoint, external flow over axisymmetric shaped objects like a sphere or a spheroid con- stitutes a classical problem in the realm of transport phenomena to uncover the various facets like flow regimes and the criteria for transition from one regime to another, loss of axisymmetry and steadiness, wake phenomena, vortex shedding, etc. These aspects influence the hydrodynamic drag and/or the convective heat transfer coefficient between an object and a non-Newtonian fluid. Also, the flow over and heat transfer from a heated object in non-Newtonian fluids denote an idealization of several industrially important processes. Typical examples include the heating and cooling of food suspensions (containing peas, potatoes, mushrooms, etc.), processing of pharma- ceutical products and drilling muds wherein the particle shape is used to modulate their viscosity, etc. Additional examples are found in blood flow and biomedical engineering applications, advanced aviation fuels which must be employed in an efficient manner in diverse settings such as their flow in porous structures employed as heat

sinks (Getachew et al., 1998), flow over surfaces (Yao and Molla, 2008; Molla and Yao, 2008; Sasmal and Chhabra, 2014; Patel and Chhabra, 2016), etc. While most multi- phase, macromolecular and/or structured fluids like worm- like micellar systems exhibit a range of non-Newtonian characteristics including shear-dependent viscosity, yield stress, time-dependent, and visco-elastic effects, but per- haps the most common aspect is the so-called shear-thinning and shear-thickening viscosity (Bird et al., 1987;

Chhabra, 2006; Chhabra and Richardson, 2008) which is frequently approximated by the simple power-law fluid model. This work is concerned with the buoyancy-assisted mixed-convection heat transfer in power-law fluids from an isothermal spheroid in the steady flow regime.

2. Previous Work

A cursory inspection of the pertinent literature shows that very little information is available on mixed-convection heat transfer even from a sphere in Newtonian fluids, let alone for spheroids in power-law fluids (Clift et al., 1978; Jaluria and Gebhart, 1998; Michaelides, 2006; Mar- tynenko and Khramstov, 2005). The pertinent limited literature for a sphere in Newtonian and power-law fluids

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

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has been reviewed, amongst others, by Chen and Mucoglu (1977), Mograbi and Bar-Ziv (2005a, 2005b), Kotouc et al. (2008, 2009), Bhattacharyya and Singh (2008) and Nirmalkar and Chhabra (2013, 2016). In particular, Chen and Mucoglu (1977) solved the boundary layer equations for convection in air from an isothermal sphere in the aiding- and opposing-buoyancy configurations including the limiting cases of the forced (zero Richardson number, i.e., Ri = 0) and the free (infinitely large Richardson number) convection regimes. They proposed that the forced convection effects begin to manifest at Ri > 1.67 for the aiding-buoyancy case and at Ri <−1.33 for the opposing- buoyancy case. Conversely, they suggested that the flow regime can be well approximated as the free convection regime for Ri > ~100. These values are, however, likely to be dependent on the value of the Prandtl number. On the other hand, Kotouc et al. (2008) have delineated different regimes for the buoyancy-assisted flow from an isothermal sphere in air. Depending upon the values of the Reyn- olds (Re≤1500) and Richardson (0 ≤Ri ≤0.7) numbers, the boundary layer could separate at or off the flow axis.

It is useful to add here that the buoyancy-induced current superimposed on the forced flow suppresses the propensity for flow separation. They identified two critical Reyn- olds numbers, Re_c and Re_s, denoting the onset of boundary layer separation on the flow axis and off the flow axis, respectively. Both values were seen to increase with the increasing Richardson number in air (Pr = 0.72) and water (Pr = 7). Further increase of the Reynolds number resulted in the recirculation only at the axis and this transition was denoted by Re_r> Re_s> Re_c. More significantly, they reported that for Re < Re_s and for Re > Re_c, the behaviour was like that for an unheated sphere. For instance, in air, up to Ri = 0.4, the value of Re_c= 153.4 is much higher than that of Re_c= 20.8 for the onset of flow separation (on flow axis) for an unheated sphere. In the region 0.4≤Ri ≤ 0.7, after meandering into the off-axis separation, the recirculation region returned to the flow axis at Re_s= 959.

Finally, the occurrence of the primary bifurcation is seen at a much higher value of 1471 for Ri = 0.7 in air in contrast to the oft-quoted value of ~212 for an unheated sphere. Based on their extensive numerical results, Kotouc et al. (2008) developed a flow regime map in Re-Ri space in terms of the regions of attached flow, recirculation at the axis and off the axis, and finally, the onset of regular bifurcation. On the other hand, Bhattacharyya and Singh (2008) numerically solved the governing equations for the aiding-buoyancy mixed convection from an isothermal sphere in air with and without considering the conduction inside the sphere over the range 1≤Re ≤200 and Gr ≤ 6×10⁴ thereby yielding the values of the Richardson number in the range 0≤Ri ≤1.5. For the two cases of with and without the conduction inside the sphere, their results are consistent with the prior literature results for Ri = 0.

Some of their results obviously (Ri < 0.7) are in the so- called attached flow and recirculation at the axis regimes according to the flow regime map of Kotouc et al. (2008).

Needless to say that depending upon the prevailing flow regime, the scaling of the drag coefficient and the Nusselt number with Re and Ri will vary from one regime to another. The corresponding case of unsteady forced convection heat transfer from a spheroid in Newtonian fluids at finite Reynolds numbers was studied by Juncu (2010) for two values of the Prandtl number of Pr = 1 and Pr = 10.

The analogous limited studies, especially for heat transfer in non-Newtonian fluids, with spherical and spheroidal particles are briefly reviewed here. Early studies of Trip- athi and co-workers (Tripathi et al., 1994; Tripathi and Chhabra, 1995) numerically solved the momentum equations for the steady axisymmetric range and provided the values of drag coefficient only in shear-thinning (n < 1) and shear-thickening (n > 1) fluids in the forced convection regime (Re≤100, 0.4 ≤n ≤1.8). Subsequently, these have been extended to forced-convection heat transfer from an unconfined (Kishore and Gu, 2011a; Alassar, 2005; Sreenivasulu et al., 2014; Srinivas and Ramesh, 2014), a confined (Kishore and Gu, 2011b; Reddy and Kishore, 2014) and two-spheroid assemblies (Kishore, 2012; Rathore et al., 2013) in Newtonian and power-law fluids in the steady flow regime. Broadly, all else being equal, shear-thinning fluid behaviour (n < 1) enhances the rate of heat transfer over and above that obtained in New- tonian fluids in all cases. This can safely be ascribed to the lowering of the effective viscosity of fluid in the vicinity of the immersed object due to intense shearing levels in this region. Indeed, it is possible to achieve enhancement in heat transfer even of the order of 60-70% under appropriate values of the Reynolds number, Prandtl number, and power-law index. On the other hand, shear-thickening fluid behaviour (n > 1) was shown to impede heat transfer by up to 15-20%. Indeed, these trends have also been reported for a sphere in power-law fluids (Nirmalkar and Chhabra, 2013; 2016; Dhole et al., 2006; Prhashanna and Chhabra, 2010) in the forced-, free-, and aiding-buoyancy mixed convection regimes. Evidently, this body of knowl- edge is nowhere near as extensive as that for spheroids in Newtonian fluids.

While a sphere denotes the limiting case of a spheroid with e = 1, as far as known to us, only Sreenivasulu and Srinivas (2015) have recently numerically studied aiding- buoyancy mixed-convection from an isothermal spheroid for two values of the Prandtl number 1 and 5, aspect ratio in the range 0.5 to 1.5, and Richardson number up to Ri = 2. They consolidated their numerical results of the surface averaged Nusselt number via a correlation which is in the form of a correction factor to be applied to the corresponding value of the Nusselt number for a sphere.

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Effect of buoyancy-assisted flow on convection from an isothermal spheroid in power-law fluids

They also reported the influence of the Reynolds and Richardson numbers on the distribution of the local Nus- selt number to be strongly modulated by the shape of the spheroid. Also, there have been a few studies on aiding- buoyancy mixed convection from axisymmetric objects in Newtonian fluids employing the boundary layer equations from which one can deduce the results for prolates and oblates (Raithby et al., 1976; Lee et al., 1991; Meissner et al., 1994; Eslami and Jafarpur, 2012). These results combined with experimental results have yielded useful predictive relations for the estimation of the average Nusselt number in Newtonian fluids for spheroids, albeit these results are neither as coherent nor as extensive as that for a sphere in Newtonian fluids (Martynenko and Khrams- tov, 2005; Morgrabi and Bar-Ziv, 2005a).

Based on the foregoing discussion, it is thus fair to con- clude that very little information is available on heat transfer from spheroidal particles in Newtonian and power-law fluids in general and in the mixed convection regime especially at high Prandtl numbers. This work endeavours to close this gap in the literature. In particular, the coupled momentum and energy equations have been solved numerically for the aiding buoyancy flow configuration for an isothermal spheroid in power-law fluids for the following ranges of parameters: Richardson number, 0≤ Ri≤5; Reynolds number, 1 ≤Re ≤100; Prandtl number, 1≤Pr ≤100; power-law index, 0.3 ≤n ≤1.8, and aspect ratio of the spheroid 0.2≤e ≤5. This work thus extends the study of Nirmalkar and Chhabra (2013, 2016) on mixed-convection from a sphere, on one hand, to show the effect of particle shape and of Sreenivasulu and Srinivas (2015) and of Gupta et al. (2014) for spheroids, on the other hand, to delineate the influence of power-law viscosity on mixed convection from an isothermal spheroid in the buoyancy-assisted regime.

3. Problem Definition and Mathematical Formu- lation

The geometric flow configuration investigated here is identical to that used in our previous studies (Gupta et al., 2014; Gupta and Chhabra, 2014; 2016) dealing with the free-convection in power-law fluids and forced-convection in visco-plastic fluids. In brief, a spheroid (with semi- axes a, b as perpendicular and parallel to the directions of flow, respectively) heated to a constant temperature T_w is immersed in a uniform stream of a power-law fluid (with free-stream velocity at temperature where T_w >

), Fig. 1a. It is thus possible to create a range of shapes by simply varying the aspect ratio, e = b/a. Also, following the strategy employed in our recent studies (Gupta et al., 2014; Gupta and Chhabra, 2014; 2016), the unconfined flow approximation is reached here by enclosing the spheroid in a fictitious concentric envelope of fluid of

diameter . In addition to the imposed external flow, the temperature difference between the spheroid and the ambient fluid induces a buoyancy current in the upward direction. In view of the small values of ΔT (< 5 K) pre- scribed here, the thermo-physical properties of the fluid (C, k, m, and n) are assumed to be temperature-invariant (The impact of this assumption on the results is revisited in a later section 5.5) except the density term in the buoyancy force in the momentum equation (i.e., ρ = is used everywhere except in the buoyant force term). Since the maximum temperature difference in the system is small, ΔT = Tw− = 5 K, it is justified to use the Boussinesq approximation to capture the linear variation of fluid density with temperature, i.e., − ρ = β(T − ) where β is the volumetric thermal expansion coefficient of the fluid. Similarly, the viscous dissipation effects are neglected as the maximum value of the Brinkman number, Br = is found to be 0.005 which is

<< 1. Furthermore, since the resulting values of the Eckert

U_∞ T_∞

T_∞

D_∞

ρ_∞

T_∞

ρ_∞ ρ_∞ T_∞

mU_∞^{n 1}⁺ /k 2a( )^{n 1}^– (T_w–T_∞)

Fig. 1. (Color online) (a) Schematics of the flow and computational domain and (b) Typical grid structure for e = 0.5.

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number, Ec = Br/Pr, are also very small, one can safely ignore the pressure work term in the energy equation.

Under these conditions, the coupled velocity and temperature fields are governed by the following differential equations (in dimensionless forms):

Continuity equation:

. (1)

Momentum equation:

. (2)

Here, δij= 1 if (i, j) = y and δij= 0 if . Energy equation:

. (3)

In Eq. (2), the body force term is approximated by an equivalent buoyant force term given by

via the Boussinesq relation. The extra stress tensor (τ) for an incompressible fluid is written as:

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. (5)

The scalar viscosity function (η) for a power-law model fluid is given by:

. (6)

In Eq. (6), m and n are the power-law consistency and flow behaviour index (or power-law index), respectively.

I₂ is the second invariant of the rate of deformation tensor (Bird et al., 1987). Eqs. (1)-(3) have been rendered dimensionless by using the free-stream velocity as the velocity scale and 2a as the length scale. Thus, the pressure is scaled using , the fluid viscosity by using , and the stress components by , etc. The fluid temperature is made dimensionless as . Dimensionless considerations suggest the coupled velocity and temperature fields to be influenced by the values of the five dimensionless numbers, namely, Reynolds number (Re), Prandtl number (Pr), Richardson number (Ri) (or the combinations thereof such as Grashof number and Peclet number), power-law index (n), and aspect ratio of the spheroid (e); these are defined as follows:

Reynolds number:

. (7)

Prandtl number:

. (8)

Richardson number:

. (9)

where Gr is the Grashof number defined as:

. (10)

Of course, the power-law index (n) and aspect ratio (e) are dimensionless on their own. In the literature, another dimensionless parameter, Peclet number (Pe = Re·Pr) is also used. It is a measure of the relative strengths of the convective to molecular diffusion (conductive) transport.

Thus, small values of Pe denote “weak” convection regime.

Finally, the problem statement is completed by prescrib- ing the suitable boundary conditions. The usual no-slip condition for flow (V = 0) and isothermal condition for temperature (ξ = 1) is used on the surface of the spheroid.

The lower half of the surrounding spherical envelope is designated as the inlet and on this surface the conditions of V_x= 0, V_y= 1, and ξ = 0 are used. Similarly, the rear half of the fluid envelope is treated as the outflow. On this plane, the usual Neumann boundary condition which assumes zero diffusive flux for all flow variables in the axial direction ( ) where ϕ = V, ξ is used. Also, for the range of conditions spanned here, the flow is expected to be symmetric about the y-axis and this condition has also been used here.

The numerical solution of Eqs. (1)-(3) maps the flow domain in terms of the primitive variables (V, p, ξ) which can be post processed to evaluate the characterizing parameters like streamlines and isotherm contours, pressure coefficient (C_p), drag coefficients (C_D, C_DP), and the local and average Nusselt numbers (Nu_θ, Nu). Since detailed discussions of their evaluation are available elsewhere (Gupta et al., 2014; Gupta and Chhabra, 2014, 2016), these are not repeated here. Suffice it to add here that, due to the coupling via the buoyancy term, both the fluid mechanical and heat transfer characteristics are governed by the values of the five dimensionless groups, namely, Ri, Re, Pr, n, and e. This work endeavours to explore the functional dependence of the drag coefficient and Nusselt number on each of these parameters.

4. Numerical Solution Methodology and Validation

4.1. Numerical details

Since the numerical solution procedure used in this

∇ V⋅ = 0

V⋅∇

( )V = ∇– p^* + 1

Re---(∇ τ⋅ ) + Riξδ_ij

( ) yi, j ≠

V⋅∇

( )ξ = 1

Re Pr⋅

( )

---∇²ξ

ρ_∞–ρ

( )g

gρ_∞β T T( – _∞)

τ = 2ηε u( )

ε u( ) = 1

2---[(∇V) ∇V+( )^T]

η = m I₂ ----2

⎝ ⎠⎛ ⎞⁽^{n 1}^– ^)/2

U_∞

ρ_∞U_∞² m U( _∞/2a)^{n 1}^–

m U( _∞/2a)ⁿ

ξ = T T( – _∞)/ T( _w–T_∞)

Re = ρ_∞( )2aⁿU_∞^{2 n}^– ---m

Pr = mC ---k U_∞

---2a

⎝ ⎠⎛ ⎞^{n 1}^–

Ri = Gr Re²

--- = gβ T( _w–T_∞) 2a( ) U_∞² ---

Gr = gβ T( _w–T_∞) 2a( )³ ρ_∞ ---m U_∞

---2a

⎝ ⎠

⎛ ⎞^{1 n}^–

⎩ ⎭

⎨ ⎬

⎧ ⎫²

∂ϕ/∂y = 0

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work is detailed elsewhere (Nirmalkar et al., 2013; Gupta et al., 2014), only the main points are recapitulated here.

The governing differential Eqs. (1)-(3), subject to the above-mentioned boundary conditions are solved numerically using the finite element method based tool Comsol Multiphysics (version 4.3a). Furthermore, “axisymmetric, steady and laminar flow” module is used together with physics of “heat transfer in fluids”. The resulting system of the algebraic equations is solved using PARDISO scheme with a relative convergence criterion of 10⁻⁶ for both the momentum and energy equations. This value was found to be sufficient as both the drag coefficient and Nusselt number values had stabilized at least up to 5 significant digits. A volumetric body force term is used to incorporate the buoyancy force via a user defined function. Very fine quadrilateral cells were used close to the surface of the spheroid to resolve the steep velocity and temperature gradients whereas non-uniform triangular elements were used away from the spheroid, Fig. 1b. In order to economize on the computational effort, Ri, n, and Pr were parameterized sequentially using the parametric sweep approach.

For the present results to be free from numerical arte- facts, an appropriate value of was chosen by system- atically varying the value of in the range 100 to 400 for e < 1 and 20 to 200 for e≥1. The values of

= 300 for oblates (e < 1) and = 100 for prolates (e≥1) were found to be adequate at the lowest values of the Reynolds number (Re = 1) and Prandtl (Pr = 1) numbers for the extreme values of the Richardson number (Ri = 0 and Ri = 5) and of the power-law index (n = 0.3 and n = 1.8). These values are comparable (and in most cases superior) to that used in the other studies avail-

able in the literature (Gupta et al., 2014; Gupta and Chhabra, 2014). Similarly, an optimum numerical mesh must meet two conflicting requirements, i.e., it must be sufficiently fine to resolve the thin boundary layers at the maximum values of Re, Pr, and Ri and at the extreme values of e and n used in this work (Table 1) without becom- ing exorbitantly computing resource intensive. Evidently, the grids labelled as G3 for e < 1 and G2 for e≥1 are found to be satisfactory, for the use of the finer mesh yielded substantially identical results at the expense of augmented computational effort. Furthermore, Fig. 2 shows the influence of the numerical mesh on the surface pressure and local Nusselt number distribution along the surface of the spheroid. On this count also, the choice of G3 for e < 1 and G2 for e≥1 is found to be satisfactory.

The smallest (largest) cell sizes ranged from δ/b = 0.022 (0.85) for e = 0.2 to δ/b = 0.007 (0.65) for e = 5. The corresponding degrees of freedom varied from 202,564 for e

= 0.2 to 153,692 for e = 5. Finally, the assumption of the steady and axisymmetric flow regime was corroborated by carrying out limited time-dependent as well as full domain three-dimensional simulations for the maximum values of Re = 100, Pr = 100, and Ri = 5 and for the extreme values of n and e. Indeed the flow was found to be steady and axisymmetric over the range of conditions spanned here.

The resulting values of the drag and Nusselt number devi- ated only by < 0.5% from those based on the assumption of the steady axisymmetric flow. Furthermore, the results of Kotouc et al. (2008) also suggest that the critical Reyn- olds numbers increase with both the Prandtl number and Richardson number. Therefore, the assumption of the steady and axisymmetric flow regime inherent in this work is justified as the values of the Prandtl and Rich- D_∞

D_∞/2b

( )

D_∞/2b

( ) (D_∞/2b)

Table 1. Grid independence study at Re = 100 and Pr = 100.

Grid N_p Elements^*

Ri = 0 Ri = 5

n = 0.3 n = 1.8 n = 0.3 n = 1.8

C_D C_DP Nu C_D C_DP Nu C_D C_DP Nu C_D C_DP Nu

e = 0.2

G1 300 47036 0.8896 0.8703 71.302 1.6498 1.1951 36.652 2.1738 2.1399 79.320 2.6038 1.9792 32.051 G2 400 74836 0.8904 0.8711 70.975 1.6492 1.1948 36.554 2.1706 2.1368 78.788 2.6034 1.9791 32.030 G3 500 97648 0.8966 0.8770 69.281 1.6484 1.1944 36.131 2.1637 2.1298 78.636 2.6017 1.9780 32.009 G4 600 144686 0.8986 0.8790 68.992 1.6483 1.1944 35.949 2.1619 2.1280 78.528 2.6010 1.9777 31.996 e = 1

G1 200 40394 0.5785 0.4259 53.500 1.6187 0.544 27.072 0.8796 0.5365 81.195 2.7807 0.9625 29.821 G2 300 79560 0.5796 0.4274 53.408 1.6185 0.5443 27.033 0.8784 0.5363 81.169 2.7802 0.9629 29.784 G3 400 111432 0.5804 0.4285 53.382 1.6184 0.5443 27.027 0.8779 0.5361 81.145 2.7798 0.9630 29.754 e = 5

G1 200 42818 0.9082 0.1731 26.697 2.1683 0.1372 15.211 2.1135 0.2318 50.227 8.2678 0.3413 18.757 G2 300 74048 0.9081 0.1732 26.758 2.1677 0.1373 15.213 2.0849 0.2324 50.471 8.2645 0.3418 18.750 G3 400 107980 0.9079 0.1732 26.767 2.1674 0.1373 15.202 2.0759 0.2326 50.536 8.2623 0.3419 18.738

*corresponds to the half computational domain.

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ardson numbers are quite large.

In summary, the new results obtained in this study are based on the following numerical parameters: = 300 and G3 for e < 1 and = 100 and G2 for e≥ 1. Further justification of these selections is presented in the next section where a few benchmark comparisons with the previous studies are reported.

4.2. Validation of results

Since detailed benchmark comparisons for the fluid- mechanical and heat transfer aspects for spheroids in Newtonian fluids in the forced- and free-convection regimes have been reported elsewhere (Gupta et al., 2014;

Gupta and Chhabra, 2014), these are not repeated here.

Suffice it to add here that in the forced-convection limit ( ), the present predictions are within 1-1.5% of the previous results available in the literature for both New- tonian and power-law fluids (Srinivas and Ramesh, 2014;

Reddy and Kishore, 2014; Gupta and Chhabra, 2014).

Therefore, only the additional comparisons are included here. Alassar and Badr (1999) employed the series trun- cation method to solve the stream function-vorticity form of the Navier-Stokes equations for low Reynolds number (0.1≤Re ≤1) flow over an oblate spheroid. Fig. 3a shows a comparison of the present values of the surface vorticity with their results, the agreement is seen to be almost per- fect for e = 0.906 but it is less good for e = 0.245, though the shape of the vorticity contour is almost the same in the two cases. To further corroborate the present results, the predictions based on grid G3 are also included here and are seen to be identical to that for grid G2. On the other hand, the present predictions of the surface pressure are virtually indistinguishable from that reported by Nir- malkar and Chhabra (2013, 2016) for a sphere (Fig. 3b).

Very recently, Sreenivasulu and Srinivas (2015) have studied aiding-buoyancy mixed-convection from a spheroid in Newtonian fluids and the present values of the local Nus- selt number are compared with their predictions in Fig. 3c.

The agreement is seen to be fair in this case also. Once

D_∞/2b

( )

D_∞/2b

( )

Ri→0

Fig. 2. Effect of grid resolution at Re = 100 and Pr = 100: (a) Pressure coefficient, C_p, and (b) local Nusselt number, Nu_θ.

Fig. 3a. (Color online) Comparison of surface vorticity (based on length a) in Newtonian fluids at Ri = 0 and Re = 1 with the results of Alassar and Badr, (1999). Filled symbols: Alassar and Badr, (1999), lines: grid G2, and hollow symbols: grid G3.

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again, the predictions based on grid G3 are also included here and are again indistinguishable from that for G2.

Similarly, the present values of the drag and Nusselt number for Ri = 0 are found to be within ± 2% of that reported in the literature (Kishore and Gu, 2011a; Sreenivasulu and Srinivas, 2015), Table 2. However, the present results for average Nusselt number deviate up to 6% from Nirmalkar and Chhabra (2013, 2016) using the present grid and computational domain size. This difference decreases to ± 0.5%

if their computational parameters (domain, grid) are used and hence such small differences. While the discrepancies

of the order seen in Figs. 3a and 3c and in Table 2 are, though not at all uncommon in such studies (Roache, 2009), probably due to the relatively less refined mesh and/or shorter domain used in Nirmalkar and Chhabra (2013, 2016) than that employed here and/or suggest the possibility of lack of accuracy and/or incomplete convergence in the previous studies (Alassar and Badr, 1999;

Sreenivasulu and Srinivas, 2015). Thus, the present results are regarded to be more accurate than that of prior studies (Alassar and Badr, 1999; Sreenivasulu and Srinivas, 2015). Based on the preceding comparisons, the new results reported herein are regarded to be reliable to within 1.5-2%.

5. Results and Discussion

The new results obtained in this work endeavour to elu- Fig. 3b. Comparison of the pressure coefficient distribution over the sphere surface with that of Nirmalkar and Chhabra (2013, 2016) at Ri = 2 and Pr = 100.

Fig. 3c. (Color online) Local Nusselt number distribution over the spheroid surface. Filled symbols: Sreenivasulu and Srinivas (2015), lines: grid G2, and hollow symbols: grid G3.

Table 2. Comparison of the average Nusselt number values for Newtonian fluids (n = 1) at Re = 100 and Pr = 5.

Ri e

*Nirmalkar and Chhabra (2013, 2016)

*Sreenivasulu and Srinivas

(2015)

Present

0

0.66 − 14.22 14.17

1 13.04 13.15 13.13

2 − 11.00 10.95

1

0.66 − 14.01 14.11

1 13.73 13.84 13.78

2 − 12.38 12.29

2

0.66 − 15.86 15.82

1 14.46 15.60 15.54

2 − 14.18 14.24

*Nu values are read off from their figures.

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cidate the effects of the pertinent dimensionless parameters over the following ranges of conditions: Richardson number, 0≤Ri≤5; Reynolds number, 1≤Re≤100; Prandtl number, 1≤Pr ≤100; power-law index, 0.3 ≤n ≤1.8, and aspect ratio, 0.2≤e ≤5. We begin with a discussion

of the streamline and isotherm contours in the next section.

5.1. Flow kinematics

Figs. 4a-d show representative streamline (left half) and

Fig. 4a. (Color online) Streamline (left half) and isotherm (right half) contours at Re = 1 and n = 0.3: (i) Pr = 1 and (ii) Pr = 100.

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isotherm (right half) contours for a range of conditions in order to delineate the influence of the Reynolds number, power-law index, Richardson number, Prandtl number, and aspect ratio. For the sake of comparison, the limiting cases of forced-convection (Ri = 0) and/or of a sphere

(e = 1) are also included here. For a sphere, superimposing the buoyancy-induced flow on the forced flow is seen to stabilize the flow, i.e., the tendency for flow separation and wake formation is increasingly suppressed with the rising value of the Richardson number irrespective of the Fig. 4b. (Color online) Streamline (left half) and isotherm (right half) contours at Re = 1 and n = 1.8: (i) Pr = 1 and (ii) Pr = 100.

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value of the power-law index, e.g., see Figs. 4c and 4d.

This is in line with the extensive study of Kotouc et al.

(2008) for air. In this case, there is a well-formed wake at Re = 100 and Ri = 0, but it quickly disappears at Ri = 1 irrespective of the values of the power-law index and Prandtl number. This is naturally due to the increasing

plume strength with the increasing Richardson number which, in turn, does not allow the adverse pressure gradient to be set up in the rear of the object even at the high- est values of Re, Pr, and Ri used here. Qualitatively, this behaviour is further augmented for prolates (e > 1) due to the increasing extent of streamlining of the shape, as the Fig. 4c. (Color online) Streamline (left half) and isotherm (right half) contours at Re = 100 and n = 0.3: (i) Pr = 1 and (ii) Pr = 100.

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flow is seen to remain attached at e = 5 even at Re = 100 whereas for e = 2, a small wake is formed at Ri = 0 but it quickly disappears at Ri = 1. Fig. 5a shows the stabilizing influence of the Richardson number on the wake length where it is seen to gradually shorten with the increasing value of Ri. In contrast, in the case of blunt shapes (e < 1),

while the wake size decreases with the increasing value of the Richardson number, much higher values of the Rich- ardson number are required to completely suppress this tendency. Broadly, the smaller the value of e and/or the larger the value of Re, the higher is the expected critical value of the Richardson number to reach this condition.

Fig. 4d. (Color online) Streamline (left half) and isotherm (right half) contours at Re = 100 and n = 1.8: (i) Pr = 1 and (ii) Pr = 100.

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Also, all else being equal, the wake length bears a positive relation with the increasing degree of bluntness (decreasing value of e), eventually reaching that for a disk. Fur- thermore, the wakes are seen to somewhat shorter in shear-thickening fluids (n > 1) than that in shear-thinning fluids (n < 1) in relation to that in Newtonian fluids otherwise identical conditions. Finally, the Prandtl number is seen to influence the streamline contours very weakly.

Therefore, from the foregoing discussion one can postu-

late that there must exist a critical value of the Richardson number (Ri_crit) for all combinations of other dimensionless parameters beyond which the flow remains attached (no wake formation) to the surface of the particle. Fig. 5b shows the dependence of the critical Richardson number (Ri_crit) on the value of power-law index (n) for a range of Reynolds numbers and for two different values of aspect ratio (i.e., e = 0.5 and e = 1) and Prandtl number (Pr = 1 and 100). It is clear that Ri_crit shows a negative dependence on the increasing value of the power-law index and Prandtl number while it increases with the Reynolds number.

The isotherm contours are seen to be more or less a mir- ror image of the corresponding streamline contours. In overall terms, the thermal boundary layer is seen to pro- gressively thin with the increasing values of the Reynolds number, Prandtl number, and Richardson number. This is also seen to be the case with the decreasing value of the power-law index which is in line with the expected trends (Chhabra, 2006; Chhabra and Richardson, 2008). Based on these considerations, it is fair to postulate that the overall heat transfer for prolates (e > 1) must be lower than that for oblates (e < 1) due to the significant bending of isotherm contours in the latter case.

Additional insights into the detailed kinematics of the flow can be gained by examining the surface vorticity and pressure profiles, Figs. 6a and 6b. An inspection of Fig. 6a suggests the following overall trends. The shape of the spheroid exerts an overwhelming influence on the vorticity contours. Thus, for instance for oblates (e < 1), at low Fig. 5a. (Color online) Variation of recirculation length with the

Richardson number (Ri) and power-law index (n) for e = 0.5 and e = 1.

Fig. 5b. (Color online) Dependence of critical Richardson number (Ri_crit) on power-law index for (i) e = 0.5 and (ii) e = 1.

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Reynolds numbers, the vorticity varies very little along the surface up to where it exhibits its maximum value as the streamlines must bend appreciably here to negotiate the body contour and after this point, it again drops to similar levels as that on the front side of the oblate. Broadly, the effect of Richardson number is seen to be positive but it is seen to be much stronger in shear-thickening fluids than that in shear-thinning media. This is simply a con- sequence of the way the dependence of the effective viscosity on shear rate is modulated by the value of the power-law index, being negative for n < 1 and positive for n > 1. As the Reynolds number is gradually increased, the flow detaches itself from the surface of the spheroid and the surface vorticity can be seen to be negative in the rear in such cases, e.g., see the results for e = 0.2 at Re = 100.

No other new features are observed in this case. In contrast, for more streamlined shaped prolates (e > 1), the sur-

face vorticity shows qualitatively different behaviour. The surface vorticity increases from its zero value at the front stagnation point up to a short distance along the surface reaching its maximum value slightly downstream from here at about followed by its continual decrease before turning upward at (depending upon the value of the Richardson number) attaining another peak (higher than the previous value) just before the rear stagnation point. At high Reynolds numbers, the role of Richardson number is somewhat diminished depending upon the value of power-law index. Under no conditions, the surface vorticity is seen to become negative for e = 5 which is consistent with the fact that no wake is formed over this range of conditions for e = 5. In this case too, the effect of the Richardson number is seen to be much more prominent in the rear of the spheroid than that in the front.

Overall, the magnitude of surface vorticity is seen to θ 90≈ ^o

θ 5≈ ^o

θ 45 135≈ – ^o

Fig. 6a. (Color online) Variation of surface vorticity (ω): (i) Pr = 1 and (ii) Pr = 100 (θ = 0 represents front stagnation point).

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increase with the inertial effects (Re), buoyancy effects (Ri) but it decreases dramatically as the fluid behaviour changes from shear-thinning (n < 1) to shear-thickening (n > 1), thereby suggesting stronger rotational component in the former. Also, the surface vorticity decreases with the increasing degree of streamlining of the spheroid, i.e., with the increasing value of e. This discussion is con- cluded by presenting typical surface pressure profiles which directly determine the form drag component of the overall hydrodynamic drag exerted on the spheroid, Fig.

6b. In this case, the nature of the distribution is strongly influenced by the shape of the spheroid. For oblate shapes (e < 1), surface pressure gradually increases from its minimum value at the front stagnation point reaching its peak value at and eventually showing some recovery hereafter even at small Reynolds numbers. While the effect of Richardson number is seen to be virtually absent

in the front half, especially in shear-thinning fluids but its impact is clearly observed in the rear for all values of power-law index. Also, its effect is seen to flip-over in the front and rear. This is clearly due to the inherently different spatial decay of the viscous forces (dominant at low Re) and the buoyancy-forces away from the spheroid. Due to high fluid inertia at Re = 100, the surface pressure is seen to be maximum at the front stagnation point which decreases all the way up to before showing some recovery in the rear. Indeed, the smaller the value of Ri, the greater is the recovery in the rear. In contrast for prolates (e > 1), at low Reynolds numbers, the surface pressure is seen to decrease rapidly from its maximum value at the front stagnation point to attain its minimum value at the rear stagnation point, though in the case of shear-thickening fluids, the bulk of the spheroid surface is exposed to nearly a uniform pressure close to zero. This trend is also θ 90= ^o

θ 90= ^o

Fig. 6b. (Color online) Distribution of pressure coefficient: (i) Pr = 1 and (ii) Pr = 100 (θ = 0 represents front stagnation point).

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seen at Re = 100, except for a weak influence of the Rich- ardson number in the rear. Finally, the surface pressure is seen to decrease with the increasing Reynolds number with weak additional dependence on the power-law index and shape of the object. It is thus clear that the form drag contributes much more to the overall drag for oblates than that in the case of prolates. This conjecture is well borne

out by the drag results reported in the next section.

5.2. Drag coefficient

Due to the prevailing shearing and normal forces, the fluid exerts a net hydrodynamic force on the spheroid in the direction of flow. Due to the coupled nature of the velocity and temperature fields, the drag coefficient is

Fig. 7. (Color online) Dependence of the total drag coefficient on dimensionless parameters: (a) Re = 1 and (b) Re = 100.

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expected to be a function of Re, Ri, Pr, n, and e. In order to delineate the influence of each of these parameters on drag, representative results are shown in Fig. 7. An inspection of this figure reveals the following trends: All else being equal, drag coefficient gradually increases with the increasing value of power-law index, i.e., drag reduction occurs in shear-thinning (n < 1) and drag enhancement in shear-thickening (n > 1) fluids with reference to Newto- nian fluids (n = 1). This is consistent with the literature findings in the context of a sphere (Chhabra, 2006). In the forced-convection limit (Ri = 0), the drag is expected to exhibit an inverse dependence on the Reynolds number and this trend is seen to persist in the mixed-convection regime also, as is evident in Fig. 7. It is useful to recall here that wake phenomenon exerts a strong influence on the drag characteristics and therefore, the role of Richard- son number here is really complex. On one hand, it suppresses the wake phenomenon but it strengthens the mean flow thereby sharpening the gradients, especially in the rear of the spheroid. This conjecture is well supported by the positive dependence of the drag coefficient on the Richardson number, Fig. 7, under all conditions. The effect of aspect ratio, e, is also seen to be far from being straight- forward here. At low Reynolds numbers, the drag is dominated by viscous forces and therefore, for slender shapes (e > 1) tangential forces act on a large surface area which give rise to large drag forces. This trend can be seen in Fig. 7 at Re = 1 wherein the drag coefficient is seen to increase with the increasing value of e, i.e., the decreasing frontal area of the spheroid. Thus, the case of a sphere is enclosed by the slender shapes above and the blunt shapes from below. On the other hand, as the Reynolds number increases, the contribution of the form drag is dominated by the normal forces and therefore the blunt configurations (e < 1) have larger projected area normal to the flow.

However, the overall value of drag is strongly influenced by wake also which, in turn, is modulated by the value of the Richardson number and e, as seen in Figs. 6 and 7.

Therefore, at Re = 100, depending upon the values of e, Ri, and n, the drag for a sphere is seen to be minimum, e.g., see Fig. 7b for n = 0.3. In this regime, a clear cut trend does not emerge with regard to the role of aspect ratio on the hydrodynamic drag. This trend can be seen in Fig. 7 at Re = 100. Therefore, the effect of shape (e) is strongly modulated by the value of the Reynolds number.

Thus, depending upon the values of n, Re, and e, the drag on a spheroid may be more or less than that on a sphere.

Finally, it needs to be recognized here that in the present case, the thickness of the momentum and thermal boundary layers is determined by a complex interplay between the kinematic parameters (Re, Ri, and Pr), liquid rheology (n), and spheroid geometry (e), as is reflected in the drag results shown in Fig. 7.

5.3. Local Nusselt number

Figs. 8a and 8b show the representative distribution of the local Nusselt number along the surface of the isothermal spheroid for a range of values of Ri, Pr, n, Re, and e.

In spite of the fact that the surface of the spheroid is at a uniform temperature, the temperature gradient normal to the surface varies from point to point thereby giving rise to the variation of the local Nusselt number. In the present context, the value of the local Nusselt number is influenced by Ri, Re, Pr, n, and e. At low Peclet numbers (Pe

= Re·Pr), the influence of the Richardson number and power-law index is very weak irrespective of the shape of the spheroid. The weak role of power-law index is simply due to feeble advection under these conditions and therefore, the viscous properties (value of n) are almost irrel- evant here. In fact, the shape seems to be a major influence under these conditions, e.g., see the results for e = 0.2, 1, and 5 in Fig. 8a. Some augmentation in heat transfer is achieved as the Prandtl number is increased from Pr = 1 to Pr = 100. This increase is roughly in line with the well- known scaling of Nu ~ Pe^1/3 at low Reynolds numbers.

Other than this, slight deterioration in heat transfer is evident in shear-thickening fluids. In contrast, at Re = 100 (Fig. 8b), while the qualitative trends remain the same but the effect of Richardson number is stronger now. Similar to the surface vorticity contours, the influence of buoyancy is seen to be reversed in the rear. Broadly, for oblates (e < 1), the local Nusselt number increases from its minimum value at the front stagnation point attaining a peak value at about , followed by a region of gradual decrease all the way up to the rear stagnation point at low Re = 1 (Fig. 8a). Such a decrease continues up to the separation point beyond which it shows some recovery (e.g., Fig. 8b for e = 0.2 and e = 1). Also, the maximum in these plots occurs at about for a sphere which is consistent with the literature results (Nirmalkar and Chhabra, 2013; 2016). For more streamlined shapes (e > 1), the maximum Nusselt number occurs very close to the front stagnation point and it monotonically decreases thereafter all the way to the rear stagnation point, for no wake formation occurs for prolates (e > 1).

5.4. Average Nusselt number

While the streamline and isotherm contours together with the variation of the local Nusselt number provide useful insights into the microscopic behaviour of the flow and temperature fields, it is the surface averaged value of the Nusselt number which is frequently needed in process engineering calculations. The preceding discussion suggests the average Nusselt number, Nu, to be determined by the values of the five dimensionless groups, namely, Ri, Pr, Re, n, and e. However, this multi-variable relationship cannot be readily ascertained through simple plots, such as

θ 90= ^o

θ 45≈ ^o

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that shown in Figs. 9a and 9b. Based on a detailed exam- ination of the results, one can infer the following trends:

the average Nusselt number bears a positive correlation with the Reynolds and Prandtl numbers at fixed values of e, n, and Ri. This can readily be explained within the framework of the gradual thinning of the thermal boundary layer. As intuitively expected, blunt shapes (e < 1) pro- mote heat transfer than the streamlined shapes. However,

this augmentation is achieved only at the expense of higher hydrodynamic drag. All else being equal, shear- thinning viscosity can enhance the value of the Nusselt number by up to ~70% under appropriate conditions of Re, Pr, and Ri. The role of Richardson number is also generally positive in the absence of flow separation, but it can have slight deleterious effect on the overall heat transfer under certain conditions, e.g., for blunt shapes when the Fig. 8a. (Color online) Local Nusselt number profiles at Re = 1: (i) Pr = 1 and (ii) Pr = 100 (θ = 0 represents front stagnation point).

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inertial forces are comparable to the buoyancy forces thereby at the point of flow separation, e.g., see results for e = 0.2 and e = 0.5 in Fig. 9a. However, the decrease in the Nusselt number over this region is rather small, especially in Newtonian and shear-thickening media.

Finally it is useful to develop a predictive correlation thereby enabling a priori estimation of the Nusselt num-

ber in a new application and/or for the intermediate values of the governing parameters. This multi-variable functional relationship can be expressed as:

Nu = f (Ri, Re, Pr, n, e). (11) Following the success of the modified Reynolds (Re^*) and Prandtl numbers (Pr^*) incorporating the influence of Fig. 8b. (Color online) Local Nusselt number profiles at Re = 100: (i) Pr = 1 and (ii) Pr = 100 (θ = 0 represents front stagnation point).

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the Richardson number (Nirmalkar and Chhabra, 2013;

2016; Meissner et al., 1994), one can re-write this functional relationship as:

Nu = f (Re^*, Pr^*, n, e) (12)

where

; . (13)

These definitions are simply based on the effective fluid Re^* = Re 1( + Ri)^{2 n}^– Pr^* = Pr 1( + Ri)^{n 1}^–

Fig. 9a. (Color online) Dependence of the average Nusselt number on the Richardson number: (i) Re = 1 and (ii) Re = 100.

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velocity being given by thereby incorporating the strength of the mixed-flow. Further reduction in the number of variables in Eq. (12) is achieved by intro- ducing the familiar Colburn j-factor for heat transfer defined as:

. (14)

Finally, the present numerical values of the average

Nusselt number are well approximated by the following expression:

(15) where the resulting values of the single constant a₀ are summarized in Table 3 together with the corresponding mean (~11-12%) and maximum (~35%) deviations. For each value of e, there are nearly 5,200 individual simula- U_∞+ ( )gβ ΔT2a

( )

j = Nu Re^*×Pr^*1/3

--- = f Re( ^*, n, e)

j = a₀ Re^*2/3 --- 3n 1+

---4n

⎝ ⎠

⎛ ⎞^0.6

Fig. 9b. (Color online) Dependence of the average Nusselt number on power-law index: (i) Re = 1 and (ii) Re = 100.

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tion results and only about 4.2% of the data points show errors greater than 25% from Eq. (15) without any dis- cernible trends. Further attempts to improve the degree of fit proved unsuccessful even when additional fitting parameters are introduced in the correlation. The fact that no satisfactory heat transfer correlation is available for a spheroid even in Newtonian fluids (Martynenko and Khram- stov, 2005) indicates the intrinsic difficulty in postulating a plausible functional form of such a correlation with general validity. Some comments about the form of this correlation, Eq. (15), are in order here. Firstly, the form of Eq. (15) retains the widely used scaling of Nu ~ Pr^1/3 and j ~ Re^2/3 in the case of Newtonian fluids. The Colburn j- factor also captures the familiar scaling of Nu ~ Pe^1/3. Sec- ondly, Eq. (15) reduces to the forced convection limit as and to Newtonian fluids in the limit of n = 1. Thus, it appears to have a wide appeal. Thirdly, the positive exponent of (3n + 1/4n) term in Eq. (15) captures the fact

that shear-thinning fluid behaviour (n < 1) fosters heat transfer. Finally, the value of the constant a₀ shows a regular variation with the aspect ratio (e), thereby enabling the interpolation of the present results for the intermediate values of the aspect ratio. However, the results for Pr≤ 1 have not been included in Eq. (15) because such low values pertain to gases and therefore are of no significance in the context of power-law fluids.

5.5. Effect of temperature-dependent viscosity

In order to ascertain the impact of the assumption of the constant viscosity with respect to temperature, limited simulations have been performed here with a temperature- dependent viscosity (power-law consistency index, m) to delineate its influence on the value of the Nusselt number.

It is generally agreed that the power-law index (n) is nearly constant over a temperature interval of 40-50ºC, e.g., see Steffe (1996), Chhabra (2006) and Peixinho et al.

(2008), etc. However, the power-law consistency, m, often shows an Arrhenius type temperature dependence, i.e., m

= m₀exp(−b0T). Naturally, the values of m₀ and b₀ vary from one fluid to another even for Newtonian fluids, therefore we have used the fluid properties reported by Peixinho et al. (2008). For a specific fluid, they reported n = 0.49 and m = 2.77e^−0.011T where m is in Pa·sⁿ and T is in ºC. In the present tests, the fluid temperature was fixed at = 300 K whereas the spheroid temperature was varied as T_w = 305, 310, 320, 330, 340, and 350 K, i.e., the ΔT ranged from 5 (the standard case in the present study) to 50 K to delineate the influence of temperature-depen- Ri→0

T_∞ Table 3. Values of fitting constant (a₀) used in Eq. (15) and cor-

responding errors.

e # Data points a₀ % Error

Average Maximum

0.2 ~5200 1.312 12.8 34.6

0.5 ~5200 1.293 11.4 34.5

1 ~5200 1.153 11.6 34.0

2 ~5200 0.946 11.5 33.7

5 ~5200 0.711 11.2 33.2

Table 4. Effect of temperature-dependent viscosity (value of m) on the Nusselt number results.

Re = 100, Pr = 10, n = 0.49

T (K) Ri = 0 Ri = 1 Ri = 3 Ri = 5

Re = 1 Re = 100 Re = 1 Re = 100 Re = 1 Re = 100 Re = 1 Re = 100

e = 0.2

Present^* 4.3914 25.068 4.4329 26.914 4.5078 27.658 4.5795 28.044

305 4.4021 25.302 4.4439 27.184 4.5198 27.936 4.5926 28.376

310 4.4125 25.536 4.4549 27.454 4.5319 28.212 4.6058 28.686

320 4.4335 25.999 4.4771 27.987 4.5564 28.754 4.6325 29.263

330 4.4548 26.456 4.4997 28.513 4.5812 29.288 4.6596 29.807

340 4.4763 26.905 4.5225 29.029 4.6065 29.812 4.6873 30.334

350 4.4982 27.346 4.5458 29.534 4.6323 30.325 4.7154 30.848

e = 5

Present^* 2.3024 10.663 2.3695 14.122 2.4832 17.188 2.5885 19.355

305 2.3108 10.797 2.3779 14.268 2.4927 17.346 2.5994 19.533

310 2.319 10.933 2.3861 14.412 2.502 17.504 2.6102 19.710

320 2.3349 11.203 2.4021 14.693 2.5204 17.816 2.6318 20.061

330 2.3503 11.470 2.4175 14.964 2.5383 18.127 2.6531 20.407

340 2.365 11.737 2.4323 15.226 2.556 18.431 2.6742 20.747

350 2.379 11.998 2.4465 15.478 2.5732 18.733 2.6951 21.081

*This denotes the base case studied in this work.

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dent viscosity on the present values of the Nusselt number.

A summary of the representative results is presented in Table 4. As expected, the resulting value of the average Nusselt number with temperature-dependent viscosity is higher than that for the constant value of m. This increase is simply due to the lowering of the fluid viscosity which augments the value of the local Reynolds number. How- ever, some of this enhancement is offset by the concomitant reduction in the value of the Prandtl number. In most cases, the net increase in the value of the Nusselt number is of the order of ~10-12%. Furthermore, some simulations have also been carried out by varying the value of the pre-exponential factor m₀= 2.77 Pa·sⁿ by ± 50%, i.e., for m₀= 4.155 Pa·sⁿ and for m₀ = 1.385 Pa·sⁿ. The increase in the value of the Nusselt number was still of the order of ~10%. The level of enhancement in heat transfer seen here is very similar to that reported for forced convection from an isothermal cylinder in power-law fluids (Soares et al., 2010). This effect is well approximated by multiplying the value of the Nusselt number estimated using Eq. (15) by the factor (m_w/ )^0.14, similar to that used to account for temperature-dependent viscosity in pipe flow heat transfer correlations for Newtonian fluids, e.g., see Kreith (2000). It is appropriate to mention here that, even in this case of temperature-dependent power-law consistency coefficient m, the Reynolds and Prandtl numbers are defined based on the physical properties evaluated at

= 300 K.

6. Conclusions

The hydrodynamic and heat transfer characteristics of a spheroid submerged in power-law fluids have been studied numerically in the steady buoyancy-assisted mixed- flow regime. The ranges of conditions spanned in this study are: Richardson number, 0≤Ri ≤5; Reynolds number, 1≤Re ≤100; Prandtl number, 1 ≤Pr ≤100; power- law index, 0.3≤n ≤1.8, and aspect ratio, 0.2 ≤e ≤5.

Extensive new results are discussed in terms of the streamline and isotherm contours, surface pressure and vorticity plots, and distribution of local Nusselt number to delineate the influence of each of the preceding parameters. Finally, the overall macroscopic characteristics are reported in terms of the drag coefficient and average Nus- selt number as functions of Ri, Re, Pr, n, and e. Overall, the propensity of flow separation is seen to be greater for oblates than for prolates due to the bluntness of the former. Aiding-buoyancy effects tend to stabilize the flow by delaying the onset of flow separation. All else being equal, the heat transfer is enhanced (by up to ~70%) in shear- thinning fluids (n < 1) over and above that in Newtonian fluids. Similarly, oblate shapes yield higher Nusselt number than the prolates. However, this is accompanied by concomitant increase in drag on oblates. Based on the

present results, a predictive heat transfer correlation has been developed in terms of the Colburn heat transfer factor as a function of the modified Reynolds and Prandtl numbers, aspect ratio, and power-law index. Under limiting conditions, the proposed correlation reduces to its expected forms for Newtonian (n = 1) and/or forced-convection (Ri = 0) conditions. The value of the average Nus- selt number is further increased by up to ~10-12% when the temperature-dependence of the power-law viscosity (through m) is incorporated into the analysis.

Acknowledgement

RPC is grateful for the award of the J.C. Bose fellow- ship (Department of Science and Technology, Govern- ment of India) for the period 2015-2020.

List of Symbols

a : Semi-axis normal to flow (m)

A_p : Projected area of the spheroid normal to flow (m²) (≡πa²)

Br : Brinkman number (dimensionless) b : Semi-axis parallel to flow (m) C : Specific heat of fluid (J·kg⁻¹·K⁻¹) C_D : Total drag coefficient (dimensionless) C_DP : Pressure drag coefficient (dimensionless)

C_p : Pressure coefficient (dimensionless)

: Diameter of the computational domain (m) e : Aspect ratio of the spheroid (dimensionless)

(≡b/a)

F_D : Total drag force (N)

F_DP : Pressure component of the drag force (N) Gr : Grashof number, dimensionless

g : Acceleration due to gravity (m·s⁻²) h : Local heat transfer coefficient (W·m⁻²·K⁻¹) I₂ : Second invariant of the rate of deformation ten-

sor (s⁻²)

j : Colburn-j factor (dimensionless)

k : Thermal conductivity of fluid, (W·m⁻¹·K⁻¹) m : Flow consistency index (Pa·sⁿ)

N_p : Number of grid points on the spheroid (dimensionless)

Nu : Average Nusselt number (dimensionless) Nu_θ : Local Nusselt number on the surface of spheroid

(dimensionless)

n : Power-law index (dimensionless) p : Pressure (Pa)

p^* : Pressure (dimensionless) Pr : Prandtl number (dimensionless) m_∞

T_∞

2F_D ρ_∞U_∞²A_p ---

⎝≡ ⎠

⎛ ⎞

2F_DP ρ_∞U_∞²A_p ---

⎝≡ ⎠

⎛ ⎞

2 p( _w–p_∞) ρ_∞U_∞² ---

⎝≡ ⎠

⎛ ⎞

D_∞

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Pr^* : Modified Prandtl number (dimensionless) Re : Reynolds number (dimensionless)

Re^* : Modified Reynolds number (dimensionless) Ri : Richardson number (dimensionless) T : Fluid temperature (K)

T_w : Temperature of the spheroid (K)

: Temperature of the fluid in free stream (K) : Free stream velocity (m·s⁻¹)

V : Velocity vector (dimensionless) Greek symbols

β : Coefficient of volumetric expansion (K⁻¹) ε(u) : Rate of deformation tensor (s⁻¹)

η : Viscosity of fluid (Pa·s)

θ : Position on the surface of spheroid (degree) ξ : Fluid temperature (≡ / ) (dimen-

sionless)

ρ : Density of fluid (kg·m⁻³) : Density of fluid at (kg·m⁻³) τ : Extra stress tensor (dimensionless)

: Del operator (dimensionless) Subscripts

i, j : Dummy arguments w : Spheroid surface References

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