International Communications in Heat and Mass Transfer 39 (2012) 336–342

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International Communications in Heat and Mass Transfer

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Steady revolving flow and heat transfer of a non-Newtonian Reiner–Rivlin fluid☆

Bikash Sahoo a,⁎, Robert A. Van Gorder b, H.I. Andersson a

a Fluids Engineering Division, Department of Energy and Process Engineering, Norwegian University of Science and Technology (NTNU), Trondheim, Norwayb Department of Mathematics, University of Central Florida, Orlando, Florida, USA

☆ Communicated by W.J. Minkowycz.⁎ Corresponding author at: National Institute of Tech

E-mail address: bikash.sahoo@ntnu.no (B. Sahoo).

0735-1933/$ – see front matter © 2011 Elsevier Ltd. Alldoi:10.1016/j.icheatmasstransfer.2011.12.007

a b s t r a c t

a r t i c l e i n f o

Available online 15 December 2011

Keywords:Reiner–Rivlin fluidRevolving flowHeat transferFinite difference methodShooting method

The steady revolving flow and heat transfer of a non-Newtonian Reiner–Rivlin fluid is studied. The momen-tum equation gives rise to a highly nonlinear boundary value problem. Attempt has been made to study theproperties of the solution of the momentum equation analytically before proceeding for numerical solution.The effects of non-Newtonian fluid characteristic on the velocity and temperature fields have been discussedin detail and shown graphically.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Von Kármán [1] considered the steady flow of a viscous incom-pressible fluid due to a rotating disk. He solved the equations of mo-tion by an approximate integral method devised by him andPolhausen [2]. There are a few minor inaccuracies in Kármán's analy-sis which were corrected by Cochran [3]. The latter was able to givean accurate numerical solution, a remarkable feat at that time. Stuart[4] studied the effects of uniform suction on the flow due to a rotatingdisk. The most accurate solution so far seems to have been reportedby Ackroyed [5]. The inverse problem arising when a viscous fluid ro-tates with a uniform angular velocity at a larger distance from a sta-tionary disk is one of the few problems in fluid dynamics for whichthe Navier–Stokes equation admits an exact solution. This problemwas studied by Bödewadt [6] by making boundary layer approxima-tions. That is why the flow is well known as Bödewadt flow. In thiscase it is observed that the fluid particles near a disk flow radially in-wards and for reasons of continuity this flow is compensated by anaxial flow upwards, away from the disk (Fig. 1). Bödewadtapproached the solution through a very laborious method consistingof a power series expansion at z=0 and an asymptotic expansionfor z→∞. A correction to this problem is worked out by A.C. Browning(unpublished) [7]. He noticed a much thicker boundary layer than inthe case of a disk rotating in a fluid at rest. The general problem of aninfinite rotating disk in fluid of which the above two problems areparticular cases has been later investigated by Hannah [8], Rogersand Lance [9,10].

It was pointed out by Schwiderski and Lugt [11] that the non-existence of a proper solution to the boundary value problems for

nology, Rourkela, India.

rights reserved.

rotating flows of Von Kármán and Bödewadt is an indication that inreality the flow is separated from the surface of the disk. The simple‘Tea cup experiment’ [7] displays very clearly a separation of thefluid from the bottom of the cup. Application of the suction is an ef-fective device to reduce the chances of separation. Nanda [12] studiedthe effects of uniform suction on the revolving flow of a viscous liquidover a stationary disk. It was found that the presence of suction intro-duces an axial inflow at infinity and the same increases with an in-crease in suction. The momentum and the displacement thicknessdecrease as the suction velocity increases. A comprehensive reviewof earlier works on flow and heat transfer due to a single and two par-allel rotating disks up to 1989 has been included in a monograph byOwen and Rogers [13].

In all of the above studies the fluid is assumed to be Newtonian.Manymaterials such as polymer solutions or melts, drilling mud, clas-tomers, certain oils and greases and many other emulsions are classi-fied as non-Newtonian fluids. For these kinds of fluids, the commonlyaccepted assumption of a linear relationship between the stress andthe rate of strain does not hold. Most of the fluids used in industriesare non-Newtonian fluids. The non-Newtonian fluids have been mod-eled by constitutive equations which vary greatly in complexity. Thenon-Newtonian fluid considered in the present paper is that forwhich the stress tensor τji is related to the rate of strain tensor eji as[14,15]

τij ¼ −pδij þ 2μeij þ 2μceike

kj ; ejj ¼ 0 ð1Þ

where p denotes the pressure, μ is the coefficient of viscosity and μc isthe coefficient of cross viscosity. This model was introduced by Reiner[14] to describe the behavior of wet sand, but was at one time consid-ered as a possible model for non-Newtonian fluid behavior. However,the model does not account for the possibility of both normal stressdifferences or shear-thinning or shear-thickening. The Von Kármánflow of different kind of non-Newtonian fluids have been studied

337B. Sahoo et al. / International Communications in Heat and Mass Transfer 39 (2012) 336–342

by various authors including diverse physical effects. A detailed dis-cussion up to 1991 regarding the flow of non-Newtonian fluids dueto rotating disks can be found in the review paper by Rajagopal[16]. Andersson and Korte [17] and Andersson et al. [18] have madenotable attempts to see effects of power-law index of non-Newtonian power-law fluid on the boundary layer arising due to Kár-mán flow. Attia [19] has studied the steady Von Kármán flow and heattransfer of Reiner–Rivlin fluid with suction or injection at the surfaceof the disk. His work has been extended by Sahoo and Sharma [20]and Sahoo [21] by incorporating the partial slip boundary condition.Sharon and MacKerrel [22] have studied the stability of Bödewadtflow. Kitchens and Chang [23] have considered the Bödewadt flowfor a non-Newtonian second-order fluid.

In the present work the steady flow and heat transfer of a non-Newtonian Reiner–Rivlin fluid (presented by (1)), which rotateswith a uniform angular velocity at a larger distance from a stationarydisk is studied. The resulting system of highly nonlinear differentialequations for the velocity and temperature field is solved by a secondorder finite difference method. Before proceeding for the numericalsolution, the attempt has been made to study the properties of the so-lution of the momentum equation analytically in Section 3.

2. Formulation of the problem

We consider a non-Newtonian Reiner–Rivlin fluid whose rheolog-ical behavior is governed by stress–strain rate law (1), occupying thespace z>0 over an infinite stationary disk, coinciding with z=0 (seeFig. 1). The motion is due to the rotation of the fluid like rigid bodywith constant angular velocity Ω at large distance from the disk. It isnatural to describe the flow in the cylindrical polar coordinates (r,ϕ,z).

In view of the rotational symmetry, ∂∂ϕ≡0. Taking V=(u,v,w) for

the steady flow, the equations of continuity and motion are

∂u∂r þ

urþ ∂w

∂z ¼ 0 ð2Þ

and

ρ u∂u∂r þw

∂u∂z−

v2

r

!¼ ∂τrr

∂r þ ∂τzr∂z þ τrr−τϕϕ

r; ð3Þ

ρ u∂v∂r þw

∂v∂z þ

uvr

� �¼ ∂τrϕ

∂r þ ∂τzϕ∂z þ 2τrϕ

r; ð4Þ

ρ u∂w∂r þw

∂w∂z

� �¼ ∂τrz

∂r þ ∂τzz∂z þ τrz

rð5Þ

The boundary conditions for the velocity field are given as

z ¼ 0; u ¼ 0; v ¼ 0; w ¼ 0; ð6aÞ

z→∞; u→0; v→rΩ; p→p∞: ð6bÞ

By using the Von Kármán transformations [1]

u ¼ rΩF ζð Þ; v ¼ rΩG ζð Þ; w ¼ffiffiffiffiffiffiffiΩν

pH ζð Þ; z ¼

ffiffiffiffiνΩ

rζ ; p−p∞ ¼ −ρνΩP

and following [10,13] Eqs. (2)–(5)take the form

dHdζ

þ 2F ¼ 0; ð7Þ

d2Fdζ2 −H

dFdζ

−F2 þ G2−12K

dFdζ

� �2−3

dGdζ

� �2−2F

d2Fdζ2

" #¼ 1 ð8Þ

d2Gdζ2 −H

dGdζ

−2FGþ KdFdζ

dGdζ

þ Fd2Gdζ2

!¼ 0; ð9Þ

d2Hdζ2 −H

dHdζ

−72KdHdζ

d2Hdζ2 þ dP

dζ¼ 0: ð10Þ

and the boundary conditions (6a)–(6b) become,

ζ ¼ 0 : F ¼ 0; G ¼ 0; H ¼ 0; ð11aÞ

ζ→∞ : F→0; G→1 ð11bÞ

where F, G, H and P are non-dimensional functions of ζ, ν is the kine-matic viscosity (ν ¼ μ

ρ) of the fluid and K ¼ μcΩμ is the parameter that

describes the non-Newtonian characteristic of the fluid. The abovesystem Eqs. (7)–(9) with the prescribed boundary conditions(11a)–(11b) are sufficient to solve for the three components of theflow velocity. Eq. (10) can be used to solve for the pressure distribu-tion at any point.

2.1. Heat transfer analysis

Due to the temperature difference between the surface of the diskand the ambient fluid, heat transfer takes place. Using the boundarylayer approximation and neglecting the dissipation terms, the energyequation takes the form,

ρcp u∂T∂r þw

∂T∂z

� �−k

∂2T∂z2

¼ 0: ð12Þ

where cp is the specific heat at constant pressure and k is the thermalconductivity of the fluid. Introducing the non-dimensional variableθ ¼ T−T∞

Tw−T∞and using the Von Kármán transformations, Eq. (12) be-

comes,

d2θdζ2 −PrH

dθdζ

¼ 0: ð13Þ

where Tw is the temperature at the surface of the disk, T∞ is the tem-perature of the ambient fluid at large distance from the disk and Pr ¼cpμk is the Prandtl number. The boundary conditions in terms of θ areexpressed as

ζ ¼ 0 : θ ¼ 1; ζ→∞ : θ→0: ð14Þ

The heat transfer from the disk surface to the fluid is computed bythe application of the Fourier's law, q ¼ −k ∂T

∂z

� �jz¼0. Introducing the

transformed variables, the expression for q becomes

q ¼ −k Tw−T∞ð ÞffiffiffiffiΩν

rdθdζ

� �ζ¼0

:

By rephrasing the heat transfer results in terms of the Nusselt

number defined as Nu ¼ qffiffiνΩ

pk Tw−T∞ð Þ, we get

Nu ¼ − dθdζ ζ¼0:���

The action of the viscosity in the fluid adjacent to the disk tends toset up tangential shear stress Tφ, which opposes the rotation of thedisk. There is also a surface shear stress Tr in the radial direction. In

338 B. Sahoo et al. / International Communications in Heat and Mass Transfer 39 (2012) 336–342

terms of the variables of the analysis, the expressions of Tφ and Tr arerespectively given as

Tφ ¼ τϕzρr

ffiffiffiffiffiffiffiffiffiνΩ3

p ¼ dGdζ ζ¼0; Tr ¼

τrzρr

ffiffiffiffiffiffiffiffiffiνΩ3

p ¼ dFdζ ζ¼0

��������

3. Mathematical properties of the solutions

Before proceeding for the numerical solution, we shall discuss twoproperties of the solutions to the boundary value problem governedby Eqs. (7)–(9) and Eqs. (11a)–(11b). First, we shall show that solu-tions are not possible in the limit K→+∞. Hence, there must existsome maximum value of K which yields physically meaningful solu-tions. Secondly, we discuss the asymptotic behavior of solutions tothe boundary value problem governed by Eqs. (7)–(9) and Eqs.(11a) – (11b) in the ζ→+∞ limit.

3.1. The K→+∞ limit

In the K→+∞ limit, Eqs. (8)–(9) yield

F ′� �2−3 G′

� �2−2FF ′′ ¼ 0; ð15Þ

F ′G′ þ FG′′ ¼ 0: ð16Þ

The latter equation implies that (FG′)′=0, and hence FG′=C,where C is a constant. Yet, F(0)=0, so C=0. Thus, either F=0 or G′=0. (Assuming that we have continuous solutions. However, thephysically meaningful solutions must be continuous.) If G′=0, we vi-olate the boundary conditions given in Eqs. (11a)–(11b), which showthat G must change over the semi-infinite interval (0,∞). Meanwhile,if F=0, the boundary conditions are satisfied. However, pluggingF=0 into (15), we see that 3(G′)2=0, which again gives G′=0thereby contradicting the boundary conditions (11a)-(11b). There-fore, in the limit K→+∞, there exist no self-similar solution of thetype governed by (7)-(9) and (11a)-(11b). We therefore must havea maximal 0bK⁎b+∞ such that solutions to the boundary valueproblem (7)-(9) and (11a)-(11b) exist only for K∈ [0,K⁎). In otherwords, only a finite bounded interval of admissible K-values permitself-similar solutions. The precise value of K⁎ likely depends on allother model parameters. We say more about this in the nextsubsection.

3.2. Solutions in the asymptotic limit ζ→+∞

In order to deduce the asymptotic behavior of a solution to theboundary value problem governed by Eqs. (7)–(9) and Eqs.(11a)–(11b), we recast Eqs. (7)–(9) as a five-dimensional dynamicalsystem:

y′1 ¼ −2y2y′2 ¼ y3

y′3 ¼ 11þ Ky2

y1y3 þ y22 þ y24 þK2y23−

3K2

y25 þ 1� �

y′4 ¼ y5

y′5 ¼ 11þ Ky5

y1y5−2y2y4−Ky3y5ð Þ:

ð17Þ

If we assume a solution satisfying H→H∞ as ζ→+∞ (where H∞ isa finite constant to be determined; such an assumption is reasonable,as F→0 and hence H′→0 as ζ→+∞), then there exists an equilibri-um (y1⁎,y2⁎,y3⁎,y4⁎,y5⁎)=(H∞,0,0,1,0) for system (17). Analysing the

Jacobian of system (17) at this equilibrium, we find that the five ei-genvalues are given by

λ∈ 0;H∞2

� 12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiH2

∞ þ 8q

;H∞2

� 12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiH2

∞−8q�

: ð18Þ

From the boundary conditions (11a)–(11b), we may exclude thesolutions which blow up as ζ→+∞. Retaining only solutions whichagree with the boundary conditions (11a)–(11b), we find that asζ→+∞, F, G and H behave like

F ζð Þ∼A1e−rζ

; G ζð Þ∼1þ A2e−rζ

; H ζð Þ∼H∞ þ A3e−rζ

; ð19Þ

where we define the rate of decay of the solutions (to their far-fieldboundary values), r, by

r ¼ r H∞ð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiH2

∞ þ 8q

−H∞

2: ð20Þ

Observe that r decreases as H∞ increases. Yet, as can be seen fromthe numerical solutions presented in the following section, H∞ de-creases as K increases. Hence, the decay rate r increases as K is in-creased. In other words, when the non-Newtonian parameter K isincreased, solutions to the boundary value problem governed byEqs. (7)–(9) and Eqs. (11a)–(11b) tend to approach their far-fieldboundary values more rapidly. Now, when K=0, we find numericallythat H∞=1.349421, so using Eq. (20), r=0.892208 when K=0.Meanwhile, for K∈(0,K⁎), physically reasonable solutions mandatethat we have H∞>0. As H∞ is decreasing in K, let us define K̂ to bethe value of K for which H∞=0. Clearly, K�≤K̂ , otherwise H∞≤0 forsome K∈(0,K⁎). Then, we see that r is bounded from above by

ffiffiffi2

p;

for any KbK�≤K̂ , H∞>0 and, thus, rbr 0ð Þ ¼ffiffiffi2

p. Hence, for

K∈(0,K⁎), r∈ 0:892208;ffiffiffi2

p� �. That is, the decay rate is increasing

in K and, furthermore, is bounded by 0:892208≤rbffiffiffi2

p.

4. The numerical solution of the problem

The system of non-linear differential Eqs. (7)–(9) and (13) issolved under the boundary conditions (11a)–(11b) and (14), respec-tively by adopting a second order finite difference scheme same as in[20,21,24]. A finite value, large enough, has been substituted for ζ∞,the numerical infinity to ensure that the solutions are not affectedby imposing the asymptotic conditions at a finite distance. Thevalue of ζ∞ has been kept invariant during the run of the program.

Now we introduce a mesh defined by

ζ i ¼ ih i ¼ 0;1;…nð Þ; ð21Þ

h being the mesh size, n is a sufficiently large finite value. TheEqs. (7)–(9) and (13) are discretized using the central difference ap-proximations for the derivatives, then the following equations areobtained.

Fiþ1−2Fi þ Fi−1

h2−Hi

Fiþ1−Fi−1

2h

� �−F2i þ G2

i −12K

Fiþ1−Fi−1

2h

� �2

−3Giþ1−Gi−1

2h

� �2−2Fi

Fiþ1−2Fi þ Fi−1

h2

� ��−1 ¼ 0

ð22Þ

Giþ1−2Gi þ Gi−1

h2−Hi

Giþ1−Gi−1

2h

� �−2FiGi

þKFiþ1−Fi−1

2h

� �Giþ1−Gi−1

2h

� �þ Fi

Giþ1−2Gi þ Gi−1

h2

� � �¼ 0

ð23Þ

θiþ1−θi þ θi−1

h2−PrHi

θiþ1−θi−1

2h

� �¼ 0 ð24Þ

Fig. 1. Geometric representation of the flow domain.

Table 1Variation of H∞, Nu, Tr, Tϕ with K.

K=0 K=0.5 K=1.5

H∞ 1.349421 1.251591 1.218600Nu 0.828328 0.774727 0.704916Tr −0.9419731 −0.799818 −0.728624Tϕ 0.7728935 0.587595 0.339843

Table 2Variations of F, G and H.

F G H

ζ Currentresult

Owen &Rogers [13]

Currentresult

Owen &Rogers [13]

Currentresult

Owen &Rogers [13]

0.0 0.000000 0.0000 0.000000 0.0000 0.000000 0.00000.5 −0.348650 −0.3487 0.383430 0.3834 0.194373 0.19441.0 −0.478766 −0.4788 0.735429 0.7354 0.624103 0.62411.5 −0.449633 −0.4496 1.013401 1.0134 1.098743 1.09872.0 −0.328745 −0.3287 1.192367 1.1924 1.492875 1.49292.5 −0.176206 −0.1762 1.272136 1.2721 1.745869 1.74593.0 −0.036086 −0.0361 1.271405 1.2714 1.849641 1.84963.5 0.066310 0.0663 1.218219 1.2182 1.830807 1.83089.5 −0.010216 −0.0102 1.011849 1.0118 1.361698 1.361710.0 −0.003282 −0.0033 1.012120 1.0121 1.368328 1.368310.5 0.001819 0.0018 1.009906 1.0099 1.368882 1.368911.0 0.004738 0.0047 1.006537 1.0065 1.365423 1.365411.5 0.005681 0.0057 1.003090 1.0031 1.360067 1.360112.0 0.005170 0.0052 1.000271 1.0003 1.354546 1.354512.5 0.003827 0.0038 0.998411 0.9984 1.350003 1.350020.0 0.000102 – 0.999893 – 1.349325 –

25.0 0.000009 – 1.000014 – 1.349457 –

25.5 0.000011 – 1.000007 – 1.349447 –

26.0 0.000010 – 1.000001 – 1.349437 –

26.5 0.000008 – 0.999997 – 1.349428 –

28.0 0.000000 0.0000 1.000000 1.0000 1.349421 1.3494

339B. Sahoo et al. / International Communications in Heat and Mass Transfer 39 (2012) 336–342

Hiþ1 ¼ Hi−h Fi þ Fiþ1� ð25Þ

Note that in Eqs. (8)–(9) and (13), which are written at jth meshpoint, the first and second derivatives are approximated by the cen-tral differences centered at jth mesh point, while in Eq. (7), which iswritten at jþ 1

2

� th mesh point, the first derivative is approximated

by the difference quotient at jth and (j+1)th mesh points, and theright hand sides are approximated by the respective averages at thesame two mesh points. This scheme ensures that the accuracy ofO(h2) is preserved in the discretization.

Eqs. (22)–(23) and (24) are three term recurrence relations in F, Gand θ respectively. Hence, in order to start the recursion, besides thevalues of F0, G0 and θ0, the values of F1, G1 and θ1 are also required.These values can be obtained by Taylor series expansion near ζ=0for F, G and θ.

If

F ′ 0ð Þ ¼ s1; G′ 0ð Þ ¼ s2 and θ′ 0ð Þ ¼ s3 ð26Þ

we have

F1 ¼ F 0ð Þ þ hF ′ 0ð Þ þ h2

2F″ 0ð Þ þ O h2

� �G1 ¼ G 0ð Þ þ hG′ 0ð Þ þ h2

2G″ 0ð Þ þ O h2

� �θ1 ¼ θ 0ð Þ þ hθ′ 0ð Þ þ h2

2θ″ 0ð Þ þ O h2

� �ð27Þ

The values H(0), G(0) and θ(0) are given as boundary conditions in(11a)–(11b) and (14). The values F″(0), G″(0) and θ″(0) can beobtained directly from Eqs. (8)–(9) and (13) and using the values inEq. (26). After obtaining the values of F1, G1 and θ1, the integrationcan now be performed as follows. H1 can be obtained from Eq. (25).Using the values of H1 in Eqs. (22), (23) and (24), the values of F2,G2 and θ2 are obtained. At the next cycle, H2 is computed from Eq.(25) and is used in Eqs. (22), (23) and (24) to obtain F3, G3 and θ3 re-spectively. The order indicated above is followed for the subsequentcycles. The integration is carried out until the values of F, G, H and θare obtained at all the mesh points.

Note that we need to satisfy the three asymptotic boundary condi-tions (11a)–(11b) and (14). In fact s1, s2 and s3 must be found by ashooting method so as to fulfil the boundary conditions(11a)–(11b) and (14). We have adopted Broyden's [25] method asour zero finding algorithm. With reasonably close trial values tostart the iterations, the convergence to the actual values within an ac-curacy of O(10−6) could be attained in 9–11 iterations.

5. Results and discussion

The value of ζ∞, the numerical infinity has been taken larger ascompared to the Kármán flow problem [20,21] and kept invariantthrough out the run of the program. Although, the results areshown only from the disk surface ζ=0 to ζ=14.0, the numerical in-tegrations were performed over a substantially larger domain inorder to assure that the outer asymptotic boundary conditions aresatisfied. To see if the program runs correctly, the values of F, G andH are compared with (see Table 2) those reported by Owen and Rog-ers [13] for a viscous fluid (K→0), and have been plotted graphicallyin Fig. 2.

Fig. 2 depicts that near the disk, the radial component of the veloc-ity F is radially inwards. It may be of interest to note that this radiallyinward flow is the cause of the accumulation of the tea leaves at thecenter of a stirred cup of tea.

Figs. 3–5 present, respectively, the steady state profiles of F, G, andH, plotted against ζ for various values of K. Fig. 3 depicts that the ra-dial component of velocity is negative near the disk and reverses

direction away from the disk. It is clear that the increase in the valuesof K decreases the radial component of velocity F in magnitude up to asignificant distance from the disk and then increases and eventuallyreaches the asymptotic value 0. Fig. 4 presents, the steady state

0 2 4 6 8 10 12 14-0.5

0

0.5

1

1.5

2

ζ

F

G

H

Fig. 2. Velocity profile for the Newtonian flow.

0 2 4 6 8 10 12 14

-0.1

-0.2

-0.3

-0.4

-0.5

0

0.1

0.2

ζ

F

K=0.0K=1.0K=2.0K=3.0

Fig. 3. Variation of F with K.

0 2 4 6 8 10 12 14

ζ

G

0

0.2

0.4

0.6

0.8

1

1.2

1.4

K=0.0K=1.0K=2.0K=3.0

Fig. 4. Variation of G with K.

0 2 4 6 8 10 12 14

ζ

H

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2K=0.0K=1.0K=2.0K=3.0

Fig. 5. Variation of H with K.

0 2 4 6 8 10−2.5

−2

−1.5

−1

−0.5

0

K

CM

Fig. 6. Variation of CM with K.

0 10 20 30 40 50−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

K

Tr

Fig. 7. Variation of Tr with K.

340 B. Sahoo et al. / International Communications in Heat and Mass Transfer 39 (2012) 336–342

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

K

τ φ

Fig. 8. Variation of Tφ with K.

0 10 20 30 40 500.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

K

Nu

Fig. 10. Variation of Nu with K.

341B. Sahoo et al. / International Communications in Heat and Mass Transfer 39 (2012) 336–342

profile of the transverse component of velocity Gwith K. It is interest-ing to find that the magnitude of G decreases with increasing K nearthe disk and increases away from the disk. This accounts for a cross-over of the profiles of G. The steady state profile of the axial compo-nent of velocity H with various values of K is shown in Fig. 5. It isobserved that increasing K, decreases H for all values of ζ. It is alsoclear that the profiles of F, G and H becomes flatter as K is increased.In other words, when the non-Newtonian parameter K is increased,solutions to the momentum equations tend to approach their far-field boundary values more rapidly, supporting the analysis inSection 3.2.

Another interesting quantity is the turning moment for the disk.The expression of the dimensionless moment coefficient CM is givenby:

CM ¼ −πG′ 0ð ÞffiffiffiffiffiffiRe

p ð28Þ

with Re=ΩR2/ν the rotational Reynolds number based on the diskradius R and the maximum velocity (ΩR). This definition of CM isthe extension of the finite disk problem, which supposes that thedisk radius is large enough. Fig. 6 shows the variation of CM with K

0 2 4 6 8 10 12 14

ζ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

θ

K=0.0K=1.0K=2.0K=3.0

Fig. 9. Variation of θ with K.

for Re=1. It's clear that whatever the flow parameters, CM exhibitsnegative values. The value of CM decreases in magnitude with an in-crease in K and approaches the asymptotic value zero for sufficientlyhigh value of the non-Newtonian parameter K. The Von Kármán flowconsidered by Sahoo [21] is precisely the inverse problem, which ex-plains the different sign.

Figs. 7 and 8 show the variations of the radial and tangential shearstresses with K. It is interesting to observe that Tr always remain neg-ative as was expected. Its value decreases in magnitude with an in-crease in K, surprisingly except for a small interval K∈(1.5,2.5),where it slightly increases with K. On the other hand, Tφ decreases ex-ponentially with K up to K=4.0 and then the rate of decrease be-comes significantly slow.

Fig. 9 depicts the variation of the non-dimensional temperature θwith K for Pr=1.0. It is clear from the figure that the thermal bound-ary layer thickens with increasing K. This is because K reduces theaxial flux H and thereby the axial thermal convection PrH dθ

dζ in the en-ergy Eq. (13). Fig. 10 shows the variation of Nusselt number Nu with Kfor Pr=3.0. It's clear from Fig. 10 and Table 1 that Nusselt number de-creases with an increasing value of K.

Table 1 presents the variation of axial velocity component at infin-ity, H∞, the Nusselt number Nu and the radial and tangential wallshear stresses Tr and Tφ for various values of the non-Newtonianfluid parameter K. It is again clear that increasing K decreases theaxial flow.

6. Conclusions

In the present paper, we have considered the steady revolvingflow and heat transfer of a particular class of non-Newtonian fluid,namely, the Reiner–Rivlin fluid. The constitutive equation of thefluid gives rise to momentum equations which, when transformedusing the similarity variables, reduce to highly non-linear system ofboundary value problem. Analysis shows that only a finite boundedinterval of admissible non-Newtonian parameter i.e. K-values (notprecisely determined) permit self-similar solutions. A second orderfinite difference technique has been used to solve the system ofresulting equations. The effects of non-Newtonian fluid parameterK on the velocity and temperature distribution has been studied indetail. It is interesting to find that the parameter K results in a cross-over of the transverse velocity profile. One of the important findingsof the present investigation is that when the non-Newtonian param-eter K is increased, solutions to the boundary value problem tend to

342 B. Sahoo et al. / International Communications in Heat and Mass Transfer 39 (2012) 336–342

approach their far-field asymptotic boundary values more rapidly.The profiles of the moment coefficient CM for the revolving (Böde-wadt) flow and the Kármán flow [21] are just opposite to eachother as was expected.

Acknowledgements

One of the authors (BS) is thankful to the Department of Science andTechnology (DST), Government of India for awarding the BOYSCAST fel-lowship to pursue this work at NTNU, Norway.

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