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Commun Nonlinear Sci Numer Simulat 16 (2011) 4284–4295
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Commun Nonlinear Sci Numer Simulat
journal homepage: www.elsevier .com/locate /cnsns
Effects of slip on steady Bödewadt flow and heat transferof an electrically conducting non-Newtonian fluid
Bikash Sahoo ⇑,1
Department of Energy and Process Engineering, Norwegian University of Science and Technology, N-7491, Trondheim, Norway
a r t i c l e i n f o a b s t r a c t
Article history:Received 21 June 2010Received in revised form 20 November 2010Accepted 23 November 2010Available online 2 December 2010
Keywords:Reiner–Rivlin fluidRotating fluidPartial slipHeat transferMagnetic fieldFinite difference method
1007-5704/$ - see front matter � 2011 Published bdoi:10.1016/j.cnsns.2010.11.023
⇑ Tel.: +47 735 93563.E-mail address: [email protected]
1 The author is on leave from National Institute of
The steady flow and heat transfer arising due to the rotation of a non-Newtonian fluid at alarger distance from a stationary disk is extended to the case where the disk surface admitspartial slip. The constitutive equation of the non-Newtonian fluid is modeled by that for aReiner–Rivlin fluid. The fluid is subjected to an external uniform magnetic field perpendicularto the plane of the disk. The momentum equation gives rise to a highly nonlinear boundaryvalue problem. Numerical solution of the governing nonlinear equations are obtained overthe entire range of the physical parameters. The effects of slip, non-Newtonian fluid charac-teristics and the magnetic interaction parameter on the momentum boundary layer and ther-mal boundary layer are discussed in detail and shown graphically. It is observed that slip hasprominent effects on the velocity and temperature fields.
� 2011 Published by Elsevier B.V.
1. Introduction
In our previous works [1,2], we have discussed the steady Von Kármán flow and heat transfer of an electrically conductingReiner–Rivlin fluid with partial slip boundary conditions. The twin problem arising when the fluid rotates with a uniformangular velocity at a larger distance from a stationary disk is one of the classical problems of fluid mechanics which has boththeoretical and practical value. In this case, the particles which rotate at a large distance from the wall are in equilibriumunder the influence of the centrifugal force which is balanced by the radial pressure gradient. Those particles close to thedisk whose circumferential velocity is retarded under the same pressure gradient are directed inwards. However, the cen-trifugal force they are subjected to is greatly decreased. This set of circumstances causes the particle near the disk to flowradially inwards, and for reasons of continuity that motion must be compensated by an axial flow upwards, as shown inFig. 1. A flow which arises in the boundary layer in this manner such that its direction deviates from that of the outer flowis generally called a secondary flow. Such type of flow can be clearly observed in a teacup: after the rotation has been gen-erated by vigorous stirring and again after the flow has been left to itself for a short while. The radial inward flow field nearthe bottom will be formed. Its existence can be inferred from the fact that tea leaves settle in a little heap near the center atthe bottom.
This problem was studied by Bödewadt [3] by making boundary layer approximations. That is why the flow problem iswidely known as the Bödewadt flow. For this problem Ackroyed’s method [4] of expansion is not so suitable. Bödewadtapproached the solution through a very laborious method similar to that used by Cochran [5] for the Von Kármán equations.The method consists of a power series expansion at z = 0 and an asymptotic expansion for z ?1. Bödewadt’s solution shows
y Elsevier B.V.
Technology, Rourkela, Orissa, India.
Fig. 1. Schematic representation of the Bödewadt flow.
B. Sahoo / Commun Nonlinear Sci Numer Simulat 16 (2011) 4284–4295 4285
that the boundary layer effects extend out to about f = 8.0. A correction to this problem is worked out by Browning (unpub-lished) [6]. He noticed a much thicker boundary layer than in the case of a disk rotating in a fluid at rest. Batchelor [7] gen-eralized theses analysis to include one parameter families of solutions having a mathematical structure very similar to thatof Von Kármán’s. It corresponds to the flow above an infinite disk rotating with certain angular velocity, with the fluid in thefar off region in the solid body rotation. The general problem of an infinite rotating fluid of which the above two problems areparticular cases has been later investigated by Hannah [8]. Subsequently, this case has been treated by Stewartson [9]. Fettis[10] had been concerned largely with the problem of Bödewadt. Rogers and Lance [11] studied numerically a similar problemwhen the disk rotates with an angular velocity x in a fluid rotating with a different speed X. When X = 0, the system reducesto the free-disk problem of Von Kármán; when x = X, there is a solid-body rotation; and when x = 0, the problem is thesame as that discussed by Bödewadt. It was pointed out by Schwiderski and Lugt [12] that the non-existence of a propersolution to the boundary value problems for rotating flows of Von Kármán and Bödewadt is an indication that in realitythe flow is separated from the surface of the disk. The simple ‘Tea cup experiment’ described above, displays very clearly aseparation of the fluid from the bottom of the cup. Application of the suction is an effective device to reduce the chancesof separation. Later, the local boundary layer approximations of first order derived in [12] have been generalized by Schw-iderski and Lugt [13] to axisymmetric motions which rotates over a rotating disk of infinite dimensions. Numerical resultsare computed and discussed for a variety of Reynolds numbers and for cases for which the disk is rotating in the same senseand in the opposite sense as the fluid far away from the disk. The critical Reynolds numbers for steady laminar motionswhich are attached to the surface of the disk are computed and displayed. Nanda [14] studied the effects of uniform suctionon the revolving flow of a viscous liquid over a stationary disk. It was found that the presence of suction introduces an axialinflow at infinity and the same increases with an increase in suction. Nydahl [15] in his doctoral thesis has extended theBödewadt flow problem by incorporating the heat transfer phenomena. The results obtained by Nydahl effectively confirmthose of Bödewadt; those of Rogers and Lance [16] give a significantly larger value of H1. The momentum and the displace-ment thickness decrease as the suction velocity increases. The spin-up process in the Bödewadt flow of a viscous fluid hasbeen studied by Chawla and Purushothaman [17]. A comprehensive review of earlier works on flow and heat transfer due toa single and two parallel rotating disks up to 1989 has been included in a monograph by Owen and Rogers [18]. Recently,Chawla and Srivastava [19] have considered the physical situation in which the disk is performing torsional oscillationsin contact with a fluid in a state of rigid-body rotation in the far field.
A literature survey indicates that there has been extensive literature available regarding the boundary layer flow over arotating disk (Karman flow) in various situations. Such studies include different fluid models, magnetohydrodynamic andhydrodynamic cases, with and without heat transfer analysis. As can be seen from the literature, there has been relativelyscarce information regarding the Bödewadt type flow. The present study is an endeavor to fill this gap. Besides the abovereason, few other curious findings which motivated the present investigation are as follows:
As reported by Owen and Rogers [18] there is a discrepancy in the numerical value of H1, which was found to be 1.3494using the method of Rogers and Lance [11] on a VAX 8530 computer with a typical precision of 16 significant figures. Thisdepicts that the accuracy achieved by Rogers and Lance [11] was not sufficient to give an accurate value of H1. Moreover, ithas been pointed out by many authors that Bödewadt’s solution implies that there is a flow out of the boundary layer every-where and no mechanism for supplying fluid to it! For an infinite disk, the problem may be overcome by assuming an infinitereservoir of fluid from which the boundary layer can draw in an unspecified way. For the more practical case of a finite disk,it must be supposed that a similarity solution does not hold near the edge of the disk. However, it is consistent with expe-rience in other fields that a similarity solution becomes valid as the boundary layer flow develops. It may then be assumed
4286 B. Sahoo / Commun Nonlinear Sci Numer Simulat 16 (2011) 4284–4295
that fluid enters the boundary layer near the edge of the disk and that this fluid is available for continuity in the similaritysolution.
2. Formulation of the problem
In this case, we consider the non-Newtonian Reiner–Rivlin fluid occupying the space z > 0 over an infinite stationary disk(Fig. 1), which coincides with z = 0. The motion is due to the rotation of the fluid like a rigid body with constant angularvelocity X at large distance from the disk. A transverse uniform magnetic field B = (0,0,B0) is applied along the z-axis, atthe surface of the disk.
2.1. Flow analysis
The flow is described in the cylindrical polar coordinates (r,/,z). In view of the rotational symmetry @@/ � 0. Taking
V = (u,v,w) for the steady flow, the equations of continuity and motion are,
rþ @[email protected]¼ 0 ð1Þ
and
q [email protected]@rþw
@[email protected]� v2
r
� �þ rB2
0u ¼ @Trr
@rþ @Trz
@zþ Trr � T//
r; ð2Þ
q [email protected]@rþw
@[email protected]þ uv
r
� �þ rB2
0v ¼@Tr/
@rþ @Tz/
@zþ 2Tr/
r; ð3Þ
q [email protected]@rþw
� �¼ @Trz
@rþ @Tzz
@zþ Trz
r; ð4Þ
where Trr, Tr/, Trz, T//, T/z are the components of the stress tensor T given by
Tij ¼ �pdij þ 2leij þ 2lceikekj; ð5Þejj ¼ 0; ð6Þ
where p is denoting the pressure, l is the coefficient of viscosity and lc is the coefficient of cross-viscosity.The no-slip boundary conditions for the velocity field are given as
z ¼ 0; u ¼ 0; v ¼ 0; w ¼ 0;z!1; u! 0; v ! rX; p! p1:
ð7Þ
The Von Kármán transformations [20]
u ¼ rXFðfÞ; v ¼ rXGðfÞ; w ¼ffiffiffiffiffiffiffiXmp
HðfÞ; z ¼ffiffiffiffimX
rf; p� p1 ¼ �qmXP ð8Þ
reduce the Navier–Stokes equations for a Newtonian fluid to a set of ordinary differential equations. The same is also true for
a non-Newtonian Reiner–Rivlin fluid. We define the magnetic interaction number Mn by Mn ¼ rB20
qX and the non-Newtonian
cross-viscous parameter L ¼ /2X/1
. With these definitions and following Owen and Rogers [18], the equations of continuity
and motion take the form,
dHdfþ 2F ¼ 0; ð9Þ
d2F
df2 � HdFdf� F2 þ G2 �MnF � 1
2L
dFdf
� �2
� 3dGdf
� �2
� 2Fd2F
df2
" #¼ 1; ð10Þ
d2G
df2 � HdGdf� 2FG�MnGþ L
dFdf
dGdfþ F
d2G
df2
!¼ 0; ð11Þ
d2H
df2 � HdHdf� 7
2L
dHdf
d2H
df2 þdPdf¼ 0 ð12Þ
B. Sahoo / Commun Nonlinear Sci Numer Simulat 16 (2011) 4284–4295 4287
and the boundary conditions (7) become,
f ¼ 0 : F ¼ 0; G ¼ 0; H ¼ 0;f!1 : F ! 0; G! 1: ð13Þ
The fluid adheres to the surface partially and the motion of the fluid exhibits partial slip conditions. A generalization ofNavier’s partial slip condition gives [21], in the radial direction,
ujz¼0 ¼ k1Trzjz¼0 ð14Þ
and in the azimuthal direction
vjz¼0 ¼ k2T/zjz¼0; ð15Þ
where k1, k2 are respectively the slip coefficients, and Trz, T/z are the physical components of the stress tensor. Let
k ¼ k1
ffiffiffiffiXm
rl; g ¼ k2
ffiffiffiffiXm
rl: ð16Þ
With the help of the transformations (8) the corresponding partial slip boundary conditions (14) and (15) become
Fð0Þ ¼ k½F 0ð0Þ � LFð0ÞF 0ð0Þ�; Gð0Þ ¼ g½G0ð0Þ � 2LFð0ÞG0ð0Þ�; Hð0Þ ¼ 0; Fð1Þ ! 0; Gð1Þ ! 1: ð17Þ
The governing equations are still Eqs. (9)–(11). The boundary conditions (13) are replaced by the partial slip boundaryconditions (17). It is clear that the boundary conditions at infinity remain unaltered.
2.2. Heat transfer analysis
Due to the temperature difference between the surface of the disk and the ambient fluid, heat transfer takes place. Theenergy equation with viscous dissipation and Joule heating, takes the form,
qcp [email protected]@rþw
� �¼ j
@[email protected] þ l
� �2
� �2" #
þ rB20ðu2 þ v2Þ: ð18Þ
Introducing the non-dimensional variable h ¼ T�T1Tw�T1
and using the Von Kármán transformations (8), Eq. (18) becomes,
Hdhdf¼ 1
Prd2h
df2 þ EcðF 02 þ G02Þ þMnEcðF2 þ G2Þ; ð19Þ
where Tw is the temperature at the surface of the disk, T1 is the temperature of the ambient fluid at large distance from thedisk, Pr ¼ cpl
j is the Prandtl number and Ec ¼ r2X2
ðTw�T1Þcpis the Eckert number. The boundary conditions in terms of the non-
dimensional parameter h are expressed as
f ¼ 0 : h ¼ 1;
f!1 : h! 0: ð20Þ
The heat transfer from the disk surface to the fluid is computed by the application of the Fourier’s law, q ¼ �jð@[email protected] Þw. Intro-
ducing the transformed variables, the expression for q becomes
q ¼ �jðTw � T1ÞffiffiffiffiXm
rdhð0Þ
df: ð21Þ
By rephrasing the heat transfer results in terms of the Nusselt number defined as Nu ¼ qffiffimX
pjðTw�T1Þ, we get
Nu ¼ �dhð0Þdf
: ð22Þ
In terms of the variables of the analysis, the expressions of the tangential shear stress Tu, radial shear stresses Tr and thedimensionless moment coefficient Cm are,
T/
qrffiffiffiffiffiffiffiffiffimX3
p ¼ T/ ¼dGð0Þ
df� 2LFð0ÞdGð0Þ
df;
Tr
qrffiffiffiffiffiffiffiffiffimX3
p ¼ Tr ¼dFð0Þ
df� LFð0ÞdFð0Þ
df;
Cm ¼�2p½G0ð0Þ � 2LFð0ÞG0ð0Þ�ffiffiffiffi
Rp ð23Þ
where R ¼ R2Xm is the Reynolds number based on the radius and the tip velocity.
4288 B. Sahoo / Commun Nonlinear Sci Numer Simulat 16 (2011) 4284–4295
3. Numerical solution of the problem
We solve the system of nonlinear differential equations (9)–(11) and (19) under the boundary conditions (17) and (20) byadopting the same second order numerical scheme described in the previous works [1,2,22]. The semi-infinite integrationdomain f 2 [0,1) is replaced by a finite domain f 2 [0,f1). In practice, f1 should be chosen sufficiently large so that thenumerical solution closely approximates the terminal boundary conditions.
Now, suppose we introduce a mesh defined by
Table 1Variatio
f
0.00.51.01.52.02.53.03.59.5
10.010.511.011.512.012.520.025.025.526.026.01
fi ¼ ih ði ¼ 0;1; . . . nÞ; ð24Þ
h being the mesh size, and discretize Eqs. (9)–(11) and (19) using the central difference approximations for the derivatives,then the following equations are obtained.
Fiþ1 � 2Fi þ Fi�1
h2 � HiFiþ1 � Fi�1
2h
� �� F2
i þ G2i �MnFi �
12
LFiþ1 � Fi�1
2h
� �2
� 3Giþ1 � Gi�1
2h
� �2"
�2FiFiþ1 � 2Fi þ Fi�1
h2
� ��� 1 ¼ 0; ð25Þ
Giþ1 � 2Gi þ Gi�1
h2 � HiGiþ1 � Gi�1
2h
� �� 2FiGi �MnGi þ L
Fiþ1 � Fi�1
2h
� �Giþ1 � Gi�1
2h
� �þ Fi
Giþ1 � 2Gi þ Gi�1
h2
� �� �¼ 0;
ð26Þ
Hihiþ1 � hi�1
2h
� �� 1
Prhiþ1 � 2hi þ hi�1
h2
� �� Ec
Fiþ1 � Fi�1
2h
� �2
þ Giþ1 � Gi�1
2h
� �2" #
�MnEcðF2i þ G2
i Þ ¼ 0; ð27Þ
Hiþ1 ¼ Hi � hðFi þ Fiþ1Þ: ð28Þ
Rest of the solution procedure is same as described in our previous works [1,2,22]. A noteworthy fact is that for the samevalues of the flow parameters, the value of f1, the numerical infinity should be chosen larger as compared to the previouslystudied Karman flow problem, so that the numerical solution closely approximates the terminal boundary conditions at f1.
4. Results and discussion
The value of f1, the numerical infinity has been taken larger as compared to the previous problem and kept invariantthrough out the run of the program. Although, the results are shown only from the disk surface f = 0 to f = 14.0, the numer-ical integrations were performed over a substantially larger domain in order to assure that the outer asymptotic boundary
ns of F, G, H for L = 0, Mn = 0, and k(= g) = 0.
F G H
Current result Owen & Rogers [18] Current result Owen & Rogers [18] Current result Owen & Rogers [18]
0.000000 0.0000 0.000000 0.0000 0.000000 0.0000�0.348650 �0.3487 0.383430 0.3834 0.194373 0.1944�0.478766 �0.4788 0.735429 0.7354 0.624103 0.6241�0.449633 �0.4496 1.013401 1.0134 1.098743 1.0987�0.328745 �0.3287 1.192367 1.1924 1.492875 1.4929�0.176206 �0.1762 1.272136 1.2721 1.745869 1.7459�0.036086 �0.0361 1.271405 1.2714 1.849641 1.8496
0.066310 0.0663 1.218219 1.2182 1.830807 1.8308�0.010216 �0.0102 1.011849 1.0118 1.361698 1.3617�0.003282 �0.0033 1.012120 1.0121 1.368328 1.3683
0.001819 0.0018 1.009906 1.0099 1.368882 1.36890.004738 0.0047 1.006537 1.0065 1.365423 1.36540.005681 0.0057 1.003090 1.0031 1.360067 1.36010.005170 0.0052 1.000271 1.0003 1.354546 1.35450.003827 0.0038 0.998411 0.9984 1.350003 1.35000.000102 – 0.999893 – 1.349325 –0.000009 – 1.000014 – 1.349457 –0.000011 – 1.000007 – 1.349447 –0.000010 – 1.000001 – 1.349437 –0.000008 – 0.999997 – 1.349428 –0.000000 0.0000 1.000000 1.0000 1.349424 1.3494
0 2 4 6 8 10 12 14−0.5
0
0.5
1
1.5
2
ζ
H
G
F
Fig. 2. Velocity profile for the Newtonian flow at Mn = 0 and k (= g) = 0.
0 2 4 6 8 10 12 14−0.3
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
ζ
F (ζ
)
L=0.0L=1.0L=2.0L=3.0
Fig. 3. Variation of F with L at k(= g) = 1.0.
0 2 4 6 8 10 12 14−0.14
−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
ζ
F (ζ
)
λ(=η)=1.0=2.0=3.0=4.0
Fig. 4. Variation of F with k(= g) at L = 2.0.
B. Sahoo / Commun Nonlinear Sci Numer Simulat 16 (2011) 4284–4295 4289
0 2 4 6 8 10 12 140.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
ζ
G (ζ
)
L=0.0L=1.0L=2.0L=3.0
Fig. 5. Variation of G with L at k(= g) = 1.0.
G (ζ
)
0 2 4 6 8 10 12 140.5
0.6
0.7
0.8
0.9
1
1.1
1.2
ζ
λ(=η)=1.0=2.0=3.0=4.0
Fig. 6. Variation of G with k(= g) at L = 2.0.
0 2 4 6 8 10 12 140
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
ζ
H (ζ
)
L=0.0L=1.0L=2.0L=3.0
Fig. 7. Variation of H with L at k(= g) = 1.0.
4290 B. Sahoo / Commun Nonlinear Sci Numer Simulat 16 (2011) 4284–4295
0 2 4 6 8 10 12 140
0.1
0.2
0.3
0.4
0.5
0.6
0.7
ζ
H (ζ
)
λ(=η)=1.0=2.0=3.0=4.0
Fig. 8. Variation of H with k(= g) at L = 2.0.
0 2 4 6 8 10 12 140
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ζ
θ (ζ
)
L=0.0L=1.0L=2.0L=3.0
Fig. 9. Variation of h with L at k(= g) = 1.0, Pr = 2.0 & Ec = 0.5.
θ (ζ
)
0 2 4 6 8 10 12 140
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ζ
λ(=η)=1.0=2.0=3.0=4.0
Fig. 10. Variation of h with k(= g) at L = 2.0, Pr = 2.0 & Ec = 0.3.
B. Sahoo / Commun Nonlinear Sci Numer Simulat 16 (2011) 4284–4295 4291
0 2 4 6 8 10 12 140
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ζ
θ (ζ
)
Pr=1.0Pr=2.0Pr=3.0Pr=4.0
Fig. 11. Variation of h with Pr at L = 2.0, k(= g) = 1.0 & Ec = 0.5.
θ (ζ
)
0 2 4 6 8 10 12 140
0.2
0.4
0.6
0.8
1
1.2
1.4
ζ
Ec=0.0Ec=2.0Ec=4.0Ec=6.0
Fig. 12. Variation of h with Ec at L = 2.0, k(= g) = 1.0 & Pr = 2.0.
Table 2Variations of F, G, H, h with Mn at L = 1.0 & k(= g) = 1.
f Mn F G H h
3.0 0.0 0.008211 1.037577 0.619340 0.0459110.1 �0.039069 1.043229 0.915031 0.2708980.2 �0.086390 1.048465 1.215490 0.3428610.3 �0.134339 1.053833 1.522010 0.380127
5.0 0.2 �0.079588 0.983112 1.505396 0.2406470.3 �0.115963 0.975023 1.973284 0.2627900.4 �0.147374 0.969996 2.445377 0.275198
9.0 0.6 �0.287108 0.875253 5.349876 0.1329620.7 �0.291781 0.853310 6.034170 0.1328700.8 �0.294788 0.849983 6.802002 0.132007
4292 B. Sahoo / Commun Nonlinear Sci Numer Simulat 16 (2011) 4284–4295
conditions are satisfied. To see if the program runs correctly, the values of F, G and H, in absence of the external magneticfield and for no-slip condition are compared with (see Table 1) those reported by Owen and Rogers [18] for a viscous fluid,and have been plotted graphically in Fig. 2. The new and accurate numerical results presented herein confirm the high qual-ity of the earlier calculations performed by Owen and Rogers.
Table 3Variations of H1, Tr, T/, Cm, & Nu with different flow parameters.
L Mn k(=g) H1 Tr T/ Cm Nu
0.0 1.790177 �0.107366 0.379735 �2.385943 0.3838382.0 0.1 2.0 1.624113 �0.068741 0.384583 �2.416408 0.2682504.0 1.571665 �0.050187 0.364340 �2.289215 0.222767
0.0 0.275792 �0.039122 0.365226 �2.294782 0.3521573.0 0.1 2.0 1.591987 �0.057852 0.375863 �2.361620 0.241134
0.2 2.939624 �0.077181 0.385003 �2.419044 0.188326
1.0 1.814946 �0.151520 0.541268 �3.400885 0.3331272.0 0.1 3.0 1.550346 �0.041316 0.283332 �1.780225 0.229701
5.0 1.484915 �0.021019 0.182252 �1.145125 0.188021
0 2 4 6 8 10−4.5
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
λ
CM
Fig. 13. Variation of the moment coefficient CM with k for L = 1 and Re = 1.
CM
0 20 40 60 80 100 120−6
−5.5
−5
−4.5
−4
−3.5
−3
L
Fig. 14. Variation of the moment coefficient CM with L for k = 1 and Re = 1.
B. Sahoo / Commun Nonlinear Sci Numer Simulat 16 (2011) 4284–4295 4293
The Fig. 2 depicts that near the disk, the radial component of the velocity F is radially inwards. It may be of interest to notethat this radially inward flow is the cause of the accumulation of the tea leaves at the center of a stirred cup of tea. In order tohave an insight of the flow and heat transfer characteristics, results are plotted graphically and tabulated, for the uniformroughness (k = g), and different values of the flow parameters.
The variations of the radial component of the velocity, F with the non-Newtonian cross-viscous parameter L and the slipparameter k(= g) are shown in Figs. 3 and 4, respectively, when the other flow parameters are kept constant. Fig. 3 revealsthat L has a prominent effect on F, near the disk. The magnitude of the radial inflow, near the disk, decreases with an increasein L. The radial component of velocity, near the disk remains negative for all values of the non-Newtonian parameter L,reverses the direction away from the disk, and finally approaches its asymptotic value. Thus, cross-overs are found in thevelocity profile. Fig. 4 shows the variation of F with the slip parameter k(= g), when other flow parameters are kept constant.
4294 B. Sahoo / Commun Nonlinear Sci Numer Simulat 16 (2011) 4284–4295
It is clear that the effect of slip on F is also prominent near the disk. The velocity profiles reverse the direction away from thedisk and approach the asymptotic value at a shorter distance from the disk as compared to the former case.
Figs. 5 and 6 depict the variations of the azimuthal component of the velocity with L and k(= g), respectively. The non-Newtonian parameter L has a spectacular effect on G, away from the disk (near f = 2.0), as is clear from Fig. 5. An increasein L decreases the velocity profile G near the disk. The velocity component increases (see Fig. 6) near the disk with an increasein k(= g). Thus, it is interesting to find that the slip has an opposite effect to that of the cross-viscous parameter on G.
The variations of the axial component of the velocity H with L and k(= g) have been plotted in Figs. 7 and 8, respectively.The figures show that both the parameters have the similar effect on H. It is clear that the axial velocity becomes flatter withan increase in L and k(= g).
The non-dimensional temperature h increases throughout the domain of integration with an increase in both L and k(= g),as is clear from Figs. 9 and 10, respectively. Thus, both the parameters increase the thermal boundary layer thickness, anddecrease the heat transfer rate (see Table 3). Figs. 11 and 12 depict the variation of h with Pr and Ec, when other flow param-eters are kept constant. The temperature gets decreased with an increase in Pr and it gets increased with an increase in Ec, aswas expected.
Moreover, the effects of the magnetic field Mn on the velocity and temperature profiles can be readily seen from Table 2.The effects of the magnetic field start manifesting as the value of Mn increases from zero. Great care is taken to ensure that aphysically sensible solution has not been missed with the inclusion of the magnetic field. The variations of various charac-teristics of the flow pattern, like the radial and tangential shear stresses, the moment coefficient and the Nusselt number withdifferent flow parameters have been computed and tabulated in Table 3.
Another interesting quantity, which can be deduced from the profiles of F and G, is the turning moment for the disk withfluid on both sides. The expression of the dimensionless moment coefficient CM is given by:
CM ¼�2pG0ð0Þ½1� 2LFð0Þ�ffiffiffiffiffiffi
Rep ; ð29Þ
with Re = XR2/m the rotational Reynolds number based on the disk radius R and the maximum velocity (XR). This definitionof CM is the extension to the finite disk problem, which supposes that the disk radius is large enough. For L = 0, Eq. (29) re-duces to CM =� 2pG0(0). In that case, the moment coefficient depends only on G0(0). Note that the present value of G0(0) for aNewtonian fluid and no slip condition (L = k = 0) is in excellent agreement with the value G0(0) = 0.77289 obtained by Owenand Rogers [18]. Thus, we got the classical value CM = �4.86 in that basic case.
Variations of CM with the slip k and non-Newtonian parameter L are shown in Figs. 13 and 14, respectively. Whatever theflow parameters, CM exhibits negative values. It may be attributed to the flow problem: a rotating fluid at infinity over a sta-tionary disk. Thus, the axial gradient of the tangential velocity G0 is still positive and CM is then always negative. The VonKármán flow considered by Sahoo [2] is precisely the inverse problem, which explains the different signs. For L = 1 andRe = 1 (see Fig. 13), the moment coefficient CM in absolute values strongly increases (resp. decreases) with increasing valuesof the slip parameter k for k < 0.6 (resp. k > 0.6). It tends rapidly to zero for high values of k, which means that the torquerequired to maintain the disk at its original speed is almost zero when the slip parameter is high. Omitting the differentboundary conditions between the two problems, the present results confirm the previous ones of Sahoo [2] for Von Kármánflows and comparable values are obtained.
The slip parameter k is now fixed to unity and the non-Newtonian parameter L varies between 0 and 100 for Re = 1(Fig. 14). CM increases in magnitude with the parameter L. In that case, the torque required to maintain the disk at rest ismuch higher than those necessary to maintain the disk at X for the Von Kármán flow considered by Sahoo [2].
5. Conclusions
In this work, the slip flow and heat transfer due to the rotation of an electrically conducting, non-Newtonian Reiner–Riv-lin fluid near a stationary disk have been examined precisely for the first time. The resulting system of highly nonlinear dif-ferential equations have been solved by adopting a second order numerical scheme. The combined effects of the slip [k (= g)]and the non-Newtonian parameter (L) on the velocity and temperature fields are studied in detail. It is interesting to find thatk(= g) and L have similar effects on the radial, axial components of velocity, and the temperature field. On the other hand,both the parameters have opposite effects on the azimuthal component of the velocity.
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