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A note on micropolar fluid flow and heat transfer over a porous shrinking sheet
International Journal of Heat and Mass Transfer 72 (2014) 388–391
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
A note on micropolar fluid flow and heat transfer over a porous shrinking sheet M. Turkyilmazoglu ⇑ Department of Mathematics, Hacettepe University, 06532 Beytepe, Ankara, Turkey
a r t i c l e
i n f o
Article history: Received 24 November 2013 Received in revised form 13 January 2014 Accepted 13 January 2014 Available online 2 February 2014 Keywords: Micropolar fluid Shrinking surface Multiple solution Heat transfer Mass transfer
a b s t r a c t The flow of micropolar fluid and heat transfer past a porous shrinking sheet is studied in this note. The main concern, unlike the recent numerical work of Bhattacharyya et al. (2012) [1], is to determine mathematically the bounds of multiple existing solutions of purely exponential kind. The presence of dual solutions are proved for the flow field, whose closed-form formulae are then derived. The energy equation is also treated analytically yielding exact solutions beneficial to understand the rate of heat transfer. Critical values for the existence or nonexistence of unique/multiple solutions are worked out. The exact form of velocity/temperature profiles and the skin friction/couple stress/heat transfer parameters enable one to easily catch the physical processes occurring in the present model. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction
2. Problem formulation
The model of microfluid first pioneered by Eringen [2] has been the focus of past research to explain the character of certain real fluid flows. This mathematical model takes into account a class of fluids having certain microscopic characters arising from the local structure and micromotions of the fluid elements [1]. Micropolar fluids are known as part of microfluids, which are also due to Eringen [3]. A rich survey of the past and recent works done concerning the micropolar fluids and, their technological and industrial applications were recently presented in [1]. One can also refer to the relevant stretching or shrinking body studies by Fang and Zhang [4], Khan et al. [5,6] and Turkyilmazoglu [7,8], amongst many others. In the recent paper of [1], the flow of micropolar fluid and heat transfer over a permeable shrinking sheet was studied by numerical means. The deriving force of the current work is, as opposed to the aforementioned numerical treatments, to mathematically explore the physical problem under consideration. Expressions of threshold values for the nonexistence of the solutions or the existence of dual solutions are derived in closed-form. Such analytical formulae are very useful, not only to interpret the flow and temperature fields but also to view behaviour of the skin friction coefficient, couple stress coefficient and Nusselt number, which play major role in industrial applications.
We consider a steady two-dimensional flow of micropolar fluid due to an impermeable shrinking sheet, whose shrinking speed is u ¼ cx; c > 0. The temperature is assumed to vary from a constant wall value to a constant free stream temperature. Making use of the classical boundary layer approximation, the governing equations of motion for the micropolar fluid and heat transfer were successfully extracted in [1], refer to Eqs. (1)–(6) in [1]. For the sake of being concise, we only give the final similarity equations that govern the flow motion and heat transfer
⇑ Tel.: +90 03122977850; fax: +90 03122972026. E-mail address: [email protected] 0017-9310/$ - see front matter Ó 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.01.039
00
0
ð1 þ KÞf 000 þ ff f 02 þ Kh ¼ 0; 00
0
ð1 þ K=2Þh þ fh f 0 h Kð2h þ f 00 Þ ¼ 0; 00
ð2:1Þ
0
h þ Prf h ¼ 0; subject to the boundary conditions
f ðgÞ ¼ s; f 0 ðgÞ ! 0;
f 0 ðgÞ ¼ 1; hðgÞ ! 0;
00
hðgÞ ¼ mf ðgÞ;
hðgÞ ¼ 1 at g ¼ 0;
h ! 0 as g ! 1; ð2:2Þ
where g is a scaled boundary layer coordinate, f 0 ðgÞ is the similarity velocity component, hðgÞ is the similarity microrotation or angular velocity, s is the mass flux velocity with s < 0 for suction and s > 0 for injection, K is the material parameter and m represents a measurement for the concentration of microelements, respectively. Moreover, h is the scaled fluid temperature and Pr is the usual Prandtl number.
389
M. Turkyilmazoglu / International Journal of Heat and Mass Transfer 72 (2014) 388–391
(a)
(b)
2.0
1.5
h’ 0
f ’’0
1.5 1.0
1.0 0.5
0.5 2.0
2.1
2.2
2.3
2.4
0.0
2.5
2.0
2.1
2.2
2.3
s
(c)
2.4
2.5
s
0.95 0.90 0.85
’0
0.80 0.75 0.70 0.65 0.60 0.55
2.0
2.1
2.2
2.3
2.4
2.5
s Fig. 1. The physical coefficients at m ¼ 1=2 (thick curves for the first branch and dotted curves for the second ones, respectively, outer loop for K ¼ 0:1 and inner loop for 0 K ¼ 0:2) as a function of s. (a) Skin friction coefficient f 00 ð0Þ, (b) couple stress coefficient h ð0Þ and (c) heat transfer coefficient h0 ð0Þ with Pr ¼ 3=7.
whose simultaneous solutions yield the dual solutions
3. Exact solutions A numerical treatment of Eqs. (2.1), (2.2) has already been given in [1]. We instead present exact analytic solutions in this section. Branch 1 or branch 2 refers to part of multiple solutions in the sequel.
Based on the analytical solutions derived in Crane [9], Troy et al. [10], Mcleod and Rajagopal [11], Lawrence and Rao [12] and Pop and Na [13] for the stretching/shrinking sheet problems of particular type, it is desired to obtain exact solutions of the system (2.1)– (2.2), which is influenced by the material, concentration and wall suction parameters. The above literature enables us to assume a solution of the form
1 ekg ; k 00 hðgÞ ¼ mf ðgÞ ¼ mkegk :
ð3:3Þ
B @m ¼
ð3:4Þ
2
8ð1 þ KÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ; s 4 þ 4K þ 8K 2 þ s2
0
8ð1 þ KÞ2 B @m ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ; s þ 4 þ 4K þ 8K 2 þ s2
1 rffiffiffiffiffi 2C k¼ A; m
ð3:5Þ
1 rffiffiffiffiffi 2C k¼ A: m
Thus, the first two of (3.5) are valid for m ¼ 1=2 and the next two are for other values of m. It should be noticed that the structure of Eq. (3.5) puts restrictions on the solutions such that the first two holds for all physical K P 0 and
sP
Upon substitution, the entire boundary conditions in (2.2) are seen to be met by the solution (3.3). It is obvious that true solutions require the condition k > 0. As a result, the momentum and angular velocity equations in (2.1) produce the relations
1 þ sk þ ð1 þ Kð1 þ mÞÞk2 ¼ 0; 2mð1 sk þ k2 Þ þ Kð2 þ m 4 þ k2 Þ ¼ 0;
m ¼ 1=2; 0
3.1. Flow field
f ðgÞ ¼ s
m ¼ 1=2;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 4 2K þ s2 ; k¼ 2þK ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s þ 4 2K þ s2 k¼ ; 2þK s
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 þ 2K ;
whereas the third for 0 6 K 6 1=2 and
sP
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 4K 8K 2
and the fourth, in addition to the above, for K P 1=2 and all s. Therefore, the critical values for the nonexistence or existence of dual solutions are given by the formulae
390
M. Turkyilmazoglu / International Journal of Heat and Mass Transfer 72 (2014) 388–391
(a)
(b)
(c)
Fig. 2. (a) Dual velocity f 0 ðgÞ (upper first, lower second branches), (b) dual angular velocity hðgÞ (upper first, lower second branches) and (c) dual temperature hðgÞ (lower first, upper second branches). The physical parameters are m ¼ 1=2, K ¼ 0:1 and Pr ¼ 3=7 (thin curves for s ¼ 2:1, dashed curves for s ¼ 2:15 and dot-dashed curves for 2:2, respectively.
(a) m ¼ 1=2; m¼
(b)
pffiffiffi ! 2 k ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ; 2þK pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 K 2K 2 2 : ð3:6Þ s ¼ 2 1 K 2K ; k ¼ 1þK
pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi s ¼ 2 2 þ K;
2 þ 2K ; 1 2K
Since the dual solutions are based on the discriminants in (3.6), the terms in the system (2.1) involving the material constant K are responsible for the existence of dual solution, under the presence of wall suction. The above technical analysis explains the discussion given in [1] on the existence domain of the parameters obtained via numerical methods. Moreover, it is known that triple solutions exist for the classical flow over a shrinking sheet, see, for example, [7,8]. However, for the same problem with the micropolar fluid elements taken into account, only dual solutions are found. Furthermore, noting that the skin friction coefficient is defined by f 00 ð0Þ ¼ k 0 and the couple stress coefficient by h ð0Þ ¼ mk2 , these are easily evaluated from the exact formulas (3.5). 3.2. Temperature field It is straightforward to show that the energy equation in (2.1) together with the temperature boundary conditions in (2.2) leads to the exact solution
h
h i egk Pr ; C Prð1þskÞ 2 2 k k h i h i hðgÞ ¼ Prð1þskÞ Pr C Prð1þskÞ ; 0 C ; 2 2 k k k2
C
Fig. 3. The solution regions for (a) k and (b) m versus s. Thick curves for the first branch and dotted curves for the second ones, outer loop for K ¼ 0:1 and inner loop for K ¼ 0:2, respectively.
i
Prð1þskÞ ;0 k2
from which we can define the local Nusselt number by
ð3:7Þ
M. Turkyilmazoglu / International Journal of Heat and Mass Transfer 72 (2014) 388–391
Prð1þskÞ Pr k2 ek2 kPr2 k 0 h i h i: Nu ¼ h ð0Þ ¼ Prð1þskÞ Pr C Prð1þskÞ ; 0 C ; 2 2 2 k k k
391
5. Concluding remarks
ð3:8Þ
Here, C½a; z is the incomplete gamma function. 4. Results and discussion We now demonstrate that the unique/multiple solutions expressed by (3.3) and (3.7), confined to the domain of parameters given by (3.5), exactly coincide with the recently published ones in [1]. Indeed, Figs. 1 and 2 show the same results as displayed in [1] for m ¼ 1=2 (see, Figs. 1–3 and 9–11 in [1]). It is noted that an adjustment in the Prandtl number is required to have the same Prandtl numbers, so our Prandtl number of Pr ¼ 3=7 corresponds to that of Pr ¼ 1 given in [1]. Therefore, the conclusions reached in [1] using numerical methods are clarified through the exact analytic results given in this note. Moreover, for the entire values of m (different from 1/2 as given in [1]), exact formulas in (3.5) can be employed, see Fig. 3, for the domain of existence of unique/dual solutions, which are timeconsuming to get using numerical ways as in [1]. From Fig. 3, one can work out the skin-friction coefficient, the couple stress coefficient and the heat transfer coefficient for any fixed value of m and Pr. For instance, the following exact dual results are obtained with K ¼ 1=10; m ¼ 5; s ¼ 5p31ffiffiffiffi and Pr ¼ 1; 10
pffiffiffi pffi2 4 10g f 0 ðgÞ ¼ e 5g ; f 0 ðgÞ ¼ e 11 ; pffiffiffi pffiffiffiffiffiffi pffi2 20 pffiffiffiffiffiffi 4 10g hðgÞ ¼ 10e 5g ; hðgÞ ¼ 10e 11 ; 11 rffiffiffi pffiffiffiffiffiffi! 2 00 4 10 ; ; f ð0Þ ¼ f 00 ð0Þ ¼ 5 11 800 0 0 h ð0Þ ¼ 2; h ð0Þ ¼ ; 121 pffiffiffi i h h p ffi2 i 0
4 10g 1 C 35 C 35 ; 52 e 5g C 759 ; 121 e 11 C 759 800 800 160 @hðgÞ ¼
; hðgÞ ¼
A; C 35 C 35 ; 52 C 759 ; 121 C 759 800 800 160
ðh0 ð0Þ ¼ 1:34542361; h0 ð0Þ ¼ 1:56981979Þ: ð4:9Þ Although, from Fig. 2(a)–(c) it is observed that for both branch 1 and branch 2 solutions the far field boundary conditions are satisfied asymptotically, the branch 1 solution is believed to be more physically meaningful. It is, of course, necessary to do a stability analysis to give a final decision on this issue.
An analytical study has been implemented for the flow of micropolar fluid and heat transfer over a shrinking sheet under porous wall conditions. The particular attention is given to derivation of exact solutions in contrary to the numerical ones already existing in the literature. By means of such a treatment, it is shown to be quite possible to explore the physical features of microrotation and temperature fields, whether they are unique or multiple. In fact, by solving two coupled algebraic equations resulting from the exponential type solution, it is found that no solution or at most two solutions may exist depending on the working parameters considered in the physical model. The numerical results and conclusions made in a recent paper [1] can be easily reached from the analysis of formulae presented here. Further rich properties of both flow and temperature fields can also be accessed and assessed via the exact solutions. References [1] K. Bhattacharyya, S. Mukhopadhyay, G.C. Layek, I. Pop, Effects of thermal radiation on micropolar fluid flow and heat transfer over a porous shrinking sheet, Int. J. Heat Mass Transfer 55 (2012) 2945–2952. [2] A.C. Eringen, Simple microfluids, Int. J. Eng. Sci. 2 (1964) 205–217. [3] A.C. Eringen, Theory of micropolar fluids, J. Math. Mech. 16 (1966) 118. [4] T. Fang, J. Zhang, Thermal boundary layers over a shrinking sheet: an analytic solution, Acta Mech. 209 (2010) 325–343. [5] Y. Khan, Q. Wu, N. Faraz, A. Yildirim, S.T. Mohyud-Din, Heat transfer analysis on the magnetohydrodynamic flow of a non-Newtonian fluid in the presence of thermal radiation: an analytic solution, Z. Naturforsch. 67 (2012) 147–152. [6] M. Madani, Y. Khan, M. Fathizadeh, A. Yildirim, Application of homotopy perturbation and numerical methods to the magneto-micropolar fluid flow in the presence of radiation, Eng. Comput. 29 (2012) 277–294. [7] M. Turkyilmazoglu, Multiple solutions of heat and mass transfer of MHD slip flow for the viscoelastic fluid over a stretching sheet, Int. J. Therm. Sci. 50 (2011) 2264–2276. [8] M. Turkyilmazoglu, I. Pop, Exact analytical solutions for the flow and heat transfer near the stagnation point on a stretching/shrinking sheet in a Jeffrey fluid, Int. J. Heat Mass Transfer 57 (2013) 8288. [9] L. Crane, Flow past a stretching plate, Z. Angew. Math. Phys. 21 (1970) 645– 647. [10] W.C. Troy, E.A. Overman, H.G.B. Ermentrout, J.P. Keener, Uniqueness of flow of a second order fluid past a stretching sheet, Q. Appl. Math. 44 (1987) 753–755. [11] J.B. Mcleod, K.R. Rajagopal, On the uniqueness of flow of a Navier Stokes fluid due to a stretching boundary, Arch. Ration. Mech. Anal. 98 (1987) 385–395. [12] P.S. Lawrence, B.N. Rao, The non-uniqueness of the MHD flow of a viscoelastic fluid past a stretching sheet, Acta Mech. 112 (1995) 223–228. [13] I. Pop, T.Y. Na, A note on MHD flow over a stretching permeable surface, Mech. Res. Commun. 25 (1998) 263–269.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
A note on micropolar fluid flow and heat transfer over a porous shrinking sheet M. Turkyilmazoglu ⇑ Department of Mathematics, Hacettepe University, 06532 Beytepe, Ankara, Turkey
a r t i c l e
i n f o
Article history: Received 24 November 2013 Received in revised form 13 January 2014 Accepted 13 January 2014 Available online 2 February 2014 Keywords: Micropolar fluid Shrinking surface Multiple solution Heat transfer Mass transfer
a b s t r a c t The flow of micropolar fluid and heat transfer past a porous shrinking sheet is studied in this note. The main concern, unlike the recent numerical work of Bhattacharyya et al. (2012) [1], is to determine mathematically the bounds of multiple existing solutions of purely exponential kind. The presence of dual solutions are proved for the flow field, whose closed-form formulae are then derived. The energy equation is also treated analytically yielding exact solutions beneficial to understand the rate of heat transfer. Critical values for the existence or nonexistence of unique/multiple solutions are worked out. The exact form of velocity/temperature profiles and the skin friction/couple stress/heat transfer parameters enable one to easily catch the physical processes occurring in the present model. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction
2. Problem formulation
The model of microfluid first pioneered by Eringen [2] has been the focus of past research to explain the character of certain real fluid flows. This mathematical model takes into account a class of fluids having certain microscopic characters arising from the local structure and micromotions of the fluid elements [1]. Micropolar fluids are known as part of microfluids, which are also due to Eringen [3]. A rich survey of the past and recent works done concerning the micropolar fluids and, their technological and industrial applications were recently presented in [1]. One can also refer to the relevant stretching or shrinking body studies by Fang and Zhang [4], Khan et al. [5,6] and Turkyilmazoglu [7,8], amongst many others. In the recent paper of [1], the flow of micropolar fluid and heat transfer over a permeable shrinking sheet was studied by numerical means. The deriving force of the current work is, as opposed to the aforementioned numerical treatments, to mathematically explore the physical problem under consideration. Expressions of threshold values for the nonexistence of the solutions or the existence of dual solutions are derived in closed-form. Such analytical formulae are very useful, not only to interpret the flow and temperature fields but also to view behaviour of the skin friction coefficient, couple stress coefficient and Nusselt number, which play major role in industrial applications.
We consider a steady two-dimensional flow of micropolar fluid due to an impermeable shrinking sheet, whose shrinking speed is u ¼ cx; c > 0. The temperature is assumed to vary from a constant wall value to a constant free stream temperature. Making use of the classical boundary layer approximation, the governing equations of motion for the micropolar fluid and heat transfer were successfully extracted in [1], refer to Eqs. (1)–(6) in [1]. For the sake of being concise, we only give the final similarity equations that govern the flow motion and heat transfer
⇑ Tel.: +90 03122977850; fax: +90 03122972026. E-mail address: [email protected] 0017-9310/$ - see front matter Ó 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.01.039
00
0
ð1 þ KÞf 000 þ ff f 02 þ Kh ¼ 0; 00
0
ð1 þ K=2Þh þ fh f 0 h Kð2h þ f 00 Þ ¼ 0; 00
ð2:1Þ
0
h þ Prf h ¼ 0; subject to the boundary conditions
f ðgÞ ¼ s; f 0 ðgÞ ! 0;
f 0 ðgÞ ¼ 1; hðgÞ ! 0;
00
hðgÞ ¼ mf ðgÞ;
hðgÞ ¼ 1 at g ¼ 0;
h ! 0 as g ! 1; ð2:2Þ
where g is a scaled boundary layer coordinate, f 0 ðgÞ is the similarity velocity component, hðgÞ is the similarity microrotation or angular velocity, s is the mass flux velocity with s < 0 for suction and s > 0 for injection, K is the material parameter and m represents a measurement for the concentration of microelements, respectively. Moreover, h is the scaled fluid temperature and Pr is the usual Prandtl number.
389
M. Turkyilmazoglu / International Journal of Heat and Mass Transfer 72 (2014) 388–391
(a)
(b)
2.0
1.5
h’ 0
f ’’0
1.5 1.0
1.0 0.5
0.5 2.0
2.1
2.2
2.3
2.4
0.0
2.5
2.0
2.1
2.2
2.3
s
(c)
2.4
2.5
s
0.95 0.90 0.85
’0
0.80 0.75 0.70 0.65 0.60 0.55
2.0
2.1
2.2
2.3
2.4
2.5
s Fig. 1. The physical coefficients at m ¼ 1=2 (thick curves for the first branch and dotted curves for the second ones, respectively, outer loop for K ¼ 0:1 and inner loop for 0 K ¼ 0:2) as a function of s. (a) Skin friction coefficient f 00 ð0Þ, (b) couple stress coefficient h ð0Þ and (c) heat transfer coefficient h0 ð0Þ with Pr ¼ 3=7.
whose simultaneous solutions yield the dual solutions
3. Exact solutions A numerical treatment of Eqs. (2.1), (2.2) has already been given in [1]. We instead present exact analytic solutions in this section. Branch 1 or branch 2 refers to part of multiple solutions in the sequel.
Based on the analytical solutions derived in Crane [9], Troy et al. [10], Mcleod and Rajagopal [11], Lawrence and Rao [12] and Pop and Na [13] for the stretching/shrinking sheet problems of particular type, it is desired to obtain exact solutions of the system (2.1)– (2.2), which is influenced by the material, concentration and wall suction parameters. The above literature enables us to assume a solution of the form
1 ekg ; k 00 hðgÞ ¼ mf ðgÞ ¼ mkegk :
ð3:3Þ
B @m ¼
ð3:4Þ
2
8ð1 þ KÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ; s 4 þ 4K þ 8K 2 þ s2
0
8ð1 þ KÞ2 B @m ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ; s þ 4 þ 4K þ 8K 2 þ s2
1 rffiffiffiffiffi 2C k¼ A; m
ð3:5Þ
1 rffiffiffiffiffi 2C k¼ A: m
Thus, the first two of (3.5) are valid for m ¼ 1=2 and the next two are for other values of m. It should be noticed that the structure of Eq. (3.5) puts restrictions on the solutions such that the first two holds for all physical K P 0 and
sP
Upon substitution, the entire boundary conditions in (2.2) are seen to be met by the solution (3.3). It is obvious that true solutions require the condition k > 0. As a result, the momentum and angular velocity equations in (2.1) produce the relations
1 þ sk þ ð1 þ Kð1 þ mÞÞk2 ¼ 0; 2mð1 sk þ k2 Þ þ Kð2 þ m 4 þ k2 Þ ¼ 0;
m ¼ 1=2; 0
3.1. Flow field
f ðgÞ ¼ s
m ¼ 1=2;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 4 2K þ s2 ; k¼ 2þK ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s þ 4 2K þ s2 k¼ ; 2þK s
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 þ 2K ;
whereas the third for 0 6 K 6 1=2 and
sP
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 4K 8K 2
and the fourth, in addition to the above, for K P 1=2 and all s. Therefore, the critical values for the nonexistence or existence of dual solutions are given by the formulae
390
M. Turkyilmazoglu / International Journal of Heat and Mass Transfer 72 (2014) 388–391
(a)
(b)
(c)
Fig. 2. (a) Dual velocity f 0 ðgÞ (upper first, lower second branches), (b) dual angular velocity hðgÞ (upper first, lower second branches) and (c) dual temperature hðgÞ (lower first, upper second branches). The physical parameters are m ¼ 1=2, K ¼ 0:1 and Pr ¼ 3=7 (thin curves for s ¼ 2:1, dashed curves for s ¼ 2:15 and dot-dashed curves for 2:2, respectively.
(a) m ¼ 1=2; m¼
(b)
pffiffiffi ! 2 k ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ; 2þK pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 K 2K 2 2 : ð3:6Þ s ¼ 2 1 K 2K ; k ¼ 1þK
pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi s ¼ 2 2 þ K;
2 þ 2K ; 1 2K
Since the dual solutions are based on the discriminants in (3.6), the terms in the system (2.1) involving the material constant K are responsible for the existence of dual solution, under the presence of wall suction. The above technical analysis explains the discussion given in [1] on the existence domain of the parameters obtained via numerical methods. Moreover, it is known that triple solutions exist for the classical flow over a shrinking sheet, see, for example, [7,8]. However, for the same problem with the micropolar fluid elements taken into account, only dual solutions are found. Furthermore, noting that the skin friction coefficient is defined by f 00 ð0Þ ¼ k 0 and the couple stress coefficient by h ð0Þ ¼ mk2 , these are easily evaluated from the exact formulas (3.5). 3.2. Temperature field It is straightforward to show that the energy equation in (2.1) together with the temperature boundary conditions in (2.2) leads to the exact solution
h
h i egk Pr ; C Prð1þskÞ 2 2 k k h i h i hðgÞ ¼ Prð1þskÞ Pr C Prð1þskÞ ; 0 C ; 2 2 k k k2
C
Fig. 3. The solution regions for (a) k and (b) m versus s. Thick curves for the first branch and dotted curves for the second ones, outer loop for K ¼ 0:1 and inner loop for K ¼ 0:2, respectively.
i
Prð1þskÞ ;0 k2
from which we can define the local Nusselt number by
ð3:7Þ
M. Turkyilmazoglu / International Journal of Heat and Mass Transfer 72 (2014) 388–391
Prð1þskÞ Pr k2 ek2 kPr2 k 0 h i h i: Nu ¼ h ð0Þ ¼ Prð1þskÞ Pr C Prð1þskÞ ; 0 C ; 2 2 2 k k k
391
5. Concluding remarks
ð3:8Þ
Here, C½a; z is the incomplete gamma function. 4. Results and discussion We now demonstrate that the unique/multiple solutions expressed by (3.3) and (3.7), confined to the domain of parameters given by (3.5), exactly coincide with the recently published ones in [1]. Indeed, Figs. 1 and 2 show the same results as displayed in [1] for m ¼ 1=2 (see, Figs. 1–3 and 9–11 in [1]). It is noted that an adjustment in the Prandtl number is required to have the same Prandtl numbers, so our Prandtl number of Pr ¼ 3=7 corresponds to that of Pr ¼ 1 given in [1]. Therefore, the conclusions reached in [1] using numerical methods are clarified through the exact analytic results given in this note. Moreover, for the entire values of m (different from 1/2 as given in [1]), exact formulas in (3.5) can be employed, see Fig. 3, for the domain of existence of unique/dual solutions, which are timeconsuming to get using numerical ways as in [1]. From Fig. 3, one can work out the skin-friction coefficient, the couple stress coefficient and the heat transfer coefficient for any fixed value of m and Pr. For instance, the following exact dual results are obtained with K ¼ 1=10; m ¼ 5; s ¼ 5p31ffiffiffiffi and Pr ¼ 1; 10
pffiffiffi pffi2 4 10g f 0 ðgÞ ¼ e 5g ; f 0 ðgÞ ¼ e 11 ; pffiffiffi pffiffiffiffiffiffi pffi2 20 pffiffiffiffiffiffi 4 10g hðgÞ ¼ 10e 5g ; hðgÞ ¼ 10e 11 ; 11 rffiffiffi pffiffiffiffiffiffi! 2 00 4 10 ; ; f ð0Þ ¼ f 00 ð0Þ ¼ 5 11 800 0 0 h ð0Þ ¼ 2; h ð0Þ ¼ ; 121 pffiffiffi i h h p ffi2 i 0
4 10g 1 C 35 C 35 ; 52 e 5g C 759 ; 121 e 11 C 759 800 800 160 @hðgÞ ¼
; hðgÞ ¼
A; C 35 C 35 ; 52 C 759 ; 121 C 759 800 800 160
ðh0 ð0Þ ¼ 1:34542361; h0 ð0Þ ¼ 1:56981979Þ: ð4:9Þ Although, from Fig. 2(a)–(c) it is observed that for both branch 1 and branch 2 solutions the far field boundary conditions are satisfied asymptotically, the branch 1 solution is believed to be more physically meaningful. It is, of course, necessary to do a stability analysis to give a final decision on this issue.
An analytical study has been implemented for the flow of micropolar fluid and heat transfer over a shrinking sheet under porous wall conditions. The particular attention is given to derivation of exact solutions in contrary to the numerical ones already existing in the literature. By means of such a treatment, it is shown to be quite possible to explore the physical features of microrotation and temperature fields, whether they are unique or multiple. In fact, by solving two coupled algebraic equations resulting from the exponential type solution, it is found that no solution or at most two solutions may exist depending on the working parameters considered in the physical model. The numerical results and conclusions made in a recent paper [1] can be easily reached from the analysis of formulae presented here. Further rich properties of both flow and temperature fields can also be accessed and assessed via the exact solutions. References [1] K. Bhattacharyya, S. Mukhopadhyay, G.C. Layek, I. Pop, Effects of thermal radiation on micropolar fluid flow and heat transfer over a porous shrinking sheet, Int. J. Heat Mass Transfer 55 (2012) 2945–2952. [2] A.C. Eringen, Simple microfluids, Int. J. Eng. Sci. 2 (1964) 205–217. [3] A.C. Eringen, Theory of micropolar fluids, J. Math. Mech. 16 (1966) 118. [4] T. Fang, J. Zhang, Thermal boundary layers over a shrinking sheet: an analytic solution, Acta Mech. 209 (2010) 325–343. [5] Y. Khan, Q. Wu, N. Faraz, A. Yildirim, S.T. Mohyud-Din, Heat transfer analysis on the magnetohydrodynamic flow of a non-Newtonian fluid in the presence of thermal radiation: an analytic solution, Z. Naturforsch. 67 (2012) 147–152. [6] M. Madani, Y. Khan, M. Fathizadeh, A. Yildirim, Application of homotopy perturbation and numerical methods to the magneto-micropolar fluid flow in the presence of radiation, Eng. Comput. 29 (2012) 277–294. [7] M. Turkyilmazoglu, Multiple solutions of heat and mass transfer of MHD slip flow for the viscoelastic fluid over a stretching sheet, Int. J. Therm. Sci. 50 (2011) 2264–2276. [8] M. Turkyilmazoglu, I. Pop, Exact analytical solutions for the flow and heat transfer near the stagnation point on a stretching/shrinking sheet in a Jeffrey fluid, Int. J. Heat Mass Transfer 57 (2013) 8288. [9] L. Crane, Flow past a stretching plate, Z. Angew. Math. Phys. 21 (1970) 645– 647. [10] W.C. Troy, E.A. Overman, H.G.B. Ermentrout, J.P. Keener, Uniqueness of flow of a second order fluid past a stretching sheet, Q. Appl. Math. 44 (1987) 753–755. [11] J.B. Mcleod, K.R. Rajagopal, On the uniqueness of flow of a Navier Stokes fluid due to a stretching boundary, Arch. Ration. Mech. Anal. 98 (1987) 385–395. [12] P.S. Lawrence, B.N. Rao, The non-uniqueness of the MHD flow of a viscoelastic fluid past a stretching sheet, Acta Mech. 112 (1995) 223–228. [13] I. Pop, T.Y. Na, A note on MHD flow over a stretching permeable surface, Mech. Res. Commun. 25 (1998) 263–269.