Boundary layer flow and heat transfer over an exponentially shrinking vertical sheet with suction

International Journal of Thermal Sciences 64 (2013) 264e272

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Boundary layer flow and heat transfer over an exponentially shrinking vertical sheet with suction Azizah Mohd Rohni a, Syakila Ahmad b, Ahmad Izani Md. Ismail b, Ioan Pop c, * a

School of Quantitative Sciences, Universiti Utara Malaysia, 06010 Sintok, Kedah, Malaysia School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia c Department of Mathematics, Babes¸-Bolyai University, CP 253, R-400084 Cluj-Napoca, Romania b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 11 January 2012 Received in revised form 18 August 2012 Accepted 22 August 2012 Available online 27 September 2012

In this paper, we investigate theoretically the problem of steady laminar two-dimensional boundary layer flow and heat transfer of an incompressible viscous fluid in the presence of buoyancy force over an exponentially shrinking vertical sheet with suction. The shrinking velocity and wall temperature are assumed to have specific exponential function forms. The governing equations are first transformed to similarity equations using an appropriate similarity transformation. The resulting equations were then solved numerically using shooting technique involving fourth-order RungeeKutta method and Newton eRaphson method. The influence of mixed convection/buoyancy parameter l, suction parameter s and Prandtl number Pr on the flow and heat transfer characteristics is examined and discussed. Numerical results indicate that the presence of buoyancy force would contribute to the existence of triple solutions to the flow and heat transfer for particular value of pertinent parameters. It is different for the nonbuoyant flow case i.e. when the buoyancy force is absent, the problem admits only dual solutions. Further, this study also reveals that the features of flow and heat transfer characteristics are significantly affected by buoyancy parameter l, suction parameter s and Prandtl number Pr. Ó 2012 Elsevier Masson SAS. All rights reserved.

Keywords: Exponentially shrinking Mixed convection Vertical Suction Triple solutions

1. Introduction Boundary layer flow due to stretching or shrinking sheet has many applications in engineering and manufacturing processes in industry. Some examples may be found in aerodynamics extrusion of plastic sheets, continuous glass casting, glass fibre production, metal extrusion, hot rolling of textiles, wire drawing and extraction of polymer sheet. It seems that Crane [1] has made the first attempt to investigate the flow of stretching sheet which was then extended by many researchers to discuss various aspects of flow and heat transfer characteristics. Representative studies may be found in Refs. [2e5], to mention just a few. On the other hand, the investigation on boundary layer flow due to a shrinking sheet has also attracted considerable interest in recent years. A pioneering work on this problem has been made by Miklav ci c and Wang [6] and then extended in various directions and to different situations by Refs. [7e21]. The flow induced by shrinking sheet is different from the stretching case whereby the fluid is compressed and attracted towards a slot or a fixed point. From physical insight, vorticity (rotation or non* Corresponding author. Tel.: þ40 264 594315; fax: þ40 264 591906. E-mail addresses: [email protected] (A.M. Rohni), syakila.ahmad@ ymail.com (S. Ahmad), [email protected] (A.I.Md. Ismail), [email protected], [email protected] (I. Pop). 1290-0729/$ e see front matter Ó 2012 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.ijthermalsci.2012.08.016

potential) flow over the shrinking sheet is not confined within a boundary layer, and a steady flow cannot exist unless adequate suction is exerted at the sheet (Miklav ci c and Wang [6]). As discussed by Goldstein [22], the flow due to a shrinking sheet is essentially a backward flow whereas the fluid losses memory of perturbation introduced by the slot. Therefore, this type of flow exhibits quite different physical phenomena from the forward stretching case and the wall mass suction is required generally to maintain the flow. All the above mentioned studies deal with problems involving linear stretching/shrinking sheet. However, practically, the quality of the final product is determined by the rate of heat transfer at the stretching/shrinking surface where both the kinematics of the stretching/shrinking and the simultaneous heating/cooling during such processes have an important influence on the quality of the final products [23,24]. Several researchers have investigated the problem of boundary layer flow in order to obtain the thermal and kinematic behaviour by considering the different forms of stretching/shrinking velocity and temperature profiles [25e30]. Apart from that, the boundary layer flow induced by exponentially stretching/ shrinking sheet is very important and frequently appears in many engineering processes. Some studies on exponentially stretching sheet can be found in Refs. [31e37]. However, for the case of exponentially shrinking sheet there is still only a few studies. It seems that the first attempt of studying the problem of exponentially

A.M. Rohni et al. / International Journal of Thermal Sciences 64 (2013) 264e272

Nomenclature Cf Gr L Nu Pr Re T TN T0 Tw(x) Uw f g s u,v uw(x) vw(x) x,y

skin friction coefficient Grashof number characteristic length of the sheet Nusselt number Prandtl number Reynolds number fluid temperature ambient uniform temperature characteristic temperature of the sheet surface temperature characteristic velocity of the sheet, Uw > 0 non-dimensional stream function acceleration due to gravity wall mass transfer/suction parameter, s > 0 velocity component along x and y directions velocity of the shrinking sheet velocity of the mass transfer/suction Cartesian coordinates along the surface of the sheet and normal to it, respectively

shrinking was made by Bhattacharyya [38] followed by Bhattacharyya and Vajravelu [39] where they studied boundary layer flow and heat transfer over a horizontal exponentially shrinking sheet with suction and near stagnation point respectively. In the earlier study, Bhattacharyya [38] reported that the steady flow due to a horizontal exponentially shrinking sheet is possible only when the mass suction parameter s > 2.266684, and dual similarity solutions for velocity and temperature distribution are found. In the study of horizontal surfaces, the effect of buoyancy force is neglected and not taken into account. However, in practical situations, the flow over a continuous material moving through a quiescent fluid is often induced by the movement of the solid material and by the thermal buoyancy. Thus the mechanism of surface motion and buoyancy force will determine the momentum and thermal transport processes. The thermal buoyancy force arising from the heating/cooling of a continuously moving surface, under certain circumstances, can change the flow and thermal fields and hence the heat transfer behaviour in the manufacturing process (Pal [24]). Therefore, nowadays, a great deal of interest in the area of boundary layer mixed convection flow on a vertical stretching/ shrinking surface because of its many applications. Several studies have been reported investigating the effect of buoyancy forces on the boundary layer exponentially stretching sheet [24,40]. Nevertheless, to the authors’ present knowledge, the flow dynamics due to exponentially shrinking sheet with the presence of buoyancy force is still unknown. Thus, our interest in the present paper is to study the problem of steady boundary layer flow over a vertical shrinking sheet with suction involving boundary conditions of exponential velocity and temperature distribution. It is hoped that the study will contribute towards a better understanding of the flow dynamics and heat transfer behaviour due to an exponentially shrinking sheet as well as real application. Numerical results indicate that the presence of buoyancy force would contribute to the existence of triple solutions to the flow and heat transfer for particular value of pertinent parameters. 2. Mathematical formulation We consider the steady two-dimensional boundary layer flow and heat transfer of a viscous and incompressible fluid past a permeable vertically moving sheet with an exponential velocity

265

Greek letters a thermal diffusivity b thermal expansion coefficient h similarity variable l mixed convection/buoyancy parameter m dynamic viscosity n kinematic viscosity j stream function r fluid density q dimensionless temperature r fluid density Subscripts a indication at singularity point c indication at critical/turning point w condition at the surface of shrinking sheet N ambient/free stream condition Superscripts differentiation with respect to h

0

towards the origin (shrinking sheet) and an exponential surface temperature. It is assumed that the velocity of the shrinking sheet is uw(x), the surface temperature is Tw(x) and the ambient uniform temperature is TN, where Tw(x) > TN corresponds to a heated sheet (assisting flow) and Tw(x) < TN corresponds to a cooled sheet (opposing flow), respectively. The physical model and the coordinate system of the problem considered are illustrated in Fig. 1. Under the assumption of Boussinesq and boundary layer approximations, the governing equations of continuity, motion and energy are

vu vv þ ¼ 0 vx vy u

(1)

vu vu v2 u þv ¼ n 2 þ g bðT  TN Þ vx vy vy

(2)

x Boundary layer Tw ( x) T

uw ( x )

vw ( x)

T

g

Assisting flow

y

O vw ( x)

Opposing flow uw ( x )

Tw ( x) T

Fig. 1. Physical model and coordinate system.

266

A.M. Rohni et al. / International Journal of Thermal Sciences 64 (2013) 264e272

vT vT v2 T u þv ¼ a 2 vx vy vy

(3)

The boundary conditions of these equations are given as


x

; u ¼ uw ðxÞ ¼ Uw exp L   2x T ¼ Tw ðxÞ ¼ TN þ T0 exp at y ¼ 0 L v ¼ vw ðxÞ;

u/0;

T/TN

as

(4)


x

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2nLUw exp f ðhÞ; 2L rffiffiffiffiffiffiffiffi
x Uw h ¼ y exp 2nL 2L

qðhÞ ¼ ðT  TN Þ=ðTw  TN Þ (5)


x  L

f 0 ðhÞ;

rffiffiffiffiffiffiffiffiffi
x nUw exp ½f ðhÞ þ hf 0 ðhÞ 2L 2L

v ¼ 

(6)

where prime denotes differentiation with respect to h. Thus, we take

rffiffiffiffiffiffiffiffiffi
x nUw exp vw ðxÞ ¼  s 2L 2L

(7)

where s > 0 is the suction parameter. Substituting Eqs. (5) and (6) into Eqs. (2) and (3), we obtain the following ordinary differential equations: 000

f þ f f 00  2f 02 þ 2lq ¼ 0

(8)

1 00 q þ f q0  4f 0 q ¼ 0 Pr

(9)

and the boundary conditions (4) become

f 0 ð0Þ ¼ 1; qð0Þ ¼ 1 qðhÞ/0 as h/N

f ð0Þ ¼ s; f 0 ðhÞ/0;

(10)

Here Pr is the Prandtl number and l is the constant mixed convection or buoyancy parameter, which is defined as

l ¼

Gr

where Gr ¼ gbT0L /n is the Grashof number and Re ¼ UwL/n is the Reynolds number. It is worth mentioning that l > 0 corresponds to an assisting flow where the plate is heated, l < 0 corresponds to the opposing flow where the plate is cooled and finally, l ¼ 0 corresponds to forced convection flow (Tw ¼ TN) i.e. non-buoyant case. The physical parameters of interest in the present problem are the skin friction coefficient Cf and the Nusselt number Nu, which are given by

rUw2

f 0 ¼ Fp ;

f 00 ¼ Fpp ;

0 Fpp þ FFpp  2Fp2 þ 2lq ¼ 0

q0 ¼ qp ;

1 0 q þ F qp  4Fp q ¼ 0 Pr p

(14)

(15)

Fð0Þ ¼ s;

qð0Þ ¼ 1;

Fp ð0Þ ¼ 1; qp ð0Þ ¼ b

Fpp ¼ a

(16)

A fourth-order RungeeKutta integration scheme will be adopted to solve the applicable initial value problem. In order to integrate Eqs. (14) and (15) as an initial value problem, we require a value for Fpp(0) i.e. f00 (0) and qp(0) i.e. q0 (0). Since these values are not given in the boundary conditions (16), a suitable guess values for f00 (0) and q0 (0) are made and integration is carried out. Then, we compare the calculated values for f0 (h) and q(h) at hN with the given boundary conditions f0 (hN) ¼ 0 and q(hN) ¼ 0 respectively and adjust the estimated values of f00 (0), q0 (0) and hN to give a better approximation for the solution. This computation is done with the aid of shootlib function in Maple software. In this study, the boundary layer thickness hN between 3 and 35 was used in the computation, depending on the values of the parameters considered, so that the boundary condition at “infinity” is achieved. For particular value of pertinent parameters, there is a possibility that two (three) values of hN are obtained, which gives two (three) different velocity and temperature profiles that satisfy the boundary conditions. Consequently, this produces two (three) different values of f00 (0) and q0 (0) respectively. As example for Pr ¼ 1, l ¼ 0.5, s ¼ 5, hN z 4 (small boundary layer thickness), hN z 7 (medium boundary layer thickness) and hN z 10 (large boundary layer thickness) were used to obtain first, second and third solutions respectively. All the profiles in these three cases reached the infinity boundary conditions asymptotically. 4. Results and discussion

3

m

The boundary value problem (BVP) of Eqs. (8) and (9) subject to boundary conditions (10) is solved via the shooting technique (Meade et al. [41]) by converting it into an equivalent initial value problem (IVP). Therefore, we set

(11)

Re2

Cf ¼

It is worth to highlight here that when l ¼ 0, Eq. (8) becomes independent of q(h) and it reduces to Eq. (13) of paper by Bhattacharyya [38] with the same boundary conditions. Therefore, we can validate our results of f00 (0) with Ref. [38] for the case l ¼ 0.

with the boundary conditions

where j is the stream function which is defined in the classical form as u ¼ vj/vy and v ¼ vj/vx. Thus, we have

u ¼ Uw exp

(13)

3. Method of solution

y/N

where L is the characteristic length of the sheet, vw(x) < 0 is the velocity of suction, Uw(>0) is the constant velocity characteristic of the sheet and T0 is the characteristic temperature of the sheet with T0 > 0 for a hot surface of the sheet (assisting flow) and T0 < 0 for a cooled surface of the sheet (opposing flow), respectively. We introduce now the following similarity variables:

j¼

pffiffiffiffiffiffiffiffiffi 2Reexpð3x=2LÞCf ¼ f 00 ð0Þ; pffiffiffiffiffiffiffiffiffiffiffi 0 2=Reexpðx=2LÞNu ¼ q ð0Þ



2

 vu ; vy y¼0

Nu ¼

  L vT  ðTw ðxÞ  TN Þ vy y¼0

Substituting Eq. (5) into Eq. (12), we get

(12)

4.1. The non-buoyant flow case, l ¼ 0 In Fig. 2, we plot skin friction coefficient f00 (0) versus suction parameter s for Pr ¼ 1 and l ¼ 0, corresponding to forced convection or non-buoyant flow. In this case, we find the existence of dual solutions in f00 (0) with a critical value (turning point) sc z 2.2665. This result is in good agreement with those reported by Bhattacharya [38] who has shown that the steady flow due to an exponentially shrinking sheet is possible only when the mass suction parameter s > 2.266684. Fig. 2 in this present paper is qualitatively similar and consistent to Fig. 1 in Bhattacharyya [38], thus giving us confidence in our numerical approach.

A.M. Rohni et al. / International Journal of Thermal Sciences 64 (2013) 264e272

267

2.5 Pr = 1, λ = 0 2

1.5

f ''(0)

1

• s = 2.2665 c

0.5

0 first solution second solution

-0.5

-1

2.3

2.4

2.5

2.6 s

2.7

2.8

2.9

3 Fig. 4. Velocity profiles f0 (h) for several values of the suction parameter s when Pr ¼ 1 and l ¼ 0.

Fig. 2. Variation of f00 (0) with the suction parameter s when Pr ¼ 1 and l ¼ 0.

The wall temperature gradient q0 (0), which is proportional to heat transfer rate or local Nusselt number for Pr ¼ 1 and l ¼ 0, has been depicted in Fig. 3. Dual solutions are observed to exist for s > sc whereby no solution exists beyond this point, i.e. there is boundary layer separation for small suction parameter, s < sc. As can be observed from the first solution in both Figs. 2 and 3, higher suction leads to higher skin friction as well as higher heat transfer rate. In other words, mass suction increases the wall drag and heat flux. The opposite happens to second solution of f00 (0) where the increases of suction decrease the wall drag. It has also been observed that there are negatives values of the skin friction coefficient for certain values of s. These solutions indicate the occurrence of reverse flow. However, for the second solution of q0 (0), discontinuity occurs between s ¼ 2.3378 and s ¼ 2.3379 where the solutions go to negative and positive infinity respectively. This means that the graph has singularity at s between s ¼ 2.3378 and s ¼ 2.3379. It is also seen that there is an intersection point between first and second solutions of heat transfer rate. We next illustrate the velocity and temperature profiles for several values of s for Pr ¼ 1 and l ¼ 0 in Figs. 4 and 5 respectively. It

is found that the velocity profile increases as s increases in first solution. This is because suction implies an increase in skin friction coefficient which is caused by the reduction of momentum boundary layer thickness; hence enhancing the flow near the surface of the wall. However, it is observed that suction reduces the temperature in the first solution because of the increment of thermal boundary layer thickness and a different profile can be found for s ¼ 2.3 and s ¼ 2.4 as it approaches the singularity point. In Fig. 6, we further illustrate heat transfer rate q0 (0) versus s for several Pr with l ¼ 0 to show the influence of Prandtl number (the ratio of momentum diffusivity and thermal diffusivity) on heat fluxes under different suction parameters. We chose Pr ¼ 0.2, 0.7, 2, 3, 5. It is seen in the first solution branch that heat flux is enhanced by increasing Prandtl numbers but the behaviours differ in the second solution branch. Physically, an increase in Prandtl number means a decrease of fluid thermal conductivity which causes the

20 first solution second solution

15 10

- θ '(0)

5 0

• s = 2.2665 c

-5 -10 -15 Pr = 1, λ = 0 -20

2

2.5

3

3.5 s

4

4.5

Fig. 3. Variation of q0 (0) with the suction parameter s when Pr ¼ 1 and l ¼ 0.

5 Fig. 5. Temperature profiles q(h) for several values of the suction parameter s when Pr ¼ 1 and l ¼ 0.

268

A.M. Rohni et al. / International Journal of Thermal Sciences 64 (2013) 264e272 Table 2 Value of q0 (0) at turning point sc for Pr ¼ 1 and l ¼ 0.

20 7

s c = 2. 2665

Pr

•

5

0.7 0.7

• ••

0.2

-5

Pr = 0.2

-10

first solution second solution

Pr = 0.7, 2

λ= 0 -15

2

2.5

0.7

1

2

3

5

7

2

2

•

0

0.2

0.4592 1.4650 1.3798 0.8030 3.6533 8.6558 13.3536 q0 (0) at sc ¼ 2.2665

3

10

- θ ' (0)

0.2

5

•

3

3.5

4

4.5

5

s Fig. 6. Variation of q0 (0) with the suction parameter s for different Prandtl number, Pr when l ¼ 0.

reduction of thermal boundary layer thickness. The critical point sc is independent of the value of Prandtl number. It is observed from the figure that for Pr ¼ 0.2, 0.7, and 2, there are singularity points but for Pr ¼ 3, 5, 7, there is no singularity point. The singularity in the second solution for Pr ¼ 0.2, 0.7, and 2 is not fixed but movable depending on the value of Pr. Table 1 shows the discontinuity point of Figs. 3 and 6. Referring Table 1, we can see that the asymptote moves to the left for Pr < 1 and then to the right for Pr  1. Then, from Table 2, we can see that the value of q0 (0) at the turning point decreases with Pr for 0 < Pr < 1 but increases with Pr for Pr  1. The existence of dual solutions is observed for l ¼ 0 with different Prandtl number and the momentum equation is independent of the thermal equation. Therefore, the changes of Prandtl number will affect only the heat transfer rate, and not the skin friction profiles. It is worth highlighting that, between these two solutions, only one of them is stable and has physical meaning while the other is not.

value of assisting parameter l. It is seen that the domain of triple solutions decreases as the value of l increases with turning point sc ¼ 3.2126, 3.5289, 3.9311 for l ¼ 0.25, 0.5, 1 respectively as depicted in Fig. 7(b). To have a better insight of the triple solutions for q0 (0), an area in Fig. 8(a) is enlarged as shown in Fig. 8(b). From these figures, it is observed that q0 (0) increases with increasing l in the first and third solutions and decreases in the second solution. The triple solution domain decreases with increasing l and no similar behaviour to the case of l ¼ 0 is obtained where there is no singularity point in the second solution for l > 0. We then plot the variation of f00 (0) against Pr for different suction parameters in Fig. 9. It is seen that all solution branches (first, second and third) of f00 (0) decreases with Pr. However, only the first solution increases with suction parameter whilst the second and third solutions decrease with s. It is found that as Pr / 0, the value of f00 (0) in the first solution approaches positive y e axis and singularity occur at 6

2 0

λ=0

-2

λ = 0.25,0.5,1

-4 -6 first solution second solution third solution

-8 -10

4.2. The assisting flow case, l > 0

-12

a 0

0.5

1

1.5

2

2.5 s

3

3.5

4

4.5

5

-0.5 Pr = 1 -1

second solution third solution 0.25

-1.5

f '' (0)

To examine the flow behaviour for assisting flow case, we plot skin friction coefficient f00 (0) and heat transfer rate q0 (0) versus suction parameter s in Figs. 7 and 8 respectively for several positive values of l i.e. l ¼ 0.25, 0.5, 1. Different phenomena compared to non-buoyant flow are observed where the existence of triple solutions in the assisting flow case is found. It is seen that the second and third solutions merge with one another at sc (turning point) while the first solution continues to exist until s ¼ 0 for both f00 (0) and q0 (0). It is clear that for assisting flow, the value of f00 (0) and q0 (0) in the first solution still can be obtained although suction is absent (s ¼ 0). As illustrated in Fig. 7(a and b), the value of f00 (0) increases with assisting flow parameter, l for all solution branches, i.e. first, second and third solutions. Stronger suction is needed so that triple solutions are possible for the flow with greater

λ = 0, 0.25, 0.5, 1

Pr = 1

4

f '' (0)

15

-2

λ=0 0.5

-2.5

λ = 0.25, 0.5, 1

-3

-3.5 Table 1 Value of s that gives singularity point of f00 (0) and q0 (0) for Pr ¼ 1 and l ¼ 0. Pr

0.2

0.7

1

2

Singularity point, sa

Between 3.4936 and 3.4937

Between 2.3674 and 2.3675

Between 2.3378 and 2.3379

Between 2.4213 and 2.4214

-4 3.2

b

1 3.3

3.4

3.5

3.6

3.7 s

3.8

3.9

4

4.1

4.2

Fig. 7. (a) Variation of f00 (0) with the suction parameter s for different values of l (assisting flow) when Pr ¼ 1. (b) Enlargement of the area in 7(a).

A.M. Rohni et al. / International Journal of Thermal Sciences 64 (2013) 264e272

60

269

120 Pr = 1

50

λ = 0.5

λ=0

first solution second solution third solution

100

40

s=5

λ = 0.25, 0.5, 1

30

80

10

- θ ' (0)

- θ ' (0)

20

λ = 0.25, 0.5, 1

60

0

40

-10 first solution second solution third solution

-20 -30 -40

20

a 0

0.5

1

1.5

2

2.5 s

3

3.5

4

4.5

0

5

1

2

λ = 0.5

4

5

λ = 0.5 35

λ=1

6

7

first solution second solution third solution

s=5

6

30

5

25

3

- θ ' (0)

4 - θ ' (0)

3

40

λ = 0.25

λ=0

7

a 0

Pr

8

λ = 0, 0.25, 0.5, 1

2

s = 3, 4, 5

20 s=4 15

1

first solution second solution third solution

0 -1 -2

s = 3, 4, 5

s=4

λ=0

b

10 5

Pr = 1

2

2.5

3

3.5

4

0

4.5

s Fig. 8. (a) Variation of q0 (0) with the suction parameter s for different l (assisting flow) when Pr ¼ 1. (b) Enlargement of the area in 8(a).

b 0

0.5

1

1.5

2

2.5 Pr

3

3.5

4

4.5

5

Fig. 10. (a) Variation of q0 (0) with Pr for different values of s and l ¼ 0.5 (assisting flow) (b) Enlargement of the area in 10(a).

6

15

λ = 0.5 10

2

s = 3, 4, 5

λ = 0, -0.25, -0.5, -1

0

5

-2

0

f '' (0)

f '' (0)

Pr = 1

4

first solution second solution third solution

s=3

-4

λ = 0, -0.25, -0.5, -1

-6

s=4

s=4

-5

-8 first solution second solution third solution

-10

-10

s=5

-12

s=5

-14

-15

0

1

2

3

4

5

6

7

Pr Fig. 9. Variation of

f00 (0)

with Pr for different values of s and l ¼ 0.5 (assisting flow).

0

0.5

1

1.5

2

2.5 s

3

3.5

4

4.5

5

Fig. 11. Variation of f00 (0) with the suction parameter s for different values of l (opposing flow) when Pr ¼ 1.

270

A.M. Rohni et al. / International Journal of Thermal Sciences 64 (2013) 264e272

Pr ¼ 0. A profile which rarely occurs can be seen in second and third solutions for small Pr < 3 as depicted in Fig. 9. For s ¼ 4 and 5, it is seen that triple solutions exist while for s ¼ 3, only dual solution exist. Referring to Figs. 9 and 10, for s ¼ 4 and 5, the region of triple solutions is 0.7228 < Pr < 2.6484 and 0.4444 < Pr < 2.7277 respectively. It is also seen that dual solutions exist for s ¼ 3, 4 and 5 in the region Pr  2.9033, Pr  2.8348 and Pr  2.8126 respectively. Observing the graph in Fig. 10, it is clear that q0 (0) increases with increasing s in all solution branches for every Pr. We have to mention that dual solution has been also reported by Ridha [42] for the problem of mixed convection flow over the vertical flat plate in both cases of assisting and opposing flows.

30 Pr = 1

first solution second solution third solution

λ=0

20 10

- θ ' (0)

0

λ = -0.25, -0.5, -1 -10 -20

λ = -0.25, -0.5, -1 -30

4.3. The opposing flow case, l < 0

-40 -50

λ=0

a 0

0.5

1

1.5

2

2.5 s

3

3.5

4

4.5

5

7 Pr = 1

first solution second solution third solution

6

λ=0

5

Figs. 11 and 12 respectively show the variation of f00 (0) and q0 (0) against s for several opposing flows parameter l ¼ 0.25, 0.5, 1. The existence of triple solutions is also found in the opposing flow case whereby the first and second solutions merge with one another at sc (turning point) and the third solution continues to exist until s ¼ 0 for both f00 (0) and q0 (0). As plotted in Fig. 11, the value of f00 (0) decreases with opposing flow parameter, l in first and third solutions. It is shown in Fig. 11 that the domain of

4 100

- θ ' (0)

3

λ = -0.25, -0.5, -1

λ = - 0.5

80

a

5

2 60

4

1

s = 3, 4, 5

40

0

3

λ = -1

-2

λ = -0.5

b

-3 2.2

20 - θ ' (0)

λ=0

-1

λ = -0.25 2.4

2.6

2.8

3 s

3.2

0

3

-20

3.4

3.6

3.8

4

-40 -60

Fig. 12. (a) Variation of q0 (0) with the suction parameter s for different values of l (opposing flow) when Pr ¼ 1. (b) Enlargement of the area in 12(a).

first solution second solution third solution

-80 5

-100 0

1

2

3

4

5

6

7

Pr

10

25

λ = - 0.5

λ = - 0.5

s = 3, 4, 5

5

s=3 s=4

-5

-10

0

1

2

3

4

5

6

first solution second solution third solution

0

first solution second solution third solution

s = 3, 4, 5

10

5

s=5

-15

s = 3, 4, 5

15 - θ ' (0)

f '' (0)

0

-20

b

20

7

Pr Fig. 13. Variation of f00 (0) with Pr for different values of s and l ¼ 0.5 (assisting flows case).

-5

0

1

2

3

4

5

6

7

Pr Fig. 14. (a) Variation of q0 (0) with Pr for different values of s and l ¼ 0.5 (assisting flow). (b) Enlargement of the area in 14(a).

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the velocity profiles in all solution branches are negative, which indicates the occurrence of reverse flow in the boundary layer. It is also presented in both Fig. 15(a) and (b), the existence of triple solutions of velocity and temperature profiles with different boundary layer thickness and hence contribute triple solutions of skin friction coefficient and heat transfer rate. Finally, it should be mentioned that Merkin [44], Weidman et al. [45], Harris et al. [46] and Postelnicu and Pop [47] have presented the mathematical proof of the conjecture of dual numerical solutions. They have performed a stability analysis and revealed that the solutions along the upper branch (first solution) are linearly stable, whilst those on the lower branch (second solution) are linearly unstable. Therefore, we will not repeat it in this paper. 5. Conclusions In this paper, the problem of boundary layer flow and heat transfer over an exponentially shrinking vertical permeable sheet has been considered. The governing equations for the flow and temperature fields are reduced to a system of coupled nonlinear ordinary differential equations. These nonlinear differential equations are then solved numerically via the shooting technique (Meade et al. [41]) involving RungeeKutta integration scheme together with NewtoneRaphson method. The numerical results are verified with the earlier study by Bhattacharya [38] and found to be in good agreement. The effects of suction parameter s, mixed convection/buoyancy parameter l and Prandtl number Pr on the physical quantities of interest have been examined and presented graphically and in tabular form. The existence of multiple (triple) solutions was observed and determined for some values of the governing parameters. It should be mentioned that such solutions for the present problem have not been reported before. The present paper is, therefore, original with new very interesting results. Acknowledgements Fig. 15. Triple profiles of: (a) velocity and (b) temperature for Pr ¼ 1, s ¼ 5 and l ¼ 0.5.

triple solutions decreases as the value of jlj increases with turning point sc ¼ 2.6364, 2.8255, 3.0821 for l ¼ 0.25, 0.5, 1 respectively. From Fig. 12, it is observed that the opposite trends occur in opposing flow compared to assisting flow, where the development of heat transfer rate q0 (0) is more retarded for higher jlj in first and third solutions. No singularity point is obtained for q0 (0) in the case of opposing flow (l < 0). In Figs. 13 and 14, we further plot the variation of f00 (0) against Pr for different suction parameters. For both figures, we chose l ¼ 0.5. It is seen that the first solution of f00 (0) increases with Pr and s while in the second solution, f00 (0) decreases with both Pr and s. However, in the third solution, f00 (0) decreases with s but increases with Pr up to certain point before boundary layer separate. It is found that as Pr / 0, the value of f00 (0) in the third solution goes to negative infinity and singularity occur at Pr ¼ 0. Referring Figs. 13 and 14, the region of triple solutions is 0.7781 < Pr < 3.2475, 0.2937 < Pr < 3.0157 and 0.2080 < Pr < 2.9523 for s ¼ 3, 4 and 5 respectively. Higher suction can delay the flow separation and keep the flow attached even for lower Prandtl numbers. Besides, suction also helps to enhance the heat transfer rate in opposing flow. It is worth mentioning such triple solutions were very recently reported by Lok and Pop [43] for the problem of steady axisymmetric stagnation point flow of a viscous and incompressible fluid over a shrinking circular cylinder with mass transfer (suction). Moreover, it is shown in Fig. 15(a) that

The first author would like to acknowledge the financial support received from Universiti Utara Malaysia and Malaysia Ministry of Higher Education throughout the course of her study. The financial support received in the form of research grants: RU Grant (1001/ PMATHS/811166) from Universiti Sains Malaysia (USM) and FRGS (203/PMATHS/6711234) from Malaysia Ministry of Higher Education are also gratefully acknowledged. The senior author (I. Pop) wishes to express his thanks to USM for giving him the chance to visit the university. The authors also would like to thank the reviewers for valuable comments and suggestions. References [1] L.J. Crane, Flow past a stretching plate, ZAMP 21 (1970) 645e647. [2] P.S. Gupta, A.S. Gupta, Heat and mass transfer on a stretching sheet with suction or blowing, Can. J. Chem. Eng. 55 (1977) 744e746. [3] B.K. Dutta, P. Roy, A.S. Gupta, Temperature field in flow over a stretching sheet with uniform heat flux, Int. Commun. Heat Mass Transf. 12 (1985) 89e94. [4] S.J. Liao, A new branch of solutions of boundary layer flows over a stretching flat plate, Int. J. Heat Mass Transf. 49 (2005) 2529e2539. [5] M. Hassani, M. Mohammad Tabar, H. Nemati, G. Domairry, F. Noori, An analytical solution for boundary layer flow of a nanofluid past a stretching sheet, Int. J. Therm. Sci. 50 (2011) 2256e2263. [6] M. Miklav ci c, C.Y. Wang, Viscous flow due a shrinking sheet, Q. Appl. Math. 64 (2006) 283e290. [7] T. Hayat, Z. Abbas, M. Sajid, On the analytic solution of magnetohydrodynamic flow of a second grade fluid over a shrinking sheet, ASME J. Appl. Mech. 74 (2007) 1165e1171. [8] T. Hayat, T. Javed, M. Sajid, Analytic solution for MHD rotating flow of a second grade fluid over a shrinking surface, Phys. Lett. A 372 (2008) 3264e3273. [9] T. Hayat, Z. Abbas, N. Ali, MHD flow and mass transfer of a upper-convected Maxwell fluid past a porous shrinking sheet with chemical reaction species, Phys. Lett. A 372 (2008) 4698e4704.

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