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ORIGINAL ARTICLE

Heat transfer analysis of boundary layer flow over hyperbolic stretching cylinder Abid Majeed *, Tariq Javed, Irfan Mustafa Department of Mathematics and Statistics, International Islamic University, Islamabad 44000, Pakistan Received 23 October 2014; revised 15 March 2016; accepted 19 April 2016

KEYWORDS Boundary layer flow; Hyperbolic stretching cylinder; Heat transfer analysis; Entropy generation

Abstract In the present study heat transfer and entropy generation analysis of boundary layer flow of an incompressible viscous fluid over hyperbolic stretching cylinder are taken into account. The governing nonlinear partial differential equations are normalized by using suitable transformations. The numerical results are obtained for the partial differential equations by a finite difference scheme known as Keller box method. The influence of emerging parameters namely curvature parameter and Prandtl number on velocity and temperature profiles, skin friction coefficient and the Nusselt number is presented through graphs. A comparison for the flat plate case is given and developed code is validated. It is seen that curvature parameter has dominant effect on the flow and heat transfer characteristics. The increment in the curvature of the hyperbolic stretching cylinder increases both the momentum and thermal boundary layers. Also skin friction coefficient at the surface of cylinder decreases but Nusselt number shows opposite results. Temperature distribution is decreasing by increasing Prandtl number. Moreover, the effects of different physical parameters on entropy generation number and Bejan number are shown graphically. Ó 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction The boundary layer flow and heat transfer with stretching boundaries received remarkable attention in modern industrial and engineering practice. The attributes of end product are greatly reliant on stretching and rate of heat transfer at final stage of processing. Due to this real-world importance, interest developed among scientists and engineers to comprehend this phenomenon. Common examples are the extrusion of metals into cooling liquids, food, plastic products, the reprocessing * Corresponding author. Tel.: +92 3335251806. E-mail address: [email protected] (A. Majeed). Peer review under responsibility of Faculty of Engineering, Alexandria University.

of material in the molten state under high temperature. During this phase of manufacturing process, the material passes into elongation (stretching) and a cooling process. Such types of processes are very handy in the production of plastic and metallic made apparatus, such as cutting hardware tools, electronic components in computers, rolling and annealing of copper wires. In many engineering and industrial applications, the cooling of a solid surface is a primary tool for minimizing the boundary layer. Due to these useful and realistic impacts, the problem of cooling of solid moving surfaces has turned out to be an area of concern for scientists and engineers. Recently researchers have shown deep interest in analyzing the characteristics of flow and heat transfer over stretching surfaces. In this context, analysis of flow and heat transfer phenomena over a stretching cylinder has its own importance in processes

http://dx.doi.org/10.1016/j.aej.2016.04.028 1110-0168 Ó 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: A. Majeed et al., Heat transfer analysis of boundary layer flow over hyperbolic stretching cylinder, Alexandria Eng. J. (2016), http:// dx.doi.org/10.1016/j.aej.2016.04.028

2 such as fiber and wire drawing, hot rolling. Based on these applications, the boundary layer flow and heat transfer due to a stretching cylinder was initially studied by Wang [1]. He extended the work of Crane [2] to study the flow of heat transfer analysis of a viscous fluid due to stretching hollow cylinder. During last few years many researchers gave attention to analyze the fluid flow and heat transfer over the stretching cylinder and found the similarity solutions. Nazar et al. [3–5] produced numerical solution of laminar boundary layer flow, uniform suction blowing and MHD effects over stretching cylinder in an ambient fluid. After that a lot of research problems have been done on the analysis of flow and heat transfer over stretching cylinder. Mukhopadhyay [6–9] considered chemical solute transfer, mixed convection in porous medium, and MHD slip flow along a stretching cylinder. Effect of magnetic field over horizontal stretching cylinder in the presence of source/sink with suction/injection is studied by Elbashbeshy [10]. Fang and Yao [11], Vajravelu et al. [12], Munawar et al. [13,14], Wang [15], Lok et al. [16], Abbas et al. [17], and Butt and Ali [18] have considered viscous swirling flow, axisymmetric MHD flow, unsteady vacillating flow, natural convection flow, axisymmetric mixed connection stagnation point flow, radiation effects in porous medium and entropy analysis of magnetohydrodynamics flow over stretching cylinder, respectively. More recently, Majeed et al. [19] applied Chebyshev Spectral Newton Iterative Scheme (CSNIS) and studied the heat transfer influence by velocity slip and prescribed heat flux over a stretching cylinder. Sheikholeslami [20] has calculated the effective thermal conductivity and viscosity of the nanofluid by Koo–Kleinstreuer–Li (KKL) model and provided the nanofluid heat transfer analysis over cylinder. Javed et al. [21] discussed the combined effects of MHD and radiation on natural convection flow over horizontal cylinder. Hajmohammadi et al. [22] considered flow and heat transfer of Cu-water and Ag-water nanofluid over permeable surface with convective boundary conditions. Their main focus was to discuss the effects of nano-particles volume fraction, and non-dimensional permeability parameter on skin friction and convection heat transfer coefficient. Dhanai et al. [23] investigated mixed convection nanofluid flow over inclined cylinder. They utilized Buongiorno’s model of nanofluid and found dual solutions with the velocity and thermal slip effects in the presence of MHD. Aforementioned studies reveal that vast literature is present on study of linearly stretching cylinder but to the best of our knowledge, the problem over a cylinder with hyperbolic stretching velocity and entropy generation has not been addressed yet. The governing equations related to flow problem solved numerically and the effects of involving parameters on flow behavior are shown graphically. Entropy generation is the quantification of thermodynamics irreversibility and it exists in all types of heat transfer phenomenon. Due to this fundamental importance the present work is optimized with the inclusion of entropy generation analysis. In this context the studies [24–30] are quite useful to explore many aspects of entropy generation which have not been taken into account yet.

A. Majeed et al. fixed radius R (see Fig. 1). The basic equations that govern the flow and heat transfer phenomena will take the form as @u @v þ ¼ 0; @x @r u

@u @u @ @u ¼ m r ; þ v @x @r [email protected] @r

ð2Þ

u

@T @T @ @T ¼ a r þ v @x @r [email protected] @r

ð3Þ

where u and v are be velocity components along x and r directions, T be the temperature and a ¼ k=qcp be the thermal diffusivity of the fluid. The relevant boundary conditions for the governing problem are as follows: uðr; x Þ ¼ Uðx Þ; vðr; x Þ ¼ 0; uðr; x Þ ! 0;

T ¼ Tw ¼ T1 þ AUðx Þ at r ¼ R

T ¼ T1 as r ! 1

;

ð4Þ

where A is a constant. Now introducing the following suitable transformation x ; L

r2 R 2 Re1=2 x ; 2Rx T T1 hðg; xÞ ¼ : Tw T1 x¼

w ¼ m Re1=2 x fðg; xÞ;

g¼

ð5Þ

where stream function w is the non-dimensional function defined through usual relationship as u¼

@w ; [email protected]

v¼

@w : [email protected]

ð6Þ

which satisfies the continuity Eq. (1). Upon using Eqs. (5) and (6) into Eqs. (2) and (3), we arrived at following transformed equations r2 2x x d Rex 2 x d Rex fg þ ffgg f þ f þ 1 ggg gg Rex dx 2Rex dx R2 R Rex1=2 @fg @f ; ð7Þ fgg ¼ x fg @x @x 1 r2 2x h þ fgg 2 gg Pr R R Re1=2 x @h @f ¼ x fg hg ; @x @x

! xtanhxhfg þ

x d Rex fhg 2 Rex dx ð8Þ

where Rex ¼ Uw xL=m (local Reynolds number) and Pr ¼ m=a (Prandtl number). Using the stretching velocity Uw ¼ U0 cosh x in Eqs. (7) and (8), we get

r, v

2. Problem formulation

R Consider the two-dimensional steady incompressible flow of a viscous fluid over a hyperbolic stretching circular cylinder of

ð1Þ

Uw= Uo cosh(x*/L) Figure 1

x*, u

Geometry of the flow.

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Heat transfer analysis of boundary layer flow K 2K 1 1 þ 2g pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ fggg þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ fgg þ ð1 þ xtanhxÞffgg 2 cosh x cosh x @fg @f 2 ; fgg xtanhxfg ¼ x fg @x @x

3

ð9Þ

1 K 2K 1 1 þ 2g pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ hgg þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ hg þ ð1 þ xtanhxÞfhg Pr 2 cosh x Pr cosh x @h @f xtanhxhfg ¼ x fg hg ð10Þ @x @x

values of x, until the difference in velocity and temperature profiles between the consecutive iteration is less than 105 . The accuracy of the employed numerical scheme has been established in Table 1 by comparison with the known results obtained by Rees and Pop [31] for flat plate ðx ¼ 0Þ: This comparison gives us confidence that the developed code is correct and has achieved the desired level of accuracy. Moreover, the values of Cf Re1=2 and Nu Re1=2 are given in Table 2. x x 4. Result and discussion

and boundary conditions take the new form as fð0; xÞ ¼ 0; fg ð0; xÞ ¼ 1; hð0; xÞ ¼ 1; fg ð1; xÞ ¼ 0; hð1; xÞ ¼ 0:

ð11Þ

where K represents curvature parameter. If we consider x ¼ 0 and K ¼ 0; then Eq. (9) reduces to the Sakiadis flow equation [31] 1 fggg þ ffgg ¼ 0; 2

ð12Þ

with boundary conditions fð0Þ ¼ 0; fg ð0Þ ¼ 1; fg ð1Þ ¼ 0

ð13Þ

The quantities of physical significance are the skin friction coefficient and Nusselt number which are defined as sw xqw Cf ¼ ; ð14Þ ; Nu ¼ kðTw T1 Þ qU2w where sw be the wall shear stress and qw the constant heat flux from the surface, which are defined as @u @T sw ¼ l ; qw ¼ k : ð15Þ @r r¼R @r r¼R After using Eq. (15) into Eq. (14), the local skin friction coefficient and local Nusselt number take the form 1=2 ¼ hg ð0; xÞ: Cf Re1=2 x ¼ fgg ð0; xÞ; Nu Rex

ð16Þ

3. Solution methodology To solve nonlinear system of partial differential Eqs. (9), (10) subject to the set of boundary conditions (11), we employed a very accurate and efficient implicit finite difference method commonly known as Keller box which is explained in detail in the book of Cebeci and Bradshaw [32]. In present problem, the solution is obtained through following steps. Step-1: Eqs. (9), (10) are written in terms of first order differential equations. Then functions and derivatives are replaced by central difference form in both coordinate directions. This results in a set of nonlinear difference equations for the number of unknown equal to the number of difference equations. Step-2: obtained nonlinear difference equations are then linearized by using Newton’s quasi-linearization technique. Step-3: and are solved using block tri-diagonal algorithm. Now to initiate the process at x ¼ 0, it is first provided guess profile for all five dependent variables arising after reduction to first order system and then obtained the solution outlined in steps 1-3. Iteration process is continued for all

An inspection of the governing partial differential equations indicates the presence of two emerging parameters: (a) the curvature parameter K, (b) and the Prandtl number Pr. The effects of these parameters are discussed in the forthcoming figures. In Figs. 2 and 3, effects of curvature parameter K on velocity and temperature profile are plotted. It is observed that curvature has significant effects on velocity and temperature profiles. In Fig. 2 the trend for velocity profile is rapidly decreasing in the region (0 < g < 0:42) and then increasing after g ¼ 0:42 for increasing values of curvature parameter K and consequently as a result the boundary layer thickness increases with the increase in curvature of the cylinder. This behavior of the fluid velocity is due to the fact that curvature K and radius of cylinder have reciprocal relationship. In other words increase in K tends to decrease in radius of cylinder. In this way, due to lesser surface area of the cylinder, the increase in velocity gradient at the surface increases and consequently enhances the shear stress per unit area. Fig. 2 also depicts that an increase in the curvature of the cylinder leads to augment in boundary layer thickness, as compared to the flat plate case (K ¼ 0). In Fig. 3 the temperature profiles decrease near the surface of the cylinder as K increases, and afterward rise significantly and the thermal boundary-layer thickness increases. This variation of the temperature profile is due to the reason that as the curvature increases, the radius of cylinder reduces the surface area which is intact with the fluid also decreases. It is also important to mention that heat is transferred into the fluid in modes: conduction at the cylinder surface and convection for the region g > 0. Now, as the area of surface of the cylinder decreases, a slender reduction in the temperature profile occurs close to the surface of the cylinder, since a smaller amount of heat energy is transferred from the surface to the fluid through conduction phenomena. On the other hand, the thermal boundary-layer thickness increases, because of the heat transport in the fluid due to enhanced convection process all around the cylinder, which is evident from Fig. 3. Fig. 4 is plotted to observe the consequence of curvature parameter K on the coefficient of skin friction Cf Re1=2 x . It is noticed that as K increases, the wall skin friction coefficient Cf Re1=2 x decreases. This is due to the reason that, for larger curvature of the cylinder, the velocity gradient at the surface becomes increased rapidly as compared to the

Table 1

Comparison with Rees and Pop [31].

x¼0

Cf Rex1=2

Nu Re1=2 x

Ref. [31] Present

0.4439 0.4439

0.3509 0.3509

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4

A. Majeed et al. 0

Table 2 Numerical values of Cf Re1=2 and Nu Re1=2 for x x various values of K when Pr ¼ 0:7 at x = 0.5. Nu Re1=2 x

0 0.5 1 1.5

0.55387 0.71911 0.88994 1.05690

0.41012 0.60043 0.78894 0.96676

−1

−1.5

−2

1

0

0.5

1

1.5

2

x

2.5

3

Figure 4 Variations in Skin-friction coefficient for different values of K against x.

0.8 fη (η, x )

K=0, 0.5, 1, 1.5

−0.5

f

Cf Rex1=2

C Re1/2 x

K

0.6

1.5

0.4 −1/2

K = 1.5, 1, 0.5, 0.0

0 0

Figure 2 at x ¼ 3.

2

4

η

6

8

10

1

Nu Rex

0.2

K = 0, 0.5, 1, 1.5

0.5

Effects on velocity profile due to different values of K

0

0.5

1

1.5

x

2

2.5

3

Figure 5 Variations in Nusselt number for different values of K against x while Pr ¼ 0:7.

1

θ (η, x )

0.8 1

0.6

0.8

0.4 fη (η, x )

K = 1.5, 1, 0.5, 0 0.2 0 0

2

4

η

6

8

10

0.6 x = 0.5, 1.0, 1.5, 3 0.4 0.2

Figure 3 Effects on temperature profile due to different values of K while Pr ¼ 0:7 and x ¼ 3.

flat plate (K ¼ 0). Fig. 5 is plotted to observe the consequence . It of curvature parameter K on the Nusselt number Nu Re1=2 x 1=2 is noticed that as K increases the Nusselt number Nu Rex also increases. In Fig. 6 and 7, it is observed that both velocity and temperature boundary layer thickness decrease by increasing the value of x. The effects of different values of Prandtl number Pr on temperature profile have been discussed in Fig. 8. It is observed that temperature is rapidly decreasing for increasing Pr and thermal boundary layer thickness decreases. It is obvious from the graph that fluids with low Prandtl number slow down the cooling process and fluids with high Prandtl number speed up the cooling process. The behavior of temperature profile shows that Prandtl number can be used to control the cooling process. This could be managed by using high Prandtl number fluids such as oils and lubri-

0 0

Figure 6

2

4

6 η

8

10

12

Velocity profile for different values of x at K ¼ 0:5.

cants. Fig. 9 shows that Nusselt number Nu Re1=2 increases x along the surface of cylinder and with the increase of Pr. This is because of fact that by increasing the value of Pr thermal boundary layer decreases and heat transfer rate enhances. This finding also testify the results shown in Fig. 8. 5. Entropy generation analysis Using the boundary layer approximation, the local volumetric rate of entropy generation EG for a Newtonian fluid over a hyperbolic stretching cylinder is defined as follows:

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Heat transfer analysis of boundary layer flow

5

1

NE ¼

0.8 θ (η, x )

0.6 x = 0.5, 1, 1.5, 3 0.4 0.2 0 0

2

4

6 η

8

10

12

Figure 7 Temperature profile for different values of x at K ¼ 0:5 and Pr ¼ 0:7.

1

θ (η, x )

Pr = 0.025, 0.7, 2, 3.6, 5.5

NEHT NEHT þ NEFF

Here

0.4

NEHT ¼ ð1 þ 2KgÞReL

cosh x ðhg ðg; xÞÞ2 x

ð21Þ

NEFF ¼ ð1 þ 2KgÞReL

cosh3 x BrX1 ðfgg ðg; xÞÞ2 x

ð22Þ

0 0

1

2

3 η

4

5

6

Temperature profile for different value of Pr at x ¼ 3.

Figure 8

2 k @T T21 @r |ﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄ}

EG ¼

2 l @u T1 @r |ﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄ}

þ

Entropy effects due to heat transfer

ð17Þ

Entropy effects due to fluid friction

After using results of Eq. (5) and Eq. (6) in Eq. (17), following form is obtained kr2 ðTw T1 Þ2 U0 cosh xðhg ðg; xÞÞ2 LR2 T21 xm þ

ð20Þ

0.6

0.2

EG ¼

ð19Þ

where NE is the entropy generation number which is the ratio of local volumetric rate of entropy generation EG and characteristic entropy generation rate E0 ¼ kðTw T1 Þ2 =L2 T21 , ReL ¼ U0 L=m is the Reynolds number, Br ¼ lU20 =kðTw T1 Þ is the Brinkman number, and X ¼ ðTw T1 Þ=T1 is the dimensionless temperature difference and the ratio BrX1 is group parameter. The Bejan number Be serves as a substitute of entropy generation parameter and it represents the ratio between the entropy generation due to heat transfer and the total entropy generation due to heat transfer and fluid friction. It is defined by Be ¼

0.8

EG ð1 þ 2KgÞReL cosh x ¼ E0 x " # Brðcosh xÞ2 ðfgg ðg; xÞÞ2 2 ðhg ðg; xÞÞ þ X

r2 lðU0 cosh xÞ3 ðfgg ðg; xÞÞ2 LR2 T1 xm

ð18Þ

Above equation can be written as

Bejan number Be can also be written as Be ¼ 1=1 þ U , where U ¼ NEFF =NEHT is called irreversibility ratio. Figs. 10–16 are drawn to discuss the influence of Prandtl number Pr, curvature parameter K, group parameter BrX1 and Reynolds number ReL on entropy generation number NE and Bejan number Be. In Figs. 10 and 11, the overall effect of Prandtl number is the enhancement in NE and Be. This is because the temperature gradient increases with the larger values of Pr. It is also noted that when Pr < 1 a small variation in values of Be at surface of the cylinder is observed and it increases for larger g and attains maximum value i.e., Be ! 1. On the other hand when Pr > 1, large values of Be at surface of the cylinder are reported. These effects start decreasing and Be ! 0 far away from the surface. Figs. 12 and 13 are plotted to see the effects of K on NE and Be. Both figures depict that entropy generation is more dominant for stretching cylinder case K > 0 in comparison with flat plat case K ¼ 0. In Fig. 13, Be decrease at the surface stretching cylinder (g ¼ 0) and reverse behavior is observed far away from the

5

25 20 Pr = 0.1, 0.3, 0.7, 3, 5, 7

Pr = 8, 6, 4, 2

3

15

NE*

Nu Rex

−1/2

4

2

10

1

5

0

0

0.5

1

1.5

x

2

2.5

3

Figure 9 Variation in Nusselt number for different values of Pr against x.

0

0

1

2

η

3

Figure 10 Effects of Pr K ¼ 0:5; ReL ¼ 2; BrX1 ¼ 1 at x ¼ 0:2.

4

on

5

NE

when

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6

A. Majeed et al. 1

1 Pr = 0.1, 0.3, 0.7, 3, 5, 7

0.8

0.6

0.6

BrΩ* −1 = 0.1, 0.3, 0.7, 1

Be

Be

0.8

0.4

0.4

0.2

0.2

0

0

2

4

η

6

8

Figure 11 Effects of Pr K ¼ 0:5; ReL ¼ 2; BrX1 ¼ 1 at x ¼ 0:2.

0

10

on

when

Be

0

1

2

η

3

4

Figure 15 Effects of BrX1 K ¼ 0:5; ReL ¼ 2; Pr ¼ 7 at x ¼ 0:2.

5

on

Be

when

3

3.5

4

40 80 30

N*

E

60 20

N*E

K = 0, 0.3, 0.7, 1

40

ReL = 1, 2, 3, 5

10 20 0

0

1

2

3

η

4

5 0

Figure 12 Effects of K ReL ¼ 2; BrX1 ¼ 1; Pr ¼ 7 at x ¼ 0:2.

NE

on

0

0.5

when

1

1.5

2

η

Figure 16 Effects of ReL K ¼ 0:5; Pr ¼ 7; BrX1 ¼ 1 at x ¼ 0:2.

2.5

on

NE

when

1

surface. The variations of group parameter BrX1 on NE and Be are depicted in Figs. 14 and 15. The enhancement in BrX1 results in augmentation of NE and these effects are opposite on Be. The values of Be are decreasing with increase in BrX1 because the fluid friction dominates when BrX1 increases and this results in decrement of Be. The effects of ReL on entropy generation number are expressed in Fig. 16. This figure reveals that the entropy generation number has gained an increasing trend with increase in ReL .

0.8 0.6

Be

K = 0, 0.3, 0.7, 1

0.4 0.2 0

0

1

2

η

3

4

5

Effects of K on Be when Pr ¼ 7; BrX1 ¼ 1 at

Figure 13 x ¼ 0:2.

A study is performed to investigate the flow and heat transfer characteristics over a nonlinear stretching cylinder. The nonlinear stretching velocity is considered as hyperbolic function. An analysis of entropy generation is also presented and results are computed numerically with Keller box method. From above study it is observed that:

35 30

*

NE

25 20

BrΩ

−1

= 0.1, 0.3, 0.7, 1

15 10 5 0 0

6. Concluding remarks

0.5

1

1.5 η

Figure 14 Effects of BrX1 Pr ¼ 7; ReL ¼ 2; K ¼ 0:5 at x ¼ 0:2.

2

2.5

on

NE

3

when

Increase in the values of curvature parameter K leads to increase in velocity and temperature distribution. skin-friction coefficient reduces and Nusselt number enhances with an increase in curvature parameter K. The temperature reduces and Nusselt number enhances by increasing Prandtl number Pr. Increasing trend of curvature parameter, Prandtl number and group parameter results in enhancement of entropy generation.

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