J Braz. Soc. Mech. Sci. Eng. DOI 10.1007/s40430-014-0165-8

TECHNICAL PAPER

Mixed convection flow of an Oldroyd-B fluid with power law heat flux and heat source T. Hayat • M. Bilal Ashraf • S. Al-Mezel S. A. Shehzad

•

Received: 16 May 2013 / Accepted: 4 March 2014  The Brazilian Society of Mechanical Sciences and Engineering 2014

Abstract This article looks at the mixed convection flow of an Oldroyd-B fluid bounded by a porous stretching surface. Mathematical formulation is developed in the presence of heat source and power law heat flux. Velocity and temperature are computed. Plots for different parameters are analyzed. Numerical values of skin friction coefficient are examined. Keywords Mixed convection flow  Oldroyd-B fluid  Heat source  Power law heat flux

1 Introduction The flows of non-Newtonian fluids with heat transfer have extensive applications in engineering. The classification of non-Newtonian fluids is presented through three main classes, namely the differential, integral and rate. Numerous attempts have been made in the past for the flows of differential type fluids. Although this class of fluids is important for the prediction of shear thinning/shear thickening properties, relaxation and retardation times effects cannot be explored by these fluids. A subclass of rate type fluids, namely an Oldroyd-B captures such features [1–10].

Technical Editor: Monica Feijo Naccache. T. Hayat  M. B. Ashraf (&)  S. A. Shehzad Department of Mathematics, Quaid-i-Azam University, 45320, Islamabad 44000, Pakistan e-mail: [email protected] T. Hayat  S. Al-Mezel Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O.Box 80203, Jeddah 21589, Saudi Arabia

Polymer solutions of high-molecular-weight is the example of an Oldroyd-B fluid. This model explores the elastic and memory effects exhibited by the polymeric and biological liquids which are used in the results of simulations fit to experimental data [11]. Less attention is paid to the twodimensional flow boundary layer flow of an Oldroyd-B fluid induced by a stretching surface. Bhatnagar et al. [12] investigated the flow of an Oldroyd-B fluid over a stretched surface in the presence of free stream velocity. Perturbation along with numerical method is implemented for the solutions. Both cases of suction and injection are discussed. Stagnation point flow of an Oldroyd-B fluid near a stretching surface is studied by Sajid et al. [13]. Hayat et al. [14] examined the Oldroyd-B fluid over the surface with convective type boundary conditions. To be more specific, it is significant in transpiration cooling and gaseous diffusion. Further, investigation of heat generation or absorption is important in problems concerned with dissociating fluids. The temperature distribution may be altered by the possible heat generation effects and as a result the particle deposition rates in nuclear reactors, electronic chips and semi conductors. The aim of this paper was to address the mixed convection flow of rate type fluid over a porous sheet. The consideration of mixed convection flow has importance in technological natural and engineering applications. These applications include lubrication and drying processes, food processing, float glass production, cooling of electronic devices, etc. Hence this study is organized in the following fashion: Section two consists of mathematical formulation. Series solutions by homotopy analysis method (HAM) [15–21] are constructed in section three. Convergence analysis is pointed out in section four. Section five presents the influences of various parameters on the velocity and temperature fields. Section six contains main points.

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J Braz. Soc. Mech. Sci. Eng.

2 Development of problems

u ¼ 0; T ¼ T1 as y ! 1;

ð5Þ

We consider the two-dimensional flow of an incompressible Oldroyd-B fluid over a porous surface. A Cartesian coordinate system is chosen in such a way that x-axis is taken along the flow direction and the y-axis perpendicular to the x-axis. The fluid fills the half-space y [ 0. Heat source is present and power law heat flux is imposed. Flow diagram is shown as

where B is the temperature coefficient and T1 is the ambient temperature. The similarity transformations are given by rffiffiffi pffiffiffiffiffi m 0 u ¼ cxf ðgÞ; v ¼  cmf ðgÞ; T ¼ T1 þ B x2 hðgÞ; c rffiffiffi c g¼ y ð6Þ m in which c is a constant and prime denotes differentiation with respect to g. Equation (1) is automatically satisfied and the Eqs. (2–5) are reduced as follows: f 000 þ ff 00  f 02 þ b1 ð2ff 0 f 00  f 2 f 000 Þ  b2 ðf 002  ff 0000 Þ þ kh ¼ 0;

ð7Þ h00 þ Prf h0  2Prf 0 h þ Prb3 h ¼ 0;

ð8Þ

f ¼ S; f 0 ¼ 1; h0 ¼ 1 at g ¼ 0;

ð9Þ

f 0 ¼ 0; h ¼ 0 as g ! 1;

ð10Þ

in which b1 ¼ k1 c and b2 ¼ k2 c are the Deborah numbers, Grx the mixed convection parameter with Grx ¼ k ¼ Re 2 x

gbT ðTT1 Þx3 m2

the local Grashof number and Rex ¼ Ux m the local 0 the suction/injection parameter, Reynolds number, S ¼ pvffiffiffi mc

Pr ¼ The present boundary layer flow is governed by the following expressions: ou ov þ ¼ 0; ox oy

ð1Þ 

u

ou ou o2 u o2 u o2 u o2 u þ v ¼ m 2  k1 u2 2 þ v2 2 þ 2uv ox oy oy ox oy oxoy   3 3 2 o u o u ou o u ou o2 v þ mk2 u þv 3   oxoy2 oy ox oy2 oy oy2 þ gbT ðT  T1 Þ;

u



oT oT k o2 T Q þv ¼ þ ðT  T1 Þ: 2 ox oy qcp oy qcp

123

oT ¼ Bx2 at y ¼ 0; oy

the Prandtl number and b3 ¼ jcQp c a heat genera-

tion/absorption parameter. Expression of local Nusselt number Nux is   xqw oT Nux ¼ ; qw ¼ k : oy y¼0 kðT  T1 Þ

¼ Nu=Re1=2 x

1 : hð0Þ

ð12Þ

3 Homotopy analysis solutions

ð3Þ

Choosing the following set of base functions fgk expðngÞk  0; n  0g

ð4Þ

ð11Þ

Dimensionless form of Eq. (11) is

ð2Þ

In the above equations u and v are the velocity components in the x- and y-directions, k1 and k2 are the relaxation and retardation times, respectively, g the gravitational acceleration, bT the thermal expansion coefficient, m ¼ ðl=qÞ the kinematic viscosity, T the fluid temperature, q the density of fluid, k the thermal conductivity of fluid, cp the specific heat at constant pressure and Q the heat source coefficient. The subjected boundary conditions are u ¼ cx; v ¼ v0 ;

qmcp k

ð13Þ

we express f and h as follows: f ðgÞ ¼ a00;0 þ

1 X 1 X

akm;n gk expðngÞ;

ð14Þ

n¼0 k¼0

hðgÞ ¼

1 X 1 X

bkm;n gk expðngÞ;

ð15Þ

n¼0 k¼0

where akm;n and bkm;n are the coefficients. Initial approximations and auxiliary linear operators are taken in the following forms:

J Braz. Soc. Mech. Sci. Eng.

f0 ðgÞ ¼ S þ 1  expðgÞ; h0 ðgÞ ¼  expðgÞ;

ð16Þ

Lf ¼ f 000  f 0 ; Lh ¼ h00 þ h0 ;

ð17Þ

g

g

g

Lf ðC1 þ C2 e þ C3 e Þ ¼ 0; Lh ðC4 þ C5 e Þ ¼ 0;

ð18Þ

f ðgÞ ¼ f0 ðgÞ þ

h i   ^ pÞ ; ð1  pÞLf f^ðg; pÞ  f0 ðgÞ ¼ phf N f f^ðg; pÞ; hðg;

ð19Þ h i h i ^ pÞ  h0 ðgÞ ¼ ph N h f^ðg; pÞ; hðg; ^ pÞ ; ð1  pÞLh hðg; h ð20Þ

f^ð0; pÞ ¼ S; f^0 ð0; pÞ ¼ 1; f^0 ð1; pÞ ¼ 0; h^0 ð0; pÞ ¼ 1; ^ hð1; pÞ ¼ 0; ð21Þ !2 of^ðg; pÞ og

o3 f^ðg; pÞ ^ o2 f^ðg; pÞ  f ðg; pÞ  3 og og2 " # 3^ of^ðg; pÞ o2 f^ðg; pÞ ^ðg; pÞÞ2 o f ðg; pÞ þ þ b1 2f^ðg; pÞ  ð f og og2 og3

hðgÞ ¼ h0 ðgÞ þ

b2 4

o2 f^ðg; pÞ og2

f^ðg; pÞ

^ pÞ; f^ðg; pÞ ¼ N h ½hðg;

^ pÞ o4 f^ðg; pÞ5 ohðg; ; þk 4 og og

1 X

hm ðgÞ:

ð30Þ

Lf ½fm ðgÞ  vm fm1 ðgÞ ¼ hf Rm f ðgÞ;

ð31Þ

Lh ½hm ðgÞ  vm hm1 ðgÞ ¼ hh Rm h ðgÞ;

ð32Þ

fm ð0Þ ¼ fm0 ð0Þ ¼ fm0 ð1Þ ¼ 0; h0m ð0Þ  chm ð0Þ ¼ hm ð1Þ ¼ 0; ð33Þ 000 Rm f ðgÞ ¼ fm1 ðgÞ þ

m1h i X 00 0 fm1k fk00  fm1k fk0 k¼0

þ b1

m1 X

fm1k

k¼0 0

þ b2 fm1k

k X 0 0 ð2fkl fl00  fkl fl000  kfm1 ðgÞÞ l¼0

m1 X

ð23Þ

0

fk  fm1k

k¼0

m1 X

! fkiv

þ khm  1;

k¼0

ð34Þ

2^

^ pÞ o hðg; pÞ ohðg; þ Prf^ðg; pÞ 2 og og ^ of ðg; pÞ ^ ^ pÞ;  2Pr hðg; pÞ þ b3 hðg; og ð24Þ

in which p is an embedding parameter, hf and hh the nonzero auxiliary parameters and N f and N h the nonlinear operators. For p ¼ 0 and p ¼ 1 we have ^ 0Þ ¼ h0 ðgÞ and fðg; ^ 1Þ ¼ fðgÞ; f^ðg; 0Þ ¼ f0 ðgÞ; hðg; ^ 1Þ ¼ hðgÞ; hðg;

ð29Þ

The mth-order deformation problems are constructed by the following expressions:

3

!2

fm ðgÞ;

m¼1

ð22Þ 2

1 X m¼1

where Ci ði ¼ 1  5Þ are the arbitrary constants. The corresponding zeroth order deformation problems are developed as

N f ½f^ðg; pÞ ¼

selected in such a way that (26) and (27) converge at p ¼ 1 and hence

00 Rm h ðgÞ ¼ hm1 ðgÞ þ Pr

m1 X

h0m1k fk  2Pr

k¼0

m1 X k¼0

þ b3 hm1 ðgÞ;  vm ¼

0; m  1; 1; m [ 1:

ð35Þ ð36Þ

If fm and hm are the special solutions, then the general solutions are fm ðgÞ ¼ fm ðgÞ þ C1 þ C2 eg þ C3 eg ;

ð25Þ

hm1k fk0

hm ðgÞ ¼

hm ðgÞ

þ C4 þ C5 eg :

ð37Þ ð38Þ

and when p increases from 0 to 1 then f ðg; pÞ and hðg; pÞ vary from f0 ðgÞ; h0 ðgÞ to f ðgÞ and hðgÞ: Taylor’s series yields f ðg; pÞ ¼ f0 ðgÞ þ

1 X

4 Convergence of the homotopy solutions m

fm ðgÞp ;

ð26Þ

m¼1

hðg; pÞ ¼ h0 ðgÞ þ

1 X

hm ðgÞpm ;

ð27Þ

m¼1

1 om f ðg; pÞ 1 om hðg; pÞ ; hm ðgÞ ¼ ; fm ðgÞ ¼ m! ogm p¼0 m! ogm p¼0 ð28Þ where the convergence of above series strongly depends upon hf and hh : The auxiliary parameters hf and hh are

The auxiliary parameters  hf and  hh have significant role in the convergence of developed series solutions. Here the  hcurves are potrayed for 18th-order of approximations to find the values of  hf and  hh ensuring convergence. Figures 1 and 2 depict that the range of admissible values of  hf and  hf B -0.2 and -0.75 B  hh B -0.2. The hh are -0.8 B  series solution converge in the whole region of g when  hh = -0.4. Table 1 depicts that 25th order hf = -0.4 and  deformations are sufficient for both the velocity and temperature expressions.

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J Braz. Soc. Mech. Sci. Eng.

0

Fig. 1  h-curve for the function f.

Fig. 3 Variation of S on f ðgÞ

Fig. 2  h-curve for the function h

Fig. 4 Variation of Pr on f ðgÞ

0

Table 1 Convergence of homotopy solutions for different order of approximations when b1 ¼ 0:2; b2 ¼ b3 ¼ 0:3; Pr ¼ S ¼ 0:5; k ¼ hh = -0.4 0:1 ,  hf = -0.4 and  Order of approximation

f 00 ð0Þ

h00 ð0Þ

1

1.10000

1.22000

5

1.19801

1.34636

10

1.20255

1.35611

15

1.20268

1.35640

20

1.20271

1.35642

25

1.20272

1.35643

30

1.20272

1.35643

5 Graphical results and discussion The purpose of this section is to highlight the variations of interesting parameters through Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 for velocity and temperature. Figures 3, 4, 5, 6, 7, 8 show the behaviors of suction/injection parameter S, Prandtl number Pr, Deborah numbers b1 and b2 , heat

123

source/sink b3 and mixed convection parameter k on the 0 velocity f ðgÞ. Figure 3 shows that effects of suction/ 0 injection parameter S on the velocity profile f ðgÞ: Here S [ 0 corresponds to suction and S\0 is for injection case. 0 We observed that the velocity f ðgÞ is lower for suction case in comparison to injection phenomenon. From physical point of view suction is an agent that resists the fluid flow. Such resistance in fluid flow creates a reduction in the velocity field and associated boundary layer thickness. Figure 4 illustrates that the larger Prandtl number shows lower velocity and thinner momentum boundary layer thickness while smaller Prandtl number corresponds to higher velocity and thicker momentum boundary layer thickness. A comparative study of Figs. 3 and 4 shows that the variation in velocity for increasing S is larger than Pr. 0 The effects of Deborah numbers b1 and b2 on f ðgÞ are seen in the Figs. 5 and 6. Here we have noted that an increase in Deborah number b1 leads to a reduction in fluid velocity but enhancing values of Deborah number b2 give rise to fluid velocity and momentum boundary layer thickness. In

J Braz. Soc. Mech. Sci. Eng.

0

Fig. 5 Variation of b1 on f ðgÞ

0

Fig. 8 Variation of Pr on f ðgÞ:

0

Fig. 9 Variation of S on hðgÞ

0

Fig. 10 Variation of b1 on hðgÞ

Fig. 6 Variation of b2 on f ðgÞ

Fig. 7 Variation of b3 on f ðgÞ:

fact the Deborah numbers b1 and b2 are dependent on the relaxation and retardation times, respectively. So an increase in Deborah numbers b1 and b2 corresponds to higher relaxation and retardation times. The fluid velocity

is lower for higher relaxation time while higher for higher retardation time. An increase in heat source parameter b3 [ 0 yields a decrease in the velocity while in case of heat sink parameter b3 \0 both the momentum boundary

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J Braz. Soc. Mech. Sci. Eng.

Fig. 11 Variation of k on hðgÞ

Fig. 12 Variation of Pr on hðgÞ

Fig. 13 Variation of b2 on hðgÞ

layer thickness and velocity enhance (see Fig. 7). Figure 8 illustrates that both the fluid velocity and boundary layer thickness decrease when mixed convection parameter is increased. Note that the mixed convection parameter involves the buoyancy force. Buoyancy force is stronger

123

Fig. 14 Variation of b3 on hðgÞ

for the larger mixed convection parameter and weaker for the smaller mixed convection parameter. This stronger buoyancy acts as an agent to create a reduction in the velocity profile and momentum boundary layer thickness. Figures 9, 10, 11, 12, 13, 14 are displayed to examine the impacts of arising parameters on dimensionless temperature profile hðgÞ. Figure 9 presents the variations in temperature hðgÞ for different values of suction/injection parameter S. From this Fig. it is seen that the temperature is higher for injection case when we compare it with suction case. The effects of suction parameter S on hðgÞ is qualitatively similar to that of the velocity. Figure 10 shows that both temperature and thermal boundary layer thickness are increasing functions of b1 . Both the temperature hðgÞ and thermal boundary layer thickness increase when mixed convection parameter k increases (see Fig. 11). Here buoyancy force is an agent that creates an enhancement in the temperature and thermal boundary layer thickness. Figure 12 depicts the variations of Prandtl number on hðgÞ. The temperature and thermal boundary layer thickness are reduced when we increase the values of Prandtl number Pr. Prandtl number depends on thermal diffusivity. Larger Prandtl fluids have smaller diffusivity and lower Prandtl fluids have higher diffusivity. Hence the increasing values of Prandtl number correspond to a reduction in thermal diffusivity. Such reduction in thermal diffusivity is responsible for a decrease in temperature and thermal boundary layer thickness. Figure 13 depicts that the temperature decreases by increasing b2 . Influence of heat source b3 on hðgÞ is presented in Fig. 14. Physically, b3 [ 0 means that Tw [ T1 and in this case heat is supplied to the flow region from the wall. The temperature rises with heat source parameter b3 [ 0 while reduction in thermal boundary layer thickness and temperature is seen with heat sink parameter b3 \0. Table 2 shows that the local Nusselt number has quite opposite behavior for b1 and b2 : The values of Nusselt

J Braz. Soc. Mech. Sci. Eng. Table 2 Values of local Nusselt number Nu/Rex1=2 for the parameters b1 ; b2 ; b3 ; Pr, S when k ¼ 0:1 b1

b2

b3

Pr

S

k

Nu=Re1=2 x

0.0

0.3

0.3

0.5

0.5

0.1

1.20133

0.2

1.17496

0.4

1.14936 0.4

1.18647

0.5 0.6

1.19622 1.20463 -0.1

1.02500

-0.2

1.09798

-0.3

differential equations. Effects of different involved parameters such as Deborah numbers b1 and b2 ; mixed convection parameter k; heat source/sink b3 and Prandtl number Pr on the flow field and temperature are analyzed. The main observations are summarized as follows: •

1.16099 0.6

1.27668

0.9

1.60523

1.2

1.91014 0.8

1.23081

1.0

1.30323

1.5

1.45103 0.2

1.15381

0.4

1.13613

0.6

1.11294

Table 3 Comparison of f 00 ð0Þ for different values of Maxwell parameter b1 when b2 ¼ S ¼ k ¼ 0 b1

Abel et al. [22] f 00 ð0Þ

Present results f 00 ð0Þ

0.0

1.00000

1.00000

0.2

1.051948

1.051889

0.4 0.6

1.101850 1.150163

1.101903 1.150137

0.8

1.196692

1.196711

1.2

1.285257

1.285363

1.6

1.368641

1.368758

2.0

1.447617

1.447651

•

• •

•

• •

The behaviors of Deborah numbers b1 and b2 on the velocity profile f 0 ðgÞ are opposite. This is due to the fact that as b1 depends on the relaxation time and b2 depends on the retardation time. As relaxation time increases it requires more time to the fluid to attain equilibrium position thats why velocity decreases while opposite phenomena took place in case of retardation time. Velocity profile and momentum boundary layer thickness reduces in case of suction S [ 0 while enhances in case of injection S\0: Effect of heat source/sink b3 on the velocity and temperature are qualitatively opposite. Thermal boundary layer thickness is increasing function of heat source parameter b3 [ 0 while reduces with heat absorption parameter b3 \0: Increase in mixed convection parameter k yields an enhancement in the temperature and thermal boundary layer thickness while reduction in heat transfer rate at wall is noted. Temperature profile decreases by increasing Prandtl number Pr. Behaviors of Deborah numbers b1 and b2 on the heat transfer rate at the wall are opposite: Heat transfer rate at wall increases with the heat absorption b3 \0 and suction parameters S [ 0:

Acknowledgments We are grateful to the reviewers for the useful suggestions. The first and second authors are grateful to the Higher Education Commission of Pakistan (HEC) for the financial support.

References number increases by increasing b3 ; Pr and S. However, it decreases by increasing k. Table 3 ensures the validity of present results for f 00 ð0Þ in a limiting sense.

6 Concluding remarks This study deals with the mixed convection flow of a rate type Oldroyd-B fluid over a stretching sheet along with suction/injection and in the presence of heat source/sink. The stretching is assumed to be stretched linearly and be maintained at power law heat flux. Analytical technique homotopy analysis method is employed to obtain the convergence of series solutions of the resulting ordinary

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