J. Cent. South Univ. (2015) 22: 3180−3188 DOI: 10.1007/s11771-015-2855-5

Flow of Burgers’ fluid over an inclined stretching sheet with heat and mass transfer T. Hayat1, 2, Sadia Asad1, A. Alsaedi2 1. Department of Mathematics, Quaid-i-Azam University, 45320, Islamabad 44000, Pakistan; 2. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80257, Jeddah 21589, Saudi Arabia © Central South University Press and Springer-Verlag Berlin Heidelberg 2015 Abstract: Effects of heat and mass transfer in the flow of Burgers fluid over an inclined sheet are discussed. Problems formulation and relevant analysis are given in the presence of thermal radiation and non-uniform heat source/sink. Thermal conductivity is taken temperature dependent. The nonlinear partial differential equations are simplified using boundary layer approximations. The resultant nonlinear ordinary differential equations are solved for the series solutions. The convergence of series solutions is obtained by plotting the  -curves for the velocity, temperature and concentration fields. Results of this work describe the role of different physical parameters involved in the problem. The Deborah numbers corresponding to relaxation time (β1 and β2) and angle of inclination (α) decrease the fluid velocity and concentration field. Concentration field decays as Deborah numbers corresponding to retardation time (β3) and mixed convection parameter (G) increase. Large values of heat generation/absorption parameters A/B, and the temperature distribution across the boundary layer increase. Numerical values of local Nusselt number, −θ′(0), and local Sherwood number, −f′(0), are computed and analyzed. It is found that θ′(0) increases with an increase in β3. Key words: Burgers’ fluid; thermal radiation; inclined stretching sheet; non-uniform heat source; variable thermal conductivity

1 Introduction Non-Newtonian fluids have common place in physiology, engineering and industry sector. However, one constitutive equation between shear stress and rate of strain cannot describe the properties of all such materials. It is in view of the fact that such fluids have great diversity in their characteristics. Therefore, the classification of non-Newtonian fluids is presented through differential, integral and rate types. Much attention in the past has been given to the flows of differential type fluid. Some advancements have been made to the subclasses of rate type fluids. For example, FETECAU et al [1] established the unsteady flow of an Oldroyd-B fluid driven by the transverse motion of an infinite plate subject to a time-dependent shear stress. The change of the kinetic energy with time is also obtained from energetic balance. Exact solutions corresponding to the second problem of stokes for Maxwell fluids have been investigated by FETECAU et al [2]. Time-dependent flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates have been reported by TAN et al [3]. XUE et al [4] developed an exact solution of strat-up flow for the

fractional generalized Burger’s fluid in a porous half-space. HAYAT et al [5−6] examined the three-dimensional flow of a rate type fluids on linearly stretching surface. Peristaltic motion of a Burgers’ fluid in a planar channel has been reported by HAYAT et al [7]. JAMIL and FETECAU [8] analyzed the exact solution of rotating flow of a generalized Burgers’ fluid in cylindrical domains. KHAN and HAYAT [9] examined the exact solutions for fractional generalized Burgers’ fluid in a porous space. RASHIDI et al [10] investigated the approximate solutions of Burgers’ fluid and regularized long wave equations by means of the homotopy analysis method. The flow induced by a stretching surface is encountered in applications like paper production, hot rolling, drawing of plastic films, annealing and tinning of copper wires and metal spinning. Moreover, the effects of heat and mass transfer on the flow caused by a stretching surface are prominent in view of several physical problems, electrical power generation, solar power technology, space vehicle re-entry and nuclear reactors. Also, radiative heat mass transfer occurs in many geophysical and engineering applications, such as nuclear reactors, migration of moisture through air contained in fibrous insulations, nuclear waste disposal

Received date: 2014−06−26; Accepted date: 2015−03−10 Corresponding author: Sadia Asad, PhD Candidate; Tel: +92−51−90642172; E-mail: [email protected]

J. Cent. South Univ. (2015) 22: 3180−3188

and dispersion of chemical pollutants through water-saturated soil. HELAL and SAIF [11] developed the problem of heat and mass transfer in a power-law fluid past an inclined sheet with thermophoresis. PA and MONDAL [12] have included the effect of variable viscosity on mixed convection flow and heat transfer. They considered nonlinear stretching sheet embedded in a saturated porous medium. KHAN et al [13] studied mass transfer in the MHD flow of a couple stress fluids over a nonlinear stretching sheet. HAYAT and HINA [14] discussed the influence of wall properties on the MHD peristaltic flow of a Maxwell fluid with heat and mass transfer. In all the above mentioned attempts, the thermal conductivity of fluid is taken as constant. Very limited information is even available about flows of viscous fluids with variable thermal conductivity. The concept of variable thermal conductivity is important for granular agricultural products, namely spring oat and soybean and were measured in dependence on moisture content from the dry state to the water fully saturated state in a qualified assessment of optimal modes of technological processes and the development of modern fully automatic agricultural equipment. CHIAM [15−16] examined the heat transfer with variable conductivity in stagnation point flow towards a stretching sheet. AHMAD et al [17] have studied the boundary layer flow and heat transfer past a stretching plate with variable thermal conductivity. Recently the variable thermal conductivity effects in compressible boundary layer flow of viscous fluid by a circular cylinder are analyzed by OYEM and KORIKO [18]. The aim of this work is to venture further in this regime for we consider the heat and mass transfer effects in the flow of Burgers’ fluid over an inclined stretching sheet. Examples of Burgers’ fluids are earth’s mantle, asphalt and asphalt mixes, food products and soil, dilute polymeric solutions, hydrocarbons, paints and several other industrial and geomechanical fluids. Here, relevant problems of velocity temperature and concentration fields are constructed and computed by homotopy analysis method (HAM) [19−24].

2 Problem formulation We have considered the effects of heat and mass transfer in the flow of Burgers’ fluid over an inclined stretching sheet. Thermal radiation, variable thermal conductivity and non-uniform heat source/sink effects are taken into account. The stretching sheet makes an angle α with the horizontal axis, i.e., x-axis. The y-axis is taken normal to the x-axis. The extra stress tensor S for Burgers’ fluid is defined as

3181

S  1

DA1  DS D2 S   2    A1  3 2 Dt Dt  Dt 

(1)

where μ is the dynamic viscosity; A1 is the rate of strain tensor; λ1 and λ2 are the relaxation time; λ3 is the retardation time; D/Dt denotes the upper convected derivative defined by

DS d S   LS  SLT Dt dt

(2)

in which d/dt is the material time derivative and L is the velocity gradient. Burgers’ fluid model reduces to the special cases of the Oldroyd-B model, Maxwell model and the Newtonian fluid model when λ2=0, λ2=λ3=0 and λ1=λ2=λ3=0, respectively. Under the boundary layer approximation, the velocity, temperature and the concentration fields are governed by the following equations, respectively. u v  0 x y u

(3)

  2u u u  2u  2 u  v  1  u 2 2  v 2 2  2uv   x x y xy  y  2 2 2  u  u  u  v  u  3  x x 2 y x 2 y 3   x 2  2  v  u u  2 u  v  u   2  3v 2    y y 2 y xy  x xy    3 3  u  2 u   u  u  2 3uv u 2  v 2 uv  2   y x 2   x y x y      2 2  2   3u v  u v  u   u u       3  xy 2  x y 2 y xy  y 2   3u u  2 u u  2 v  v 3   g 0  T T  T  cos  (4) x y 2 y y 2  y



2 u 3

 3u

 v3

 3u

 T T    T  q r  k  v  q  y  y  y  y  x

c p  u u

C C  2C v  De 2 x y y

(5)

(6)

u  u w ( x)  ax, v  0, T  Tw , C  C w at y  0

(7)

u  0, T  T , C  C as y  

(8)

where u and v represent the velocity components in the x and y directions, respectively; T is the fluid temperature; T∞ is the ambient fluid temperature; C∞ is the ambient fluid concentration; v=(μ/ρ) is the kinematic viscosity; ρ is the density of fluid; h is the convective heat transfer coefficient; C is the concentration of fluid; q″ is the non-uniform heat generated (q″>0) or absorbed (q″<0) per unit volume; D is the effective diffusion coefficient

J. Cent. South Univ. (2015) 22: 3180−3188

3182

and the variable thermal conductivity of the fluid k is defined as

when

T  T   k  k  1    T  

qw   k

(9)

where ε is the small parameter; k∞ is the thermal conductivity of the fluid far away from the surface; ΔT=Tw−T∞. The non-uniform heat source/sink q″ is modeled by q  

ku s ( x, t ) A(Ts  T ) f   (T  T ) B x

(10)

where A and B are the coefficients of space and temperature-dependent heat source/sink, respectively. Here, two cases arise. For internal heat generation A>0 and B>0 and for internal heat absorption, we have A<0 and B<0. We introduce the change of variables as follows:

  xa 1/ 2 f ( ),  ( ) 

T  T , Tw  T (11)

And the velocity components are   , v y x

(12)

where ψ is the stream function. Now, Eq. (3) is identically satisfied and Eqs. (4)−(8) yield f   ff   f 2  1 (2 ff f   f 2 f )   2 ( f 3 f   2 ff 2 f   3 f 2 f 2 )   3 ( f 2  ff  )  G cos   0

(13) 4 (1  R   )   Pr f    2  1    Af   B   0 3 (14)

   Sc f   f    0

(15)

f  0, f   1,   1   1 at   0

(16)

f   0,   0   0 as   

(17)

where β1=λ1a and β2=λ2a are the Deborah numbers in terms of relaxation time respectively; β3=λ3a is the Deborah number in terms of retardation time; Pr=vρcp/k∞ g  (T  T ) x 3 / v is the Prandtl number; G  0 T 2w 2  is the uw x / v 3

mixed convection parameter; R 

4 *T *

is the radiation

k k parameter; Sc=v/D is the Schmidt number. The local Nusselt number Nux and local Sherwood number Sh are Nu x 

, Mw   D y 0

C y

(19) y 0

Dimensionless form of local Nusselt number Nux and local Sherwood number Sh are Nu / Re1x/ 2   (0), Sh / Re1x/ 2   (0)

(20)

where the local Reynolds number Rex 1/ 2 

 xu s

.

3 Homotopy analysis solutions Initial approximations and auxiliary linear operators are chosen as follows: f 0 ( )  (1  e  ),  0 ( )  e  , 0  e 

(21)

 f  f   f ,        ,       

(22)

with properties

1/ 2

C  C a ,   y   ( )  C w  C  

u

T y

xq w tM w , Sh  k (Ts  T ) Dw (Cs  C )

(18)

L f (C1  C2 e  C3e  )  0     L (C4 e  C5e )  0    L (C6 e  C7 e )  0

(23)

where Ci (i=1−7) are the arbitrary constants. If p  [0, 1] denotes an embedding parameter and hf, hθ and hf are the non zero auxiliary parameters, then the zeroth order deformation problems are

1  p  f  fˆ ( ; p)  f 0 ( )  p f N f  fˆ ( ; p)

(24)

1  p   ˆ( ; p)   0 ( )  p  N  fˆ ( ; p),ˆ(; p)

(25)

1  p   ˆ( ; p)  0 ( )  p  N  fˆ ( ; p), ˆ( ; p) (26) fˆ (0; p)  0, fˆ  (0; p)  1, fˆ  (; p)  0, ˆ (0; p)  1, ˆ(; p )  0, ˆ(0; p)  1, ˆ(; p )  0 (27)

where the nonlinear operators Nf, Nθ and Nf are  3 fˆ ( , p) ˆ  2 fˆ ( , p )  f ( , p )  N f [ fˆ ( , p )]  3   2 2

2 ˆ ˆ   fˆ ( , p)    1 2 fˆ ( , p) f ( , p)  f ( , p)        2    4 ˆ  2  3 fˆ ( , p )  ˆ ( , p) 3  f ( , p )  fˆ ( , p) f     2  3   4 







2







 fˆ ( , p)   2 fˆ ( , p )  2   3 fˆ ( , p)   2 fˆ ( , p) 2         2 2     2 fˆ ( , p )   2ˆ     3   f ( , p )      2     2     

J. Cent. South Univ. (2015) 22: 3180−3188

3183

 4 fˆ ( , p )  fˆ ( , p )   Gˆ( , p ) cos   4   4   ˆ( , p )  N [ˆ( , p ), fˆ ( , p )]  1  R     3   2  2 ˆ( , p )    ˆ( , p)  1     Pr fˆ ( , p )       Af   B 

(28)

2



N  [ˆ( , p), fˆ ( , p )] 

 2ˆ( , p )  2

(29)

 ˆ( , p )   Sc fˆ ( , p )   

(30)

f ( , p )  f 0 ( )   f m ( ) p m m 1 

 ( , p)   0 ( )    m ( ) p m m 1



 ( , p)  0 ( )   m ( ) p m m 1



 ( )  0 ( )   m ( ) m 1

m 1

R mf ( )  f m1 ( )    f m 1 k f k  f m 1 k f k   k 0

m 1 

k

k



k 0 

l 0

l 0



m 1 

k

1   2 f m1 k  f kl f l f m 1 k  f k l f l  

(31)

 l 0

(32) (33) (34)

l 0

f kl f p  3 f m 1 k

l

 f l p f p 2 f m1k 

p 0

k

 l 0

  f k l fl  3  f m1 k  

m 1

 f 

k 0

m 1  f m 1 k  f k  G m 1 cos  k 0 

4 R ) m 1   m 1 k 3

m 1

 m 1  Pr    m 1 k f k    m 1 k  k 0 

m 1

R m ( )  (1 

B m1 )  A m1k

  k 

k 0

  k  ( Af m 1 

k 0

m 1

m 1

k 0

k 0

 f k  B m1k   k

m 1

R m ( )  m 1  Sc  m 1 k f k  f m 1 k  k  0, m  1 1, m  1

m  

p 0

(35) p 0

p 0

(36) (37) (38)

The m-th-order deformation problems can be provided by

 k

(43)

k 0

f ( )  f 0 ( )   f m ( )

m 1

(42)

where

k

where the convergence depends upon hf, hθ and hf. By proper choices of hf, hθ and hf, the series (Eqs. (32)−(35)) converge for p=1, and hence



(41)





 ( )   0 ( )    m ( )

L [m ( )   mm 1 ( )]    R m ( )

k 0 

When p variation is taken from 0 to 1, f(η, p), θ(η, p) and f(η, p) approach f0(η, p), θ0(η, p) and f0(η, p) to f(η), θ(η) and f(η). Now f, θ and f in Taylor's series can be expanded in the forms as

m 1

(40)

 2   f m 1 k  f k l

 fˆ ( ; 0)  f 0 ( ), ˆ( ; 0)   0 ( ), ˆ( ; 0)  0 ( )  ˆ ˆ ˆ  f ( ;1)  f ( ),  ( ;1)   ( ),  ( ;1)   ( )



L [ m ( )   m m 1 ( )]    R m ( )

 m ( 0)   m (  )  0

When p=0 and p=1,

 m  f m ( )  1  f ( ; p)  m!  m   1  m ( ; p )   m ( )  m!  m    1  m ( ; p)   m ( )  m!  m 

(39)

f m (0)  f m (0)  f m ( )  0,  m (0)   m ( )  0,



 fˆ ( , p) ˆ  ( , p)   

L f [ f m ( )   m f m 1 ( )]   f R mf ( )

(44) (45) (46)

The general solutions of Eqs. (38)−(40) can be written as f m ( )  f m ( )  C1  C 2 e  C3e 

(47)

 m ( )   m ( )  C 4 e  C5 e 

(48)

 m ( )   m ( )  C6 e  C7 e 

(49)

where f m ,  m and  m are the particular solutions.

4 Convergence of solutions It is fairly understandable that the resultant series solutions contain the auxiliary parameters  f ,   and   . Such parameters have key role in the convergence of series solutions. For permissible values of auxiliary parameters, we plot the   curves at the 14-th order of approximations (see Figs. 1−3). It is clearly seen from these figures that the meaningful values of  f ,   and

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3184

Table 1 Convergence of HAM solution for different orders of approximations

Fig. 1 Curve for velocity field

Fig. 2 Curve for temperature field

Order of approximation

−f″(0)

−θ″(0)

−f″(0)

1

1.13084

0.412012

1.04613

5

1.18308

0.365947

1.01901

10

1.18324

0.365564

1.01677

15

1.18324

0.365553

1.01657

20

1.18324

0.365551

1.01654

25

1.18324

0.365550

1.01653

30

1.18324

0.365550

1.01653

35

1.18324

0.365550

1.01653

40

1.18324

0.365550

1.01653

45

1.18324

0.365550

1.01653

50

1.18324

0.365550

1.01653

5.1 Dimensionless velocity field Figure 4 presents the influence of α on the velocity field. It is seen that large values of α cause a reduction in the velocity profile. In fact the increase in α corresponds to more gravity effect. Figures 5−7 are plotted for various values of Deborah numbers (β1, β2 and β3). Increases in β1 and β2 cause a reduction in velocity profile and associated boundary layer thickness. Effect of β3 on the fluid motion is quite opposite to that of β1 and β2. It is also clear from Fig. 7 that the velocity field is more pronounced near the wall but after some large distance it vanishes. Figure 8 illustrates that an increase in G yields an increase in the velocity and boundary layer thickness. 5.2 Dimensionless temperature field Figure 9 depicts the variation of ε on the temperature profile. In fact the increase in variable thermal conductivity parameter ε results in the increase of the temperature and thermal boundary layer thickness. Figures 10 and 11 illustrate the effects of β1 and β2 on temperature profile. Deborah numbers related to relaxation time (β 1 and β 2 ) increase the temperature

Fig. 3 Curve for concentration field

  are [−1.3, −0.6], [−1.25, −0.3] and [−1.3, −0.6], respectively. Moreover Table 1 depicts that the series solutions are convergent up to six decimal places.

5 Discussion Consider that problem contains different physical parameters like β1, β2, β3, ε, G, α, Pr, R, A, B and Sc. Graphical results of such parameters are displayed and discussed in this section.

Fig. 4 Influence of α on velocity field

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Fig. 5 Influence of β1 on velocity field

3185

Fig. 9 Influence of ε on temperature field

Fig. 6 Influence of β2 on velocity field Fig. 10 Influence of β1 on temperature field

Fig. 7 Influence of β3 on velocity field Fig. 11 Influence of β2 on temperature field

Fig. 8 Influence of G on velocity field

profile and thermal boundary layer thickness. Influence of β3 (Deborah number related to retardation time) and G (mixed convection parameter) are qualitatively similar (see Figs. 12 and 13). Both parameters decrease the temperature and thermal boundary layer thickness. Figure 14 illustrates the effect of thermal radiation R on the temperature profile. Temperature profile decreases rapidly when R increases. Physically the larger values of R imply larger surface heat flux, making the fluid warmer which enhances the temperature profile. The

3186

Fig. 12 Influence of β3 on temperature field

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Fig. 15 Influence of Pr on temperature field

Fig. 16 Influence of A on temperature field Fig. 13 Influence of G on temperature field

Fig. 17 Influence of B on temperature field Fig. 14 Influence of R on temperature field

effect of Prandtl number Pr on the temperature distribution is displayed in Fig. 15. It can be seen that the temperature decreases with an increase of Pr. It implies that the momentum boundary layer is thicker than the thermal boundary layer. This is due to the fact that for higher Prandtl number, the fluid has a relatively low thermal conductivity which reduces conduction. For large values of heat generation/absorption parameters A and B, the temperature distribution across the boundary layer increases (see Figs. 16 and 17).

5.3 Dimensionless concentration field Figures 18−20 show the variation of Deborah numbers (β1, β2 and β3) on the concentration field. Here, concentration field shows gradual increase when β1 and β2 increase. However, the concentration field is decreasing function of β3 (see Fig. 20). Figure 21 illustrates the effect of mixed convection parameter G on the concentration field. For larger values of G number the concentration field decreases. Figure 22 depicts the influence of α on the concentration field. It is seen that larger values of α decrease the concentration field. Figure 23 shows the variation of Sc on f(η). Concentration field decreases rapidly when Sc increases.

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Fig. 18 Influence of β1 on concentration field

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Fig. 22 Influence of α on concentration field

Fig. 19 Influence of β2 on concentration field Fig. 23 Influence of Sc on concentration field Table 2 Values of local Nusselt number Nu / Re1x/ 2 for different parameters ε

β1

β2

β3

R

Pr

Nu / Re1x/ 2

0

0.3

0.2

π/4

0.1

1.0

0.763096

0.2

0.672509

0.3

0.634519

0.2

Fig. 20 Influence of β1 on concentration field

Fig. 21 Influence of G on concentration field

0

0.650943

0.1

0.642600

0.3

0.626717 0

0.644663

0.1

0.639651

0.2

0.634519 0.1

0.626474

0.2

0.634519

0.3

0.641885 0

0.712749

0.1

0.678452

0.2

0.634519 1.1

0.694574

1.3

0.791845

1.4

0.880650

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3188 Table 3 Values of local Sherwood number Sh for different parameters β1

β2

β3

Sc

−f′(0)

0

0.2

0.2

1.1

1.1135

0.3

1.0914

0.6

1.0716 0

1.1066

0.3

1.0944

0.5

1.0862 0

1.0748

0.1

1.0836

0.2

1.0913 0

0.82259

0.1

1.0295

0.3

1.2071

[6]

[7] [8]

[9]

[10]

[11]

[12]

6 Conclusions 1) Deboarh number decreases the velocity and concentration field. 2) Velocity field is decreasing function of angle of inclination. 3) Deborah number corresponding to retardation time has opposite behavior on the velocity and temperature fields. 4) Prandtl number Pr reduces the temperature field. 5) In view of an increase in Biot number, both the temperature and thermal boundary layer thickness increase. 6) The effects of β3 and G on concentration field are qualitatively similar. 7）Effects ε, β1, β2 and R on local Nusselt number are qualitatily similar.

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