J. Cent. South Univ. (2015) 22: 1114−1123 DOI: 10.1007/s11771-015-2623-6
Convective heat and mass transfer in MHD mixed convection flow of Jeffrey nanofluid over a radially stretching surface with thermal radiation M. BILAL ASHRAF1, T. HAYAT2, 3, A. ALSAEDI3, S. A. SHEHZAD4
1. Department of Mathematics, Comsats Institute of Information Technology, Wancantt 47040, Pakistan; 2. Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan; 3. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80207, Jeddah 21589, Saudi Arabia; 4. Department of Mathematics, Comsats Institute of Information Technology, Sahiwal, Pakistan © Central South University Press and Springer-Verlag Berlin Heidelberg 2015 Abstract: Mixed convection flow of magnetohydrodynamic (MHD) Jeffrey nanofluid over a radially stretching surface with radiative surface is studied. Radial sheet is considered to be convectively heated. Convective boundary conditions through heat and mass are employed. The governing boundary layer equations are transformed into ordinary differential equations. Convergent series solutions of the resulting problems are derived. Emphasis has been focused on studying the effects of mixed convection, thermal radiation, magnetic field and nanoparticles on the velocity, temperature and concentration fields. Numerical values of the physical parameters involved in the problem are computed for the local Nusselt and Sherwood numbers are computed. Key words: Jeffrey nanofluid; mixed convection flow; radially stretching surface; convective boundary conditions; magnetic field
1 Introduction The boundary layper flows by stretching surface occur in polymer technology, extrusion process, paper production, glass fiber production, manufacturing of plastic, rubber sheets and etc. Initially, the boundary layer flow over a linear stretching surface was presented by CRANE [1]. Later on, investigators extended the work of CRANE [1] for linear, non-linear and exponential stretching surfaces [2−6] under different aspects and fluid models. Jeffrey fluid is one of the rate type non-Newtonian fluids which exhibits the stress relaxation and retardation times. Many investigators considered Jeffrey fluid under various characteristics just to predict the relaxation and retardation effects [7−10]. However, less attention is given for the flows induced by radially stretching surfaces. WANG [11] studied the natural convection flow over a vertical radially stretching sheet. Later on, AHMAD et al [12] provided the analytical solution of unsteady axisymmetric flow of second grade fluid over a radially stretching surface. KHAN and SHEHZAD [13] discussed the axisymmetric flow of Sisko fluid over a radially stretching surface. Boundary layer stretched flows in the presence of transverse magnetic field are very important due to their applications in many engineering problems such as in plasma studies, petroleum industries, geothermal energy
extractions. Solution of the MHD Falkner−Skan flow by homotopy analysis method was presented by ABBASBANDY and HAYAT [14]. Heat and mass transfer characteristics in magnetohydrodynamic (MHD) viscous flow over a permeable stretching surface were analyzed by TURKYILMAZOGLU [15]. MUKHOPHADHYAY et al [16] studied the boundary layer flow of an electrically conducting liquid through variable free stream temperature past a porous stretching non-isothermal surface with a power-law stretching velocity. Incompressible fluid saturates the porous medium. MOTSA et al [17] obtained the solutions for upper-convected Maxwell fluid over porous stretching sheet. Here, the authors employed the successive Taylor series linearization method. Radiative mixed convection has gained much importance amongst the recent researchers due to the number of applications in geophysical and energy storage problems such as in furnaces, ovens and boilers and the interest in our environment and in no conventional energy sources, such as the use of salt gradient solar ponds for energy collection and storage. MUKHOPHADHYAY [18] studied the effects of thermal radiation on unsteady boundary layer mixed convection flow with heat transfer over a vertical porous stretching surface embedded in porous medium. CHEN [19] analyzed the magnetohydrodynamic mixed convection flow of power law fluid over a stretching surface with
Received date: 2014−05−06; Accepted date: 2014−08−11 Corresponding author: M. BILAL ASHRAF, Assistant Professor, PhD; Tel: +92−321−5155751; E-mail:
[email protected]
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thermal radiation and heat source/sink. HAYAT et al [20] presented the analytical solutions of radiative mixed convection stagnation point flow over a vertical stretching sheet in the presence of porous medium and transverse magnetic field. Heat and mass transfer effects in unsteady natural convection flow of nanofluids past a vertical infinite flat plate with radiation effect are investigated by TURKYILMAZOGLU and POP [21]. The enhancement in thermal conductivity of conventional fluids via suspensions of solid particles is a modern development in engineering technology. Nanofluids have been proposed as a means for enhancing the performance of heat transfer liquids currently available, such as water, toluene, oil and ethylene glycol mixture. CHOI and EASTMAN [22] were probably the first researchers to combine a mixture of nanoparticles and base fluid (i.e. nanofluid). RASHIDI et al [23] presented the analytical solutions for the laminar axisymmetric mixed convection boundary layer flow of nanofluid past a vertical cylinder. SHEIKHOLESLAMI and GANJI [24] studied the heat transfer of Cu−water nanofluid flow between parallel plates. TURKYILMAZOGLU [25] examined the unsteady mixed convection flow of nanofluids over a moving vertical flat plate with heat transfer. MUSTAFA et al [26] presented the analysis of nanofluid over a linearly stretching sheet. ALSAEDI et al [27] provided an analysis to discuss the stagnation point flow of nanofluid near a permeable stretched surface with thermal convective condition. SHATEYI and MAKINDE [28] presented the numerical solution of hydro magnetic stagnation point flow towards a radially stretching surface with convective condition. The aim of the present work is to discuss the MHD flow of mixed convection nanofluid over a radially stretching sheet with thermal radiation. Jeffrey fluid is taken as a base fluid in the present study. Analysis is carried out by taking convective conditions through both heat and mass transfer on the radially stretching sheet. Series solutions of the governing problem are developed by using HAM [29−36]. Discussion section is made to analyze the impacts of all physical parameters on the flow, temperature and concentration fields. Numerical values of local Nusselt and Sherwood numbers for mixed convection, concentration buoyancy parameter, thermal radiation, Biot numbers, Deborah numbers and Hartman number are given in discussion section. Conclusion section contains the main findings of the study.
where u and w are the velocity components in the r- and z-direction, respectively; λ1 and λ2 are the ratios of relaxation to retardation times and retardation time, respectively; βT is the thermal expansion coefficient; βC is the concentration expansion coefficient; is the electrical conductivity; B0 is the magnitude of applied magnetic field; is the density of fluid; g is the gravitational acceleration; is the kinematic viscosity; k is the thermal conductivity; qr the radiative heat flux; T is the fluid temperature; cp is the specific heat of the fluid; is the density of nanoparticles; c p is the specific heat of the nanoparticles; C is the concentration field; D is the mass diffusivity and prime denotes differentiation with respect to η. By using the Rosseland approximation, the radiative heat flux qr [18−24] is given by
2 Mathematical modelling
qr
Mixed convection boundary layer flow of Jeffrey nanofluid over a radially stretching sheet with thermal radiation is considered. The sheet is stretched linearly with velocity u U w (r ) ar (with a as a real number).
where s is the Stefan−Boltzmann constant and ke is the mean absorption coefficient. If the temperature differences are sufficiently small, then Eq. (5) can be linearized by expanding T4 into the Taylor series about
The surface coincides with the plane z=0 and the flow is confined in the region z>0. Transverse magnetic field of strength B0 is applied to the normal direction of the flow i.e parallel to the z-axis. Electric and induced magnetic fields are neglected. It is also assumed that the sheet is heated by convection from a hot fluid placed at the bottom of sheet at temperature T with heat transfer coefficient h [27−28]. By using this assumption, convective boundary conditions for both heat and mass transfer are imposed on the surface of stretching sheet. The governing equations of flow analysis can be expressed as follows [9−10]:
(ru ) (rw) 0 z r u
u u w z 1 1 r
(1) 2u 3u u 2 u 2 2 u 2 r z 2 rz z
u 2u 3u w 3 g T (T T ) r rz z g C (C C )
u
B02 u
k 2T T T 1 qr w 2 r z c p z c p z 2 cp C T DT T DB z z T z c p
u
( 2)
C C 2C D 2T w DB 2 T 2 T z r z z
4 s T 4 3ke z
( 3)
(4)
(5)
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T∞, and after neglecting higher order terms it takes the form as T 4 4T3T 3T4
(6)
The boundary conditions of the present flow configuration are: T u U w ar , w 0, k h(Tf T ), z T D h (Cf C ) at z 0; z
(7)
u (r , z ) arf ( ), w(r , z ) 2 a f ( ), T T C C a , ( ) , z Tw T Cw C
(9)
Equation (1) is identically satisfied and Eqs. (2)−(8) give f ( f 2 2 ff ) (1 1 )(2 ff f 2 ) (1 1 )( N ) (1 1 ) Mf 0
(10) (11)
N 2 Scf t 0, Nb
(12)
(13)
f 0, 0, 0 as
(14)
where β is the Deborah number; λ is the mixed convection parameter; Grx is the local Grashof number; N is the concentration buoyancy parameter; M is the Hartman number; R is the radiation parameter; Pr is the Prandtl number; Nb is the Brownian motion parameter; Nt is the thermophoretic parameter; Sc is the Schmidt number; γ1 is the heat transfer Biot number; γ2 is the mass transfer Biot number. These can be defined as
N
, Grx 2
Rex
g T (Tf T )r 3
2
Sh / Re1r / 2 (0)
(17)
Nt
h c DT (Tw T ) , Sc , 1 , D k a cT
2
h D a
f 0 ( ) (1 e ), 0 ( )
0 ( )
1 exp( ) , 1 1
2 exp( ) 1 2
(18)
Lf f f , L , L
(19)
The above operators satisfy the following properties: Lf (C1 C2e C3e ) 0, L (C4e C5e ) 0,
(20)
where Ci (i=1−7) are the arbitrary constants. Deformation problems at zeroth order are written below:
ˆ( ; p )]
(21)
(1 p) L [ˆ( ; p) 0 ( )] p N [ fˆ ( ; p),ˆ( ; p), ˆ( ; p)] (22) (1 p ) L [ˆ( ; p ) 0 ( )] p N [ fˆ ( ; p ),ˆ( ; p ), ˆ( ; p )] (23) fˆ (0; p ) 0, fˆ (0; p ) 1, fˆ (; p ) 0, ˆ (0, p ) [1 (0, p)], ˆ(, p ) 0, 1
ˆ (0, p) 2 [1 ˆ(0, p )], ˆ(, p ) 0 N f [ fˆ ( ; p),ˆ( ; p ), ˆ( ; p)]
(1 1 ) fˆ ( , q )
C (Cw C ) B02 4 T 3 ,M , R ( s ), ke k T (Tf T ) a
c DB (Cw C ) , Nb , c
Initial guesses and auxiliary linear operators for the series solutions are selected in the forms as
2 fˆ ( , q)
3 fˆ ( , q) 3
(24)
2
fˆ ( , q ) ( 1 ) 1 2 2 fˆ ( , p) fˆ ( , p) 2 fˆ ( , p) ( fˆ ( , p)) 2 2
,
Pr
3 Series solutions
(1 p ) L f [ fˆ ( ; p) f 0 ( )] p f N f [ fˆ ( ; p ),ˆ( ; p ),
f 0, f 1, 1 (1 (0)), 2 (1 (0)), at 0;
Grx
(16)
L (C 6 e C7 e ) 0,
4 2 1 R 2 Pr f Pr ( N b N t ) 0 3
2c,
Nu / Re1r / 2 (0)
where Rer=Uwr/v is the local Reynold number.
u 0, T T , C C as z (8) h where subscript w corresponds to the wall condition; is the heat transfer coefficient; h* is the concentration transfer coefficient; Tf is the ambient fluid temperature; Cf the ambient fluid concentration. Using
( )
The non-dimensionless expressions of the local Nusselt and Sherwood numbers can be written as follows:
3 fˆ ( , p) fˆ ( , p) (1 1 )ˆ( , p) M (1 1 ) 3 (25)
(15)
2 4 ˆ( , p ) N [ fˆ ( ; p ),ˆ( ; p ), ˆ( ; p )] 1 R 2 3
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ˆ( , p) ˆ( , p) Pr N b 2 Pr (( fˆ ( , p))
ˆ( ; p )
ˆ( , p) Nt
N [ fˆ ( ; p),ˆ( ; p),ˆ( ; p)] 2Sc( fˆ ( , p))
ˆ( ; p)
2
2ˆ( ; p)
2
(26)
N 2ˆ( , p) t N b 2
(27)
and p=1, we obtain fˆ ( ; 0) f 0 ( ), ˆ( , 0) 0 ( ), ˆ( ; 0) 0 ( ); fˆ ( ;1) f ( ), ˆ( ,1) ( ), ˆ( ;1) ( )
(28)
As p rises from 0 to 1, f(η, p), θ(η, p) and (η, p) change from f0(η), θ0(η) and 0(η) to f(η), θ(η) and (η). Using Taylor’s expansion, we have m
(29) p 0
( , p ) 0 ( ) m ( ) p m , m 1
(30) p 0
(37)
m 1
(31) p 0
The convergence of above series highly depends upon f , and . Considering that f , and are selected properly, Eqs. (29)−(31)
converge at
p=1. Therefore,
f ( ) f 0 ( ) f m ( )
(32)
m 1
( ) 0 ( ) m ( )
(33)
m 1
( ) 0 ( ) m ( ) m 1
Deformation problems at mth-order are
(38)
m 1 R mf ( ) f m1 ( ) (1 1 ) 2 f m 1 k f k k 0 m 1 f m 1 k f k M (1 1 ) f m 1 ( ) k 0 m 1 m 1 f m 1 k f k 2 f m 1 k f k k 0 k 0 (1 1 )( m 1 Nm 1 )
(39)
m 1 4 R m ( ) 1 R m 1 2 Pr m 1 k f k 3 k 0 m 1 N Pr b m1 k k k 0 N t m 1 k k
(40)
m 1
N t m 1 Nb
(41)
Solving the corresponding mth-order deformation problems, we get f m ( ) f m ( ) C1 C2e C3e
(42)
m ( ) m ( ) C4e C5e
(43)
m ( ) m ( ) C6e C7 e where the special solutions are
(44) f m ,
m
and
m
.
4 Convergence analysis and discussion
( , p ) 0 ( ) m ( ) p m , 1 m ( ; p) m ( ) m! p m
L [m ( ) mm 1 ( )] R m ( )
k 0
m 1
1 m ( ; p ) m ( ) m! p m
(36)
R m ( ) m 1 2Sc m 1 k f k
f ( , p) f 0 ( ) f m ( ) p , 1 m f ( ; p) m! p m
L [ m ( ) m m 1 ( )] R m ( )
m () 0, m (0) 2m (0) m () 0
nonlinear operators are N f , N and N . Setting p=0
f m ( )
(35)
f m (0) f m (0) f m ( ) 0, m (0) 1 m (0)
where p is an embedding parameter; the non-zero auxiliary parameters are f , and and the
L f [ f m ( ) m f m 1 ( )] f R mf ( )
(34)
Equations (32)−(34) depend on the auxiliary parameters f , and . These parameters play significant role to obtain the convergent solutions. For suitable ranges of these parameters, the - curves have been sketched at 18th order of approximations. The acceptable values of f , and are those where - curves are parallel to η-axis. Figure 1 shows that the admissible values of f , and are 0.9 f 0.2, 1.0 0.2 and 1.1 0.1. Table 1 confirms that homotopic solutions of the problems converge in the whole region of η when f 0.6. Effects of the ratio of relaxation to retardation times λ1, Deborah number β, Hartman number M, mixed convection parameter λ and concentration buoyancy parameter N on the velocity profile f '(η) are depicted in Figs. 2−6. Figure 2 exhibits that as the ratio of relaxation to retardation times λ1 enhances, the velocity profile f '(η)
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boundary layer thickness reduce with an increase in M. In fact, applied magnetic field generates a force (Lorentz force) opposite to the flow direction which tends to oppose the flow. Figures 5 and 6 are plotted to analyze the behavior of mixed convection parameter λ and concentration buoyancy parameter N on the velocity profile f '(η). These figures depict that the velocity profile f ( ) and momentum boundary layer thickness are increasing functions of λ and N. It is due to the fact that buoyancy forces are much more dominant than the viscous forces.
Fig. 1 -curves for functions f(η), θ(η) and φ(η) Table 1 Convergence of series solutions for different order of approximations when K=M=0.1, A=N=β=α=0.2, λ=N=k=0.3, γ1=γ2=0.5, Pr=0.7, Sc=0.8, α=0.2 and f g 0.6 Order of approximations
−f"(0)
−θ'(0)
−'(0)
1
1.2695
0.22020
0.20627
5
1.2861
0.20900
0.17385
10
1.2791
0.20816
0.16320
15
1.2780
0.20825
0.16053
20
1.2778
0.20828
0.16005
25
1.2778
0.20828
0.16002
30
1.2778
0.20828
0.16002
Fig. 3 Influence of β on f ′(η)
Fig. 4 Influence of M on f ′(η)
Fig. 2 Influence of λ1 on f ′(η)
and momentum boundary layer thickness reduce. This is due to the fact that relaxation time is more dominant than the retardation time. As Deborah number depends on the retardation time, with an increase in Deborah number β both the velocity profile f '(η) and momentum boundary layer thickness are enhanced. Effect of β on f '(η) is opposite to the effect of the ratio of relaxation to retardation times λ1 (see Fig. 3). Variation of Hartman number M on the velocity profile f '(η) is seen in Fig. 4. It is found that the velocity profile f '(η) and momentum
Fig. 5 Influence of λ on f ′(η)
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the fluid shoot up, so that the fluid temperature increases. Figures 8 and 9 exhibit the effect of thermophoretic parameter Nt on the temperature θ(η) and concentration (η). Thermophoresis is a mechanism which causes small particles to be driven away from hot surface toward a cold one, due to the fact that both temperature θ(η) and concentration profiles (η) are increasing functions of thermophoretic parameter Nt. Also, the associated boundary layer thicknesses increases with an enhancement in thermophoretic parameter Nt. Outcome
Fig. 6 Influence of N on f ′(η)
Figures 7−19 are illustrated to show the effects of the involved parameters on the temperature θ(η) and concentration (η). Figure 7 shows that temperature θ(η) increases with an increase in radiation parameter R. Also, thermal boundary layer thickness enhances with R, which is due to the fact that as thermal radiation parameter increases, the mean absorption coefficient ke, decreases, which results in rise to the divergence of radiative heat flux. Hence, the rate of radiative heat transferred to
Fig. 9 Influence of Nt on (η)
Fig. 7 Influence of R on θ(η)
Fig. 10 Influence of Nb on θ(η)
Fig. 8 Influence of Nt on θ(η)
Fig. 11 Influence of Nb on (η)
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of Brownian motion parameter Nb on the temperature θ(η) and concentration (η) are depicted in the Figs. 10 and 11. With rise in Brownian motion parameter Nb, the temperature θ(η) and thermal boundary layer thickness boost while reverse behavior is noted for concentration profile (η). Figures 12 and 13 show the influence of mixed convection parameter λ on the temperature θ(η) and concentration (η). Reduction in associated boundary layer thicknesses is noted with an increase in mixed convection parameter λ. Impact of concentration
Fig. 15 Influence of N on (η)
Fig. 12 Influence of λ on θ(η)
Fig. 16 Influence of γ1 on θ(η)
Fig. 13 Influence of λ on (η)
Fig. 17 Influence of γ2 on (η)
Fig. 14 Influence of N on θ(η)
buoyancy parameter on the temperature θ(η) and concentration (η) is similar to that of mixed convection parameter (see Figs. 14 and 15). Variation of heat transfer Biot number γ1 enhances the temperature θ(η) and thermal boundary layer thickness (see Fig. 16). Figure 17 is drawn to analyze the behavior of mass transfer Biot number γ2 on the concentration profile (η) Here, concentration profile (η) and associated boundary layer thickness are increasing functions of γ2. Impact of
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Prandtl number Pr on the temperature θ(η) is plotted in Fig. 18. Thermal boundary layer thickness and temperature θ(η) are decreasing functions of Pr. This is due to the fact that with an enhancement in Prandtl number Pr, thermal diffusivity decreases, which leads to a reduction in temperature θ(η). Figure 19 exhibits that concentration profile (η) and associated boundary layer thickness decay for larger Sc. It is due to the fact that Sc is the ratio of momentum to mass diffusivities which means that when Sc increases, mass diffusivity decreases and there is a reduction in concentration (η).
1121
transfer at the wall −θ'(0) (local Nusselt number) reduces while mass transfer at the wall −'(0) (Sherwood number) enhances. Also, it is noted that heat transfer at the wall −θ'(0) (local Nusselt number) is increasing function of γ1. However, mass transfer at the wall −'(0) (Sherwood number) is decreasing function of γ1. Table 2 Local Nusselt number Rex1 / 2 Nux and Sheerwood number Rex1 / 2 Shx for various values of λ1, β, R, M, Nt, Nb when λ=N=γ1=γ2=0.3, Pr=1.0, Sc=0.7, and f 0.6
λ1
β
R
M
Nt
Nb
−θ'(0)
−'(0)
0.1
0.2
0.2
0.2
0.2
0.2
0.21139
0.16576
0.3
0.20975
0.16270
0.5
0.20828
0.16002
0.0
0.20678
0.15736
0.2
0.20975
0.16270
0.4
0.21187
0.16678
0.0
0.21883
0.15931
0.2
0.20975
0.16270
0.5
0.19825
0.16696
0.0
0.20963
0.16244
0.2
0.20828
0.16002
0.5
0.20632
0.15663
0.0
0.21085
0.20706
0.2
0.20975
0.16270
0.4
0.20861
0.12022
0.1
0.21086
0.11836
0.2
0.20975
0.16270
0.3
0.20888
0.17762
0.3
0.3
0.5
0.2
0.2
0.2
Fig. 18 Influence of Pr on θ(η) 0.3
0.3
0.2
0.2
0.2
0.2
0.2
0.2
0.2
Table 3 Local Nusselt number Rex1 / 2 Nux and Sheerwood number Rex1 / 2 Shx for values of λ, N, γ1, γ2 when λ= 0.3, β=R=M=Nt=Nb=0.2, Pr=1.0, Sc=0.7, and f 0.6
Fig. 19 Influence of Sc on (η)
Tables 2 and 3 computed the numerical values of ratio of relaxation to retardation times λ1, Deborah number β, radiation parameter R, Hartman number M, thermophoretic parameter Nt, Brownian motion parameter Nb, mixed convection parameter λ, concentration buoyancy parameter N, heat transfer Biot number γ1 and mass transfer Biot number γ2 on the local Nusselt and Sherwood numbers. It is found that both heat transfer at the wall −θ'(0) (local Nusselt number) and mass transfer at the wall −'(0) (Sherwood number) are decreasing functions of λ1, M and Nt. However, λ, N and β show increasing behaviors of local Nusselt and Sherwood numbers. With rise in Nb, R and γ2, the heat
λ
N
γ1
γ2
−θ'(0)
−'(0)
0.0
0.3
0.3
0.3
0.20806
0.15950
0.3
0.20975
0.16270
0.5
0.21066
0.16438
0.0
0.20903
0.16136
0.3
0.20975
0.16270
0.5
0.21018
0.16356
0.1
0.087484
0.18764
0.2
0.15546
0.17366
0.3
0.20975
0.16270
0.1
0.21029
0.068073
0.2
0.20999
0.12074
0.3
0.20975
0.16270
0.3
0.3
0.3
0.3
0.3
0.3
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5 Conclusions 1) Variation of ratio of relaxation to retardation times λ1 on the velocity profile f '(η) is opposite to that of Hartman number M or Deborah number β. 2) Momentum boundary layer thickness enhances with an increase in mixed convection parameter λ and concentration buoyancy parameter N. However, thermal boundary layer and concentration boundary layer thicknesses are decreasing functions of mixed convection parameter λ and concentration buoyancy parameter N. 3) Thermophoretic parameter Nt enhances both the temperature θ(η) and concentration profiles (η). However, temperature θ(η) and thermal boundary layer thickness rise with an increase in Brownian motion parameter Nb; while reverse behavior is noted for concentration profile (η). 4) Thermal boundary layer thickness and temperature θ(η) increase with an increase in thermal radiation R and heat transfer Biot number γ1, while, these quantities reduce with an increase in Prandtl number Pr. 5) Concentration (η) and associated boundary layer thickness reduce with an increase in Schmidt number Sc. 6) Heat transfer rate at wall −θ'(0) (local Nusselt number) and mass transfer rate at wall −'(0) (Sherwood number) reduce with the increase in ratio of relaxation to retardation times λ1, Hartman number M and thermophoretic parameter Nt. However, the local Nusselt and Sherwood numbers boost up with an increase in mixed convection parameter λ, concentration buoyancy parameter N and Deborah number β.
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