Meccanica DOI 10.1007/s11012-015-0205-1

Dual solutions of heat and mass transfer of nanofluid over a stretching/shrinking sheet with thermal radiation Poulomi De . Hiranmoy Mondal . Uttam Kumar Bera

Received: 8 October 2013 / Accepted: 16 May 2015 Ó Springer Science+Business Media Dordrecht 2015

Abstract The present paper deals with the effects of nanofluids and thermal radiation over a stretching/ shrinking sheet. The governing differential equations are transformed into a set of non-linear coupled ordinary differential equations which are then solved using numerical technique with appropriate boundary conditions for various values of physical parameters. The effects of various physical parameters on the dimensionless velocity, temperature, and concentration profiles are depicted graphically and analyzed in detail. Favorable comparisons with previously published work on various special cases of the problem are obtained. The effects of various physical parameters on the local Nusselt number and local Sherwood number are also presented in the tabular form. Keywords Thermal radiation  Dual solutions  Shrinking sheet  Nanofluids  Mixed convection

P. De  U. K. Bera Department of Mathematics, National Institute of Technology, Agartala 799055, Tripura, India e-mail: [email protected] U. K. Bera e-mail: [email protected] H. Mondal (&) Department of Mathematics, Bengal Institute of Technology and Management, Santiniketan 731236, West Bengal, India e-mail: [email protected]

1 Introduction Mixed convective heat and mass transfer of a nanofluid over a stretching/shrinking sheet with thermal radiation has important industrial applications, for example, in metallurgical processes, such as drawing of continuous filaments through quiescent fluids, annealing and tinning of copper wires, glass blowing, manufacturing of plastic and rubber sheets, crystal growing, and continuous cooling and fiber spinning, in addition to wide-ranging applications in many engineering processes, such as polymer extrusion, wire drawing, continuous casting, manufacturing of foods and paper, glass fiber production, stretching of plastic films, and many others. One of the common applications of shrinking sheet problems is shrinking film, packing of bulk products. A nanofluid is a new class of heat transfer fluids that contain a base fluid and nanoparticles. Nanofluids are suspensions whose distinctive feature is an unusually small size of particles suspended in a base fluid (which can be water or an organic solvent). The nanoparticles used in nanofluids are typically made of metals ðAl; CuÞ, oxides Al2 O3 , carbides ðSiC Þ, nitrides ðAlN; SiN Þ or nonmetals (graphite, carbon nanotubes) and the base fluid is usually a conductive fluid, such as water or ethylene glycol. Other base fluids are oil and other lubricants, bio-fluids and polymer solutions. Nanoparticles range in diameter between 1 and 100 nm. Kuznetsov and Nield [1] have examined the influence of nanoparticles on natural

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convection boundary-layer flow past a vertical plate using a model in which Brownian motion and thermophoresis are accounted for. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis for the porous medium. Khan and Pop [2] studied forced convective boundary layer flow of a nanofluid past a stretching surface with the effect of Brownian diffusion and thermophoresis. Rana and Bhargava [3] analyzed the flow and heat transfer of a nanofluid over a nonlinear stretching sheet. Rahman and Eltayeb [4] studied the radiative heat transfer in a hydromagnetic nanofluid past a nonlinear stretching surface with convective boundary condition. Mahapatra et al. [5] analyzed the oblique stagnation point flow and heat transfer towards a shrinking sheet with thermal radiation. Recently, the boundary layer flow due to a shrinking sheet has attracted considerable interest. Bachok et al. [6] studied the flow and heat transfer of a nanofluid over a shrinking sheet. Das [7] analyzed the nanofluid flow over a shrinking sheet with surface slip. Rohni et al. [8] have investigated the flow over an unsteady shrinking sheet with suction in a nanofluid. Akbar et al. [9] studied the dual solutions in MHD stagnation-point flow of Prandtl fluid impinging on shrinking sheet. Rajotia and Jat [10] analyzed the dual solutions of three dimensional MHD boundary layer flow and heat transfer due to an axisymmetric shrinking sheet with viscous dissipation and heat generation/absorption. Ali et al. [11] investigated the dual solutions in MHD flow on a nonlinear porous shrinking sheet in a viscous fluid. Radiative heat transfer in which heat is transmitted from one point to another without heating the intervening medium has found very important in the design of reliable equipments, nuclear plants, gas turbines and various propulsion devices for aircraft, missiles, satellites and space vehicles. Also, the effect of thermal radiation on the forced and free convection flows are important in the content of space technology and processes involving high temperature. Based on these applications, Rosseland diffusion approximation had been utilized in this investigation of convection flow with radiation. Pal and Mondal [12] studied the influence of thermal radiation on hydromagnetic Darcy-Forchheimer mixed convection flow past a stretching sheet embedded in a porous medium. Olanrewaju et al. [13] studied the boundary layer flow of nanofluids over a moving surface in a flowing fluid

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in the presence of radiation. Ahmad and Pop [14] analyzed the mixed convection boundary layer flow from a vertical flat plate embedded in a porous medium filled with nanofluids. Bachok et al. [15] examined the boundary layer flow of nanofluids over a moving surface in a flowing fluid. Prasad et al. [16] studied the mixed convection heat transfer over a nonlinear stretching surface with variable fluid properties. In view of the above investigations, authors envisage to analyzed the effect of nanofluids and thermal radiation over a continuous nonlinear stretching/shrinking sheet are described in details. The combined effects of Brownian motion, thermophoresis and thermal radiation parameters are incorporated in the mathematical formulation of the problem. The governing equations are solved numerically using fifth-order Runge– Kutta Fehlberg method with shooting technique.

2 Mathematical formulations The steady, incompressible, two- dimensional boundary layer flow heat and mass transfer of a nanofluid past a flat sheet coinciding with the plane y = 0 and the flow being confined to y [ 0 The flow is generated, due to non-linear stretching of the sheet, caused by the simultaneous application of two equal and opposite forces along the x-axis. Keeping the origin fixed, the sheet is then stretched with a velocity uw ¼ axn where a is constant, n is a nonlinear stretching/shrinking parameter and x is the coordinate measured along the stretching surface, varying nonlinearly with the distance from the slit (see Fig. 1). The basic steady conservation of mass, momentum, thermal energy and nanoparticles equations for nanofluids can be written in cartesian coordinates x and y as (see Rana and Bhargava [3]):

v y u

Slit x

Fig. 1 Physical model and co-ordinate system

Meccanica

ou ou þ ¼ 0; ox oy u

ð1Þ

ou ou o2 u þv ¼m 2 ox oy oy

ð2Þ

"   # oT oT oC oT DT oT 2 þv ¼ am rT þ s DB þ u ox oy oy oy T1 oy 

1 oqr qcp oy

ð3Þ

  oC oC o2 C D T o2 T u þv ¼ DB 2 þ ox oy oy T1 oy2

ð4Þ

where am ¼

km ; ðqcÞf

s¼

ðqcÞp ðqcÞf

ð5Þ

subject to the boundary conditions u ¼ a xn ; v ¼ V0 xn1=2 ; C ¼ Cw at y ¼ 0 u ¼ v ¼ 0; as y ! 1

T ¼ T1 ;

T ¼ Tw ;

qr ¼ 

4 r oT 4 ; 3k oy

ð8Þ

where r is the Stefan-Boltzmann constant k is the Rosseland mean absorption coefficient. Let us introduce the wall temperature excess ratio parameter (see Ref. [17]) as hw ¼ TT1w : Thus using (8), T 4 may be expressed as 4 T 4 ¼ T1 f1 þ ðhw  1Þhg4

ð9Þ

Substituting Eq. (8) into the governing Eqs. (2)–(4) and using Eq. (9) we finally obtain a system of nonlinear ordinary differential equations with appropriate boundary conditions: 2n 02 f 000 þ f 00  f ¼0 ð10Þ nþ1  i0 1 h 1 þ Nr f1 þ ðhw  1Þhg3 h0 þ f h0 Pr þ Nbh0 /0 þ Nth02 ¼ 0 /00 þ Lef /0 þ

ð11Þ

Nt 00 h ¼0 Nb

ð12Þ

The transformed boundary conditions are

C ¼ C1 ð6Þ

where u, v are the velocity components along the axes x, y, respectively. qf is the density of the base fluid, am is the thermal diffusivity, m is the kinematic viscosity, DB is the Brownian diffusion coefficient, DT is the ðqcÞ

thermophoretic diffusion coefficient, s ¼ ðqcÞp is the f

ratio between the effective heat capacity of the nanoparticle material and heat capacity of the fluid, and c is the volumetric volume expansion coefficient, qp is the density of the particles and V0 is the porosity of the sheet. The governing Eqs. (1)–(4) subject to the boundary conditions (6) can be expressed in a similar form by introducing the following transformation: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aðn þ 1Þ n1 x 2 ; u ¼ axn f 0 ðgÞ; g¼y 2m rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     amðn þ 1Þ n1 n1 ð7Þ x2 fþ v¼ gf 0 ; 2 nþ1 T  T1 C  C1 hðgÞ ¼ ; /ðgÞ ¼ Tw  T1 Cw  C1 where g is the similarity variable. Employing the Rosseland diffusion approximation the radiation heat flux is given by

g ¼ 0; g ! 1;

f ¼ S;

f 0 ¼ 1;

h ¼ 1;

h ¼ 0;

/¼0

f 0 ¼ 0;

/¼1

ð13Þ ð14Þ

V0 ffi is the constant mass transfer paramwhere S ¼ pffiffiffiffiffiffiffiffiffi a m ðnþ1Þ 2

eter with S [ 0 for suction and S \ 0 for injection. Pr ¼ m m am is the Prandtl number, Le ¼ DB is the Lewis number, Nb ¼

ðqcÞp DB ðCw C1 Þ ðqcÞf m

is the Brownian motion parameter,

Nt ¼

ðqcÞp DT ðTw T1 Þ ðqcÞf T1 m

is the thermophoresis parameter,

3 16r T1 3k k

Nr ¼ is the thermal radiation parameter. The quantities of physical interest in this problem are the local skin-friction coefficient, the local Nusselt number, and the local Sherwood numbers, which are defined by sw xqw ; Cf ¼ 2 ; Nux ¼ quw kðTw  T1 Þ ð15Þ xqm Shx ¼ DB ðCw  C1 Þ where

     ou 16r T 3 oT ; qw ¼  þ k ; oy y¼0 oy y¼0 3k   oC ; ð16Þ qm ¼ DB oy y¼0 sw ¼ l

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Using non-dimensional variables (8) and relation (17) into (16), we obtain rffiffiffiffiffiffiffiffiffiffiffi n þ 1 00 1=2 Cf Rex ¼ f ð0Þ; 2 rffiffiffiffiffiffiffiffiffiffiffi

nþ1 0 3 ¼  1 þ Nrh ð17Þ Nux Re1=2 h ð0Þ; w x 2 rffiffiffiffiffiffiffiffiffiffiffi nþ1 0 1=2 ¼ Shx Rex / ð0Þ; 2 where Rex ¼ uwm x is the local Reynolds number.

interest that have significant effects. Comparisons of the present results (linear stretching surface in absence of thermal radiation) with previously works are performed and excellent agreement has been obtained. Comparison of our results of h0 ð0Þ and /0 ð0Þ with those obtained by Khan and Pop [2] (see Table 1) show an excellent agreement. Figure 2 shows that the effect of suction parameter on the velocity profile. It is seen from Fig. 2 that the dimensionless velocity profile f 0 ðgÞ increases with the increasing value of S for the first solution while for second solution its also increases with increasing of suction parameter S. Figure 3 observed the effect of nonlinearly stretching/ shrinking sheet parameter n on velocity distribution

3 Results and discussion 0.0

-0.2

Pr = 2.0, Nt = 0.5, Nb = 0.5 Nr= 0.5, Le = 2.0,θ w = 2.0 -0.4

f ' (η)

The Eqs. (10)–(12) constitute highly non-linear coupled boundary value problem of third order of velocity and second-order of energy and concentration equations respectively. Thus we have developed most effective numerical shooting technique with fifthorder Runge–Kutta–Fehlberg integration algorithm. The steady flow is possible over the axisymmetric stretching/shrinking sheet for various values of the parameters involved such as suction parameter S, nonlinear stretching/shrinking sheet parameter n, Prandtl number Pr, Brownian motion parameter Nb, thermophoresis parameter Nt, temperature ratio parameter hw and Lewis number Le. The results are presented graphically in Figs. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and 13 and conclusions are drawn for flow field and other physical quantities of

n = 0.5, 0.8, 1.0 -0.6

-0.8

First Solution Second Solution -1.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

η

Fig. 3 Dual velocity profile for various values of n

0.0

1.0

-0.2

0.8

First Solution Second Solution

S = 2.0, 2.2, 2.5

θ (η)

f ' (η)

θw = 1.2, 2.0, 3.0

0.6

-0.4

0.4

Pr = 2.0, Nt = 0.5, Nb = 0.5 Le = 2.0, Nr= 0.5, θ w = 2.0

-0.6

First Solution Second Solution

-0.8

n = 1.0, Pr = 2.0, Nt = 0.5, Nb = 0.5, Le = 2.0, S = 2.0 Nr = 0.5

0.2

0.0

-1.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

η

Fig. 2 Dual velocity profile for various values of S

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3.5

4.0

0

1

2

3

4

5

6

7

8

η

Fig. 4 Dual temperature profile for different values of hw

9

Meccanica 1.0

1.0

First Solution Second Solution

First Solution Second Solution

0.8

0.8

n = 1.0, Pr = 2.0, Nt = 0.5, Nb = 0.5, Le = 2.0, Nr = 0.5, S = 2.0

0.6

Nt = 0.2, 0.5, 0.8, 1.2, 2.0

θ (η)

φ (η)

0.6

0.4

0.4

n = 1.0, Nb= 0.5, Le = 2.0 Nr = 0.5, S = 2.0, Pr = 2.0 θw = 2.0

θw = 1.2, 2.0, 3.0 0.2

0.2

0.0 0.0

0 0

1

2

3

4

5

7

6

8

1

2

3

9

4

5

6

7

8

η

η

Fig. 8 Dual temperature profile for different values of Nt

Fig. 5 Dual concentration profile for various values of hw 1.0 First Solution Second Solution 0.8

1.0

First Solution Second Solution

n = 1.0, Nt = 0.5, Nb = 0.5, Le = 2.0, Nr = 0.5, S= 2.0, θw =2.0

θ (η)

0.6

0.8

0.6

Nt = 0.2, 0.5, 0.8, 1.2, 2.0

φ (η)

Pr = 0.72, 1, 2 0.4

n=1.0, Nb= 0.5 Le = 2.0, Nr = 0.5 S = 2.0, Pr = 2.0 θw = 2.0

0.4

0.2 0.2

0.0 0

1

2

3

4

5

6

7

8

η

0.0 0

1

2

3

4

5

6

7

8

η

Fig. 6 Dual temperature profile for various values of Pr Fig. 9 Dual concentration profile for different values of Nt 1.0

First Solution Second Solution

0.8

0.6

φ (η)

n = 1.0, Nt = 0.5, Nb = 0.5, Le = 2.0 Nr = 0.5, S = 2.0, θw = 2.0

0.4

Pr = 0.72, 1, 2

0.2

0.0 0

1

2

3

4

η

5

6

7

8

Fig. 7 Dual concentration profile for different values of Pr

9

f 0 ðgÞ. In Fig. 3 the dual velocity profile show that the velocity decreases with increase in stretching/shrinking sheet parameter n for the first solution and it increases with increase in n for the second solution. Figure 4 shows that the dual temperature profile hðgÞ increases for first and second solution as the temperature ratio parameter hw increases for different values of physical parameters. This result is expected because as the value of hw increases there is increase in the value of the temperature difference Tw  T1 and so the dual temperature of the fluid increases for both solutions. Figure 5 depict the dual concentration profile /ðgÞ for the effect of temperature ratio parameter hw . As the temperature ratio parameter hw

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Meccanica 1.0

1.0

First Solution Second Solution 0.8

0.9

First solution

0.8

Second solution

0.7

n = 1.0, Pr = 2.0, Nt = 0.5, Nb = 0.5 Nr = 0.5, S = 2.0, θ w= 2.0

0.6

0.6

0.4

θ (η)

θ (η)

Nb = 0.2, 0.5, 0.8, 1.2, 2.0

0.5 0.4

Nt = 0.5, n =1.0, Le = 2.0, Nr = 0.5, S = 2.0, θw = 2.0

Le = 2, 6, 10

0.3 0.2

0.2

0.1 0.0

0.0 0

1

3

2

η

4

5

6

0

7

1

2

3

η

4

5

6

7

Fig. 12 Dual temperature profile for various values of Le

Fig. 10 Dual temperature profiles for different values of Nb

1.0

1.0

First Solution Second Solution

First Solution Second Solution

φ (η)

0.6

φ (η)

Nt = 0.5, n =1.0, Le = 2.0, Nr = 0.5, S = 2.0, Pr = 2.0, θw = 2.0

0.8

Nb = 0.2, 0.5, 0.8, 1.2, 2.0

n = 1.0, Pr = 2.0, Nt = 0.5, nb = 0.5, Nr = 0.5, θ w = 2.0, S = 2.0

0.5

0.4

Le = 2, 6, 10

0.2

0.0

0.0 0

1

2

3

4

5

6

η

0

2

η

4

6

Fig. 11 Dual concentration profile for different values of Nb

Fig. 13 Dual concentration profile for various values of Le

increases solutal boundary layer thickness decreases for first and second solution for different values of physical parameters. The variations in the dual solutions of temperature and concentrations distributions for several values of Prandtl number Pr are depicted in Figs. 6 and 7. Prandtl number signifies the relative contribution of momentum diffusion to thermal diffusion in the boundary layer regime. For Pr [ 1, momentum diffusion rate exceeds thermal diffusion rate. With increase in Prandtl number Pr, thermal boundary layer thickness reduces for both solutions. In short, an increase in the Prandtl number means the slow rate of thermal diffusion. The effects of Prandtl number on a nanofluids is similar to what has already been

observed in common fluids qualitatively but they are different quantitatively. Therefore, these properties are inherited by nanofluids. Crossovers of concentration profiles are found in Fig. 7 for both solutions. When Prandtl number increases, the concentration initially increases for small g and for large g it decreases. This nature is visualized in second solution. Whereas when Prandtl number increases, the concentration initially decreases for small g and for large g it increases which can be seen in first solution. Figures 8 and 9 depicts dual profiles for temperature hðgÞ and concentration /ðgÞ for various values of thermophoretic parameter Nt. It is observed that an increase in the thermophoretic parameter Nt leads to increase the fluid temperature and concentration of

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Meccanica Table 1 Comparison of the present results of Nux and Shx with Khan and Pop [2] when Nr ¼ 0:0 and hw ¼ 1:0

Nb

Nt

Pr

Le

N

Khan and Pop [2] 0

Present results

0

h ð0Þ

/ ð0Þ

h0 ð0Þ

/0 ð0Þ

0.1

0.1

10

10

1.0

0.9524

2.1294

0.947477

2.092820

0.2

0.2

10

10

1.0

0.3654

2.5152

0.365202

2.474027

0.3

0.3

10

10

1.0

0.1355

2.6088

0.136212

2.567439

0.4 0.5

0.4 0.5

10 10

10 10

1.0 1.0

0.0495 0.0179

2.6038 2.5731

0.049905 0.018006

2.565047 2.540617

first and second solution. It is noticed that increasing the value of Nt, results in increasing temperature and the concentration differences between the stretching surface and the ambient fluid. Therefore, with increase in thermophoretic parameter, Nt, temperature distribution increase for both solutions. The effect of Brownian motion parameter Nb on dual temperature profiles hðgÞ and dual concentration profiles /ðgÞ are shown in Figs. 10 and 11. The boundary layer profiles for the temperature are of the same form as in the case of regular heat transfer fluids. The temperature in the boundary layer increases with the increase in the Brownian motion parameter Nb for both solutions. This is due to enhanced molecular activity at higher values of Nb, which increases fluid motion and transport of heat through thermal diffusion. But, the nanoparticle volume fraction profile decreases for first and second solution with the increase in the Brownian motion parameter Nb. Brownian motion serves to warm the boundary layer and simultaneously it impairs particle deposition away from the fluid regime (onto the surface), thereby accounting for the reduced concentration magnitudes in Fig. 11. The Brownian motion of nanoparticles can enhance thermal conduction via one of two mechanisms, either a direct effect owing to nanoparticles that transport heat or alternatively via an indirect contribution due to micro-convection of fluid surrounding individual nanoparticles. For small particles, Brownian motion is strong and the parameter Nb will have high values, the converse is the case for large particles and clearly Brownian motion does exert a significant enhancing influence on both temperature and concentration profiles. Figures 12 and 13 exhibits the dual temperature profiles hðgÞ and dual concentration profiles /ðgÞ for various values of Lewis number Le. As expected, the boundary layer profiles for the temperature function hðgÞ are essentially the same form as in case of a

regular fluid. It is observed from Fig. 12 that the temperature decreases for both solutions with the increase in Lewis number Le. It can be seen from Fig. 13 that as Lewis number, Le increases, the concentration decreases for both first and second solutions. The decrease in concentration profiles are accompanied by simultaneous reductions in concentration boundary layers thicken which is analogous to the effect of increasing Prandtl number on the thickness of a thermal boundary layer. Lewis number defines the ratio of thermal diffusivity to mass diffusivity. Effectively, it is also the ratio of Schmidt number and the Prandtl number. However, the thickness of the boundary layer concentration profile is found to be smaller than the thermal boundary layer thickness for Le greater than one. It is also evident from the Fig. 13 that the concentration, C takes its limiting value,C1 , for higher values of the dimensionless distance, g. From this figure, it is also observed that when the concentration difference DC is maintained constant, the dimensionless concentration profile decreases due to increase in Lewis number Le.

4 Conclusions A numerical model is developed to study the influence of thermal radiation and nanofluids on mixed convection past a nonlinear stretching/shrinking sheet. Graphical results were obtained to illustrate the details of flow, heat and mass transfer characteristics and their dependence on some physical parameter. The results in summary have shown that: 1.

The Brownian motion parameter Nb enhance temperature in the boundary layer region. Also, increasing Brownian motion Nb decreases concentration.

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2.

3. 4.

Increasing the nonlinear stretching/shrinking sheet parameter n decrease the flow velocity profile in the boundary layer. The concentration decreases with an increasing in Lewis number Le. The thermophorotic parameter Nt leads to increase the fluid temperature and concentration profiles of first and second solutions.

References 1. Kuznetsov AV, Nield DA (2010) Natural convective boundary-layer flow of a nanofluid past a vertical plate. Int J Therm Sci 49:243–247 2. Khan WA, Pop I (2010) Boundary-layer flow of a nanofluid past a stretching sheet. Int J Heat Mass Transf 53:2477– 2483 3. Rana P, Bhargava R (2012) Flow and heat transfer of a nanofluid over a nonlinearly stretching sheet: a numerical study. Commun Nonlinear Sci Numer Simul 17:212–226 4. Rahman MM, Eltayeb IA (2013) Radiative heat transfer in a hydromagnetic nanofluid past a non-linear stretching surface with convective boundary condition. Meccanica 48(3):601–615 5. Mahaptra TR, Nandi SK, Gupta AS (2012) Oblique stagnation-point flow and heat transfer towards a shrinking sheet with thermal radiation. Meccanica 47:1325–1335 6. Bachok N, Aleng NL, Arifin NM, Ishak A, Senu N (2014) Flow and heat transfer of a nanofluid over a shrinking sheet. Int J Mech Aerosp Ind Mechatron Eng 8(9):1578–1582 7. Das MK (2014) Nanofluid flow over a shrinking sheet with surface slip. Microfluid Nanofluid 16:391–401

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8. Rohni AM, Ahmad S, Ismail AIMD, Pop I (2012) Flow over an unsteady shrinking sheet with suction in a nanofluid. Int J Mod Phys 9:512–519 9. Akbar NS, Khan ZH, Haq RU, Nadeem S (2014) Dual solutions in MHD stagnation-point flow of Prandtl fluid impinging on shrinking sheet. Appl Math Mech 35(7): 813–820 10. Rajotia D, Jat RN (2014) Dual solutions of three dimensional MHD boundary layer flow and heat transfer due to an axisymmetric shrinking sheet with viscous dissipation and heat generation/absorptionand. Indian J Pure Appl Phys 52:812–820 11. Ali FM, Nazar R, Arifin NM, Pop I (2013) Dual solutions in MHD flow on a nonlinear porous shrinking sheet in a viscous fluid. Bound Value Probl 32:1–7 12. Pal D, Mondal H (2011) The influence of thermal radiation on hydromagnetic Darcy-Forchheimer mixed convection flow past a stretching sheet embedded in a porous medium. Meccanica 46:739–753 13. Olanrewaju PO, Olanrewaju MA, Adesanya AO (2012) Boundary layer flow of nanofluids over a moving surface in a flowing fluid in the presence of radiation. Int J Appl Sci Technol 2(1):274–285 14. Ahmad S, Pop I (2010) Mixed convection boundary layer flow from a vertical flat plate embedded in a porous medium filled with nanofluids. Int Commun Heat Mass Transf 37:987–991 15. Bachok N, Ishak A, Pop I (2010) Boundary layer flow of nanofluids over a moving surface in a flowing fluid. Int J Therm Sci 49:1663–1668 16. Prasad KV, Vajravelu K, Datti PS (2010) Mixed convection heat transfer over a non-linear stretching surface with variable fluid properties. Int J Non-linear Mech 45:320–330 17. Pal D, Mondal H (2011) MHD non-Darcian mixed convection heat and mass transfer over a non-linear stretching sheet with Soret–Dufour effects and chemical reaction. Int Commun Heat Mass Transf 38:463–467