Mixed convection from a wavy surface embedded in a ...

International Communications in Heat and Mass Transfer 77 (2016) 78–86

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Mixed convection from a wavy surface embedded in a thermally stratified nanofluid saturated porous medium with non-linear Boussinesq approximation夽 Peri K. Kameswaran a , B. Vasu b , P.V.S.N. Murthy c , Rama Subba Reddy Gorla d, * a

Department of Mathematics, School of Advanced Sciences, VIT University, Vellore 632014, India Department of Mathematics, Motilal Nehru National Institute of Technology, Allahabad 211004, India c Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, India d Department of Mechanical Engineering, Cleveland State University, Cleveland, OH 44115, USA b

A R T I C L E

I N F O

Available online xxxx Keywords: Nanofluid flow Heat transfer Non-linear Boussinesq Thermal stratification Convective boundary conditions

A B S T R A C T In this article, we have made an attempt to study the convective heat transfer in the influence of nonlinear Boussinesq approximation, thermal stratification and convective boundary conditions on non-Darcy nanofluid flow over a vertical wavy surface. The surface of the vertical wavy plate, put up at convective temperature and concentration. A coordinate transformation is used to transform the wavy surface into smooth surface. Using local similarity and non-similarity method the governing nondimensional equations are transformed into coupled nonlinear ordinary differential equations. Effects of thermal convective parameter, thermal stratification parameter, Lewis number, Sherwood number on the wave geometry on the heat and mass transfer characteristics have been studied. Numerical results have been obtained for various physical parameters. A comparison of the present results is made with the earlier published results and is found to be in good agreement. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction The mixed convection heat transfer is a common phenomenon in both environmental processes and engineering systems, particularly in production and manufacturing processes. Extrusion of plastic sheets, cooling of electrical components, paper production, glass blowing, metal spinning and drawing plastic films, are some of practical applications of such a system. A comprehensive review of buoyancy induced flows is given in the books by Gebhart et al. [1], Schlichting and Gersten [2] and Pop and Ingham [3]. Convective heat transfer in a saturated porous medium occurs in nature, science and engineering which are important in theoretical as well as experimental point of view and has been an area of intensive study for the last several decades. A wide applications of porous media in many practical applications and the achievements as an excellent review can be found in the well-known books by Nield and Bejan [4] and Vafai [5]. In the fast growing technology, many technically advanced industries like microelectronics, manufacturing, and transportation are

夽 Communicated by W.J. Minkowycz. * Corresponding author. E-mail address: [email protected] (R. Gorla).

http://dx.doi.org/10.1016/j.icheatmasstransfer.2016.07.006 0735-1933/© 2016 Elsevier Ltd. All rights reserved.

facing a problem of limiting cooling efficiency of heat transfer of conventional fluids such as water, ethylene glycol, lubricants and oil. The efficiencies of thermal devices and systems are related to heat transfer rates which in turn depends on the thermal conductivity of the working fluids. Despite considerable previous research and development efforts on heat transfer enhancement, the demand is growing for more efficient and robust heat transfer fluids with significantly higher thermal conductivities than traditional ones. Hence, the enhancement of heat transfer of conventional fluids can be improved by taking dilute suspension of nanoparticles into them. The suspension of nanoparticles is called “Nanofluid” is a moderately new class of fluids which consist of a base fluid with nano-sized particles (1–100 nm) suspended within them, which is first termed by Choi [6]. Nanofluids are commonly have two mathematical models for describing the convective transport of nanofluids; one is single component (homogeneous) model reported by Das et al. [7], where the thermophysical properties of base fluid are enhanced and modified with the nanoparticle influenced correlations of viscosity and thermal conductivity and another is two component (non-homogeneous) model, described by Buongiorno [8], which consists of seven slip mechanisms: inertia, Brownian diffusion, thermophoresis, diffusiophoresis, Magnus effect, fluid drainage, and gravity. These can produce a relative velocity between the

P. Kameswaran, et al. / International Communications in Heat and Mass Transfer 77 (2016) 78–86

nanoparticles and the base fluid. Of all of these mechanisms, only Brownian diffusion and thermophoresis were found to be important for nanoparticle transport mechanism, this hold well for nanoparticles of any material and size. Also, he has stated that nanofluids have higher thermal conductivity and single-phase heat transfer coefficients than their base fluids. Some of the recent investigations show the transport process in nanofluid with different physical occurring situations by considering single as well as two component models Khan and Pop [9], Kuznetsov and Nield [10], Murthy et al. [11], Gorla and Hossain [12], Rashad et al. [13], Parvin and Chamkha [14], Beg et al. [15]. An extension to the case of a nanofluid, based on a twocomponent model presented by Buongiorno [8] has been presented by Kuznetsov and Nield [16], which fully accounted for the effect of distribution of the nanoparticle volume fraction at the boundary. Series of articles by Kuznetsov and Nield [17], Sheremet et al. [18], Moshizi et al. [19] presents the revised Buongiorno model for the convective heat transfer in nanofluid. All these studies have assumed that there is no nanoparticle flux at the boundary and that the particle fraction value there adjusts accordingly. Thus it is advisable to replace their boundary conditions by a set that are more realistic physically. Recently, Dhanai et al. [20], have studied numerical a mixed convection slip flow and heat transfer of uniformly conducting nanofluid past an inclined cylinder under the influence of Brownian motion, thermophoresis and viscous dissipation. Saeed et al. [21] have been developed the transient MHD mixed convection stagnation-point flow and heat transfer of an electrically conducting nanofluid over a vertical permeable stretching/shrinking sheet by means of Tiwari–Das nanofluid model. Faroogh et al. [22] has been reported a numerical study of mixed convection heat transfer of nanofluid (Al2 O3 −−water) in a lid driven cavity flow is carried out by using a Buongiorno model. The study of heat and mass transfer near irregular surfaces is of fundamental importance because it is often found in many industrial applications. The presence of irregular surface not only alters the flow field but also alters the heat and mass transfer characteristics. Natural or mixed convection along a roughened surface occurs often in problems involving the enhancement of heat transfer. These irregularities come across in several heat transfer devices such as microelectronic devices, flat plate solar collectors and flat plate condensers in refrigerators. Moulic and Yao [23] studied mixed convection along a wavy surface. Cheng [24] reported a study of the phenomenon of natural convection boundary layer flow near an inclined wavy surface in a fluid saturated porous medium with Soret and Dufour effects using boundary layer approximations. Combined effects of heat and mass transfer were studied numerically by Hossain and Rees [25] over a semi-infinite vertical wavy surface. Siddiqa et al. [26] analyzed the natural convection flow induced by horizontal triangular wavy surface embedded in a viscous fluid. In this analysis, consideration was given to explore the impact of amplitude on the flow structure and heat transfer and established it increases heat transfer rate. Rathish Kumar and Shalini [27] investigated the natural convection heat transfer from a vertical wavy surface in a thermally stratified fluid saturated porous medium under Forchheimer based non-Darcian assumptions. Kameswaran et al. [28] studied the effects of nonlinear temperature, concentration and thermophoresis on steady mixed convection boundary layer flow over a vertical impermeable wall in a non-Darcy porous medium. Mahdy and Ahmed [29] has been examined laminar free convective of a nanofluid along a vertical wavy surface saturated porous medium. Srinivasacharya and Vijay Kumar [30] have reported the mixed convection in a nanofluid along an inclined wavy surface embedded in a porous medium. Rashidi et al. [31] analyzed the mixed convection heat transfer of nano-fluid flow in vertical channel with sinusoidal walls under magnetic field effect. The Boussinesq approximation used for calculated the density variations. Transient natural convection in a porous wavy-walled cavity filled with a nanofluid has been

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studied numerically by Sheremet et al. [32] over vertical flat and horizontal wavy walls with constant temperatures. The vast majority of investigations have only considered the problems of free or mixed convection of nanofluid over the flat surfaces, wavy walls and been confined to the basic Buongiorno’s model. Clearly much remains to be explored in this topic, in the way of revised Buongiorno’s model for nanofluid transport and thermal stratification in a non-linear mixed convection over a sinusoidal wall saturated in a non-Darcian porous media. The objective of the present work is therefore to investigate the effect of thermally stratification on nonlinear convection of nanofluid past a vertical wavy surface with convective boundary conditions. The model, which includes the effects of Brownian motion and thermophoresis, is revised so that the nanofluid particle fraction on the boundary is passively rather than actively controlled. In this respect the model is more realistic physically than that employed by previous authors. A coordinate transformation is used to transform the wavy surface into smooth surface. Using local similarity and non-similarity method the governing nondimensional equations are transforming into coupled ordinary non-linear differential equations. The influence of pertinent parameters on the flow characteristics are examined and presented through graphs and tables. 2. Mathematical analysis We consider a non-linear Boussinesq approximation on the boundary layer flow near a vertical wavy surface in a porous medium saturated with nanofluid in a non-Darcy porous medium. The fluid flow is considered to be steady and two-dimensional. Further we consider a cartesian coordinate system (x, y), where x and y are coordinates measured along the wavy surface and normal  to it,  respectively. The wavy surface profile is given by s(x) = a¯ sin pxl , where a¯ is the amplitude of the wavy surface and 2l is the characteristic length of the wavy surface. At the wall surface, the fluid is heated via convection from a hot fluid temperature Tf and heat transfer coefficient h. It is assumed that nanoparticle flux is zero there. Sufficiently for away from the wall, porous medium temperature is considered as T∞,x . The ambient values are denoted by T∞,0 and C∞ respectively. Using the scale analysis and following Buongiorno [8] and Kuznetsov and Nield [16, 17]. The governing equations for steady laminar flow heat and mass transfer can be written as follows:

∂ u¯ ∂v + = 0, ∂x ∂y

u¯ +

(1)

√ (1 − C∞ )Kgqf∞  Cf K 2 K ∂p b0 (T − T∞,0 ) u¯ = − − l /qf∞ l ∂x l  (qp − q )Kg f∞ (C − C∞ ), + b1 (T − T∞,0 )2 − l

√ Cf K 2 K ∂p v+ v =− l /qf∞ l ∂y 

u¯

(3)







∂T ∂T ∂ 2T ∂ 2T ∂C ∂T ∂C ∂T + t DB + +v =a + ∂x ∂y ∂x ∂x ∂y ∂y ∂ x2 ∂ y2    ∂T 2 ∂T 2 DT + + T∞,0 ∂x ∂y 

u¯

(2)







(4)



DT ∂C ∂C ∂ 2C ∂ 2C ∂ 2T ∂ 2T + + + . +v = DB 2 2 T ∂x ∂y ∞,0 ∂x ∂y ∂ x2 ∂ y2

(5)

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a=

e(qc)p k and t = . (qc)f (qc)f

The boundary conditions for Eqs. (1)–(5) are given in the following form: v = 0, −K

∂T ∂ C DT ∂ T = hf [Tf − T(x, 0)] , DB + = 0 at ∂y ∂ y T∞ ∂ y

px y = s(x) = a¯ sin u¯ → 0, l 

T → T∞,x

C → C∞

where the non-dimensional constants in Eqs. (9)–(11) are the Grashof number Gr ∗ , modified Rayleigh number Ra, thermal convective parameter a 1 , thermal stratification parameter S, buoyancy ratio parameter Nr, coefficient of Brownian motion parameter Nb, thermophoresis parameter Nt, the Lewis number Le. These are defined as

as y → ∞ (6)

¯ v) are the volume averaged velocity components along the where (u, (x, y) axes, Cf is the Forchheimer empirical constant, K is the permeability constant, l is the viscosity of nanofluid, qf∞ is the density of the base fluid, p is the average pressure, g is the acceleration due to gravity, b0 and b1 volumetric thermal expansion coefficients, T and C are the volume averaged temperature and concentration, respectively. a is the thermal diffusivity of the base fluid, t is ratio of the effective heat capacity of the nanoparticle material and the heat capacity of the fluid, DB is the Brownian diffusion coefficient, DT is the thermophoretic diffusion coefficient, k is the thermal conductivity, (qc)f is the heat capacity of the base fluid and (qc)p is the effective heat capacity of the nanoparticle material, Under these assumptions, eliminating the pressure gradient, invoking the Boussinesq approximations, Eqs. (2) and (3) become √

Cf K ∂ ∂ u¯ ∂ v ∂ 2 ¯ 2− − + (u) (v) ∂ y ∂ x l /qf∞ ∂ y ∂x (1 − C∞ )Kgqf∞ ∂T [b0 + 2b1 (T − T∞,0 )] l ∂y   qp − qf∞ Kg ∂ C − l ∂y

Nb =

b0

(7)

∂x ∂x and v = − ∂y ∂x

,

tDB (Cw − C∞ ) , a

K

,

Ra =

(1 − C∞ )Kgqf∞ lb0 (Tf − T∞,0 ) la

,

S=

Nt =

tDT (Tf − T∞,0 ) , T∞,0 a

Le =

a . DB

The boundary conditions for equations are given by hf l ∂h ∂0 ∂h = + Nt = 0 on [h − 1 + Sx] , Nb k ∂y ∂y ∂y ∂x → 0, h → 0, 0 → 0 as y → ∞ (12) y = s(x) = a sin(px) ∂y

x = 0,

The effect of the wavy surface can be transformed from the boundary conditions into the governing equations by means of the coordinate transformation given by x˜ = x,

We use the boundary layer approximation and introduce the stream function x(x, y) such that u¯ =

2b1 (Tf − T∞,0 )

√

l dT∞,x , Tf − T∞,0 dx (qp − qf∞ )(Cw − C∞ ) , Nr = (1 − C∞ )b0 qf∞ (Tf − T∞,0 ) a1 =

=

(1 − C∞ )Kgb0 (Tf − T∞,0 )Cf  2 l /qf∞

Gr∗ =

y˜ =

√

Ra [y − s(x)] ,

x x˜ = √ Ra

(13)

Substituting Eq. (13) into Eqs. (9)–(11) and letting Ra → ∞, we obtain the following boundary layer equations:   ∂ 2 x˜   ∂ x˜ ∂ 2 x˜  ∂h  + 2Gr ∗ 1 + sx˜3 = 1 + a1 (h + S x˜ ) 1 + sx˜2 2 2 ˜ ˜ ˜ ∂y ∂y ∂ y˜ ∂y ∂0 − Nr , (14) ∂ y˜

We, then introduce the nondimensional variables x= T∞,x

x a¯ , x= , l a C − C∞ = T∞,0 + Ax 0 = , C w − C∞ x , l

y=

y , l

a=

s=

s , l

h=

T − T∞,x , Tf − T∞,0 (8)

the Eqs. (4)–(7) reduces into the following form: 



2



∂ 2 x ∂ 2 x Gr∗ ∂ ∂ x ∂ ∂x + + − Ra ∂ y ∂ y ∂x ∂x ∂ x2 ∂ y2

 ∂h  ∂0 − Nr , = Ra 1 + a1 (h + Sx) ∂y ∂y

 ∂ 2h    ∂ h 2    ∂ x˜ 2 ∂h ∂0 2 − S + Nb 1 + s + Nt 1 + s 1 + sx˜2 ˜ ˜ x x ∂ y˜ ∂ y˜ ∂ y˜ ∂ y˜ ∂ y˜ 2 ˜ ˜ ∂x ∂h ∂x ∂h = − , (15) ∂ y˜ ∂ x˜ ∂ x˜ ∂ y˜    ∂ 20  2  x ∂0 Nt  ∂ x ∂ 0 ∂ 2 ∂ h − + = Le . 1 + s 1 + sx2  x Nb y ∂ x x ∂ y ∂ ∂ y2 y2 ∂ ∂

2 

(9)

(16)

The corresponding boundary conditions are

∂h ∂0 ∂h = Bi [h − 1 + S x] , N b + Nt = 0 on ∂ ∂ ∂ y y y x ∂  ( → 0, h → 0, 0 → 0 as  y→∞ y=s x) = a sin(p x) y ∂  x = 0,







∂ 2h ∂ 2h ∂h ∂0 ∂h ∂0 + + Nb +S + + Nt ∂x ∂x ∂y ∂y ∂ x2 ∂ y2

∂x ∂h ∂x ∂h = +S − , ∂y ∂x ∂x ∂y 







∂h +S ∂x



2 +

∂h ∂y

2 

(17)

(10)

where

∂ 2 0 ∂ 2 0 Nt ∂ 2 h ∂ 2 h ∂x ∂0 ∂x ∂0 = Le + + + . − Nb ∂ x2 ∂y ∂x ∂x ∂y ∂ x2 ∂ y2 ∂ y2

(11)

hf l Bi = √ , −1 k Ran 2

sx˜ =

dsx d x˜

P. Kameswaran, et al. / International Communications in Heat and Mass Transfer 77 (2016) 78–86 Table 1 Comparison of

Nux 1

1

Nr −4 −4 −4 −4 −1 −2 −3

Shx

and

Ra 2 n 2

1

1

values with Cheng [33] for Gr∗ = a = a 1 = S = n = Nb = Nt = 0 and Bi → ∞.

Ra 2 n 2 Nux

Le

1

1 4 10 100 4 4 4

1

[Present]

Shx 1

1

Nux

[Present]

1

y˜

g= 

1+

sn2

 n

1

Shx

[Exact]

1

1

[Exact]

Nux 1

1

[33]

Shx 1

1

Ra 2 n 2

Ra 2 n 2

Ra 2 n 2

Ra 2 n 2

Ra 2 n 2

Ra 2 n 2

0.9923 0.7976 0.6810 0.5207 0.5585 0.6495 0.7277

0.9923 2.0549 3.2897 10.5206 1.3575 1.6244 1.8524

0.992 0.798 0.681 0.521 0.559 0.650 0.728

0.992 2.055 3.290 10.521 1.358 1.624 1.852

0.9923 0.7976 0.6811 0.5209 0.5585 0.6494 0.7278

0.9923 2.0550 3.2899 10.521 1.3575 1.6243 1.8525

We may reduce Eqs. (14)–(16) to a form more convenient for numerical solution by the transformation, n = x˜ ,

81

1 2

,

1 x˜ = n 2 f (n, g)

(18)

[33]

Substituting Eq. (18) into Eqs. (14)–(16), we obtain the following equations:

h +

(1 + sn3 )

  f  f  = 1 + a1 (h + Sn) h − Nr 0 ,

(19)

1  ∂h ∂f − h , f h + Nbh 0 + Nth2 − Sn f  = n f  2 ∂n ∂n

(20)

f  + 2Gr∗

(1 + sn2 )2

Fig. 1. Variation of velocity profile with similarity variable g for different Nr, when Gr∗ = 0.1, a = 0.2, a 1 = 1.5, S = 0.05, Nb = Nt = 0.2, Le = 2, and Bi = 0.1.

Fig. 3. Variation of concentration profile with similarity variable g for different Nr, when Gr∗ = 0.1, a = 0.2, a 1 = 1.5, S = 0.05, Nb = Nt = 0.2, Le = 2, and Bi = 0.1.

Fig. 2. Variation of temperature profile with similarity variable g for different Nr, when Gr∗ = 0.1, a = 0.2, a 1 = 1.5, S = 0.05, Nb = Nt = 0.2, Le = 2, and Bi = 0.1.

Fig. 4. Variation of velocity profile with similarity variable g for different Nt, Nb, when Gr∗ = 0.1, a = 0.3, a 1 = 1, S = 0.1, Nr = 0.1, Le = 1, and Bi = 0.1.

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P. Kameswaran, et al. / International Communications in Heat and Mass Transfer 77 (2016) 78–86

Fig. 5. Variation of temperature profile with similarity variable g for different Nt, Nb, when Gr∗ = 0.1, a = 0.3, a 1 = 1, S = 0.1, Nr = 0.1, Le = 1, and Bi = 0.1.

0 +

Fig. 7. Variation of velocity profile with similarity variable g for different a 1 , when Gr∗ = 0.1, a = 0.3, S = 0.1, Nr = 0.2, Bi = 0.1, Le = 1, and Nb = Nt = 0.2





1 Nt  ∂0 ∂f − 0 Le f 0 + h = Len f  . 2 Nb ∂n ∂n

(21)

Subject to the boundary conditions are f = 0, 

f → 0,

  h = Bi(1 + sn2 ) h − 1 + Sn , h → 0,

0 → 0 as

Nb0 + Nth = 0

g→∞

at

g=0 (22)

where k is the effective thermal conductivity of the porous medium, nˆ is the unit normal vector to the surface. Using Eq. (23) in Eq. (24) the dimensionless Nusselt number can be represented as below: Nux 1 2

Ra n

3. Heat and mass transfer coefficients The heat transfer rate at the surface flux at the wavy surface is given by   ˆ qw = −k. n∇T , y=0

(23)

The local Nusselt number is defined as Nux =

xqw .  k Tf − T∞,0

1 2

=−

h (n, 0) 1

(1 + sn2 ) 2

(25)

The mass transfer rate at the surface flux at the wavy surface is given by   ˆ qm = −D. n∇C , y=0

(26)

(24)

Fig. 6. Variation of concentration profile with similarity variable g for different Nt, Nb, when Gr∗ = 0.1, a = 0.3, a 1 = 1, S = 0.1, Nr = 0.1, Le = 1, and Bi = 0.1.

Fig. 8. Variation of temperature profile with similarity variable g for different a 1 , when Gr∗ = 0.1, a = 0.3, S = 0.1, Nr = 0.2, Bi = 0.1, Le = 1, and Nb = Nt = 0.2.

P. Kameswaran, et al. / International Communications in Heat and Mass Transfer 77 (2016) 78–86

83

Fig. 11. Effect of heat transfer profile with increasing values of Nt, for different values of a, when Gr∗ = 0.1, a 1 = 0.5, S = 0.05, Nr = 0.2, Bi = 0.1, Le = 2, and Nb = 0.3.

Using Eq. (26) in Eq. (27) the dimensionless Sherwood number can be represented as below: Fig. 9. Variation of temperature profile with similarity variable g for different n, when Gr∗ = S = 0.1, a = 0.3, a 1 = 1.5, Nr = 0.2, Bi = 0.1, Le = 1, and Nb = Nt = 0.2.

Shx 1 2

Ra n

1 2

=−

0 (n, 0) 1

(1 + sn2 ) 2

(28)

4. Results and discussion The set of nonlinear differential Eqs. (19)–(21) along with the boundary conditions (Eq. (22)) have been solved numerically by using Matlab bvp4c solver. In order to check the accuracy of the present results for the Nusselt number and Sherwood number are compared with those obtained by Cheng [33] in particular case is presented in Table 1.

Fig. 10. Variation of concentration profile with similarity variable g for different n, when Gr∗ = S = 0.1, a = 0.3, a 1 = 1.5, Nr = 0.2, Bi = 0.1, Le = 1, and Nb = Nt = 0.2.

The local Sherwood number is defined as Shx =

xqm . D (Cw − C ∞ )

(27)

Fig. 12. Effect of mass transfer profile with increasing values of Nt, for different values of a, when Gr∗ = 0.1, a 1 = 0.5, S = 0.05, Nr = 0.2, Bi = 0.1, Le = 2, and Nb = 0.3.

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P. Kameswaran, et al. / International Communications in Heat and Mass Transfer 77 (2016) 78–86

Fig. 13. Effect of heat transfer profile with increasing values of Nt, for different values of Nb and a, when Gr∗ = 0.1, a 1 = 0.5, S = 0.05, Nr = 0.2, Bi = 0.1, and Le = 2.

It is inspected from the table that current results are good in agreement with the results obtained by Cheng [33], furthermore, it can be recorded that for a fixed value of buoyancy ratio, increase in Lewis number Nusselt number increases and the Sherwood number decreases. The effects of discrete physical and fluid criterions are shown in Figs. 1–18. Figs. 1–3 emphasize the velocity, temperature and nanoparticle volume fraction fields, respectively, for different values of Nr. It can be explored from Fig. 1 that velocity profile increases near the boundary and decreases far from the boundary with an increase in the values of Nr. It can also be noticed that increase in Nr reduces the temperature profile and increases the concentration. Variations of velocity, temperature and nanoparticle volume fraction for varying values of Brownian motion (Nb) and thermophoresis parameters (Nt) are shown in Figs. 4–6. It can be observed that with an increase in Brownian motion parameter have the tendency to

Fig. 14. Effect of mass transfer profile with increasing values of Nt, for different values of Nb and a, when Gr∗ = 0.1, a 1 = 0.5, S = 0.05, Nr = 0.2, Bi = 0.1, and Le = 2.

Fig. 15. Effect of heat transfer profile with increasing values of Nt, for different values of a 1 , when Gr∗ = 0.1, a = 0.2, Nb = 0.3, S = 0.05, Nr = 0.2, Bi = 0.1, and Le = 2.

decrease the fluid velocity and increases the temperature and concentration profiles. However, the temperature and nanoparticle volume fraction decreases with an increase in thermophoresis parameters. This phenomena may be explained such a way that Brownian motion of nanoparticles can enhance thermal conduction via direct effect of nanoparticles that transport heat or micro-convection of fluid surrounding individual nanoparticles. The Brownian motion and thermophoresis parameters in this present study are small. The values of Nb are selected here 0.1, 0.3, 0.5. The thermophoresis parameter Nt values have been selected to simulate realistic applications in heat exchangers, chemical engineering, porous media flows, etc. The values of Nt are choose as 0.1, 0.3. The values follow the elaborate study of Buongiorno [8]. The effect of nonlinear temperature on non-dimensional velocity and temperature profiles are shown in Figs. 7 and 8. We observe from Figs. 7 and 8 that the tangential velocity profile increases and temperature profile decrease with an increase in values of a 1 . It may

Fig. 16. Effect of mass transfer profile with increasing values of Nt, for different values of a 1 , when Gr∗ = 0.1, a = 0.2, Nb = 0.3, S = 0.05, Nr = 0.2, Bi = 0.1, and Le = 2.

P. Kameswaran, et al. / International Communications in Heat and Mass Transfer 77 (2016) 78–86

Fig. 17. Effect of heat transfer profile with increasing values of Nt, for different values of S, when Gr∗ = 0.1, a = 0.2, Nb = 0.3, a 1 = 1, Nr = 0.2, Bi = 0.1, and Le = 2.

be physically explained due to the inertial and Nanoparticle buoyancy forces. These results are similar to the results in literature by Kameswaran et al. [28]. The temperature and concentration profiles for various values of n are shown in Figs. 9 and 10. It is clear from the figures that temperature and concentration profiles decreases with an increase in values of n. The variation of the average Nusselt number and average Sherwood number for flat surface (a = 0) and wavy surface (a = 0.1, 0.2) as an increasing function of thermophoresis parameter are depicted in Figs. 11 and 12. The results show that, in general local Nusselt number follow the geometry of the wave wall by taking the behavior inside the boundary layer region. More over the Nusselt number first increases sharply and then gradually decreases. Comparing the curves (a = 0) and (a = 0.1, 0.2), We can conclude that increasing the amplitude–wavelength ratio tends to increase the average Nusselt number. In other words we can express this result as a sinusoidal wavy surface which is constantly smaller than that of the corresponding flat plate. Mass transfer rate decreases with an increase in amplitude–wavelength parameter. Figs. 13 and 14 plot the local Nusselt number and local Sherwood number as a function of thermophoresis parameter for different values of Brownian motion and amplitude parameters. It is observed that for fixed values of Brownian motion Nusselt number increase with an increase in the amplitude ratio parameter. Increase in Brownian motion parameter heat transfer rate enhances. The opposite results observed in the mass transfer case without the wavy nature. Figs. 15–18 display the local Nusselt number and local Sherwood number along wavy surface in the presence of non-linear temperature and thermal stratification. We found that increase in non-linear temperature leads to increase in Heat transfer rate. The opposite phenomena is true in case mass transfer. But this decrease in heat transfer is less with an increase in non-linear temperature. Figs. 17 and 18 show that streamwise distribution of dimensionless heat transfer rate and mass transfers. It should be noted that the increase in thermal stratification parameter leads to decrease the heat transfer rates. This is because, higher thermal stratification parameter leads lower buoyancy force driven by thermal and solutal gradients, increasing the thermal and concentration boundary layer thicknesses leads to decrease the heat and mass transfer rates between the fluid and wall.

85

Fig. 18. Effect of mass transfer profile with increasing values of Nt, for different values of S, when Gr∗ = 0.1, a = 0.2, Nb = 0.3, a 1 = 1, Nr = 0.2, Bi = 0.1, Le = 2.

5. Conclusions In this article, we have presented a steady, two-dimensional boundary layer nanofluid flow over a vertical wavy surface saturated non-Darcy porous medium in the presence of non-linear Boussinesq approximation and thermal stratification with convective boundary conditions. Using local similarity and non-similarity method the governing non-dimensional equations are transformed into coupled non-linear equations and solved numerically. It is found that with an increase in nonlinear temperature have the trend to hike the fluid velocity. Increase in amplitude–wavelength ratio tends to increase the average Nusselt number. As the thermal stratification parameter increases, the thermal boundary layer thickness increases, then heat transfer rate decreases.

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