absorption aspects in MHD nonlinear convective flow of third grade nanofluid over a nonlinear stretching sheet with variable thickness

Accepted Manuscript Chemical reaction and heat generation/absorption aspects in MHD nonlinear convective flow of third grade nanofluid over a nonlinear stretching sheet with variable thickness Sajid Qayyum, Tasawar Hayat, Ahmed Alsaedi PII: DOI: Reference:

S2211-3797(17)30932-4 http://dx.doi.org/10.1016/j.rinp.2017.07.043 RINP 816

To appear in:

Results in Physics

Please cite this article as: Qayyum, S., Hayat, T., Alsaedi, A., Chemical reaction and heat generation/absorption aspects in MHD nonlinear convective flow of third grade nanofluid over a nonlinear stretching sheet with variable thickness, Results in Physics (2017), doi: http://dx.doi.org/10.1016/j.rinp.2017.07.043

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Chemical reaction and heat generation/absorption aspects in MHD nonlinear convective flow of third grade nanofluid over a nonlinear stretching sheet with variable thickness Sajid Qayyum a,1 > Tasawar Hayat a,b and Ahmed Alsaedib d

Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan e

Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, P. O. Box 80207, Jeddah 21589, Saudi Arabia

Abstract:

Nonlinear thermal radiation and chemical reaction in magnetohydrodynamic

(MHD) flow of third grade nanofluid over a stretching sheet with variable thickness are addressed. Heat generation/absorption and nonlinear convection are considered. The sheet moves with nonlinear velocity. Sheet is convectively heated. In addition zero mass flux condition for nanoparticle concentration is imposed. Results for velocity, temperature, concentration, skin friction and local Nusselt number are presented and examined. It is found that velocity and boundary layer thickness are increasing for Reynolds number. Temperature is a increasing function of the heat generation/absorption parameter while it causes a decrease in the heat transfer rate. Moreover eect of Brownian motion and chemical reaction on the concentration are quite reverse.

Keywords: Third grade nanofluid; magnetohydrodynamic (MHD); nonlinear convection; nonlinear thermal radiation; variable thickness sheet; heat generation/absorption; chemical reaction. 1

Corresponding author. Tel.: +92 51 90642172.

email address: [email protected] (Sajid Qayyum)

1

1

Introduction

The investigation of transport problems of non-Newtonian fluids associated with energy conversion passing a stationary/moving, linearly/non-linearly moving surface is a compatible system in numerous industrial procedures including manufacture, glass fiber and paper production, drawing of plastics and rubber sheets, polymer and metal extrusion mechanisms, cooling of metallic surfaces and crystal growth (all of which utilize unnecessary energy input). It is important to cool the extrusion stretching surface when the finishing procedure at high temperature occurs. In addition, the fluids have been processed utilizing an assortment of supplementary impacts (i.e. magnetic force, thermal/mass buoyancy and mass diusion) for the system, and eectively such frameworks constitute a conjugate energy innovation system which for optimization requires both theoretical and experimental analysis. The rate of cooling/heating can be instrumental in determining the constitution of constructed materials in which a persuasive surface rises up from a slit and consequently a boundary layer flow adjacent to the sheet is generated along the movement of sheet. Sakiadis [1] first examined the surface flow past a continuous solid sheet. Thereafter Crane [2] investigated the two-dimensional boundary layer flow of a viscous liquid over a stretching surface. As pointed out by Wang [3], there have been various numerical and analytical studies imparted on stretching/shrinking sheet flows. In this context we further cite Van Gorder et al. [4], Pantokratoras [5], Hayat et al. [6] and Gireesha et al. [7]. Hayat et al. [8] further studied the melting phenomenon in stagnation point flow of Carbon nanotubes over a nonlinear stretching sheet with variable thickness. Numerical study for three-dimensional flow of nanofluid over a non-linearly stretching sheet is presented by Khan et al. [9]. Abbas et al. [10] considered stagnation point flow of Casson liquid towards a shrinking/stretching surface with chemical reaction and thermal radiation. Further the magnetohydrodynamics (MHD) are important in blood 2

pump machine, nuclear fuels treatment and cancer tumer treatment. In modern electromagnetic materials processing, phenomena of MHD transport are exploited frequently in flows from continuously moving, shrinking/stretching, cooled/heated sheets in a moving/quiescent free stream (Bataller [11]). Magnetohydrodynamic flow of nanofluid towards a bidirectional non-linear stretching surface with prescribed surface heat flux is described by Mahanthesh et al. [12]. Nonlinear radiative flow of nanoliquid with magnetic field and velocity slip is studied by Hayat et al. [13]. Hayat et al. [14] considered radiative flow of Maxwell nanofluid over a stretching surface with heat generation/absorption and magnetic field. MHD flow of viscoelastic fluid in view of Cattaneo-Christov flux is studied by Li et al. [15]. Ellahi et al. [16] computed numerical solution for magnetohydrodynamic (MHD) generalized Couette flow of Eyring-Powell liquid. Hayat et al. [17] discussed MHD stratified flow of viscous liquid between coaxial rotating stretchable disks. Rashidi et al. [18] described entropy analysis in MHD convective flow of third grade liquid bounded by a stretching sheet. The heat transfer mechanism via thermal radiation is important when the dierence between the surface and ambient temperatures is adequate when the operating temperature is significantly high. Radiation has significant role in controlling heat/mass as well as momentum transfer. It has considerable impact on the final product during manufacturing. High temperature plasmas, hypersonic, cooling of nuclear reactors, glass production, fights, space vehicles, gas turbines, nuclear power plants, gas cooled nuclear reactors are some fundamental uses of radiative heat transfer from a sheet to conductive liquids. Impact of radiation heat transfer in flow of both Newtonian and non-Newtonian liquids over linearly/non-linearly stretching/shrinking surfaces has attracted the attention of researchers extensively. Important studies in this direction include Cortel [19], Mahanthesh et al. [20], Hakeem et al. [21] and Hayat et al. [22]. Previous investigators applied a linear Rosseland diusion approxima3

tion for radiation which has limited accuracy when the temperature dierence between the surface and surrounding is very large. Recently few researchers have studied the flow with nonlinear radiation [23-25]. Uddin et al. [26] studied MHD flow bounded by a non-linearly stretching surface with radiation. Brownian motion and thermophoresis in manetohydrodynamic (MHD) bioconvection flow of nanoliquid via nonlinear thermal radiation is addressed by Makinde and Animasaun [27]. Shehzad et al. [28] investigated the thermophoresis and Brownian motion in three-dimensional flow of Jerey nanofluid with non-linear thermal radiation. Moreover, analysis of the heat transfer with convective boundary condition on the stretched flow is very important due to its practical applications. In particular, it is used in thermal energy storage, nuclear plants, gas turbines etc. Aziz [29] initially developed similarity solution for laminar thermal boundary layer towards a flat surface with convective boundary condition. Similarity solutions for flow and heat transfer towards a permeable surface with convective boundary condition is reported by Ishak [30]. Stagnation point flow of tangent hyperbolic fluid with convective conditions is deliberated by Hayat et al. [31]. Mahanthesh et al. [32] analyzed the three-dimensional radiative flow of water based magneto nanofluid towards a non-linearly stretching sheet with convective boundary condition. Non-Newtonian materials have been extensively acknowledged by the researchers due to their massive industrial and technological applications like coal slurries, paper production, polymers, cosmetics, coating of wires, oil recovery, drawing of plastics and rubber sheets and mixture of clays etc. Non-Newtonian materials cannot be depicted by single constitutive relationship. Various fluid models have been proposed to predict the salient features of non Newtonian materials [33-35]. The non-Newtonian materials at present are argued through three main classifications namely the dierential, the rate and the integral. Third grade material is one of the subclasses of dierential type materials. Third grade fluid in comparison 4

to second grade fluid not only explains the behavior of normal stresses but also illustrates the impact of shear thickening/thinning property. Farooq et al. [36] examined heat and mass transfer in two-layer flows of third-grade nanofluids in a vertical channel. Wang et al. [37] studied MHD third grade fluid flow with heat transfer due to parallel plates. Hayat et al. [38] computed analytical results for magnetohydrodynamic (MHD) squeezing flow of third grade liquid. The main theme of present article is to model the MHD and nonlinear radiation eect in nonlinear convective flow of third grade nanofluid over a nonlinear stretching sheet with variable thickness. Brownian motion and thermophoresis eects are present. Chemical reaction and heat generation/absorption are also present. The nonlinear system is computed through the implementation of homotopic analysis technique (HAM). Convergence analysis of obtained solutions developed. The homotopic solutions [39-48] for nonlinear systems are computed. The velocity, thermal field and nanoparticle concentration are graphically sketched for various governing parameters. Skin friction coe!cient and local Nusselt number are tabulated and examined.

2

Modeling

Let us consider nonlinear convective and MHD flow of an incompressible third grade fluid. Flow caused is by nonlinear stretching sheet of variable thickness at | =  W ({ + e) Adopted magnetic field in |direction E({) = E0 ({ + e)

qW 31 2

13qW 2

=

(see Fig. 1). Induced mag-

netic field is omitted for small magnetic Reynolds number. Nanofluid for Brownian motion and thermophoresis is considered. Stretching sheet is convectively heated. Heat transfer is through nonlinear thermal radiation and heat generation/absorption. The stretching velocity W

is xz ({) = d({ + e)q (where d and e denotes the dimensional constants). Zero mass flux 5

condition is imposed. The relevant problems satisfy

Fig. 1. Physical sketch and coordinate system. Cx Cy + = 0> C{ C|

(1)

 ¸ C3x Cx C 2 x W1 C 3 x Cx C 2 x Cx C 2 x Cx x +y =  2+ +y 3 + +3 x C{ C| C|  C{C| 2 C| C{ C| 2 C| C{C| µ ¶ 2 W Cx C 2 x E 2 W Cx C 2 x +6 3  x +2 2  C| C{C|  C| C| 2 i +j{\1 (W  W" ) + \2 (W  W" )2 } + j{\3 (F  F" ) + \4 (F  F" )2 }> (2) µ ¶ µ 2 ¶ µ ¶2 CW CF C W  GW CW ni + +  GE = (fs )i C| 2 C| C| W" C| W 1 Ctu T  + (W  W" )> (fs )i C| (fs )i ¶ µ C 2 F GW C 2 W CF CF nWW (F  F" )> +y = GE 2 + x C{ C| C| W" C| 2

CW CW +y x C{ C|

W

x = xz ({) = d({ + e)q > y = 0>  ni GE

(3) (4)

CW = ki (Wi  W )> C|

13qW CF GW CW + = 0 at | =  W ({ + e) 2 > C| W" C|

x $ 0> W $ W" > F $ F" as | $ 4= 6

(5)

In the above expressions the velocity components along the ({> |) directions are indicated by (x> y)>  = (@)i denotes the kinematic viscosity, W1 > W2 and W3 the material parameters,  the electrical conductivity, j the gravitational acceleration, \1 and \2 the linear and nonlinear thermal expansion coe!cients, \3 and \4 the linear and nonlinear concentration expansion coe!cients, i the fluid density, (fs )i the fluid heat capacity, s the particle density, (fs )s the particle heat capacity, ni the thermal conductivity, tu the radiative heat flux,  = (fs )s @(fs )i the capacity ratio, GE the Brownian diusion coe!cient, GW the themophoretic diusion coe!cient, TW the coe!cient of heat generation/absorption, nWW the reaction rate whereas nWW A 0 leads to destructive and n WW ? 0 for generative reactions, W the fluid temperature, F the fluid concentration, Wi the convective fluid temperature, Fi the convective fluid concentration, ki = k ({ + e)

qW 31 2

the non-uniform heat transfer coe!cient,

W" the ambient fluid temperature, F" the nanoparticle concentration far away from the surface and qW the power index. Radiative heat flux is tu = 

4 W CW 4 16 W 3 CW =  W > 3n W C| 3nW C|

(6)

where  W indicate the Stefan-Boltzmann constant and nW for mean absorption coe!cient. Now Eq. (3) has the following form [47]: CW CW x +y C{ C|

¶ µ µ ¶2  GW CW ni C 2 W CW CF + = +  GE (fs )i C| 2 C| C| W" C| µ ¶ W W 1 16 C T 3 CW + W + (W  W" ) = (fs )i 3nW C| C| (fs )i

7

(7)

The transformations are r

(qW + 1) ³ d ´ W  W" > or W = W" (1 + (Xz  1)X)> ({ + e)qW 31 |> X() =  = 2  Wi  W" sµ ¶ F  F" 2 > #() = d({ + e)qW +1 I ()> x () = F" qW + 1 sµ  ¶ ¸ qW + 1 qW  1 0 W W 31 0 q q I () +  W x = d({ + e) I () > y =  d({ + e) I () >(8) 2 q +1 Eq. (1) is trivially satisfied by variables defined in previous equation and other equations yield  3 (3qW  1) 002 2qW 00 02 I + I I + 1 (3qW  1) I 0 I 000 + 2(qW  1)I 00 I 000 + I I  W q +1 2 ¸ h i qW + 1 002 000 qW + 1 2 ly + 2 (3qW  1) I 00 + (qW  1)I 00 I 000 + 63 Re{ II I I  2 2 2 2 2 Kd2 I 0 + W (1 +  1 X)X + W Q W (1 +  2 x)x = 0> (9)  W q +1 q +1 q +1 000

µ ¶ ³ ´ 4 4 h 2 1 + U X00 + U (Xz  1)3 3X2 (X0 ) + X3 X00 + 3 (Xz  1)2 3 3 ³ ´ ³ ´i 2 2 2X (X0 ) + X2 X00 + 3 (Xz  1) (X0 ) + XX00 + Pr I X0 2 Pr X = 0> +1

(10)

2 Qw 00 Vfx + X = 0> +1 Qe

(11)

2

+ Pr Qe X0 x0 + Pr Qw (X0 ) + x00 + VfI x0  µ

1  qW 1 + qW

qW

qW

¶

> I 0 (W ) = 1> X0 (W ) = Ew (1  X(W ))> r d (qW + 1) Qe x0 (W ) + Qw X0 (W ) = 0 at W =  W > 2y W

W

I ( ) = 

I (W ) $ 0> X(W ) $ 0> x(W ) $ 0 as W $ 4> W

where  = 

W

q

d(qW +1) 2y

(12)

is a variable related to the thickness of the wall (i.e., wall thickness

parameter). Defining I () = i(  W ) = i ()> X() = (  W ) = () and x() =

8

!(  W ) = !()> Eqs. (9  12) are reduced to the forms:  3 (3qW  1) 002 2qW 02 00 i + i i + 1 (3qW  1) i 0 i 000 + 2(qW  1)i 00 i 000 + i i  W q +1 2 ¸ h i qW + 1 ly qW + 1 002 000 W 002 W 00 000 i i + 2 (3q  1) i + (q  1)i i + 63 Re{ i i  2 2 2 2 2 Kd2 i 0 + W (1 +  w ) + W Q W (1 +  f !)! = 0> (13)  W q +1 q +1 q +1 000

µ ¶ ³ ´ ³ ´ 4 4 h 3 2 00 2 0 2 3 00 0 2 2 00 1 + U  + U (z  1) 3 ( ) +   + 3 (z  1) 2 ( ) +   3 3 ³ ´i 2 2 0 2 00 Pr  = 0>(14) +3 (z  1) ( ) +  + Pr i0 + Pr Qe 0 !0 + Pr Qw (0 ) + W q +1 !00 + Vfi !0  W

i() = 

µ

1  qW 1 + qW

¶

qW

Qw 00 2 Vf! +  = 0> +1 Qe

(15)

> i 0 () = 1> 0 (0) = Ew (1  ())>

Qe !0 () + Qw 0 () = 0 at  = 0> i 0 () $ 0> () $ 0> !() $ 0 as  $ 4>

(16)

where 1 > 2 and 3 denotes third grade fluid parameters, Kd for magnetic parameter/Hartmann number,  for mixed convection parameter,  w for nonlinear convection parameter due to temperature, Q W for ratio of concentration to thermal buoyancy forces,  f for nonlinear convection parameter due to concentration, U for radiation parameter, z for temperature parameter, Pr for Prandtl number, Qe for Brownian motion variable, Qw for thermophoresis variable,  for heat generation/absorption variable, Vf for Schmidt number,  for chemical reaction parameter, Ju for Grashof number in term of temperature and JuW for Grashof number in term of concentration and Ew for Biot number. These variables are

9

described as follows: W

1 = Kd2 = QW = z =  =

W

W

W d({ + e)q 31 W d({ + e)q 31 W1 d({ + e)q 31 > 2 = 2 > 2 = 3 >    E 2 ({) \2 (Wi  W" ) Ju > W 31 >  = 2 > w = q d({ + e) \1 Re{ 3 \3 F" JuW 4W W" \4 F" > U= = > f = > Ju \1 (Wi  W" ) \3 ni nW Wi (fs )i  GE F"  GW (Wi  W" ) > Pr = > Qe = > Qw = > W" ni  W" nWW ki  TW > Vf = > = > EW = p > qW 31 d (fs )i d GE d ({ + e) 2 ni y

j\1 (Wi  W" ) ({ + e)3 j\3 F" ({ + e)3 W > Ju = = Ju = 2 2

(17)

It should be noted that the case for viscous fluid can be recovered by putting 1 = 0> 2 = 0> 3 = 0 and Re{ = 0 in Eq. (13). Skin friction coe!cient Fi{ and local Nusselt number Qx{ are Fi{ =

z > 1 x2z 2

Qx{ =

({ + e) tz > ni (Wi  W" )

(18)

in which  z denotes surface shear stress and tz the surface heat flux i.e. z tz

µ ¶3 # µ ¶ 2 2 Cx Cx C x x Cx Cx C + W1 x +2 + y 2 + 2W3 > =  C| C{C| C{ C| C| C| 13qW W |= ({+e) 2 µ ¶µ ¶ W 3 16 W CW = = ni 1 + W W 3ni n C| |=W ({+e) 13q 2 "

(19)

Dimensionless variables finally give

Re{30=5 Qx{ in which Re{ =

r

µµ W  ¶ qW + 1 00 7q  1 2 i (0) + 1 i 0 (0)i 00 (0) + (qW  1) (i 00 (0)) 2 2 µ W ¸ µ W ¶ ¶ ¶ q +1 q +1 3 000 00  i(0)i (0) + 23 Re{ (i (0)) > 2 2 r ¶ µ 4 qW + 1 3 (20) 1 + U (1 + (z  1)(0)) 0 (0) > =  2 3

1 0=5 Re Fi{ = 2 {

xz ({+e) i

denotes the local Reynolds number.

10

3

Homotopic expressions

Initial guesses (i0 ()> 0 ()> !0 ()) and auxiliary linear operators (Li > L > L! ) for homotopic solutions are taken in the forms ¶ qW  1 Ew > 0 () = exp () > i0 () = (1  exp ())   W q +1 1 + Ew µ µ ¶ ¶ Ew Qw !0 () = exp   > 1 + Ew Qe W

Li (i ) =

µ

gi g2  g2 ! g3 i  > L () =  > L (!) =  !>  ! g 3 g g2 g2

(21) (22)

with Li [l1 + l2 exp() + l3 exp()] = 0>

(23)

L [l4 exp() + l5 exp()] = 0>

(24)

L! [l6 exp() + l7 exp()] = 0>

(25)

in which ll (l = 1  7) indicates the constants.

3.1

Zeroth-order systems

The relevant problems are i h i h b (; s˘) > (1  s˘) Li ib(; s˘)  i0 () = s˘Ki ~i Ni ib(; s˘) > b  (; s˘) > ! i h i h b b b b (1  s˘) L  (; s˘)  0 () = s˘K ~ N i (; s˘) >  (; s˘) > ! (; s˘) >

h i h i b b b b (1  s˘) L! ! (; s˘)  !0 () = s˘K! ~! N! i (; s˘) >  (; s˘) > ! (; s˘) > b0

W

i (0; s˘) = 

µ

1  qW 1 + qW

¶

(26) (27) (28)

> ib0 (0; s˘) = 1 and ib0 (4; s˘) $ 0 as  $ 4>

0 b  (0; s˘) = Ew (1  b (0; s˘)) and b  (; s˘) $ 0 as  $ 4>

0 b (; s˘) $ 0 as  $ 4> b0 (0; s˘) + Qwb Qe !  (0; s˘) = 0 and !

11

(29)

where s˘ 5 [0> 1] is the embedding variable, (~i > ~ > ~! ) the convergence control variables and (Ni > N > N! ) represent the nonlinear operators given by à !2 h i W 2b 3b b s˘) i (; s ˘ ) 2q C i(; C b s˘) C i(; s˘) b (; s˘) =  + i(; Ni ib(; s˘) > b  (; s˘) > ! qW + 1 C C 3 C 2 " b s˘) C 3 ib(; s˘) b s˘) b s˘) C 3 i(; C 2 i(; C i(; W + 2(q  1) +1 (3qW  1) C C 3 C 2 C 3 6 à !2 2b 4b W W C i(; s˘) 8 3 (3q  1) C i (; s˘) q +1b i(; s˘) +  2 2 2 C C 4 6 5 !2 à 2b 3b 2b C i(; s˘) C i(; s˘) 8 C i (; s˘) + (qW  1) +2 7(3qW  1) 2 C C 2 C 3 !2 à b ˘) 2 C 3 ib(; s˘) qW + 1 C 2 ib(; s˘) 2 C i(; s  +63 Re{ K 2 qW + 1 d C C 2 C 3 ³ ´ 2 2 b (; s˘))! b (;(30)  1 +  wb (; s˘) + W Q W (1 +  f ! (; s˘) b s˘) > + W q +1 q +1 µ ¶ 2b i h 4 C (; s˘) 4 £ b b b N i (; s˘) >  (; s˘) > ! (; s˘) = + U (z  1)3 1+ U 2 3 3 C 3 4 à !2 2b b C (; s˘) C (; s˘) D C3(b + (b (; s˘))3 (; s˘))2 C C 2 4 3 !2 à 2b b C (; s˘) D C (; s˘) + (b (; s˘))2 +3 (z  1)2 C2b (; s˘) C C 2 3à 46 !2 2 Cb (; s˘) C b (; s˘) D8 +b (; s˘) +3 (z  1) C C C 2 b b b b s˘) C (; s˘) + Pr Qe C (; s˘) C !(; s˘) + Pr i(; C C C !2 à Cb (; s˘) 2 + W Pr b (; s˘)> + Pr Qw C q +1

(31)

i h 2b b b s˘) b (; s˘) = C !(; s˘) + Vfib(; s˘) C !(; s˘)  2 Vf !(; N! ib(; s˘) > b  (; s˘) > ! C qW + 1 C 2 (; s˘) Qw C 2b + > (32) Qe C 2 12

3.2

pth-order deformation systems

The corresponding problems satisfy Li [ip ()  "p ip31 ()] = ~i Rip () >

(33)

L [p ()  "p p31 ()] = ~ Rp () >

(34)

£ ¤ L! !p ()  "p !p31 () = ~! R!p () >

(35)

0 0 ip (0) = 0> ip (0) = 0 and ip () $ 0 as  $ 4>

0p (0)  Ew p (0) = 0 and p () $ 0 as  $ 4> Qe !0p (0) + Qw 0p (0) = 0 and !p () $ 0 as  $ 4> Rip

(36)

p31 X

¸ p31 X£ 2qW 0 0 0 ip313n in + 1 (3qW  1) ip313n in000  W () = () + q + 1 n=0 ¸ ¶ µ n=0 W qW + 1 3 (3q  1) 00 ly W 00 000 00 ip313n in +2(q  1)ip313n in + ip313n in  2 2 n p31 X X ¤ £ 00 00 00 000 +2 (3qW  1) ip313n in3o io00 in00 + (qW  1) ip313n in000 + 63 Re{ ip313n ip313n in00

000 ip31

n=0

o=0

2 2 0  W Kd2 ip31 () + W (1 +  w p31 ())p31 () q +1 q +1 2 + W Q W (1 +  f !p31 ())!p31 ()> q +1

(37)

" ¶ µ p31 n o X X X ¢ ¡ 0 0 4 4 3 00  (z  1) p313n n3o 3o3v v + o3v 00v Rp () = 1 + U p31 () + U 3 3 n=0 v=0 o=0 +3 (z  1)2 p313n + Pr

p31 X

n X o=0

ip313n 0n + Pr

=

!00p31

p31 X n=0

n=0

R!p ()

# ¢ ¢ ¡ ¡ 0 0 n31 2n3o o + n3o 00o + 3 (z  1) 0p313n 0n + p313n 00n

() + Vf

p31 X n=0

¢ ¡ Qe 0p313n !0n + Qw 0p313n 0n +

ip313n !0n 

"p =

; A A ? 0>

A A = 1>

13

qW

2 Pr p31 > qW + 1

Qw 00 2  > Vf!p31 () + +1 Qe p31

p1 pA1

>

(39)

(40)

(38)

b (; 0) = !0 () when s˘ = 0> ib(; 0) = i0 () > b  (; 0) = 0 () > !

b (; 1) = ! () when s˘ = 1= ib(; 1) = i () > b  (; 1) =  () > !

(41)

b (; s˘) vary from the initial solutions When s˘ increases from 0 to 1 then ib(; s˘) > b  (; s˘) and !

i0 () > 0 () and !0 () to the desired solutions i () > () and !() respectively. The solutions can be written through Taylor’s series as follows: ib(; s˘) = i0 () +

" X

ip () s˘p

p=1

b  (; s˘) = 0 () + b (; s˘) = !0 () + !

" X

p () s˘p

p=1 " X

p=1

!p () s˘p

¯ 1 C p ib(; s˘) ¯¯ with ip () = ¯ p! C s˘p ¯

>

(42)

s˘=0

¯  (; s˘) ¯¯ 1 C pb with p () = ¯ > p! C s˘p ¯ s˘=0 ¯ b (; s˘) ¯¯ 1 C p! with !p () = ¯ = p! C s˘p ¯

(43)

(44)

s˘=0

The series of i>  and ! are convergent for s˘ = 1 and thus i () = i0 () +  () = 0 () + ! () = !0 () +

" X

p=1 " X

p=1 " X

ip () > p () > !p () =

(45)

p=1

W The general solution (ip > p > !p ) in the frame of special functions (ip > Wp > !Wp ) are given

by B ip () = ip () + l1 + l2 exp() + l3 exp()>

p () = Bp () + l4 exp() + l5 exp()> !p () = !Bp () + l6 exp() + l7 exp()>

14

(46)

where the constant ll (l = 1  7) after using boundary conditions have values l1 l4 l6

4

¯ ¯ B B Cip () ¯¯ Cip () ¯¯ W  ip (0)> l2 = > l3 = 0> =  C ¯=0 C ¯=0 " # ¯ CBp () ¯¯ 1 =  Ew Bp (0) > l5 = 0> l7 = 0> Ew + 1 C ¯=0 à " ¯ ¯ ! ¯ # B B ¯ ¯ C!Bp () ¯¯ C! () Qw C! () 1 p p ¯ ¯ = Ew !Bp (0)  + + = (47) ¯ ¯ ¯ C Q E + 1 C C e w =0 =0 =0

Convergence analysis

The convergence control variables }i > } and }! have key role for convergence of the analytic solutions. The }curves are plotted at 9wk order of approximations for the convergence intervals (see Figs. 2). Appropriate values of convergence control variables are 1=25 6 }i 6 0=25> 1=85 6 } 6 0=35 and 1=9 6 }! 6 0=3=

1

I ' +0/ 0

T ' +0/

f '' +0/

2

1

1

0

00

0

1

³

0

Fig. 2. }curves for i (0)>  (0) and ! (0)= Table 1: Solutions convergence when qW = 1=5> 1 = 0=1> 2 = 0=1> 3 = 0=1> Re{ = 0=1> Kd = 0=1>  = 0=1>  w = 0=1> Q W = 0=5>  f = 0=1> U = 0=3> z = 1=1> Pr = 1=5> Qe = 0=7> Qw = 0=2>  = 0=1> Vf = 1=5>  = 0=9> W = 0=2 and Ew = 0=3=

15

Order of approximation i 00 (0) 0 (0)

!0 (0)

1

0=8096

0=2132

0=0609

5

0=7856

0=1954

0=0558

10

0=7828

0=1907

0=0545

12

0=7823

0=1900

0=0542

15

0=7821

0=1893

0=0542

20

0=7821

0=1893

0=0542

25

0=7821

0=1893

0=0542

30

0=7821

0=1893

0=0542

Table 1 is computed to ensure the convergence analysis of the homotopic solutions. It is analyzed that computations are enough for 15wk order iterations for momentum and energy equations while 12wk order iterations of the concentration is acceptable for the convergent series solutions.

5

Analysis

Here behavior of dierent involved physical variables on velocity, temperature, nanoparticles concentration, skin friction coe!cient and local Nusselt number is analyzed. This objective is achieved through plots 3-17 and Tables 2 and 3.

5.1

Dimensionless velocity

Variation of 1 and 2 (i.e., third grade fluid parameters) for velocity is considered in Fig. 3. The other parameters are kept fixed. One can see that velocity and layer thickness are increased via larger 1 and 2 = Since material variable are inversely proportional to the 16

viscosity. Thus for larger 1 and 2 the fluid viscosity decreases and hence the velocity enhances. Fig. 4 demonstrates the features of fluid variable 3 and Reynolds number Re on the velocity. Obviously larger fluid variable 3 and Reynolds number Re{ show an increase in the momentum layer thickness and velocity. As Reynolds number is the inertial to viscous forces ratio. For larger the inertial forces are more dominant than viscous forces and so velocity enhances. Fig. 5 provides impact of magnetic field Kd on velocity. Both velocity and layer thickness are decreased for higher Kd = In fact for higher values of magnetic variable the Lorentz force enhances which produce more resistance to liquid motion and thus velocity decays. Fig. 6 disclosed variation of wall thickness parameter W on velocity. Both velocity and layer thickness are enhanced via larger W A 1= Since stretching velocity enhances when qW A 1 therefore velocity increases. f '+ [ /

f '+ [ /

1.0

n O R Nt

0.8

1.0

1.5, D3 0.1, Re x 0.1, Ha 0.1, 0.1, Et 0.1, N 0.5, Ec 0.1, 0.3, Tw 1.1, Pr 1.5, Nb 0.7, 0.2, G 0.1, Sc 1.5, J 0.9, D 0.2, Bt 0.3

n O R Nt

0.8

0.6

1.5, D1 0.1, D2 0.1, Et 0.1, N 0.3, Tw 1.1, Pr 0.2, G 0.1, Sc D 0.2, Bt

0.1, H a 0.1, 0.5, Ec 0.1, 1.5, Nb 0.7, 1.5, J 0.9, 0.3

0.6

0.4

0.4

D1 D2

0.2

1

2

3

D3 Re x

0.0, 0.3, 0.6 0.0, 0.3, 0.6 0.2

4

5

[

1

0

2

3

0.0, 0.5, 1.0 0.0, 0.5, 1.0

4

5

[

0

Fig. 3. Variation of 1 and 2 on i ()=

Fig. 4. Variation of 3 and Re on i ()=

f '+[ /

f '+[ /

1.0

1.0

1.5, D1 0.1, D2 0.1, D 3 0.1, Re x 0.1, O 0.1, Et 0.1, N 0.5, Ec 0.1, R 0.3, Tw 1.1, Pr 1.5, Nb 0.7, Nt 0.2, G 0.1, Sc 1.5, J 0.9, D 0.2, Bt 0.3

n

n 0.8

0.6

Re x

N

0.8

Pr

1.5, D1 0.1, Ha 0.5, Ec 1.5, Nb Sc 1.5,

0.6

0.4

0.1, D2 0.1, D 3 0.1, 0.1, O 0.1, Et 0.1, 0.1, R 0.3, Tw 1.1, 0.7, Nt 0.2, G 0.1, J 0.9, Bt 0.3

0.4

0.2

0.2

Ha 1

2

3

D

0.0, 0.5, 1.0, 1.5

4

5

6

[

1

2

3

4

W

0

Fig. 5. Variation of Kd on i ()=

0.0, 1.5, 3.0, 4.5 5

0

Fig. 6. Variation of  on i ()= 17

6

[

5.2

Dimensionless temperature

Radiation parameter U on temperature is described in Fig. 7. It is observed that temperature and thermal layer thickness are increasing functions of U= In fact radiation parameter is inversely proportional to mean absorption coe!cient. Higher value of radiation parameter leads to decay nW which results in temperature enhancement. Temperature and thermal layer thickness are increasing functions of temperature variable z (see Fig. 8). Physically higher z lead to enhance the convective surface temperature. Ultimate there is increase in temperature. It is seen from Fig. 9 that both temperature and thickness layer diminish for higher Prandtl number Pr = Physically higher Pr lead to lower thermal diusivity which ensures decrease in temperature. Fig. 10 clearly illustrates that the thermal field and associated layer thickness are increased for larger thermophoresis parameter Qw = In thermophoresis phenomenon heated fluid particles are pulled away from hot surface to the cold region. Due to this fact the fluid temperature rises. Behavior of heat generation/absorption variable  on the thermal field is elucidated in Fig. 11. The temperature and thermal layer are increased for larger heat generation parameter = Obviously in heat generation process more heat is delivered to working liquid which result in the enhancement of thermal field. Eect of Biot number Ew on thermal field is shown in Fig 12. Both thermal field and layer thickness are enhanced for higher Ew = This is because higher thermal Biot number demonstrates an upgrade

18

in heat transfer coe!cient. T +[ / 0.35

T +[ /

n

1.5, D1 0.1, D2 0.1, D 3 0.1, Re x 0.1, Ha 0.1, O 0.1, Et 0.1, N 0.5, Ec 0.1, Tw 1.1, Pr 1.5, Nb 0.7, Nt 0.2, G 0.1, Sc 1.5, J 0.9, D 0.2, Bt 0.3

n 0.30 0.25

0.30

0.20

0.15

0.15

0.10

0.10

R 1

2

3

4

0.0, 0.2, 0.4, 0.6

5

N Nb

0.25

0.20

0.05

Re x

6

1.5, D1 0.1, D2 0.1, D 3 0.1, 0.1, Ha 0.1, O 0.1, Et 0.1, 0.5, Ec 0.1, R 0.3, Pr 1.5, 0.7, Nt 0.2, G 0.1, Sc 1.5, J 0.9, D 0.2, Bt 0.3

Tw

0.05 [

1

Fig. 7. Variation of U on ()=

2

3

0.0, 0.5, 1.0, 1.5

4

5

6

[

Fig. 8. Variation of z on ()=

T +[ /

T +[ /

0.35

n

0.30

Re x

N Nb

0.25

1.5, D1 0.1, D2 0.1, D 3 0.1, 0.1, Ha 0.1, O 0.1, Et 0.1, 0.5, Ec 0.1, R 0.3, Tw 1.1, 0.7, Nt 0.2, G 0.1, Sc 1.5, J 0.9, D 0.2, Bt 0.3

0.20

0.15

0.15

0.10

0.10

Pr

1

2

3

4

0.1, 0.5, 1.0, 1.5

5

Re x

N

0.25

0.20

0.05

n

0.30

6

Pr

Nt

0.05 [

1

Fig. 9. Variation of Pr on ()= T +[ / 0.4

1.5, D1 0.1, D2 0.1, D 3 0.1, Re x 0.1, Ha 0.1, O 0.1, Et 0.1, N 0.5, Ec 0.1, R 0.3, Tw 1.1, Pr 1.5, Nb 0.7, Nt 0.2, Sc 1.5, J 0.9, D 0.2, Bt 0.3

0.25

3

4

0.1, 1.0, 2.0, 3.0 5

6

[

n

1.5, D1 0.1, D2 0.1, D 3 0.1, 0.1, Ha 0.1, O 0.1, Et 0.1, N 0.5, Ec 0.1, R 0.3, Tw 1.1, Pr 1.5, Nb 0.7, Nt 0.2, G 0.1, 1.5, J 0.9, D 0.2 Sc

n

0.30

2

0.1, D3 0.1, 0.1, Et 0.1, 0.3, Tw 1.1, 0.1, Sc 1.5, Bt 0.3

Fig. 10. Variation of Qw on ()=

T +[ /

0.35

1.5, D 1 0.1, D 2 0.1, H a 0.1, O 0.5, Ec 0.1, R 1.5, Nb 0.7, G J 0.9, D 0.2,

Re x 0.3

0.20

0.2

0.15 0.10

0.1

G

0.05

1

2

3

 0.4,  0.2, 0.0, 0.2, 0.4

4

5

6

Fig. 11. Variation of  on ()=

5.3

Bt [

1

2

3

4

0.1, 0.2, 0.3, 0.4

5

Fig. 12. Variation of Ew on ()=

Dimensionless concentration

Nanoparticle concentration for Brownian motion variable Qe is drawn in Fig. 13. There is a reduction in concentration while layer thickness enhances for Qe = For higher Qe the collision 19

6

[

of fluid particles enhances and the nanoparticle concentration field diminishes. Opposite behavior of nanoparticle concentration and layer thickness is observed due to thermophoresis parameter Qw (see Fig. 14). Thermal conductivity of fluid increases within the frame of nanoparticles. Larger Qw assist in an increase of fluid thermal conductivity. Higher thermal conductivity leads to more concentration. Analysis of Schmidt number Vf on nanoparticle concentration is addressed in Fig. 15. One may see from the Fig. that concentration enhances for higher Schmidt number Vf= Physically Schmidt number is the viscous to molecular diusion rates ratio. Thus for larger viscous diusion rate due to higher Vf there is an increment in the fluid concentration and its layer thickness diminishes. Characteristic of generative/destructive chemical reaction  on concentration distribution is delineated in Fig. 16. There is an enhancement in fluid concentration in view of destructive chemical reaction variable ( A 0)= However reverse situation is noticed for generative chemical reaction ( ? 0)= Impact of Biot number Ew on the nanoparticle concentration is drawn in Fig. 17. Here an increase in Ew shows decay in the nanoparticle concentration field. However concentration layer thickness increases. I + [/

I + [/

0.00 0.00

Nb

 0.05

n

 0.10

Re x

N Pr

 0.15

 0.10

1.5, D1 0.1, D2 0.1, D 3 0.1, 0.1, Ha 0.1, O 0.1, Et 0.1, 0.5, E2 0.1, R 0.3, Tw 1.1, 1.5, Nt 0.2, G 0.1, Sc 1.5, J 0.9, D 0.2, Bt 0.3

4

6

8

10

n Re x

N

 0.15

[

2

Nt

 0.05

0.1, 0.2, 0.3, 0.4

Pr

Fig. 13. Variation of Qe on !()=

1.5, D1 0.1, D2 0.1, Ha 0.1, O 0.5, Ec 0.1, R 1.5, Nb 0.7, G J 0.9, D 0.2,

0.1, D 3 0.1, 0.1, Et 0.1, 0.3, Tw 1.1, 0.1, Sc 1.5, Bt 0.3

 0.20

5

12

0.1, 0.15, 0.20, 0.25

10

15

20

25

30

Fig. 14. Variation of Qw on !()=

20

35

[

I +[/ 0.0

I + [/ 0.00

 0.05

Sc

J

 0.1

0.0, 0.5, 1.0, 1.5

 0.2

 0.10

n

1.5, D1 0.1, D2 0.1, D 3 0.1, Re x 0.1, Ha 0.1, O 0.1, Et 0.1, N 0.5, Ec 0.1, R 0.3, Tw 1.1, 1.5, Nb 0.7, Nt 0.2, G 0.1, Pr J 0.9, D 0.2, Bt 0.3

n  0.15

Re x Pr

4

6

8

10

12

1.5, D 1 0.1, H a 0.5, Ec 1.5, Nb Sc 1.5,

 0.4

 0.20

2

N

 0.3

 1.0,  0.5, 0.0, 0.5, 1.0

[

2

Fig. 15. Variation of Vf on !()=

4

6

0.1, D 2 0.1, O 0.1, R 0.7, Nt D 0.2,

0.1, D3 0.1, 0.1, Et 0.1, 0.3, Tw 1.1, 0.2, G 0.1, Bt 0.3

8

Fig. 16. Variation of  on !()=

I + [/

0.00

 0.02

Bt

0.1, 0.2, 0.3, 0.4

 0.04

n

1.5, D1 0.1, D2 0.1, D 3 0.1, 0.1, Ha 0.1, O 0.1, Et 0.1, N 0.5, Ec 0.1, R 0.3, Tw 1.1, Pr 1.5, Nb 0.7, Nt 0.2, G 0.1, 1.5, J 0.9, D 0.2 Sc

 0.06

Re x

 0.08  0.10

5

10

15

20

[

Fig. 17. Variation of Ew on !()=

5.4

Skin friction coe!cient and local Nusselt number

Impact of power index qW > third grade fluid parameters 1 and 2 > magnetic parameter/Hartmann number Kd > mixed convection parameter  and wall thickness variable W on skin friction coe!cient are depicted in Table 2. Skin friction coe!cient increases for higher values of qW > 1 and Kd while it decays for larger 3 >  and W = In Table 3 some numerical measurements of heat transfer rate are given via power index qW > radiation parameter U> temperature variable z > Prandtl number Pr> heat generation/absorption variable  and thermal Biot number Ew = It is recognized that heat transfer rate enhances via qW > U> z > Pr and Ew while it decreases for = W Table 2: Skin friction coe!cient 12 Re0=5 { Fi{ when 3 = 0=1> Re{ = 0=1>  w = 0=2> Q = 0=5>

21

10

[

 f = 0=1> U = 0=3> z = 1=1> Pr = 1=5> Qe = 0=7> Qw = 0=2>  = 0=1> Vf = 1=5>  = 0=9 and Ew = Parameters (fixed values)

Parameters  21

1 = 0=1> 2 = 0=1> Kd = 0=1>  = 0=1> W = 0=2

qW

qW = 1=5> 2 = 0=1> Kd = 0=1>  = 0=1> W = 0=2

qW = 1=5> 1 = 0=1> Kd = 0=1>  = 0=1> W = 0=2

qW = 1=5> 1 = 0=1> 2 = 0=1>  = 0=1> W = 0=2

1

2

Kd

qW = 1=5> 1 = 0=1> 2 = 0=1> Kd = 0=1> W = 0=2 

qW = 1=5> 1 = 0=1> 2 = 0=1> Kd = 0=1>  = 0=1

W

q

2 qW +1

0=0

0=5199

1=0

1=0695

1=5

1=2912

0=1

1=2912

0=2

1=4795

0=3

1=6413

0=1

1=2912

0=2

1=2087

0=3

1=1356

0=1

1=2912

0=2

1=3043

0=5

1=3984

0=1

1=2912

0=2

1=2777

0=3

1=2306

0=2

1=2912

0=3

1=2787

0=5

1=2498

Re0=5 { Fi{

Table 3: Local Nusselt number Uh{30=5 Qx{ when 1 = 0=1> 2 = 0=1> 3 = 0=1> Re{ = 0=1> Kd = 0=1>  = 0=1>  w = 0=1> Q W = 0=5>  f = 0=1> Qe = 0=7> Qw = 0=2> Vf = 1=5>  = 0=9 and W = 0=2=

22

Parameters (fixed values)

Parameters

U = 0=3> z = 1=1> Pr = 1=5>  = 0=1> Ew = 0=3

qW

qW = 1=5> z = 1=1> Pr = 1=5>  = 0=1> Ew = 0=3

qW = 1=5> U = 0=3> Pr = 1=5>  = 0=1> Ew = 0=3

qW = 1=5> U = 0=3> z = 1=1>  = 0=1> Ew = 0=3

6

2 qW +1

0=2740

1=5

0=3049

0=1

0=2533

0=3

0=3049

0=5

0=3578

1=1

0=3049

1=2

0=3147

1=5

0=3435

Pr 1=0

0=2901

1=5

0=3049

2=0

0=3206

0=1

0=3049

0=2

0=2861

0=5

0=1677

0=2

0=2322

0=3

0=3049

0=5

0=4126

U

z

Ew

Re30=5 Qx{ {

0=2002

1=0

qW = 1=5> U = 0=3> z = 1=1> Pr = 1=5> Ew = 0=3 

qW = 1=5> U = 0=3> z = 1=1> Pr = 1=5>  = 0=1

0=0

q

Concluding remarks

The main outcomes of present study are: • The variations of 1 > 2 > 3 > Re{ and W are quantitatively similar for velocity field.

23

• Increasing thermal Biot number Ew enhances the temperature and related layer. • Nanoparticle concentration decays when Brownian motion parameter Qe is enhanced. • Concentration and related layer have reverse eect for chemical reaction parameter. • Skin friction coe!cient via 3 >  and W has similar qualitative eect. • Reduction in local Nusselt number is observed for heat generation/absorption parameter =

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1. 2. 3. 4. 5.

Magnetohydrodynamic (MHD) nonlinear radiative flow of third grade nanofluid modeled. Effects of heat generation/absorption and nonlinear convection are considered. Aspects of chemical reaction and zero mass flux condition are introduced. Brownian motion and thermophoresis phenomena are taken due to nanofluid. Skin friction coefficient and local Nusselt number are tabulated and examined.