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absorption aspects in third grade magneto-nanofluid over a slendering stretching sheet with Newtonian conditions
Accepted Manuscript Thermal radiation and heat generation/absorption aspects in third grade magnetonanofluid over a slendering stretching sheet with Newtonian conditions Sajid Qayyum, Tasawar Hayat, Ahmed Alsaedi PII:
S0921-4526(18)30073-5
DOI:
10.1016/j.physb.2018.01.043
Reference:
PHYSB 310690
To appear in:
Physica B: Physics of Condensed Matter
Received Date: 25 October 2017 Revised Date:
16 January 2018
Accepted Date: 17 January 2018
Please cite this article as: S. Qayyum, T. Hayat, A. Alsaedi, Thermal radiation and heat generation/ absorption aspects in third grade magneto-nanofluid over a slendering stretching sheet with Newtonian conditions, Physica B: Physics of Condensed Matter (2018), doi: 10.1016/j.physb.2018.01.043. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Thermal radiation and heat generation/absorption aspects in third grade magneto-nanofluid over a slendering stretching sheet with Newtonian conditions Sajid Qayyum a, 1 , Tasawar Hayat a, b and Ahmed Alsaedib
1
M AN U
Abstract: Mathematical modeling for magnetohydrodynamic (MHD) radiative flow of third grade nano-material bounded by a nonlinear stretching sheet with variable thickness is introduced. The sheet moves with nonlinear velocity. Definitions of thermal radiation and heat generation/absorption are utilized in the energy expression. Intention in present investigation is to develop a model for nanomaterial comprising Brownian motion and thermophoresis phenomena. Newtonian conditions for heat and mass species are imposed. Governing equations of the locally similar flow are attempted through a homotopic technique and behaviors of involved variables on the flow fields are displayed graphically. It is revealed that increasing values of thermal conjugate variable corresponds to high temperature. Numerical investigation are explored to obtain the results of skin friction coefficient and local Nusselt and Sherwood numbers. It is revealed that velocity field reduces in the frame of magnetic variable while reverse situation is observed due to mixed convection parameter. Here qualitative behaviors of thermal field and heat transfer rate are opposite for thermophoresis variable. Moreover nanoparticle concentration and local Sherwood number via Brownian motion parameter are opposite. Keywords: Third grade nanofluid; magnetohydrodynamic (MHD); mixed convection; thermal radiation; heat generation/absorption; Newtonian conditions.
Introduction
AC C
EP
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The study of non-Newtonian materials have attained the special importance in recent technological and industrial processes. Such material are involved in geophysics, biotechnology, chemical and petroleum engineering. NonNewtonian materials which explore a nonlinear relation among stress and strain rate, possessing a coefficient of variable viscosity and have the features that cannot be elucidated by the Navier-Stokes expressions. Materials such as salt solutions, apple sauce, drilling muds, shampoos, sugar solution pastes, clay coating, tooth pastes, chocolates in liquefies form, hydrogenated caster oils, lubricant, colloidal and suspension solutions interpret the features of nonNewtonian fluids [1-5]. The non-Newtonian fluids at present are argued through three main classifications namely the rate, the differential and the integral. The second-grade liquid model is a simplest description (i.e., differential type) of the non-Newtonian liquids. This description is able to predict the normal stress differences yet it cannot properly explain the phenomena of shear thickening and thinning due to its constant apparent shear viscosity. The third grade liquid theory is sufficient to foresee such phenomena. Therefore, theory of the third grade liquid is appropriated in the present analysis. Some prominent literature for this liquid model can be found in the refs. [6-10]. The diffusion of solid particles of diameter less than 100nm in fluids is known as ”nanofluid”. It is first considered by Maxwell [11]. Since the thermal conductivity of the solid particles is higher than base fluid, Maxwell expected 1
SC
44000, Pakistan Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, P. O. Box 80207, Jeddah 21589, Saudi Arabia
b
RI PT
a Department of Mathematics, Quaid-I-Azam University 45320 , Islamabad
Corresponding author. Tel.: +92 51 90642172.
email address: [email protected] (Sajid Qayyum)
1
TE D
SC
M AN U
the nanofluids could have a better thermal performance. However, it was until that Choi [12] attempted to utilize nanofluids as working agents in heat transfer. Since then, nanofluids have attracted wide attention from industrial cooling, drag delivery, automotive, nuclear power generation, fuel cell, cancer therapy, detergency, dynamic sealing etc. The recent modern technology is a clear observe that the nanoliquids have potential part in manufacturing of cars, microreactors, airplanes, micromachines and many more. The unique physical and chemical aspects of nano-sized material further upgrade the utilizations of nanoliquids in the industrial measurements which cover the meltspinning, petroleum reservoirs, manufacturing of plastic and rubber sheets, drying and cooling of papers, nuclear reactors, extrusion of plastic materials, chemical catalytic reactors, rotors of gas turbines, electronic devices, air cleaning machines and medical equipments. Buongiorno [13] presented that the thermal performance of base liquid is enhanced due to Brownian motion and thermophoretic factors. Some impressive studies on the theory of nanofluids under various circumstances can be addressed in [14-20]. The present work focuses on the modelling and analysis for the magnetohydrodynamic mixed convective flow of third grade nanofluid passing over a nonlinear stretching surface with variable thickness. Concept of heat generation/absorption and thermal radiation are utilized in the energy expression. Our main concern here is to scrutinize the radiative flow of third grade nanofluid in frame of Newtonian conditions. In existing literature the researchers only used the Newtonian heating [21-25]to scrutinize the aspects of different liquid theories under various characteristics and flow geometries. Here we introduced the Newtonian mass condition for nonlinear stretching flow phenomenon of non-Newtonian material. Features of nanoparticles is assumed by considering Brownian motion and thermophoresis. Suitable variables are implementation to transform the nonlinear partial differential systems as a group of nonlinear ordinary differential systems and then resulting systems are solved by homotopy analysis approach [26-35]. Graphs are captured and examined for the consequences of sundry physical variables on the velocity, thermal field and nanoparticles concentration. Skin friction coefficient and local Nusselt and local Sherwood numbers are tabulated and examined. The solutions acquired by HAM are preferred than the numerical solutions in perspective of the following points. (i) HAM gives the solutions within the domain of interest at each point while the numerical solutions hold just for discrete points in the domain. (ii) Algebraically developed approximate solutions require less effort and having a sensible measure of precision when compared to numerical solution which are more convenient for the scientist, an engineer or an applied mathematician. (iii) Although most of the scientific packages required some initial approximations for the solution are not generally convergent. In such conditions approximate solutions can offer better initial guess that can be readily advanced to the exact numerical solution in the limited iterations. Finally an approximate solution, if it is analytical, is most pleasing than the numerical solutions.
RI PT
ACCEPTED MANUSCRIPT
2
Mathematical description
We consider the steady two-dimensional mixed convective flow of third grade nanofluid by a nonlinear stretching surface with variable thickness at y =
1−n∗ 2 . Here δ ∗ is the surface thickness coefficient. Stretching sheet ∗ has velocity uw (x) = a(x + b)n (where a and b denotes the dimensional n∗ −1 constants). A non-uniform magnetic field B(x) = B0 (x + b) 2 is utilized
EP
δ ∗ (x + b)
∂u ∂x
∂v
+
∂y
u
∂u ∂x
AC C
(see Fig. 1). Induced magnetic field for low Reynolds number is neglected. The heat transfer phenomenon has been examined in the frame of heat generation/absorption and thermal radiation. Besides this viscous dissipation and joule heating are not accounted. Newtonian conditions for heat and mass transportation are introduced. Features of nanoparticles is taken in the version of Brownian motion and thermophoresis. In such assumption, the governing boundary layer structure for flow of third grade nanofluid are = 0,
+v
∂u ∂y
= ν
∂2u ∂y 2
+2
+
α∗ 1 ρ
"
u
∂3u ∂x∂y 2
2 α∗ 2 ∂u ∂ u
ρ ∂y ∂x∂y
+6
+v
α∗ 3 ρ
(1)
∂3u
+
∂u ∂ 2 u
+3
∂u ∂ 2 u
∂y 3 ∂x ∂y 2 ∂y ∂x∂y 2 2 ∂ u σB 2 (x) − u ∂y ∂y 2 ρf
#
∂u
+g (βT (T − T∞ ) + βC (C − C∞ )) ,
u
∂T ∂x
+v
∂T ∂y
=
kf
∂2T
(ρcp )f
∂y 2
−
u
∂C ∂x
+v
∂C ∂y
= DB
∂2C ∂y 2
1
∂qr
(ρcp )f ∂y
+
! +
DT
∂2T
T∞
∂y 2
+ τ DB Q∗ (ρcp )f
!
∂T ∂C ∂y ∂y
(2)
+
τ DT T∞
∂T ∂y
2
(T − T∞ ),
(3)
,
(4)
2
ACCEPTED MANUSCRIPT
u = uw (x) = a(x + b)
∂y
= −ht T,
1−n∗ ∗ y = δ (x + b) 2 ,
u → 0, T → T∞ , C → C∞ as y → ∞.
(5)
In the aforestated expressions the velocity components along (x, y) directions ∗ ∗ are denoted by (u, v), ν = (μ/ρ)f for kinematic viscosity, α∗ 1 , α2 and α3 for material variables, σ for electrical conductivity, g for gravitational acceleration, βT and βC for thermal and concentration expansion coefficients, ρf and ρp for fluid and particle densities, (cp )f and (cp )p for fluid and particle heat capacities, kf for thermal conductivity, qr for radiative heat flux, τ = (ρcp )p /(ρcp )f for capacity ratio, DB and DT for Brownian and themophoretic ∗
diffusion coefficients, Q∗ = Q0 (x + b)n −1 for heat generation/absorption coefficient, T and C for fluid temperature and concentration, T∞ and C∞ for ambient temperature and concentration, ht = h∗ t (x + b) n∗ −1
n∗ −1 2 for non-uniform
2 heat transfer coefficient, hc = h∗ for non-uniform mass diffusion c (x + b) coefficient and n∗ denote the power index. Rosseland relation gives radiative heat flux as [35]:
qr = −
4σ ∗ ∂T 4 3k∗ ∂y
,
(6)
4
3
4
T u 4T∞ T − 3T∞ .
(7)
So Eq. (6) becomes
qr = −
3 ∂T 16σ ∗ T∞
3k∗
.
∂y
(8)
Now Eqs. (3) and (8) yield ∂T ∂x
+v
∂T
kf ∂2T + τ DB ρcp f ∂y 2
=
∂y
+
ψ(η) =
(n∗ + 1) 2
T − T∞
T∞ s
a ν
2
n∗
+
τ DT T∞
∂T
∂y
2
Q∗ (T − T∞ ) . ρcp f
+
∂y 2
3k∗
(9)
∗ (x + b)n −1 y,
, Φ (η) =
n∗ + 1
∂y ∂y
C − C∞ C∞
,
∗ aν(x + b)n +1 F (η),
EP
θ(η) =
s
∂T ∂C
3 16σ ∗ T∞ ∂2T
1
(ρcp )f
The suitable transformations are
η =
TE D
u
M AN U
where σ ∗ and k∗ presents the Stefan-Boltzmann constant and coefficient of mean absorption. We expect that temperature difference in the flow behavior is such that the term T 4 may be expanded in a Taylor theorem. Hence expending T 4 about T∞ and neglecting higher order terms we obtain
0
AC C
u = a(x + b) F (η) , s # " n∗ − 1 0 n∗ + 1 ∗ F (η) , v = − aν(x + b)n −1 F (η) + η ∗ 2 n +1
(10)
equation (1) is trivially satisfied and other equations yield
F
000
+
−
2n∗
n∗ + 1
F
3 3n∗ − 1 2
∗
02
F
+FF
00
002
n∗ + 1
00
+ (n − 1)ηF F −
1+
00
4 3
2 n∗ + 1
2
−
000
0
Ha F +
#
+ α1
2
"
0 000 ∗ ∗ 00 000 3n − 1 F F + 2(n − 1)ηF F
FF
+ 6α3 Rex
2 n∗ + 1
iv
#
+ α2
n∗ + 1 2
F
"
002
002 ∗ 3n − 1 F
F
(11) 2 n∗ + 1
Pr δθ = 0,
(12)
Nt 00 θ = 0,
0
Φ + ScF Φ +
(13)
Nb
∗
F (α ) = α
0
000
∗ λ θ + N Φ = 0,
2 00 0 0 0 0 R θ + Pr F θ + Pr Nb θ Φ + Pr Nt θ +
∗
1 − n∗ 1 + n∗
∗
!
0
∗
0
∗
∗
Φ (α ) = −Bc (1 + Φ(α )) at α = δ ∗
∗
∗
∗
, F (α ) = 1, θ (α ) = −Bt (1 + θ(α )),
∗
∗
s
a (n∗ + 1) 2v
RI PT
= −hc C at
∂y
∂T
, v = 0,
SC
∂C
n∗
,
∗
F (α ) → 0, θ(α ) → 0, Φ(α ) → 0 as α → ∞,
(14)
3
ACCEPTED MANUSCRIPT
q a(n∗ +1) where α∗ = δ ∗ is a variable related to the thickness of wall (i.e., wall 2v thickness parameter). Defining F (η) = f (η − α∗ ) = f (ξ), θ(η) = θ(η − α∗ ) = θ(ξ) ∗ and Φ(η) = φ(η − α ) = φ(ξ) then Eqs. (11-14) yield " 2 2n∗ 000 f 0 + f f 00 + α1 3n∗ − 1 f 0 f 000 + 2(n∗ − 1)ξf 00 f 000 f − ∗ n +1 # " # 2 3 3n∗ − 1 002 n∗ + 1 f f iv + α2 3n∗ − 1 f 00 + (n∗ − 1)ξf 00 f 000 f + − 2 2 n∗ + 1 002 000 2 2 2 0 ∗ f − f Ha f + λ θ + N φ = 0, 2 n∗ + 1 n∗ + 1
1+
4 3
2 00 0 0 0 0 + R θ + Pr f θ + Pr Nb θ φ + Pr Nt θ
2
n∗ + 1
Pr δθ = 0,
(16)
Nt 00 00 0 θ = 0, φ + Scf φ + Nb
f (ξ) = α
∗
(17) 1 − n∗ 1 + n∗
!
0
0
, f (ξ) = 1, θ (ξ) = −Bt (1 + θ(ξ)),
0
φ (ξ) = −Bc (1 + φ(ξ)) at ξ = 0, 0
f (ξ) → 0, θ(ξ) → 0, φ(ξ) → 0 as ξ → ∞,
(18)
α1 =
α3 = λ = Pr =
n∗ −1 α∗ 1 a(x + b)
μ n∗ −1 α∗ 3 a(x + b)
μ Gr Re2 x
∗
, N =
(μcp )f kf
Gr
Gr
, Nb =
τ DT
∗
, α2 =
n∗ −1 α∗ 2 a(x + b)
2
, Ha = =
μ 2 σB0
u0 ρ
βC C ∞ βT T ∞
τ D B C∞ ν
Q0
,
,
, R=
3 4σ ∗ T∞
kf k ∗
,
,
M AN U
where α1 , α2 and α3 presents third grade fluid variables, Ha for magnetic parameter/Hartmann number, λ for mixed convection variable, N ∗ for ratio of concentration to thermal buoyancy forces, R for radiation parameter, Nb for Brownian motion variable, Nt for thermophoresis variable, Pr for Prandtl number, δ for heat generation/absorption variable, Sc for Schmidt number, Bt and Bc for thermal and solutal conjugate variables, Gr and Gr ∗ for Grashof numbers in the frame of temperature and concentration. These variables are described as follows:
Gr =
ν2
∗
, Gr =
gΛ3 C∞ (x + b)3 ν2
.
(19)
Physical quantities
EP
2.1
gΛ1 T∞ (x + b)3
TE D
ν , δ= , , Sc = νT∞ (ρcp )f a DB r r ν ν ∗ ∗ Bt = ht , B c = hc , a a
Nt =
The skin friction coefficient Cfx , local Nussult N ux and Sherwood Shx numbers are describe as τw (x + b) qw (x + b) jw C fx = 1 , N ux = , Shx = , ρu2 kf (T − T∞ ) DB (C − C∞ ) w 2
(20)
AC C
where τw for surface shear stress, qw for surface heat flux, qm for surface mass flux. The appropriate expressions for these quantities are as follows ! " 3 # ∂2u ∂u ∂u ∂2u ∂u ∗ ∗ ∂u τw = μ + 2α3 + α1 u +2 +v 1−n∗ , 2 ∂y ∂x∂y ∂x ∂y ∂y ∂y 2 y=δ ∗ (x+b) ! 3 16σ ∗ T∞ ∂T ∂C . (21) qw = −kf 1 + 1−n∗ , jw = −DB 3kf k∗ ∂y y=δ∗ (x+b) 2 ∂y Skin friction coefficient Cfx and local Nusselt N ux and local Sherwood Shx numbers in dimensionless form are s " ! 7n∗ − 1 1 n∗ + 1 0.5 00 0 00 f (0) + α1 f (0)f (0) Rex Cfx = 2 2 2 ! ! 2 n∗ + 1 00 ∗ ∗ 000 f (0)f (0) + α n − 1 f (0) − 2 # ! 3 n∗ + 1 0.5 00 , ξ f (0) +2α3 Rex 2 s ! 4 1 n∗ + 1 −0.5 Rex N ux = Bt 1 + R , 1+ 2 3 θ (0) s ! n∗ + 1 1 −0.5 , (22) Bc 1 + Shx = Rex 2 φ(0) in which Rex =
RI PT
(15)
SC
+ 6α3 Rex
uw (x+b) denote the local Reynolds number. νf
4
ACCEPTED MANUSCRIPT
3
Solutions expressions
Linear operators Lf , Lθ and Lφ and base function f0 (ξ), θ0 (ξ) and φ0 (ξ) are taken in the form dξ 3
−
df dξ
, Lθ (θ) =
d2 θ dξ 2
f0 (ξ) = (1 − exp (−ξ)) − α∗ φ0 (ξ) =
Bc 1 − Bc
d2 φ
− θ, Lφ (φ) = n∗ − 1 n∗ + 1
!
dξ 2
− φ,
, θ0 (ξ) =
(23)
Bt
exp (−ξ)
1 − Bt
exp (−ξ) ,
(24)
satisfying the following properties Lf [Λ1 + Λ2 exp(−ξ) + Λ3 exp(ξ)] = 0, Lθ [Λ4 exp(−ξ) + Λ5 exp(ξ)] = 0, Lφ [Λ6 exp(−ξ) + Λ7 exp(ξ)] = 0,
(25)
in which Λi (i = 1 − 7) denote the arbitrary constants.
3.1
Zeroth-order systems
M AN U
If embedding variable is denoted by p ˘ ∈ [0, 1] and nonzero auxiliary parameters by ~f , ~θ and ~φ then the relevant problems are i h h i b(ξ; p) b (ξ; p) ˘ f ~f Nf fb(ξ; p) (1 − p) ˘ Lf fb(ξ; p) ˘ − f0 (ξ) = pH ˘ , θ ˘ , φ ˘ , (26)
i h h i b(ξ; p) b(ξ; p) b (ξ; p) ˘ θ ~θ Nθ fb(ξ; p) ˘ − θ0 (ξ) = pH ˘ , θ ˘ , φ ˘ , (1 − p) ˘ Lθ θ i h h i b (ξ; p) b(ξ; p) b (ξ; p) ˘ φ ~φ Nφ fb(ξ; p) ˘ − φ0 (ξ) = pH ˘ , θ ˘ , φ ˘ , (1 − p) ˘ Lφ φ 0 ∗ ˘ =α fb (0; p)
1 − n∗ 1 + n∗
!
(27)
(28)
0 0 ˘ = 1 and fb (ξ; p) ˘ → 0 as ξ → ∞, , fb (0; p)
b0 (0; p) b p)) b(ξ; p) θ ˘ = −Bt (1 + θ(0; ˘ and θ ˘ → 0 as ξ → ∞,
b b (ξ; p) b0 (0; p) ˘ = −Bc (1 + φ(0; p)) ˘ and φ ˘ → 0 as ξ → ∞. φ
(29)
Nonlinear operators N f , Nθ and Nφ are
−
2n∗
n∗ + 1
+ α1
"
∂ fb(ξ; p) ˘ ∂ξ
∗
3n − 1
∗
+ 2(n − 1)ξ 3 3n∗ − 1
+ fb(ξ; p) ˘
−
2
n∗ + 1
∂ξ 3
∂ξ
∂ξ 2
∂ 2 fb(ξ; p) ˘ ∂ξ 2
∗
3n − 1
∗
+ (n − 1)ξ + 6α3 Rex
− +
2
n∗ + 1 2
n∗ + 1
∂ξ 2
˘ ∂ 3 fb(ξ; p) ˘ ∂ 2 fb(ξ; p)
fb(ξ; p) ˘
AC C
+ α2
2 "
˘ ∂ 2 fb(ξ; p)
˘ ∂ 3 fb(ξ; p) ˘ ∂ fb(ξ; p)
EP
+
!2
TE D
h i ∂ 3 fb(ξ; p) ˘ b(ξ; p) b (ξ; p) Nf fb(ξ; p) ˘ , θ ˘ , φ ˘ = ∂ξ 3
˘ ∂ 4 fb(ξ; p)
∂ξ 4
∂ξ 3 !2 #
∂ 2 fb(ξ; p) ˘ ∂ξ 2
!2
˘ ∂ 3 fb(ξ; p) ˘ ∂ 2 fb(ξ; p) ∂ξ 2
∗
n +1 2
∂ξ 3
˘ ∂ 2 fb(ξ; p) ∂ξ 2
#
!2
˘ ∂ 3 fb(ξ; p) ∂ξ 3
b ˘ 2 ∂ f (ξ; p)
Ha
∂ξ ∗b b p) λ θ(ξ; ˘ +N φ (ξ; p) ˘ ,
(30)
2b h i b p) ˘ 4 ∂ θ(ξ; p) ∂ θ(ξ; ˘ b(ξ; p) b (ξ; p) ˘ , θ ˘ , φ ˘ = 1+ R Nθ fb(ξ; p) + Pr fb(ξ; p) ˘ 3 ∂ξ 2 ∂ξ + Pr Nb + Pr Nt
b p) b p) ∂ θ(ξ; ˘ ∂ φ(ξ; ˘ ∂ξ
b p) ∂ θ(ξ; ˘ ∂ξ
∂ξ !2
+
RI PT
d3 f
SC
Lf (f ) =
2 n∗ + 1
b p), Pr δ θ(ξ; ˘
(31)
h i ∂ 2 φ(ξ; b p) b p) b p) ˘ Nt ∂ 2 θ(ξ; ˘ ∂ φ(ξ; ˘ b(ξ; p) b (ξ; p) Nφ fb(ξ; p) + + Scfb(ξ; p) ˘ , ˘ , θ ˘ , φ ˘ = 2 ∂ξ ∂ξ Nb ∂ξ 2
(32)
5
ACCEPTED MANUSCRIPT
3.2
mth-order deformation systems
We can express f Lf fm (ξ) − χm fm−1 (ξ) = ~f Rm (ξ) ,
(33)
θ
Lθ θm (ξ) − χm θm−1 (ξ) = ~θ Rm (ξ) , φ Lφ φm (ξ) − χm φm−1 (ξ) = ~φ Rm (ξ) ,
(34) (35)
0
0
fm (0) = 0, fm (0) = 0 and fm (ξ) → 0 as ξ → ∞, 0
θm (0) + Bt θm (0) = 0 and θm (ξ) → 0 as ξ → ∞, 0
φm (0) + Bc φm (0) = 0 and φm (ξ) → 0 as ξ → ∞, " m−1 X k=0
" m−1 X k=0
3 3n∗ − 1
+
+ 6α3 Rex 2 n∗ + 1
θ
Rm (ξ) =
00
n∗ + 1
00
fm−1−k fk −
m−1 X
!
4
1+
l=0
3
2
n∗ + 1
m−1 X 00 0 R θm−1 (ξ) + Pr fm−1−k θk k=0
m−1 X
0
0
0
2
Pr δθm−1 (ξ) ,
n∗ + 1
m−1 X
0
fm−1−k φk +
k=0
0, 1,
m ≤ 1, m > 1,
0
Nb θm−1−k φk + Nt θm−1−k θk
,
(38)
Nt 00 θm−1 (ξ) ,
TE D
00
Rm (ξ) = φm−1 (ξ) + Sc (
#
∗ λ θm−1 (ξ) + N φm−1 (ξ) , (37)
k=0
+
χm =
#
iv
fm−1−k fk
k X 00 00 fk−l fl
000
fm−1−k
2 0 Ha fm−1 (ξ) +
+ Pr
φ
0
0 ∗ 000 ∗ 00 000 3n − 1 fm−1−k fk + 2(n − 1)ξfm−1−k fk
k=0
−
n∗ + 1
0
fm−1−k fk
2 2 i 00 00 ∗ 00 ∗ 000 3n − 1 fm−1−k fk + n − 1 ξfm−1−k fk
h
+ α2
2n∗
M AN U
+ α1
00
fm−1−k fk −
SC
000
f
Rm (ξ) = fm−1 (ξ) +
(36)
(39)
Nb
(40)
b(ξ; 0) = θ0 (ξ) , φ b (ξ; 0) = φ0 (ξ) when p fb(ξ; 0) = f0 (ξ) , θ ˘ = 0,
b(ξ; 1) = θ (ξ) , φ b (ξ; 1) = φ (ξ) when p fb(ξ; 1) = f (ξ) , θ ˘ = 1.
EP
(41)
b(ξ; p) b (ξ; p) Here we noticed that when p ˘ varies from 0 to 1 then fb(ξ; p) ˘ ,θ ˘ and φ ˘ vary from the base solutions f0 (ξ) , θ0 (ξ) and φ0 (ξ) to the desired solutions f (ξ) , θ(ξ) and φ(ξ) respectively. The solutions can be written through Taylor’s series as follows: ∞ X
fm (ξ) p ˘
∞ X
θm (ξ) p ˘
m
with
fm (ξ) =
AC C
fb(ξ; p) ˘ = f0 (ξ) + b(ξ; p) θ ˘ = θ0 (ξ) +
m=1
m
m=1
b (ξ; p) φ ˘ = φ0 (ξ) +
∞ X
m
φm (ξ) p ˘
m=1
1 ∂ m fb(ξ; p) ˘ m! ∂p ˘m
,
(42)
p=0 ˘
mb ˘ 1 ∂ θ (ξ; p) with θm (ξ) = , m! ∂p ˘m p=0 ˘
(43)
mb ˘ 1 ∂ φ (ξ; p) with φm (ξ) = . m m! ∂p ˘ p=0 ˘
(44)
The series of f (ξ) , θ (ξ) and φ (ξ) are convergent for p ˘ = 1 and thus ∞ X
f (ξ) = f0 (ξ) +
fm (ξ) ,
m=1
θ (ξ) = θ0 (ξ) +
∞ X
θm (ξ) ,
m=1
φ (ξ) = φ0 (ξ) +
∞ X
RI PT
φm (ξ) .
(45)
m=1
The general solution fm (ξ) , θm (ξ) and φm (ξ) in the frame of special functions ∗ ∗ fm (ξ) , θm (ξ) and φ∗ m (ξ) are given by ∗
fm (ξ) = fm (ξ) + Λ1 + Λ2 exp(−ξ) + Λ3 exp(ξ), ∗
θm (ξ) = θm (ξ) + Λ4 exp(−ξ) + Λ5 exp(ξ), ∗
φm (ξ) = φm (ξ) + Λ6 exp(−ξ) + Λ7 exp(ξ),
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where the constant Λi (i = 1 − 7) after using boundary conditions have values ∗ ∗ ∂fm ∂fm (ξ) (ξ) ∗ , Λ3 = 0, ξ=0 − fm (0), Λ2 = ∂ξ ∂ξ ξ=0 # " ∗ ∂θm 1 (ξ) ∗ Λ4 = |η=0 + Bt θm (0) , Λ5 = 0, 1 − Bt ∂ξ # " ∂φ∗ 1 ∗ m (ξ) Λ6 = |η=0 + Bc φm (0) , Λ7 = 0. 1 − Bc ∂ξ Λ1 = −
The auxiliary parameters }f , }θ and }φ have key role for convergence of the
homotopic solutions. The }−curves are plotted at 23 th −order of iterations for the convergence intervals (see Figs. 2). Acceptable values of auxiliary parameters are −1.25 6 }f 6 −0.25, −1.51 6 }θ 6 −0.25 and −1.55 6 }φ 6 −0.3. Table 1: HAM solution convergence when n∗ = 1.5, α1 = 0.1, α2 = 0.1, α3 = 0.1, Rex = 0.2, Ha = 0.1, λ = 0.1, N ∗ = 0.1, R = 0.3, Pr = 1.5, Nb = 0.2, Nt = 0.1, δ = 0.1, Sc = 1.5, α∗ = 0.2, Bt = 0.2 and Bc = 0.2. −f 00 (0)
−θ 0 (0)
−φ0 (0)
1 5 10 15 20 25 30 33 40 45
0.7796 0.7663 0.7594 0.7566 0.7553 0.7545 0.7545 0.7545 0.7545 0.7545
0.2720 0.3105 0.3297 0.3386 0.3431 0.3455 0.3467 0.3472 0.3472 0.3472
0.2698 0.2920 0.2973 0.2988 0.2988 0.2988 0.2988 0.2988 0.2988 0.2988
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Order of approximation
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Table 1 is computed to investigate the convergence analysis for velocity, thermal field and nanoparticles concentration. Presented values explored that th 25 order of approximations are sufficient for f 00 (0) and 33th and 15th orders of approximations are sufficient for θ 0 (0) and φ0 (0).
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Graphical description
This section analyze the behavior of various variables on the velocity, thermal field and nanoparticles concentration are investigated through plots.
Dimensionless velocity
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5.1
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Behavior of power index n∗ on velocity field is seen in Fig 3. Here we noticed that velocity and associated thickness layer are enhanced via larger n∗ . In fact stretching velocity increases due to larger n∗ which produce more deformation in the liquid. Hence velocity increases. Fig. 4-6 demonstrates the features of α1 , α2 and α3 (i.e., third grade fluid parameters) on velocity. Obviously larger fluid variables α1 , α2 and α3 show an enhancement in the boundary layer thickness and velocity. Since material variables are inversely proportional to the viscosity. Thus for larger α1 , α2 and α3 the fluid viscosity decay and hence velocity enhances. Fig. 7 provides impact of magnetic field Ha on velocity. Both velocity and layer thickness are decreased for higher Ha . Physically Lorentz force enhances in the frame of higher magnetic variable which produce more resistance to liquid motion and thus velocity diminishes. Fig. 8 captures features of mixed convection λ on velocity. Both velocity and its layer thickness are increased for higher λ. In fact larger λ give rise to more buoyancy force and so velocity and related layer thickness are enhanced. Fig. 9 disclosed variation of wall thickness parameter α∗ on velocity. It is clear from this Figs. that velocity and thickness layer are increased via larger α∗ . Since stretching velocity enhances when n∗ > 1 therefore velocity increases.
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Convergence analysis
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Dimensionless temperature
Temperature and associated thickness layer are increasing behavior of n∗ (see Fig. 10). With the increase of n∗ stretching rate of the surface increases due to more energetic particle are transferred to the liquid. Thus temperature enhances. Radiation variable R on temperature is described in Fig. 11. It is observed that thermal field and associated layer thickness are increased via R. In fact radiation parameter is inversely proportional to mean absorption coefficient. Higher value of radiation parameter leads to decay k∗ which results in temperature enhancement. It is observe from Fig. 12 that both thermal field and related thickness layer decays for larger values of Prandtl number Pr .
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Dimensionless concentration
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Concentration for n∗ is captured in Fig. 17. Obviously nanoparticle concentration and layer thickness increases in the frame of n∗ . Nanoparticle concentration for Brownian motion variable Nb is drawn in Fig. 18. It is clearly seen that concentration field and related thickness layer are diminished when we give rise to values of Nb . Physically noticed that the collision of liquid particles enhances due to larger Brownian motion variable Nb which correspond to decay in concentration profile. It is disclosed that nanoparticle concentration and its related thickness layer are enhanced for larger thermophoresis variable Nt (see Fig. 19). The thermal conductivity of fluid increases in frame of nanoparticles. In fact larger Nt gives rise to liquid thermal conductivity. Due to higher thermal conductivity the nanoparticle concentration rises. Analysis of Schmidt number Sc on nanoparticle concentration is addressed in Fig. 20. There is a reduction in fluid concentration in the frame of Sc. Since Schmidt number is the ratio of momentum to mass diffusivities. For higher Sc mass diffusivity decay which results in the reduction of nanoparticle concentration distribution. Fig. 21. give the influence of Bc (i.e., solutal conjugate parameter) on the nanoparticles concentration field. This Fig. presents increasing behavior of nanoparticle concentration field and its layer thickness in front of Bc . In fact mass transfer hc increases via larger Bc which results observed that concentration enhances.
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Physically it is due fact that higher Pr lead to decay thermal diffusivity which ensures diminishes in thermal field. Fig. 13 offers the graphical interpretation for temperature versus Brownian motion variable Nb . An increment in thermal field and associated layer thickness in the frame of Nb is noticed. Obviously more heat is delivered through the random motion of liquid particles due to higher Brownian motion Nb and ultimate the temperature enhanced. Fig. 14 clearly illustrates that the thermal field and thickness of related layer are enhanced for higher thermophoresis variable Nt . In thermophoresis process heated fluid particles are pulled away from hot surface region to the cold area. Due to this reason thermal field of the liquid rises. Impact of heat generation/absorption δ on the thermal field is elucidated in Fig. 15. Thermal field and its thickness layer are enhanced for larger heat generation variable (δ > 0) while reverse observation is hold for heat absorption case (δ < 0). Obviously in case of heat generation process more heat is delivered to working liquid which result in the increment of thermal field. Temperature for thermal conjugate parameter Bt is capture in Fig. 16. Thermal field and related layer thickness are boost for larger Bt . The thermal conjugate variable depends on the coefficient of heat transfer ht . An increase in Bt leads to enhancement in coefficient of heat transfer ht . It relates to more heat transfer from heated area to cooled area of liquid. As a whole thermal field of the fluid increases due to more heat transfer from surface to liquid.
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Skin friction coefficient and local
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Nusselt and Sherwood numbers
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Consequences of sundry physical variables on skin friction coefficient is revealed in Table 2. It is examined that skin friction coefficient is increasing function of power index n∗ , third grade fluid parameter α1 , Hartmann number Ha . Skin friction coefficient decreases via third grade fluid variable α2 , mixed convection λ, wall thickness variable α∗ and thermal Bt and solutal conjugate parameters Bc . Table 3 computes for local Nusselt number. Here we see that local Nusselt number enhance for increasing values of power index n∗ , mixed convection λ, radiation R, Prandtl number Pr . However it shows reverse behavior for Brownian motion Nb , thermophoresis Nt , heat generation/absorption δ and wall thickness variable variables α∗ . Table 4 is calculated for local Sherwood number in view of sundry physical variables. Here we observed that that Sherwood number enhances for power index n∗ , mixed convection λ, Brownian motion variable Nb , Schmidt number Sc and solutal conjugate variable Bc while it reduced in view of wall thickness variable α∗ and thermophoresis variable Nt . q Table 2:
Skin friction coefficient 1 2
0.5 2 Rex Cfx when α3 = 0.1, n∗ +1
Rex = 0.2, N ∗ = 0.1, R = 0.3, Pr = 1.5, Nb = 0.2, Nt = 0.1, δ = 0.1 and Sc = 1.5.
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∗
n
n∗ = 1.5, α2 = 0.1, Ha = 0.1, λ = 0.1, α∗ = 0.2, Bt = 0.2, Bc = 0.2
α1
n∗ = 1.5, α1 = 0.1, Ha = 0.1, λ = 0.1, α∗ = 0.2, Bt = 0.2, Bc = 0.2
α2
n∗ = 1.5, α1 = 0.1, α2 = 0.1, λ = 0.1, α∗ = 0.2, Bt = 0.2, Bc = 0.2
Ha
n∗ = 1.5, α1 = 0.1, α2 = 0.1, Ha = 0.1, α∗ = 0.2, Bt = 0.2, Bc = 0.2
λ
n∗ = 1.5, α1 = 0.1, α2 = 0.1, Ha = 0.1, λ = 0.1, Bt = 0.2, Bc = 0.2
α∗
n∗ = 1.5, α1 = 0.1, α2 = 0.1, Ha = 0.1, λ = 0.1, α∗ = 0.2, Bc = 0.2
Bt
n∗ = 1.5, α1 = 0.1, α2 = 0.1, Ha = 0.1, λ = 0.1, α∗ = 0.2, Bt = 0.2
Bc
Local Nusselt number
q
0.0 1.0 1.5 0.1 0.2 0.3 0.1 0.2 0.3 0.1 0.2 0.5 0.1 0.2 0.5 0.2 0.3 0.5 0.2 0.3 0.5 0.2 0.3 0.5
2 Re0.5 x C fx n∗ +1
0.3782 1.0344 1.2534 1.2534 1.4389 1.5995 1.2534 1.1826 1.1197 1.2534 1.2678 1.3592 1.2534 1.2169 1.1110 1.2534 1.2404 1.2093 1.2534 1.1943 0.5239 1.2534 1.2503 1.2243
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α1 = 0.1, α2 = 0.1, Ha = 0.1, λ = 0.1, α = 0.2, Bt = 0.2, Bc = 0.2
q
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∗
Table 3:
−1 2
Parameters
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Parameters (fixed values)
−0.5 2 Rex N ux when α1 = 0.1, n∗ +1
α2 = 0.1, α3 = 0.1, Re = 0.2, Ha = 0.1, N ∗ = 0.1, Sc = 1.5, Bt = 0.2 and Bc = 0.2.
Parameters (fixed values)
Parameters
∗
λ = 0.1, R = 0.3, Pr = 1.5, Nb = 0.2, Nt = 0.1, δ = 0.1, α = 0.2
n∗ = 1.5, R = 0.3, Pr = 1.5, Nb = 0.2, Nt = 0.1, δ = 0.1, α∗ = 0.2
λ
R
0.0 1.0 1.5 0.1 0.2 0.5 0.1 0.3 0.5 1.0 1.5 2.0 0.2 0.4 0.7 0.1 0.2 0.5 0.1 0.2 0.5 0.2 0.3 0.5
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n∗ = 1.5, λ = 0.1, Pr = 1.5, Nb = 0.2, Nt = 0.1, δ = 0.1, α∗ = 0.2
n
∗
Pr
n∗ = 1.5, λ = 0.1, R = 0.3, Pr = 1.5, Nt = 0.1, δ = 0.1, α∗ = 0.2
Nb
n∗ = 1.5, λ = 0.1, R = 0.3, Pr = 1.5, Nb = 0.2, δ = 0.1, α∗ = 0.2
Nt
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n∗ = 1.5, λ = 0.1, R = 0.3, Nb = 0.2, Nt = 0.1, δ = 0.1, α∗ = 0.2
δ
n∗ = 1.5, λ = 0.1, R = 0.3, Pr = 1.5, Nb = 0.2, Nt = 0.1, δ = 0.1
α∗
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n∗ = 1.5, λ = 0.1, R = 0.3, Pr = 1.5, Nb = 0.2, Nt = 0.1, α∗ = 0.2
Table 4: Local Sherwood number
q ∗
−0.5 2 Rex Shx when α1 = 0.1, n∗ +1
α2 = 0.1, α3 = 0.1, Re = 0.2, Ha = 0.1, N = 0.1, R = 0.3, Pr = 1.5, δ = 0.1 and Bt = 0.2.
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2 Re−0.5 N ux x n∗ +1
0.6437 0.6945 0.7366 0.7366 0.7453 0.7663 0.6647 0.7366 0.8053 0.6340 0.7366 0.8491 0.7366 0.7010 0.6476 0.7366 0.7026 0.5933 0.7366 0.5841 0.3289 0.7366 0.7188 0.6776
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∗
∗
λ = 0.1, Nb = 0.2, Nt = 0.1, Sc = 1.5, α = 0.2, Bc = 0.2
n
n∗ = 1.5, Nb = 0.2, Nt = 0.1, Sc = 1.5, α∗ = 0.2, Bc = 0.2
λ
n∗ = 1.5, λ = 0.1, Nt = 0.1, Sc = 1.5, α∗ = 0.2, Bc = 0.2
Nb
n∗ = 1.5, λ = 0.1, Nb = 0.2, Sc = 1.5, α∗ = 0.2, Bc = 0.2
Nt
n∗ = 1.5, λ = 0.1, Nb = 0.2, Nt = 0.1, α∗ = 0.2, Bc = 0.2
Sc
n∗ = 1.5, λ = 0.1, Nb = 0.2, Nt = 0.1, Sc = 1.5, Bc = 0.2
α∗
n∗ = 1.5, λ = 0.1, Nb = 0.2, Nt = 0.1, Sc = 1.5, α∗ = 0.2
Bc
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0.0 1.0 1.5 0.1 0.2 0.5 0.2 0.4 0.7 0.1 0.2 0.5 1.0 1.5 2.0 0.2 0.3 0.5 0.2 0.3 0.5
q
2 Re−0.5 Shx x n∗ +1
0.5298 0.6160 0.6638 0.6638 0.6714 0.6833 0.6638 0.7389 0.7663 0.6638 0.5740 0.4527 0.5428 0.6638 0.7775 0.6638 0.6555 0.6311 0.6638 0.7351 0.7864
Concluding remarks
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Magnetohydrodynamics (MHD) radiative flow of third grade nanofluid over a nonlinear stretching surface with variable thickness and Newtonian heat and mass conditions has been studied. The main outcomes of present study are:
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Parameters
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Parameters (fixed values)
• Velocity enhances with the increase of third grade fluid parameters and wall thickness variable.
• Increasing thermal conjugate parameter Bt enhances the temperature and related layer.
• Opposite behavior of concentration field is noticed in view of Nb and Nt .
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• Reduction in skin friction coefficient is observed for mixed convection parameter λ.
• Qualitative behaviors of temperature and heat transfer rate are similar for radiation R.
References
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• Variation of solutal conjugate parameters Bc result in the enhancement of Sherwood number.
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[30] M. Waqas, T. Hayat, S. A. Shehzad and A. Alsaedi, Transport of magnetohydrodynamic nanomaterial in a stratified medium considering gyrotactic microorganisms, Physica B 529 (2018) 33-40.
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Figure captions Fig. 1. Flow configuration over a variable thickness. Fig. 2. }−plots for f 00 (0), θ 0 (0) and φ0 (0). Fig. 3. f 0 (η) variation via n∗ . Fig. 4. f 0 (η) variation via α1 . Fig. 5. f 0 (η) variation via α2 . Fig. 6. f 0 (η) variation via α3 . Fig. 7. f 0 (η) variation via Ha . Fig. 8. f 0 (η) variation via λ. Fig. 9. f 0 (η) variation via α∗ . Fig. 10. θ(η) variation via n∗ . Fig. 11. φ(η) variation via R. Fig. 12. θ(η) variation via Pr . Fig. 13. θ(η) variation via Nb . Fig. 14. θ(η) variation via Nt . Fig. 15. θ(η) variation via δ. Fig. 16. θ(η) variation via Bt . Fig. 17. φ(η) variation via n∗ . Fig. 18. φ(η) variation via Nb . Fig. 19. φ(η) variation via Nt . Fig. 20. φ(η) variation via Sc. Fig. 21. φ(η) variation via Bc .
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S0921-4526(18)30073-5
DOI:
10.1016/j.physb.2018.01.043
Reference:
PHYSB 310690
To appear in:
Physica B: Physics of Condensed Matter
Received Date: 25 October 2017 Revised Date:
16 January 2018
Accepted Date: 17 January 2018
Please cite this article as: S. Qayyum, T. Hayat, A. Alsaedi, Thermal radiation and heat generation/ absorption aspects in third grade magneto-nanofluid over a slendering stretching sheet with Newtonian conditions, Physica B: Physics of Condensed Matter (2018), doi: 10.1016/j.physb.2018.01.043. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Thermal radiation and heat generation/absorption aspects in third grade magneto-nanofluid over a slendering stretching sheet with Newtonian conditions Sajid Qayyum a, 1 , Tasawar Hayat a, b and Ahmed Alsaedib
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Abstract: Mathematical modeling for magnetohydrodynamic (MHD) radiative flow of third grade nano-material bounded by a nonlinear stretching sheet with variable thickness is introduced. The sheet moves with nonlinear velocity. Definitions of thermal radiation and heat generation/absorption are utilized in the energy expression. Intention in present investigation is to develop a model for nanomaterial comprising Brownian motion and thermophoresis phenomena. Newtonian conditions for heat and mass species are imposed. Governing equations of the locally similar flow are attempted through a homotopic technique and behaviors of involved variables on the flow fields are displayed graphically. It is revealed that increasing values of thermal conjugate variable corresponds to high temperature. Numerical investigation are explored to obtain the results of skin friction coefficient and local Nusselt and Sherwood numbers. It is revealed that velocity field reduces in the frame of magnetic variable while reverse situation is observed due to mixed convection parameter. Here qualitative behaviors of thermal field and heat transfer rate are opposite for thermophoresis variable. Moreover nanoparticle concentration and local Sherwood number via Brownian motion parameter are opposite. Keywords: Third grade nanofluid; magnetohydrodynamic (MHD); mixed convection; thermal radiation; heat generation/absorption; Newtonian conditions.
Introduction
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The study of non-Newtonian materials have attained the special importance in recent technological and industrial processes. Such material are involved in geophysics, biotechnology, chemical and petroleum engineering. NonNewtonian materials which explore a nonlinear relation among stress and strain rate, possessing a coefficient of variable viscosity and have the features that cannot be elucidated by the Navier-Stokes expressions. Materials such as salt solutions, apple sauce, drilling muds, shampoos, sugar solution pastes, clay coating, tooth pastes, chocolates in liquefies form, hydrogenated caster oils, lubricant, colloidal and suspension solutions interpret the features of nonNewtonian fluids [1-5]. The non-Newtonian fluids at present are argued through three main classifications namely the rate, the differential and the integral. The second-grade liquid model is a simplest description (i.e., differential type) of the non-Newtonian liquids. This description is able to predict the normal stress differences yet it cannot properly explain the phenomena of shear thickening and thinning due to its constant apparent shear viscosity. The third grade liquid theory is sufficient to foresee such phenomena. Therefore, theory of the third grade liquid is appropriated in the present analysis. Some prominent literature for this liquid model can be found in the refs. [6-10]. The diffusion of solid particles of diameter less than 100nm in fluids is known as ”nanofluid”. It is first considered by Maxwell [11]. Since the thermal conductivity of the solid particles is higher than base fluid, Maxwell expected 1
SC
44000, Pakistan Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, P. O. Box 80207, Jeddah 21589, Saudi Arabia
b
RI PT
a Department of Mathematics, Quaid-I-Azam University 45320 , Islamabad
Corresponding author. Tel.: +92 51 90642172.
email address: [email protected] (Sajid Qayyum)
1
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SC
M AN U
the nanofluids could have a better thermal performance. However, it was until that Choi [12] attempted to utilize nanofluids as working agents in heat transfer. Since then, nanofluids have attracted wide attention from industrial cooling, drag delivery, automotive, nuclear power generation, fuel cell, cancer therapy, detergency, dynamic sealing etc. The recent modern technology is a clear observe that the nanoliquids have potential part in manufacturing of cars, microreactors, airplanes, micromachines and many more. The unique physical and chemical aspects of nano-sized material further upgrade the utilizations of nanoliquids in the industrial measurements which cover the meltspinning, petroleum reservoirs, manufacturing of plastic and rubber sheets, drying and cooling of papers, nuclear reactors, extrusion of plastic materials, chemical catalytic reactors, rotors of gas turbines, electronic devices, air cleaning machines and medical equipments. Buongiorno [13] presented that the thermal performance of base liquid is enhanced due to Brownian motion and thermophoretic factors. Some impressive studies on the theory of nanofluids under various circumstances can be addressed in [14-20]. The present work focuses on the modelling and analysis for the magnetohydrodynamic mixed convective flow of third grade nanofluid passing over a nonlinear stretching surface with variable thickness. Concept of heat generation/absorption and thermal radiation are utilized in the energy expression. Our main concern here is to scrutinize the radiative flow of third grade nanofluid in frame of Newtonian conditions. In existing literature the researchers only used the Newtonian heating [21-25]to scrutinize the aspects of different liquid theories under various characteristics and flow geometries. Here we introduced the Newtonian mass condition for nonlinear stretching flow phenomenon of non-Newtonian material. Features of nanoparticles is assumed by considering Brownian motion and thermophoresis. Suitable variables are implementation to transform the nonlinear partial differential systems as a group of nonlinear ordinary differential systems and then resulting systems are solved by homotopy analysis approach [26-35]. Graphs are captured and examined for the consequences of sundry physical variables on the velocity, thermal field and nanoparticles concentration. Skin friction coefficient and local Nusselt and local Sherwood numbers are tabulated and examined. The solutions acquired by HAM are preferred than the numerical solutions in perspective of the following points. (i) HAM gives the solutions within the domain of interest at each point while the numerical solutions hold just for discrete points in the domain. (ii) Algebraically developed approximate solutions require less effort and having a sensible measure of precision when compared to numerical solution which are more convenient for the scientist, an engineer or an applied mathematician. (iii) Although most of the scientific packages required some initial approximations for the solution are not generally convergent. In such conditions approximate solutions can offer better initial guess that can be readily advanced to the exact numerical solution in the limited iterations. Finally an approximate solution, if it is analytical, is most pleasing than the numerical solutions.
RI PT
ACCEPTED MANUSCRIPT
2
Mathematical description
We consider the steady two-dimensional mixed convective flow of third grade nanofluid by a nonlinear stretching surface with variable thickness at y =
1−n∗ 2 . Here δ ∗ is the surface thickness coefficient. Stretching sheet ∗ has velocity uw (x) = a(x + b)n (where a and b denotes the dimensional n∗ −1 constants). A non-uniform magnetic field B(x) = B0 (x + b) 2 is utilized
EP
δ ∗ (x + b)
∂u ∂x
∂v
+
∂y
u
∂u ∂x
AC C
(see Fig. 1). Induced magnetic field for low Reynolds number is neglected. The heat transfer phenomenon has been examined in the frame of heat generation/absorption and thermal radiation. Besides this viscous dissipation and joule heating are not accounted. Newtonian conditions for heat and mass transportation are introduced. Features of nanoparticles is taken in the version of Brownian motion and thermophoresis. In such assumption, the governing boundary layer structure for flow of third grade nanofluid are = 0,
+v
∂u ∂y
= ν
∂2u ∂y 2
+2
+
α∗ 1 ρ
"
u
∂3u ∂x∂y 2
2 α∗ 2 ∂u ∂ u
ρ ∂y ∂x∂y
+6
+v
α∗ 3 ρ
(1)
∂3u
+
∂u ∂ 2 u
+3
∂u ∂ 2 u
∂y 3 ∂x ∂y 2 ∂y ∂x∂y 2 2 ∂ u σB 2 (x) − u ∂y ∂y 2 ρf
#
∂u
+g (βT (T − T∞ ) + βC (C − C∞ )) ,
u
∂T ∂x
+v
∂T ∂y
=
kf
∂2T
(ρcp )f
∂y 2
−
u
∂C ∂x
+v
∂C ∂y
= DB
∂2C ∂y 2
1
∂qr
(ρcp )f ∂y
+
! +
DT
∂2T
T∞
∂y 2
+ τ DB Q∗ (ρcp )f
!
∂T ∂C ∂y ∂y
(2)
+
τ DT T∞
∂T ∂y
2
(T − T∞ ),
(3)
,
(4)
2
ACCEPTED MANUSCRIPT
u = uw (x) = a(x + b)
∂y
= −ht T,
1−n∗ ∗ y = δ (x + b) 2 ,
u → 0, T → T∞ , C → C∞ as y → ∞.
(5)
In the aforestated expressions the velocity components along (x, y) directions ∗ ∗ are denoted by (u, v), ν = (μ/ρ)f for kinematic viscosity, α∗ 1 , α2 and α3 for material variables, σ for electrical conductivity, g for gravitational acceleration, βT and βC for thermal and concentration expansion coefficients, ρf and ρp for fluid and particle densities, (cp )f and (cp )p for fluid and particle heat capacities, kf for thermal conductivity, qr for radiative heat flux, τ = (ρcp )p /(ρcp )f for capacity ratio, DB and DT for Brownian and themophoretic ∗
diffusion coefficients, Q∗ = Q0 (x + b)n −1 for heat generation/absorption coefficient, T and C for fluid temperature and concentration, T∞ and C∞ for ambient temperature and concentration, ht = h∗ t (x + b) n∗ −1
n∗ −1 2 for non-uniform
2 heat transfer coefficient, hc = h∗ for non-uniform mass diffusion c (x + b) coefficient and n∗ denote the power index. Rosseland relation gives radiative heat flux as [35]:
qr = −
4σ ∗ ∂T 4 3k∗ ∂y
,
(6)
4
3
4
T u 4T∞ T − 3T∞ .
(7)
So Eq. (6) becomes
qr = −
3 ∂T 16σ ∗ T∞
3k∗
.
∂y
(8)
Now Eqs. (3) and (8) yield ∂T ∂x
+v
∂T
kf ∂2T + τ DB ρcp f ∂y 2
=
∂y
+
ψ(η) =
(n∗ + 1) 2
T − T∞
T∞ s
a ν
2
n∗
+
τ DT T∞
∂T
∂y
2
Q∗ (T − T∞ ) . ρcp f
+
∂y 2
3k∗
(9)
∗ (x + b)n −1 y,
, Φ (η) =
n∗ + 1
∂y ∂y
C − C∞ C∞
,
∗ aν(x + b)n +1 F (η),
EP
θ(η) =
s
∂T ∂C
3 16σ ∗ T∞ ∂2T
1
(ρcp )f
The suitable transformations are
η =
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u
M AN U
where σ ∗ and k∗ presents the Stefan-Boltzmann constant and coefficient of mean absorption. We expect that temperature difference in the flow behavior is such that the term T 4 may be expanded in a Taylor theorem. Hence expending T 4 about T∞ and neglecting higher order terms we obtain
0
AC C
u = a(x + b) F (η) , s # " n∗ − 1 0 n∗ + 1 ∗ F (η) , v = − aν(x + b)n −1 F (η) + η ∗ 2 n +1
(10)
equation (1) is trivially satisfied and other equations yield
F
000
+
−
2n∗
n∗ + 1
F
3 3n∗ − 1 2
∗
02
F
+FF
00
002
n∗ + 1
00
+ (n − 1)ηF F −
1+
00
4 3
2 n∗ + 1
2
−
000
0
Ha F +
#
+ α1
2
"
0 000 ∗ ∗ 00 000 3n − 1 F F + 2(n − 1)ηF F
FF
+ 6α3 Rex
2 n∗ + 1
iv
#
+ α2
n∗ + 1 2
F
"
002
002 ∗ 3n − 1 F
F
(11) 2 n∗ + 1
Pr δθ = 0,
(12)
Nt 00 θ = 0,
0
Φ + ScF Φ +
(13)
Nb
∗
F (α ) = α
0
000
∗ λ θ + N Φ = 0,
2 00 0 0 0 0 R θ + Pr F θ + Pr Nb θ Φ + Pr Nt θ +
∗
1 − n∗ 1 + n∗
∗
!
0
∗
0
∗
∗
Φ (α ) = −Bc (1 + Φ(α )) at α = δ ∗
∗
∗
∗
, F (α ) = 1, θ (α ) = −Bt (1 + θ(α )),
∗
∗
s
a (n∗ + 1) 2v
RI PT
= −hc C at
∂y
∂T
, v = 0,
SC
∂C
n∗
,
∗
F (α ) → 0, θ(α ) → 0, Φ(α ) → 0 as α → ∞,
(14)
3
ACCEPTED MANUSCRIPT
q a(n∗ +1) where α∗ = δ ∗ is a variable related to the thickness of wall (i.e., wall 2v thickness parameter). Defining F (η) = f (η − α∗ ) = f (ξ), θ(η) = θ(η − α∗ ) = θ(ξ) ∗ and Φ(η) = φ(η − α ) = φ(ξ) then Eqs. (11-14) yield " 2 2n∗ 000 f 0 + f f 00 + α1 3n∗ − 1 f 0 f 000 + 2(n∗ − 1)ξf 00 f 000 f − ∗ n +1 # " # 2 3 3n∗ − 1 002 n∗ + 1 f f iv + α2 3n∗ − 1 f 00 + (n∗ − 1)ξf 00 f 000 f + − 2 2 n∗ + 1 002 000 2 2 2 0 ∗ f − f Ha f + λ θ + N φ = 0, 2 n∗ + 1 n∗ + 1
1+
4 3
2 00 0 0 0 0 + R θ + Pr f θ + Pr Nb θ φ + Pr Nt θ
2
n∗ + 1
Pr δθ = 0,
(16)
Nt 00 00 0 θ = 0, φ + Scf φ + Nb
f (ξ) = α
∗
(17) 1 − n∗ 1 + n∗
!
0
0
, f (ξ) = 1, θ (ξ) = −Bt (1 + θ(ξ)),
0
φ (ξ) = −Bc (1 + φ(ξ)) at ξ = 0, 0
f (ξ) → 0, θ(ξ) → 0, φ(ξ) → 0 as ξ → ∞,
(18)
α1 =
α3 = λ = Pr =
n∗ −1 α∗ 1 a(x + b)
μ n∗ −1 α∗ 3 a(x + b)
μ Gr Re2 x
∗
, N =
(μcp )f kf
Gr
Gr
, Nb =
τ DT
∗
, α2 =
n∗ −1 α∗ 2 a(x + b)
2
, Ha = =
μ 2 σB0
u0 ρ
βC C ∞ βT T ∞
τ D B C∞ ν
Q0
,
,
, R=
3 4σ ∗ T∞
kf k ∗
,
,
M AN U
where α1 , α2 and α3 presents third grade fluid variables, Ha for magnetic parameter/Hartmann number, λ for mixed convection variable, N ∗ for ratio of concentration to thermal buoyancy forces, R for radiation parameter, Nb for Brownian motion variable, Nt for thermophoresis variable, Pr for Prandtl number, δ for heat generation/absorption variable, Sc for Schmidt number, Bt and Bc for thermal and solutal conjugate variables, Gr and Gr ∗ for Grashof numbers in the frame of temperature and concentration. These variables are described as follows:
Gr =
ν2
∗
, Gr =
gΛ3 C∞ (x + b)3 ν2
.
(19)
Physical quantities
EP
2.1
gΛ1 T∞ (x + b)3
TE D
ν , δ= , , Sc = νT∞ (ρcp )f a DB r r ν ν ∗ ∗ Bt = ht , B c = hc , a a
Nt =
The skin friction coefficient Cfx , local Nussult N ux and Sherwood Shx numbers are describe as τw (x + b) qw (x + b) jw C fx = 1 , N ux = , Shx = , ρu2 kf (T − T∞ ) DB (C − C∞ ) w 2
(20)
AC C
where τw for surface shear stress, qw for surface heat flux, qm for surface mass flux. The appropriate expressions for these quantities are as follows ! " 3 # ∂2u ∂u ∂u ∂2u ∂u ∗ ∗ ∂u τw = μ + 2α3 + α1 u +2 +v 1−n∗ , 2 ∂y ∂x∂y ∂x ∂y ∂y ∂y 2 y=δ ∗ (x+b) ! 3 16σ ∗ T∞ ∂T ∂C . (21) qw = −kf 1 + 1−n∗ , jw = −DB 3kf k∗ ∂y y=δ∗ (x+b) 2 ∂y Skin friction coefficient Cfx and local Nusselt N ux and local Sherwood Shx numbers in dimensionless form are s " ! 7n∗ − 1 1 n∗ + 1 0.5 00 0 00 f (0) + α1 f (0)f (0) Rex Cfx = 2 2 2 ! ! 2 n∗ + 1 00 ∗ ∗ 000 f (0)f (0) + α n − 1 f (0) − 2 # ! 3 n∗ + 1 0.5 00 , ξ f (0) +2α3 Rex 2 s ! 4 1 n∗ + 1 −0.5 Rex N ux = Bt 1 + R , 1+ 2 3 θ (0) s ! n∗ + 1 1 −0.5 , (22) Bc 1 + Shx = Rex 2 φ(0) in which Rex =
RI PT
(15)
SC
+ 6α3 Rex
uw (x+b) denote the local Reynolds number. νf
4
ACCEPTED MANUSCRIPT
3
Solutions expressions
Linear operators Lf , Lθ and Lφ and base function f0 (ξ), θ0 (ξ) and φ0 (ξ) are taken in the form dξ 3
−
df dξ
, Lθ (θ) =
d2 θ dξ 2
f0 (ξ) = (1 − exp (−ξ)) − α∗ φ0 (ξ) =
Bc 1 − Bc
d2 φ
− θ, Lφ (φ) = n∗ − 1 n∗ + 1
!
dξ 2
− φ,
, θ0 (ξ) =
(23)
Bt
exp (−ξ)
1 − Bt
exp (−ξ) ,
(24)
satisfying the following properties Lf [Λ1 + Λ2 exp(−ξ) + Λ3 exp(ξ)] = 0, Lθ [Λ4 exp(−ξ) + Λ5 exp(ξ)] = 0, Lφ [Λ6 exp(−ξ) + Λ7 exp(ξ)] = 0,
(25)
in which Λi (i = 1 − 7) denote the arbitrary constants.
3.1
Zeroth-order systems
M AN U
If embedding variable is denoted by p ˘ ∈ [0, 1] and nonzero auxiliary parameters by ~f , ~θ and ~φ then the relevant problems are i h h i b(ξ; p) b (ξ; p) ˘ f ~f Nf fb(ξ; p) (1 − p) ˘ Lf fb(ξ; p) ˘ − f0 (ξ) = pH ˘ , θ ˘ , φ ˘ , (26)
i h h i b(ξ; p) b(ξ; p) b (ξ; p) ˘ θ ~θ Nθ fb(ξ; p) ˘ − θ0 (ξ) = pH ˘ , θ ˘ , φ ˘ , (1 − p) ˘ Lθ θ i h h i b (ξ; p) b(ξ; p) b (ξ; p) ˘ φ ~φ Nφ fb(ξ; p) ˘ − φ0 (ξ) = pH ˘ , θ ˘ , φ ˘ , (1 − p) ˘ Lφ φ 0 ∗ ˘ =α fb (0; p)
1 − n∗ 1 + n∗
!
(27)
(28)
0 0 ˘ = 1 and fb (ξ; p) ˘ → 0 as ξ → ∞, , fb (0; p)
b0 (0; p) b p)) b(ξ; p) θ ˘ = −Bt (1 + θ(0; ˘ and θ ˘ → 0 as ξ → ∞,
b b (ξ; p) b0 (0; p) ˘ = −Bc (1 + φ(0; p)) ˘ and φ ˘ → 0 as ξ → ∞. φ
(29)
Nonlinear operators N f , Nθ and Nφ are
−
2n∗
n∗ + 1
+ α1
"
∂ fb(ξ; p) ˘ ∂ξ
∗
3n − 1
∗
+ 2(n − 1)ξ 3 3n∗ − 1
+ fb(ξ; p) ˘
−
2
n∗ + 1
∂ξ 3
∂ξ
∂ξ 2
∂ 2 fb(ξ; p) ˘ ∂ξ 2
∗
3n − 1
∗
+ (n − 1)ξ + 6α3 Rex
− +
2
n∗ + 1 2
n∗ + 1
∂ξ 2
˘ ∂ 3 fb(ξ; p) ˘ ∂ 2 fb(ξ; p)
fb(ξ; p) ˘
AC C
+ α2
2 "
˘ ∂ 2 fb(ξ; p)
˘ ∂ 3 fb(ξ; p) ˘ ∂ fb(ξ; p)
EP
+
!2
TE D
h i ∂ 3 fb(ξ; p) ˘ b(ξ; p) b (ξ; p) Nf fb(ξ; p) ˘ , θ ˘ , φ ˘ = ∂ξ 3
˘ ∂ 4 fb(ξ; p)
∂ξ 4
∂ξ 3 !2 #
∂ 2 fb(ξ; p) ˘ ∂ξ 2
!2
˘ ∂ 3 fb(ξ; p) ˘ ∂ 2 fb(ξ; p) ∂ξ 2
∗
n +1 2
∂ξ 3
˘ ∂ 2 fb(ξ; p) ∂ξ 2
#
!2
˘ ∂ 3 fb(ξ; p) ∂ξ 3
b ˘ 2 ∂ f (ξ; p)
Ha
∂ξ ∗b b p) λ θ(ξ; ˘ +N φ (ξ; p) ˘ ,
(30)
2b h i b p) ˘ 4 ∂ θ(ξ; p) ∂ θ(ξ; ˘ b(ξ; p) b (ξ; p) ˘ , θ ˘ , φ ˘ = 1+ R Nθ fb(ξ; p) + Pr fb(ξ; p) ˘ 3 ∂ξ 2 ∂ξ + Pr Nb + Pr Nt
b p) b p) ∂ θ(ξ; ˘ ∂ φ(ξ; ˘ ∂ξ
b p) ∂ θ(ξ; ˘ ∂ξ
∂ξ !2
+
RI PT
d3 f
SC
Lf (f ) =
2 n∗ + 1
b p), Pr δ θ(ξ; ˘
(31)
h i ∂ 2 φ(ξ; b p) b p) b p) ˘ Nt ∂ 2 θ(ξ; ˘ ∂ φ(ξ; ˘ b(ξ; p) b (ξ; p) Nφ fb(ξ; p) + + Scfb(ξ; p) ˘ , ˘ , θ ˘ , φ ˘ = 2 ∂ξ ∂ξ Nb ∂ξ 2
(32)
5
ACCEPTED MANUSCRIPT
3.2
mth-order deformation systems
We can express f Lf fm (ξ) − χm fm−1 (ξ) = ~f Rm (ξ) ,
(33)
θ
Lθ θm (ξ) − χm θm−1 (ξ) = ~θ Rm (ξ) , φ Lφ φm (ξ) − χm φm−1 (ξ) = ~φ Rm (ξ) ,
(34) (35)
0
0
fm (0) = 0, fm (0) = 0 and fm (ξ) → 0 as ξ → ∞, 0
θm (0) + Bt θm (0) = 0 and θm (ξ) → 0 as ξ → ∞, 0
φm (0) + Bc φm (0) = 0 and φm (ξ) → 0 as ξ → ∞, " m−1 X k=0
" m−1 X k=0
3 3n∗ − 1
+
+ 6α3 Rex 2 n∗ + 1
θ
Rm (ξ) =
00
n∗ + 1
00
fm−1−k fk −
m−1 X
!
4
1+
l=0
3
2
n∗ + 1
m−1 X 00 0 R θm−1 (ξ) + Pr fm−1−k θk k=0
m−1 X
0
0
0
2
Pr δθm−1 (ξ) ,
n∗ + 1
m−1 X
0
fm−1−k φk +
k=0
0, 1,
m ≤ 1, m > 1,
0
Nb θm−1−k φk + Nt θm−1−k θk
,
(38)
Nt 00 θm−1 (ξ) ,
TE D
00
Rm (ξ) = φm−1 (ξ) + Sc (
#
∗ λ θm−1 (ξ) + N φm−1 (ξ) , (37)
k=0
+
χm =
#
iv
fm−1−k fk
k X 00 00 fk−l fl
000
fm−1−k
2 0 Ha fm−1 (ξ) +
+ Pr
φ
0
0 ∗ 000 ∗ 00 000 3n − 1 fm−1−k fk + 2(n − 1)ξfm−1−k fk
k=0
−
n∗ + 1
0
fm−1−k fk
2 2 i 00 00 ∗ 00 ∗ 000 3n − 1 fm−1−k fk + n − 1 ξfm−1−k fk
h
+ α2
2n∗
M AN U
+ α1
00
fm−1−k fk −
SC
000
f
Rm (ξ) = fm−1 (ξ) +
(36)
(39)
Nb
(40)
b(ξ; 0) = θ0 (ξ) , φ b (ξ; 0) = φ0 (ξ) when p fb(ξ; 0) = f0 (ξ) , θ ˘ = 0,
b(ξ; 1) = θ (ξ) , φ b (ξ; 1) = φ (ξ) when p fb(ξ; 1) = f (ξ) , θ ˘ = 1.
EP
(41)
b(ξ; p) b (ξ; p) Here we noticed that when p ˘ varies from 0 to 1 then fb(ξ; p) ˘ ,θ ˘ and φ ˘ vary from the base solutions f0 (ξ) , θ0 (ξ) and φ0 (ξ) to the desired solutions f (ξ) , θ(ξ) and φ(ξ) respectively. The solutions can be written through Taylor’s series as follows: ∞ X
fm (ξ) p ˘
∞ X
θm (ξ) p ˘
m
with
fm (ξ) =
AC C
fb(ξ; p) ˘ = f0 (ξ) + b(ξ; p) θ ˘ = θ0 (ξ) +
m=1
m
m=1
b (ξ; p) φ ˘ = φ0 (ξ) +
∞ X
m
φm (ξ) p ˘
m=1
1 ∂ m fb(ξ; p) ˘ m! ∂p ˘m
,
(42)
p=0 ˘
mb ˘ 1 ∂ θ (ξ; p) with θm (ξ) = , m! ∂p ˘m p=0 ˘
(43)
mb ˘ 1 ∂ φ (ξ; p) with φm (ξ) = . m m! ∂p ˘ p=0 ˘
(44)
The series of f (ξ) , θ (ξ) and φ (ξ) are convergent for p ˘ = 1 and thus ∞ X
f (ξ) = f0 (ξ) +
fm (ξ) ,
m=1
θ (ξ) = θ0 (ξ) +
∞ X
θm (ξ) ,
m=1
φ (ξ) = φ0 (ξ) +
∞ X
RI PT
φm (ξ) .
(45)
m=1
The general solution fm (ξ) , θm (ξ) and φm (ξ) in the frame of special functions ∗ ∗ fm (ξ) , θm (ξ) and φ∗ m (ξ) are given by ∗
fm (ξ) = fm (ξ) + Λ1 + Λ2 exp(−ξ) + Λ3 exp(ξ), ∗
θm (ξ) = θm (ξ) + Λ4 exp(−ξ) + Λ5 exp(ξ), ∗
φm (ξ) = φm (ξ) + Λ6 exp(−ξ) + Λ7 exp(ξ),
(46)
6
ACCEPTED MANUSCRIPT
where the constant Λi (i = 1 − 7) after using boundary conditions have values ∗ ∗ ∂fm ∂fm (ξ) (ξ) ∗ , Λ3 = 0, ξ=0 − fm (0), Λ2 = ∂ξ ∂ξ ξ=0 # " ∗ ∂θm 1 (ξ) ∗ Λ4 = |η=0 + Bt θm (0) , Λ5 = 0, 1 − Bt ∂ξ # " ∂φ∗ 1 ∗ m (ξ) Λ6 = |η=0 + Bc φm (0) , Λ7 = 0. 1 − Bc ∂ξ Λ1 = −
The auxiliary parameters }f , }θ and }φ have key role for convergence of the
homotopic solutions. The }−curves are plotted at 23 th −order of iterations for the convergence intervals (see Figs. 2). Acceptable values of auxiliary parameters are −1.25 6 }f 6 −0.25, −1.51 6 }θ 6 −0.25 and −1.55 6 }φ 6 −0.3. Table 1: HAM solution convergence when n∗ = 1.5, α1 = 0.1, α2 = 0.1, α3 = 0.1, Rex = 0.2, Ha = 0.1, λ = 0.1, N ∗ = 0.1, R = 0.3, Pr = 1.5, Nb = 0.2, Nt = 0.1, δ = 0.1, Sc = 1.5, α∗ = 0.2, Bt = 0.2 and Bc = 0.2. −f 00 (0)
−θ 0 (0)
−φ0 (0)
1 5 10 15 20 25 30 33 40 45
0.7796 0.7663 0.7594 0.7566 0.7553 0.7545 0.7545 0.7545 0.7545 0.7545
0.2720 0.3105 0.3297 0.3386 0.3431 0.3455 0.3467 0.3472 0.3472 0.3472
0.2698 0.2920 0.2973 0.2988 0.2988 0.2988 0.2988 0.2988 0.2988 0.2988
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Order of approximation
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Table 1 is computed to investigate the convergence analysis for velocity, thermal field and nanoparticles concentration. Presented values explored that th 25 order of approximations are sufficient for f 00 (0) and 33th and 15th orders of approximations are sufficient for θ 0 (0) and φ0 (0).
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Graphical description
This section analyze the behavior of various variables on the velocity, thermal field and nanoparticles concentration are investigated through plots.
Dimensionless velocity
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5.1
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Behavior of power index n∗ on velocity field is seen in Fig 3. Here we noticed that velocity and associated thickness layer are enhanced via larger n∗ . In fact stretching velocity increases due to larger n∗ which produce more deformation in the liquid. Hence velocity increases. Fig. 4-6 demonstrates the features of α1 , α2 and α3 (i.e., third grade fluid parameters) on velocity. Obviously larger fluid variables α1 , α2 and α3 show an enhancement in the boundary layer thickness and velocity. Since material variables are inversely proportional to the viscosity. Thus for larger α1 , α2 and α3 the fluid viscosity decay and hence velocity enhances. Fig. 7 provides impact of magnetic field Ha on velocity. Both velocity and layer thickness are decreased for higher Ha . Physically Lorentz force enhances in the frame of higher magnetic variable which produce more resistance to liquid motion and thus velocity diminishes. Fig. 8 captures features of mixed convection λ on velocity. Both velocity and its layer thickness are increased for higher λ. In fact larger λ give rise to more buoyancy force and so velocity and related layer thickness are enhanced. Fig. 9 disclosed variation of wall thickness parameter α∗ on velocity. It is clear from this Figs. that velocity and thickness layer are increased via larger α∗ . Since stretching velocity enhances when n∗ > 1 therefore velocity increases.
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Convergence analysis
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Dimensionless temperature
Temperature and associated thickness layer are increasing behavior of n∗ (see Fig. 10). With the increase of n∗ stretching rate of the surface increases due to more energetic particle are transferred to the liquid. Thus temperature enhances. Radiation variable R on temperature is described in Fig. 11. It is observed that thermal field and associated layer thickness are increased via R. In fact radiation parameter is inversely proportional to mean absorption coefficient. Higher value of radiation parameter leads to decay k∗ which results in temperature enhancement. It is observe from Fig. 12 that both thermal field and related thickness layer decays for larger values of Prandtl number Pr .
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Dimensionless concentration
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Concentration for n∗ is captured in Fig. 17. Obviously nanoparticle concentration and layer thickness increases in the frame of n∗ . Nanoparticle concentration for Brownian motion variable Nb is drawn in Fig. 18. It is clearly seen that concentration field and related thickness layer are diminished when we give rise to values of Nb . Physically noticed that the collision of liquid particles enhances due to larger Brownian motion variable Nb which correspond to decay in concentration profile. It is disclosed that nanoparticle concentration and its related thickness layer are enhanced for larger thermophoresis variable Nt (see Fig. 19). The thermal conductivity of fluid increases in frame of nanoparticles. In fact larger Nt gives rise to liquid thermal conductivity. Due to higher thermal conductivity the nanoparticle concentration rises. Analysis of Schmidt number Sc on nanoparticle concentration is addressed in Fig. 20. There is a reduction in fluid concentration in the frame of Sc. Since Schmidt number is the ratio of momentum to mass diffusivities. For higher Sc mass diffusivity decay which results in the reduction of nanoparticle concentration distribution. Fig. 21. give the influence of Bc (i.e., solutal conjugate parameter) on the nanoparticles concentration field. This Fig. presents increasing behavior of nanoparticle concentration field and its layer thickness in front of Bc . In fact mass transfer hc increases via larger Bc which results observed that concentration enhances.
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Physically it is due fact that higher Pr lead to decay thermal diffusivity which ensures diminishes in thermal field. Fig. 13 offers the graphical interpretation for temperature versus Brownian motion variable Nb . An increment in thermal field and associated layer thickness in the frame of Nb is noticed. Obviously more heat is delivered through the random motion of liquid particles due to higher Brownian motion Nb and ultimate the temperature enhanced. Fig. 14 clearly illustrates that the thermal field and thickness of related layer are enhanced for higher thermophoresis variable Nt . In thermophoresis process heated fluid particles are pulled away from hot surface region to the cold area. Due to this reason thermal field of the liquid rises. Impact of heat generation/absorption δ on the thermal field is elucidated in Fig. 15. Thermal field and its thickness layer are enhanced for larger heat generation variable (δ > 0) while reverse observation is hold for heat absorption case (δ < 0). Obviously in case of heat generation process more heat is delivered to working liquid which result in the increment of thermal field. Temperature for thermal conjugate parameter Bt is capture in Fig. 16. Thermal field and related layer thickness are boost for larger Bt . The thermal conjugate variable depends on the coefficient of heat transfer ht . An increase in Bt leads to enhancement in coefficient of heat transfer ht . It relates to more heat transfer from heated area to cooled area of liquid. As a whole thermal field of the fluid increases due to more heat transfer from surface to liquid.
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Skin friction coefficient and local
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Nusselt and Sherwood numbers
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Consequences of sundry physical variables on skin friction coefficient is revealed in Table 2. It is examined that skin friction coefficient is increasing function of power index n∗ , third grade fluid parameter α1 , Hartmann number Ha . Skin friction coefficient decreases via third grade fluid variable α2 , mixed convection λ, wall thickness variable α∗ and thermal Bt and solutal conjugate parameters Bc . Table 3 computes for local Nusselt number. Here we see that local Nusselt number enhance for increasing values of power index n∗ , mixed convection λ, radiation R, Prandtl number Pr . However it shows reverse behavior for Brownian motion Nb , thermophoresis Nt , heat generation/absorption δ and wall thickness variable variables α∗ . Table 4 is calculated for local Sherwood number in view of sundry physical variables. Here we observed that that Sherwood number enhances for power index n∗ , mixed convection λ, Brownian motion variable Nb , Schmidt number Sc and solutal conjugate variable Bc while it reduced in view of wall thickness variable α∗ and thermophoresis variable Nt . q Table 2:
Skin friction coefficient 1 2
0.5 2 Rex Cfx when α3 = 0.1, n∗ +1
Rex = 0.2, N ∗ = 0.1, R = 0.3, Pr = 1.5, Nb = 0.2, Nt = 0.1, δ = 0.1 and Sc = 1.5.
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∗
n
n∗ = 1.5, α2 = 0.1, Ha = 0.1, λ = 0.1, α∗ = 0.2, Bt = 0.2, Bc = 0.2
α1
n∗ = 1.5, α1 = 0.1, Ha = 0.1, λ = 0.1, α∗ = 0.2, Bt = 0.2, Bc = 0.2
α2
n∗ = 1.5, α1 = 0.1, α2 = 0.1, λ = 0.1, α∗ = 0.2, Bt = 0.2, Bc = 0.2
Ha
n∗ = 1.5, α1 = 0.1, α2 = 0.1, Ha = 0.1, α∗ = 0.2, Bt = 0.2, Bc = 0.2
λ
n∗ = 1.5, α1 = 0.1, α2 = 0.1, Ha = 0.1, λ = 0.1, Bt = 0.2, Bc = 0.2
α∗
n∗ = 1.5, α1 = 0.1, α2 = 0.1, Ha = 0.1, λ = 0.1, α∗ = 0.2, Bc = 0.2
Bt
n∗ = 1.5, α1 = 0.1, α2 = 0.1, Ha = 0.1, λ = 0.1, α∗ = 0.2, Bt = 0.2
Bc
Local Nusselt number
q
0.0 1.0 1.5 0.1 0.2 0.3 0.1 0.2 0.3 0.1 0.2 0.5 0.1 0.2 0.5 0.2 0.3 0.5 0.2 0.3 0.5 0.2 0.3 0.5
2 Re0.5 x C fx n∗ +1
0.3782 1.0344 1.2534 1.2534 1.4389 1.5995 1.2534 1.1826 1.1197 1.2534 1.2678 1.3592 1.2534 1.2169 1.1110 1.2534 1.2404 1.2093 1.2534 1.1943 0.5239 1.2534 1.2503 1.2243
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α1 = 0.1, α2 = 0.1, Ha = 0.1, λ = 0.1, α = 0.2, Bt = 0.2, Bc = 0.2
q
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∗
Table 3:
−1 2
Parameters
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Parameters (fixed values)
−0.5 2 Rex N ux when α1 = 0.1, n∗ +1
α2 = 0.1, α3 = 0.1, Re = 0.2, Ha = 0.1, N ∗ = 0.1, Sc = 1.5, Bt = 0.2 and Bc = 0.2.
Parameters (fixed values)
Parameters
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λ = 0.1, R = 0.3, Pr = 1.5, Nb = 0.2, Nt = 0.1, δ = 0.1, α = 0.2
n∗ = 1.5, R = 0.3, Pr = 1.5, Nb = 0.2, Nt = 0.1, δ = 0.1, α∗ = 0.2
λ
R
0.0 1.0 1.5 0.1 0.2 0.5 0.1 0.3 0.5 1.0 1.5 2.0 0.2 0.4 0.7 0.1 0.2 0.5 0.1 0.2 0.5 0.2 0.3 0.5
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n∗ = 1.5, λ = 0.1, Pr = 1.5, Nb = 0.2, Nt = 0.1, δ = 0.1, α∗ = 0.2
n
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Pr
n∗ = 1.5, λ = 0.1, R = 0.3, Pr = 1.5, Nt = 0.1, δ = 0.1, α∗ = 0.2
Nb
n∗ = 1.5, λ = 0.1, R = 0.3, Pr = 1.5, Nb = 0.2, δ = 0.1, α∗ = 0.2
Nt
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n∗ = 1.5, λ = 0.1, R = 0.3, Nb = 0.2, Nt = 0.1, δ = 0.1, α∗ = 0.2
δ
n∗ = 1.5, λ = 0.1, R = 0.3, Pr = 1.5, Nb = 0.2, Nt = 0.1, δ = 0.1
α∗
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n∗ = 1.5, λ = 0.1, R = 0.3, Pr = 1.5, Nb = 0.2, Nt = 0.1, α∗ = 0.2
Table 4: Local Sherwood number
q ∗
−0.5 2 Rex Shx when α1 = 0.1, n∗ +1
α2 = 0.1, α3 = 0.1, Re = 0.2, Ha = 0.1, N = 0.1, R = 0.3, Pr = 1.5, δ = 0.1 and Bt = 0.2.
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2 Re−0.5 N ux x n∗ +1
0.6437 0.6945 0.7366 0.7366 0.7453 0.7663 0.6647 0.7366 0.8053 0.6340 0.7366 0.8491 0.7366 0.7010 0.6476 0.7366 0.7026 0.5933 0.7366 0.5841 0.3289 0.7366 0.7188 0.6776
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∗
∗
λ = 0.1, Nb = 0.2, Nt = 0.1, Sc = 1.5, α = 0.2, Bc = 0.2
n
n∗ = 1.5, Nb = 0.2, Nt = 0.1, Sc = 1.5, α∗ = 0.2, Bc = 0.2
λ
n∗ = 1.5, λ = 0.1, Nt = 0.1, Sc = 1.5, α∗ = 0.2, Bc = 0.2
Nb
n∗ = 1.5, λ = 0.1, Nb = 0.2, Sc = 1.5, α∗ = 0.2, Bc = 0.2
Nt
n∗ = 1.5, λ = 0.1, Nb = 0.2, Nt = 0.1, α∗ = 0.2, Bc = 0.2
Sc
n∗ = 1.5, λ = 0.1, Nb = 0.2, Nt = 0.1, Sc = 1.5, Bc = 0.2
α∗
n∗ = 1.5, λ = 0.1, Nb = 0.2, Nt = 0.1, Sc = 1.5, α∗ = 0.2
Bc
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0.0 1.0 1.5 0.1 0.2 0.5 0.2 0.4 0.7 0.1 0.2 0.5 1.0 1.5 2.0 0.2 0.3 0.5 0.2 0.3 0.5
q
2 Re−0.5 Shx x n∗ +1
0.5298 0.6160 0.6638 0.6638 0.6714 0.6833 0.6638 0.7389 0.7663 0.6638 0.5740 0.4527 0.5428 0.6638 0.7775 0.6638 0.6555 0.6311 0.6638 0.7351 0.7864
Concluding remarks
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Magnetohydrodynamics (MHD) radiative flow of third grade nanofluid over a nonlinear stretching surface with variable thickness and Newtonian heat and mass conditions has been studied. The main outcomes of present study are:
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Parameters
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Parameters (fixed values)
• Velocity enhances with the increase of third grade fluid parameters and wall thickness variable.
• Increasing thermal conjugate parameter Bt enhances the temperature and related layer.
• Opposite behavior of concentration field is noticed in view of Nb and Nt .
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• Reduction in skin friction coefficient is observed for mixed convection parameter λ.
• Qualitative behaviors of temperature and heat transfer rate are similar for radiation R.
References
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• Variation of solutal conjugate parameters Bc result in the enhancement of Sherwood number.
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Figure captions Fig. 1. Flow configuration over a variable thickness. Fig. 2. }−plots for f 00 (0), θ 0 (0) and φ0 (0). Fig. 3. f 0 (η) variation via n∗ . Fig. 4. f 0 (η) variation via α1 . Fig. 5. f 0 (η) variation via α2 . Fig. 6. f 0 (η) variation via α3 . Fig. 7. f 0 (η) variation via Ha . Fig. 8. f 0 (η) variation via λ. Fig. 9. f 0 (η) variation via α∗ . Fig. 10. θ(η) variation via n∗ . Fig. 11. φ(η) variation via R. Fig. 12. θ(η) variation via Pr . Fig. 13. θ(η) variation via Nb . Fig. 14. θ(η) variation via Nt . Fig. 15. θ(η) variation via δ. Fig. 16. θ(η) variation via Bt . Fig. 17. φ(η) variation via n∗ . Fig. 18. φ(η) variation via Nb . Fig. 19. φ(η) variation via Nt . Fig. 20. φ(η) variation via Sc. Fig. 21. φ(η) variation via Bc .
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