Simulation of magnetohydrodynamics and radiative heat transport in convectively heated stratified flow of Jeffrey nanofluid

Accepted Manuscript Simulation of magnetohydrodynamics and radiative heat transportation in convectively heated stratified flow of Jeffrey nanomaterial M. Waqas, S.A. Shehzad, T. Hayat, M. Ijaz Khan, A. Alsaedi PII:

S0022-3697(18)32487-9

DOI:

https://doi.org/10.1016/j.jpcs.2019.03.031

Reference:

PCS 8963

To appear in:

Journal of Physics and Chemistry of Solids

Received Date: 18 September 2018 Revised Date:

23 March 2019

Accepted Date: 29 March 2019

Please cite this article as: M. Waqas, S.A. Shehzad, T. Hayat, M.I. Khan, A. Alsaedi, Simulation of magnetohydrodynamics and radiative heat transportation in convectively heated stratified flow of Jeffrey nanomaterial, Journal of Physics and Chemistry of Solids (2019), doi: https://doi.org/10.1016/ j.jpcs.2019.03.031. 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

Simulation of magnetohydrodynamics and radiative heat transportation in convectively heated stratified flow of Jeffrey nanomaterial

a

RI PT

M. Waqasa,1 , S.A. Shehzadb , T. Hayatc,d , M. Ijaz Khanc and A. Alsaedid

NUTECH School of Applied Sciences and Humanities, National University of Technology, Islamabad, 44000, Pakistan

Department of Mathematics, COMSATS University Islamabad, Sahiwal Campus, 57000, Pakistan

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

d

M AN U

c

SC

b

Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80257, Jeddah

D

21589, Saudi Arabia

TE

Abstract: The rising requirement regarding energy worldwide necessitates that consideration be devoted to formulating and functioning thermal mechanisms and heat exchangers to utilize and

EP

resuscitate thermal energy. Hence new heat transportation liquids subjected to improved heat

AC C

transport characteristics are required to rise convection heat transportation, and nanoliquids have been proved effective substitutes to standard heat transportation liquids. With such intention, here we formulated mixed convective Jeffrey nanoliquid stratified flow considering magnetohydrodynamics. Heat absorption and heat generation aspects in addition to convective conditions and thermal radiation are considered for formulation. Mathematical modeling is based on theory of boundary-layer. Ordinary systems are acquired from partial ones via implementation of apposite variables. Effectiveness of significant variables is reported through graphical outcomes. It is vi1

Corresponding author:

[email protected], mw [email protected] (M. Waqas)

1

ACCEPTED MANUSCRIPT sualized that thermal radiation consideration corresponds to higher temperature in comparison to stratification phenomenon.

Keywords: Magnetohydrodynamics; Jeffrey nanoliquid; Heat absorption and heat gener-

AC C

EP

TE

D

M AN U

SC

RI PT

ation; Thermal radiation; Stratified flow.

2

ACCEPTED MANUSCRIPT

Nomenclature velocity components

T∞

ambient liquid temperature

x, y

space coordinates

C∞

ambient liquid concentration

ρf

density of base fluid

T0

reference temperature

σ

electrical conductivity

C0

reference concentration

B0

magnetic field strength

g

gravitational acceleration

ν

kinematic viscosity

Q0

k

thermal conductivity

(ρc)f

fluid specific heat

(ρc)p

SC

RI PT

u, v

uniform volumetric heat generation

M AN U

and heat absorption coefficient Stefan-Boltzmann constant

nanoparticles specific heat

uw

stretching velocity

DB

coefficient of Brownian diffusion

c, a1 , a2

dimensional

DT

coefficient of thermophoresis

d1 , d2

constants

Λ1

thermal expansion

D

σ∗

τ=

(ρc)p (ρc)f

TE

diffusion heat capacity ratio temperature

C

nanoparticles concentration

B0 k∗

non-uniform heat generation

Grx

non-uniform magnetic field

heat transfer coefficient

hg

mass transfer coefficient

Tf

hot liquid temperature

Cf

hot liquid concentration

thermal Grashof number

Grx∗

mean absorption coefficient

hf

solutal expansion coefficient

AC C

Q0

Λ2

EP

T

coefficient

solutal Grashof number

λ1

relation of relaxation and retardation times

λ2

3

relaxation time

ACCEPTED MANUSCRIPT

β

Deborah number

γ1

thermal Biot number

Ha

Hartman number

γ2

solutal Biot number

δ

thermal buoyancy parameter

α=

N

buoyancy ratio parameter

Rex

local Reynolds number

R

thermal radiation parameter

η

dimensionless variable

Pr

Prandtl number

f

dimensionless velocity

Nt

thermophoresis parameter

θ

Nb

Brownian motion variable

S1

SC

RI PT

thermal diffusivity

dimensionless temperature

M AN U

k (ρc)f

dimensionless concentration

thermal stratification parameter

Cf

skin-friction coefficient

S2

solutal stratification parameter

Nu

Nusselt number

Sc

Schmidt number

Sh

Sherwood number

S

heat generation (S > 0) and

TE

D

φ

heat absorption (S < 0)

Introduction

EP

1

AC C

The base fluids such as ethanol fuel, oil, water, glycol etc., in presence of solid tiny sized particles are commonly known as “nanofluids”. The features of heat transportation of such fluids generally depend on the thermo-physical nature of nanoparticles, volume fraction of these particles and thermophysical behavior of base fluid. Choi [1] was first who reported the term nanofluid and proved that these fluids have implications in computer devices, vehicle transformers, nuclear reactors, electronic devices, energy production, safer surgery, cancer therapy and biomedical imaging. Buongiorno [2] presented the comprehensive detail of nanofluids heat transportation phenomenon and discussed the key aspects which are responsible in aug4

ACCEPTED MANUSCRIPT mentation of heat transportation performance. Stagnation-point laminar nanofluid stretched flow subjected to variable conductivity is formulated by Zargartalebi et al. [3]. Pal and Mandal [4] elaborated porous medium significance in mixed convective nanoliquid stretching

RI PT

flow considering viscous dissipation. Thermally radiating Eyring-Powell nanoliquid dissipative flow towards unsteady moving surface is modeled by Mahanthesh et al. [5]. Waqas et al. [6] evaluated magneto-Carreau nanoliquid radiating flow with Joule dissipation. So-

SC

lar energy importance in generalized magneto-Burgers nanoliquid flow considering chemical reactions is addressed by Khan et al. [7]. Mahanthesh et al. [8] scrutinized convective con-

M AN U

ditions impact in mixed convective three-dimensional radiated Oldroyd-B nanoliquid flow by stretched surface. Analysis for gyrotactic microorganisms in non-Newtonian (Oldroyd-B) nanoliquid stretched flow by stratified surface is produced by Waqas et al. [9]. Qayyum et al. [10] described Newtonian conditions effectiveness in third-grade magneto nanoliquid flow

D

by radiated surface. A revised nanoliquid model featuring rate type (Maxwell) model sub-

TE

jected to radiation and heat source/sink is presented by Khan et al. [11]. Thermal radiation

EP

characteristics in magneto squeezed nanoliquid flow are scrutinized Ullah et al. [12]. Ashlin and Mahanthesh [13] established exact solutions for radiating Cu − Al2 O3 − H2 O hybrid

AC C

nanoliquids by vertical infinite plate. Nonlinear radiation impact in non-Newtonian (Jeffrey and Carreau) nanoliquid flow is reported by Khan et al. [14, 15]. Flows with mixed convection (free convection and combined forced) has implication in numerous transportation processes for illustration nutrients diffusion, solidification, food processing, reverse osmosis and defroster system [16]. Keeping aforesaid utilizations in attention, some studies covering mixed convection aspect has been reported. For illustration, nonlinear radiation influence in Walter-B nanomaterial mixed convective flow is evaluated by Khan et al. [17]. Mahanthesh et al. [18] formulated boundary-layer mixed convective 5

ACCEPTED MANUSCRIPT nanoliquids radiated flow by melting surface. Convective condition and radiation impacts on non-Newtonian (Cross) fluid considering mixed convection are analyzed by Manzur et al. [19]. Gireesha et al. [20] scrutinized two-phase Oldroyd-B nanoliquid radiated flow subjected

RI PT

to convective condition and mixed convection. Double stratification and nonlinear convection impacts in non-Darcian Maxwell nanoliquid convective flow are reported by Hayat et al. [21] Irfan et al. [22, 23] and Khan et al. [24] modeled and elaborated mixed convective

SC

flows towards radiated stretched surface.

The consideration of stratification aspect is influential regarding analysis of thermal as

M AN U

well as solutal transportation phenomena. It refers to layers deposition subjected to alteration in concentration and temperature variations or by means of liquids combination having diverse densities. The investigations subjected to stratified flows have achieved momentum owing to its considerable utilizations in advanced industrial and technological activities.

D

Particular examples include atmospheric heat rejection for illustration solar ponds, food,

TE

seas, manufacturing, rivers, reservoirs, canals etc. Considering such implications, Hayat et

EP

al. [25] explored mixed convective laminar thixotropic nanoliquid stratified flow with heat source/sink. Characteristics of chemically reacting stratified Williamson liquid by stretched

AC C

cylinder are reported by Rehman et al. [26]. Daniel et al. [27] and Mahanthesh et al. [28] considered stratification aspect to formulate nonlinear convective magnetic fluid towards stretchable surface. Application of improved Fourier-Fick formulas for analysis of non-Newtonian (generalized Burger) fluid with double stratification is presented by Waqas et al. [29]. Waqas et al. [30] scrutinized gyrotactic microorganisms and magnetic field characteristics in Jeffrey nanoliquid stratified flow. The aim here is to formulate the mixed convective Jeffrey nanoliquid flow towards moving stratified surface. The salient aspects like thermal radiation, thermophoretic and Brownian 6

ACCEPTED MANUSCRIPT diffusions, heat generation and heat absorption and convective conditions are considered for modeling. Homotopy method [31-40] is employed for simulations of dimensionless equations.

2

RI PT

Besides, significance of physical constraints is addressed via plots.

Formulation

SC

Two-dimensional incompressible Jeffrey nanoliquid stretching flow is formulated. Liquid is conducting electrically where induced magnetism is overlooked considering smaller Reynolds

M AN U

number. Polarization features are not present owing to electric field absence. Modeling subjected to thermal radiation, thermophoretic and Brownian diffusions, heat source/sink, convective conditions and double stratification is presented. Dissipation phenomenon is not considered. Fig. 1 elaborates physical configuration of modeled problem. Governing

TE

D

expressions subject to aforesaid assumptions are [21, 30]:

(1)

EP

∂u ∂v + = 0, ∂x ∂y

AC C

 2   ∂u ∂u ν ∂ u ∂u ∂ 2 u ∂ 3u ∂u ∂ 2 u ∂ 3u u +v = + λ2 +u − +v 3 ∂x ∂y 1 + λ1 ∂y 2 ∂y ∂x∂y ∂x∂y 2 ∂x ∂y 2 ∂y 2 σB − 0 u + g {Λ1 (T − T∞ ) + Λ2 (C − C∞ )} , ρf

(2)

"  2 # ∂T ∂T ∂ 2T ∂C ∂T DT ∂T u +v = α 2 + τ DB + ∂x ∂y ∂y ∂y ∂y T∞ ∂y 3 16σ ∗ T∞ ∂ 2T Q0 + ∗ + (T − T∞ ) , 2 3k (ρc)f ∂y (ρc)f

u

∂C ∂C ∂ 2 C DT ∂ 2 T +v = DB 2 + , ∂x ∂y ∂y T∞ ∂y 2

7

(3)

(4)

ACCEPTED MANUSCRIPT with conditions [21]: u = uw (x) = cx, v = 0, − k

∂T ∂C = hf (Tf − T ) , − DB = hg (Cf − C) at y = 0,(5) ∂y ∂y

Mathematical forms of Tf and Cf are [21]:

M AN U

Introducing [21]:

SC

Tf = T0 + a1 x, Cf = C0 + a2 x.

√ c , u = cxf 0 (η), v = − cνf (η), ν C − C∞ T − T∞ , φ(η) = , θ(η) = Tf − T0 Cf − C0 r

η = y

(6)

RI PT

u → 0, T → T∞ = T0 + d1 x, C → C∞ = C0 + d2 x when y → ∞.

(7)

(8)

TE

D

Eq. (1) is justified automatically while Eqs. (2)-(6) have the forms

f 000 + (1 + λ1 ) f f 00 − f 02 + β f 002 − f f iv − Ha2 (1 + λ1 ) f 0

EP

+δ (1 + λ1 ) [θ + N φ] = 0,

AC C



(9)

 4 1 + R θ00 + Pr f θ0 − Pr f 0 θ 3

+ Pr Nb φ0 θ0 + Pr Nt θ02 − Pr S1 f 0 + Pr Sθ = 0, φ00 + Scf φ0 − Scf 0 φ +

Nt 00 θ − ScS2 f 0 = 0, Nb

(10) (11)

f = 0, f 0 = 1, θ0 = −γ 1 (1 − S1 − θ (η)) , φ0 = −γ 2 (1 − S2 − φ (η)) at η = 0,

(12)

f 0 → 0, θ → 0, φ → 0 as η → ∞.

(13)

8

ACCEPTED MANUSCRIPT The variables appearing in Eqs. (9)-(12) are defined as follows: Grx σB02 gΛ1 (Tf − T0 )x3 xuw , δ= , , Gr = , Re = x 2 2 x cρf ν ν Rex Grx∗ gΛ2 (Cf − C0 ) x3 d2 ν d1 N= , Grx∗ = , S = , Pr = , , S = 2 1 Grx ν2 a1 a2 α 3 τ DB (Cf − C0 ) 4σ ∗ T∞ τ DT (Tf − T0 ) , Nb = , R= Nt = , T∞ ν ν kk ∗ r r ν hf ν hg ν Q , Sc = , γ1 = , γ = . S= (ρc)f c DB k c 1 DB c

RI PT

β = λ2 c, Ha =

Simplifying Eqs. (15)-(17) we have 1/2

M AN U

SC

Expressions of physical quantities (Cf , N u, Sh) are     2τ w ∂ 2v ∂ 2u µ ∂u ∂ 2u Cf = +u 2 +v 2 , τw = + λ2 u , ρf Uw2 1 + λ1 ∂y ∂x∂y ∂x ∂y    3 16σ ∗ T∞ xqw ∂T , qw = − k + Nu = , ∗ k (Tf − T∞ ) 3k ∂y y=0   xqm ∂C Sh = . , qm = −DB DB (Cf − C∞ ) ∂y y=0

2 (f 00 (0) + βf 00 (0)) , x 1 + λ1
− 12 1 + 34 R 0 N u Re = − θ (0) , x 1 − S1

TE

D

Cf Re =

− 21

Sh Re = −

(16) (17)

(18) (19) (20)

Convergence and results

AC C

3

(15)

EP

x

1 φ0 (0) . 1 − S2

(14)

We opted HAM for nonlinear computations. HAM certifies governing expressions convergence conveniently. Thus, we portrayed Fig. 1 along with Table 1 for convergent solutions authentication. The utilized values for Figs. and Table 1 are β = N = γ 2 = R = 0.4, δ = S2 = γ 1 = 0.3, λ1 = 0.5, S = S1 = Ha = 0.2, Nt = Nb = 0.1, Sc = 1.1 and Pr = 1.2. From Fig. 2, we found 1.35 ≤ }f ≤ −0.25, −1.30 ≤ }θ ≤ −0.20 and −1.32 ≤ }φ ≤ −0.21. Table 1 elaborates that 20th order approximations are adequate for Eqs. (9)-(11) convergence. Besides, features of influential variables is described via Figs. (3)-(20). 9

ACCEPTED MANUSCRIPT Importance of R on θ is described via Fig. 3. Increasing R corresponds to a boost in θ. Larger R diminishes k ∗ which yields θ enhancement. Fig. 4 reveals θ variations subject to Nt . Here, θ augments via higher Nt . Physically, thermophoretic force rises when Nt is

RI PT

increased. Such force assists to abscond nanoparticles by hotter towards colder part and ultimate θ boosts. Influence of S1 is interpreted in Fig. 5. Clearly θ dwindles. Such scenario is observed because difference of temperature between surfaces (heated, away) decreases. Fig.

SC

6 portrays Nb impact on θ. As expected, θ increases. In nanoliquids, the Brownian movement arises due to nanoparticles size (nanometer scale) and at this point, particles movement and

M AN U

its impact against liquid has a significant contribution regarding heat transportation. A rise in Nb yields effective nanoparticles movement inside flow. The strength of this disordered movement augments the nanoparticles kinetic energy and ultimate θ rises. Characteristics of S > 0 and S < 0 versus θ are elaborated via Figs. 7. These Figs. witness that θ increments

D

for S > 0 while it dwindles for S < 0. Physically, extra heat into liquid is produced when

TE

S > 0 while heat is absorbed when S < 0. Consequently, θ increases for S > 0 and diminishes

EP

for S < 0. Fig. 8 highlights γ 1 significance versus θ. According to this Fig., θ enhances. Biot number (γ 1 ) signifies the convective heating strength. Consequently higher γ 1 indicates

AC C

strong convection heating at surface which corresponds to θ increment. Effect of Pr on θ is disclosed in Fig. 9. Here, θ diminishes via larger Pr estimations. The curves of φ for Sc estimations are designed in Fig. 10. Larger Sc estimations yield φ reduction. Physically, mass diffusivity (ν) diminishes when Sc is augmented. Thus, φ decreases. Fig. 11 elucidates Nt impact on φ. We noticed an increment in φ subjected to higher Nt estimations. From physical viewpoint, a rise in thermophoresis force is witnessed through larger Nt which often moves nanoparticles (higher towards lower temperature region). Hence φ augments. Variations of S2 for scrutinization of φ are disclosed in Fig. 12. 10

ACCEPTED MANUSCRIPT Larger S2 corresponds to a reduction in volumetric fraction between surface and reference nanoparticles. So φ dwindles. Fig. 13 elaborates Nb impact on φ. Clearly an augmentation in the Nb magnitude upsurges the speed through which nanoparticles move with diverse

RI PT

velocities in random path owing to Brownian aspect. Consequently, larger Nb estimations creates a decrease in φ and related concentration layer. Outcome of γ 2 on φ is addressed

increments due to an increment in γ 2 . Thus φ enhances.

SC

via Fig. 14. This Fig. illustrates enhancement in φ. In fact coefficient of mass transfer (hg )

  −1/2 −1/2 Influences of sundry variables against physical quantities Cf Re1/2 , N u Re , Sh Re x x x

M AN U

are elaborated via Figs. 15-20. Figs. 15 and 16 report λ1 , β, δ and Ha impacts on Cf Re1/2 x . These Figs. highlight a decay in Cf Re1/2 for larger λ1 and δ estimations while Cf Re1/2 x x boosts when β and Ha are augmented. Characteristics of Nb , Nt , S1 and R on N u Re−1/2 x are interpreted through Figs. 17 and 18. Clearly N u Re−1/2 rises when S1 and R are inx

D

creased whereas opposing situation is noticed for Nb and Nt . Figs. 19 and 20 visualize

TE

Sh Re−1/2 analysis subjected to Nb , Nt , S2 and Sc. As expected Sh Re−1/2 augment when x x

EP

Nb , S2 and Sc are augmented however opposing scenario is noted for Nt . Table 1: Series solutions convergence analysis for several order approximations when

AC C

β = N = γ 2 = R = 0.4, δ = S2 = γ 1 = 0.3, λ1 = 0.5, S = S1 = Ha = 0.2, Nt = Nb = 0.1,

11

ACCEPTED MANUSCRIPT Sc = 1.1 and Pr = 1.2. −f 00 (0) −θ0 (0) −φ0 (0)

1

1.0123

0.1935

0.2142

5

1.0351

0.2003

0.2201

10

1.0382

0.2012

15

1.0387

0.2014

20

1.0387

0.2014

25

1.0387

0.2014

1.0387

Conclusions

0.2198 0.2196 0.2196

SC

M AN U

30

4

RI PT

Order of approximations

0.2014

0.2196 0.2196

Two-dimensional incompressible magneto Jeffrey nanoliquid mixed convective flow by mov-

TE

D

ing stratified surface is described. This research further comprises heat generation and heat absorption, double stratification and convective conditions. Present research facilitates us

EP

to elucidate following conclusions:

AC C

• Thermal field (θ) diminishes when thermal stratification parameter (S1 ) , heat absorption parameter (S < 0) and Prandtl number (Pr) are augmented. • Similar characteristics are witnessed qualitatively for larger radiation parameter (R) , thermophoretic parameter (Nt ) , thermal Biot number (γ 1 ) , Brownian motion variable (Nb ) and heat generation parameter (S > 0) against θ. • Stratification variables (S1 , S2 ) correspond to thermal (θ) and solutal (φ) fields reduction while Biot numbers (γ 1 , γ 2 ) augment thermal (θ) and solutal (φ) fields. • An increment in Brownian motion variable (Nb ) and thermophoretic parameter (Nt ) 12

ACCEPTED MANUSCRIPT yield higher thermal (θ) field whereas these parameters portray opposing characteristics versus solutal (φ) field.   • Larger Deborah number (β) and Hartman number (Ha) yields skin-friction Cf Re1/2 x

and thermal buoyancy parameter (δ) are augmented.

RI PT

enhancement while it diminishes when relation of relaxation and retardation times (λ1 )

SC

  −1 • Nusselt number N ux Rex 2 and thermal (θ) field have reverse impacts for Brownian motion variable (Nb ) , thermophoretic parameter (Nt ) and thermal stratification

M AN U

parameter (S1 ) .

• Increasing solutal stratification parameter (S2 ) and Schmidt number (Sc) diminish   −1 solutal (φ) field however Sherwood number Shx Rex 2 improves when solutal strati-

References

TE

D

fication parameter (S2 ) and Schmidt number (Sc) are increased.

EP

[1] S.U.S. Choi and J.A. Eastman, Enhancing thermal conductivity of fluids with nanoparti-

AC C

cles: The Proceedings of the 1995 ASME International Mechanical Engineering Congress and Exposition, San Francisco, USA, ASME, FED 231/MD, 66 (1995) 99 − 105. [2] J. Buongiorno, Convective transport in nanofluids, ASME Journal of Heat Transfer 128 (2006) 240-250.

[3] H. Zargartalebi, M. Ghalambaz, A. Noghrehabadi and A. Chamkha, Stagnation-point heat transfer of nanofluids toward stretching sheets with variable thermo-physical properties, Advanced Powder Technology 26 (2015) 819-829.

13

ACCEPTED MANUSCRIPT [4] D. Pal and G. Mandal, Mixed convection–radiation on stagnation-point flow of nanofluids over a stretching/shrinking sheet in a porous medium with heat generation and viscous dissipation, Journal of Petroleum Science and Engineering 126 (2015) 16-25.

RI PT

[5] B. Mahanthesh, B. J. Gireesha and R. S. R. Gorla, Unsteady three-dimensional MHD flow of a nano Eyring-Powell fluid past a convectively heated stretching sheet in the

SC

presence of thermal radiation, viscous dissipation and Joule heating, Journal of the Association of Arab Universities for Basic and Applied Sciences 23 (2017) 75-84.

M AN U

[6] M. Waqas, M. I. Khan, T. Hayat and A. Alsaedi, Numerical simulation for magneto Carreau nanofluid model with thermal radiation: A revised model, Computer Methods in Applied Mechanics and Engineering 324 (2017) 640-653.

[7] W. A. Khan, M. Irfan, M. Khan, A. S. Alshomrani, A.K.Alzahrani and M. S. Alghamdi,

D

Impact of chemical processes on magneto nanoparticle for the generalized Burgers fluid,

TE

Journal of Molecular Liquids 234 (2017) 201-208.

EP

[8] B. Mahanthesh, B. J. Gireesha, S. A. Shehzad, F. M. Abbasi and R. S. R. Gorla, Nonlinear three-dimensional stretched flow of an Oldroyd-B fluid with convective con-

AC C

dition, thermal radiation, and mixed convection, Applied Mathematics and Mechanics 38 (2017) 969-980.

[9] M. Waqas, T. Hayat, S. A. Shehzad and A. Alsaedi, Transport of magnetohydrodynamic nanomaterial in a stratified medium considering gyrotactic microorganisms, Physica B: Condensed Matter 529 (2018) 33-40.

14

ACCEPTED MANUSCRIPT [10] S. Qayyum, T. Hayat and A. Alsaedi, Thermal radiation and heat generation/absorption aspects in third grade magneto-nanofluid over a slendering stretching sheet with Newtonian conditions, Physica B: Condensed Matter 537 (2018) 139-149.

RI PT

[11] M. Khan, M. Irfan and W. A. Khan, Impact of heat source/sink on radiative heat transfer to Maxwell nanofluid subject to revised mass flux condition, Results in Physics

SC

9 (2018) 851-857.

[12] I. Ullah, M. Waqas, T. Hayat, A. Alsaedi and M. I. Khan, Thermally radiated squeezed

M AN U

flow of magneto-nanofluid between two parallel disks with chemical reaction, Journal of Thermal Analysis and Calorimetry (2018) DOI: 10.1007/s10973-018-7482-6. [13] T. S. Ashlin and B. Mahanthesh, Exact solution of non-coaxial rotating and non-linear convective flow of Cu − Al2 O3 − H2 O hybrid nanofluids over an infinite vertical plate

TE

D

subjected to heat source and radiative heat, Journal of Nanofluids 8 (2019) 781-794. [14] M. I. Khan, T. Hayat, M. Waqas, A. Alsaedi and M. I. Khan, Effectiveness of radiative

EP

heat flux in MHD flow of Jeffrey-nanofluid subject to Brownian and thermophoresis

AC C

diffusions, Journal of Hydrodynamics (2019) DOI: 10.1007/s42241-019-0003-7. [15] M. I. Khan, A. Kumar, T. Hayat, M. Waqas and R. Singh, Entropy generation in flow of Carreau nanofluid with homogeneous-heterogeneous reactions, Journal of Molecular Liquids (2019) DOI: 10.1016/j.molliq.2018.12.109. [16] S. Ayub, T. Hayat, S. Asghar and B. Ahmad, Thermal radiation impact in mixed convective peristaltic flow of third grade nanofluid, Results in Physics 7 (2017) 36873695.

15

ACCEPTED MANUSCRIPT [17] M. I. Khan, M. Waqas, T. Hayat, A. Alsaedi and M. I. Khan, Significance of nonlinear radiation in mixed convection flow of magneto Walter-B nanoliquid, International Journal of Hydrogen Energy 42 (2017) 26408-26416.

RI PT

[18] B. Mahanthesh, B. J. Gireesha and I. L. Animasaun, Exploration of non-linear thermal radiation and suspended nanoparticles effects on mixed convection boundary layer flow

SC

of nanoliquids on a melting vertical surface, Journal of Nanofluids 7 (2018) 833-843. [19] M. Manzur, M. Khan and M. U. Rahman, Mixed convection heat transfer to cross fluid

M AN U

with thermal radiation: Effects of buoyancy assisting and opposing flows, International Journal of Mechanical Sciences 138-139 (2018) 515-523.

[20] B. J. Gireesha, B. Mahanthesh and K. L. Krupalakshmi, Numerical investigation of two-phase mixed convection flow of particulate Oldroyd-B fluid with non-linear thermal

D

radiation and convective boundary condition, Defect and Diffusion Forum 388 (2018)

TE

204-222.

EP

[21] T. Hayat, S. Naz, M. Waqas and A. Alsaedi, Effectiveness of Darcy-Forchheimer and nonlinear mixed convection aspects in stratified Maxwell nanomaterial flow induced by

AC C

convectively heated surface, Applied Mathematics and Mechanics 39 (2018) 1373-1384.

[22] M. Irfan, W. A. Khan, M. Khan and M. M. Gulzar, Influence of Arrhenius activation energy in chemically reactive radiative flow of 3D Carreau nanofluid with nonlinear mixed convection, Journal of Physics and Chemistry of Solids 125 (2019) 141-152.

16

ACCEPTED MANUSCRIPT [23] M. Irfan, M. Khan and W. A. Khan, Behavior of stratifications and convective phenomena in mixed convection flow of 3D Carreau nanofluid with radiative heat flux, Journal of the Brazilian Society of Mechanical Sciences and Engineering 40 (2018) 521.

RI PT

[24] M. I. Khan, T. Hayat, M. I. Khan, M. Waqas and A. Alsaedi, Numerical simulation of hydromagnetic mixed convective radiative slip flow with variable fluid properties: A

SC

mathematical model for entropy generation, Journal of Physics and Chemistry of Solids 125 (2019) 153-164.

M AN U

[25] T. Hayat, M. Waqas, M. I. Khan and A. Alsaedi, Analysis of thixotropic nanomaterial in a doubly stratified medium considering magnetic field effects, International Journal of Heat and Mass Transfer 102 (2016) 1123-1129.

[26] K. U. Rehman, A. A. Khan, M. Y. Malik, U. Ali and M. Naseer, Numerical analysis

D

subject to double stratification and chemically reactive species on Williamson dual con-

TE

vection fluid flow yield by an inclined stretching cylindrical surface, Chinese Journal of

EP

Physics 55 (2017) 1637-1652.

[27] Y. S. Daniel, Z. A. Aziz, Z. Ismail and F. Salah, Double stratification effects on unsteady

AC C

electrical MHD mixed convection flow of nanofluid with viscous dissipation and Joule heating, Journal of Applied Research and Technology 15 (2017) 464-476. [28] B. Mahanthesh, B. J. Gireesha and C. S. K. Raju, Cattaneo-Christov heat flux on UCM nanofluid flow across a melting surface with double stratification and exponential space dependent internal heat source, Informatics in Medicine Unlocked 9 (2017) 26-34.

17

ACCEPTED MANUSCRIPT [29] M. Waqas, T. Hayat, S. A. Shehzad and A. Alsaedi, Analysis of forced convective modified Burgers liquid flow considering Cattaneo-Christov double diffusion, Results in Physics 8 (2018) 908-913.

RI PT

[30] M. Waqas, M. I. Khan, T. Hayat, S. Farooq and A. Alsaedi, Interaction of thermal radiation in hydromagnetic viscoelastic nanomaterial subject to gyrotactic microorganisms,

SC

Applied Nanoscience (2019) DOI: 10.1007/s13204-018-00938-7.

[31] S. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput.

M AN U

147 (2004) 499-513.

[32] T. Hayat, S. Ali, A. Alsaedi and H. H Alsulami, Influence of thermal radiation and Joule heating in the Eyring–Powell fluid flow with the Soret and Dufour effects, Journal

D

of Applied Mechanics and Technical Physics 57 (2016) 1051-1060.

TE

[33] M. Waqas, M. Farooq, M. I. Khan, A. Alsaedi, T. Hayat and T. Yasmeen, Magnetohydrodynamic (MHD) mixed convection flow of micropolar liquid due to nonlinear

EP

stretched sheet with convective condition, International Journal of Heat and Mass Trans-

AC C

fer 102 (2016) 766-772.

[34] T. Hayat, S. Ali, M. Awais and A. Alsaedi, Joule heating effects in MHD flow of Burgers’ fluid, Heat Transfer Research 47 (2016) 1083-1092. [35] M. Waqas, T. Hayat, M. Farooq, S. A. Shehzad and A. Alsaedi, Cattaneo-Christov heat flux model for flow of variable thermal conductivity generalized Burgers fluid, Journal of Molecular Liquids 220 (2016) 642-648. [36] T. Hayat, M. I. Khan, M. Farooq, A. Alsaedi, M. Waqas and T. Yasmeen, Impact of Cattaneo–Christov heat flux model in flow of variable thermal conductivity fluid over 18

ACCEPTED MANUSCRIPT a variable thicked surface, International Journal of Heat and Mass Transfer 99 (2016) 702-710. [37] M. Waqas, M. I. Khan, T. Hayat and A. Alsaedi, Stratified flow of an Oldroyd-B

RI PT

nanoliquid with heat generation, Results in Physics 7 (2017) 2489-2496.

[38] T. Hayat, H. Khalid, M. Waqas, A. Alsaedi and M. Ayub, Homotopic solutions for stag-

SC

nation point flow of third-grade nanoliquid subject to magnetohydrodynamics, Results in Physics 7 (2017) 4310-4317.

M AN U

[39] M. Waqas, T. Hayat and A. Alsaedi, A theoretical analysis of SWCNT–MWCNT and H2 O nanofluids considering Darcy-Forchheimer relation, Applied Nanoscience (2018) DOI: 10.1007/s13204-018-0833-6.

D

[40] M. Irfan, M. Khan and W. A. Khan, Impact of non-uniform heat sink/source and con-

TE

vective condition in radiative heat transfer to Oldroyd-B nanofluid: A revised proposed

AC C

EP

relation, Physics Letters A 383 (2019) 376-382.

19

ACCEPTED MANUSCRIPT Figure captions Fig. 1. Physical configuration. Fig. 2: }-curves for f, θ and φ.

RI PT

Fig. 3. θ via R. Fig. 4. θ via Nt . Fig. 5. θ via S1 .

SC

Fig. 6. θ via Nb . Fig. 7. θ via S.

M AN U

Fig. 8. θ via γ 1 . Fig. 9. θ via Pr . Fig. 10. φ via Sc. Fig. 11. φ via Nt .

EP

Fig. 14. φ via γ 2 .

TE

Fig. 13. φ via Nb .

D

Fig. 12. φ via S2 .

Fig. 15. Cf Re1/2 via λ1 and β. x

AC C

Fig. 16. Cf Re1/2 via δ and Ha. x

Fig. 17. N u Re−1/2 via Nb and Nt . x Fig. 18. N u Re−1/2 via S1 and R. x Fig. 19. Sh Re−1/2 via Nb and Nt . x Fig. 20. Sh Re−1/2 via S2 and Sc. x

20

ACCEPTED MANUSCRIPT

1. Mixed convection flow of Jeffrey nanomaterial is modeled. 2. Novel idea of combine convective and stratification phenomena is introduced.

energy expression.

AC C

EP

TE D

M AN U

SC

4. Nonlinear convective flow is scrutinized.

RI PT

3. Heat generation and thermal radiation aspects are accounted for modeling