Mixed convection peristaltic motion of copper-water nanomaterial with velocity slip effects in a curved channel

Accepted Manuscript

Mixed convection peristaltic motion of Copper-water nanomaterial with velocity slip effects in a curved channel T. Hayat, S. Farooq, A. Alsaedi PII: DOI: Reference:

S0169-2607(16)30995-6 10.1016/j.cmpb.2017.02.006 COMM 4350

To appear in:

Computer Methods and Programs in Biomedicine

Received date: Revised date: Accepted date:

19 September 2016 20 January 2017 8 February 2017

Please cite this article as: T. Hayat, S. Farooq, A. Alsaedi, Mixed convection peristaltic motion of Copper-water nanomaterial with velocity slip effects in a curved channel, Computer Methods and Programs in Biomedicine (2017), doi: 10.1016/j.cmpb.2017.02.006

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

Highlights • Copper-water nanomaterial in peristalsis is modeled. • First order velocity slip conditions are considered.

CR IP T

• Heat generation/absorption is presented.

AC

CE

PT

ED

M

AN US

• Heat transfer rate at upper wall of the channel is also discussed.

1

ACCEPTED MANUSCRIPT

Mixed convection peristaltic motion of Copper-water nanomaterial with velocity slip effects in a curved channel T. Hayata,b , S. Farooqa,1 and A. Alsaedib Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan

b

Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of

CR IP T

a

Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia Abstract:

Background and objective: The primary objective of present analysis is to model

AN US

the peristalsis of copper-water based nanoliquid in the presence of first order velocity and thermal slip conditions in a curved channel. Mixed convection, viscous dissipation and heat generation/absorption are also accounted. Method:

Mathematical formulation is simplified under the assumption of small

Reynolds number and large wavelength. Regular perturbation technique is employed to

M

find the solution of the resulting equations in terms of series for small Brinkman number. The final expression for pressure gradient, pressure rise, stream function, velocity and tem-

ED

perature are obtained and discussed through graphs. Mathematica software is utilized to compute the solution of the system of equations and to plot the graphical results.

PT

Results: Results indicates that insertion of 30% copper nanoparticles in the basefluid (water) velocity and temperature reduces by almost 3% and 40% respecively. Moreover it

CE

is seen that size of the trapped bolus also reduces almost 20% with the insertion of 20% nanoparticles (copper) in the basefluid (water). Conclusion: It is noted that velocity and temperature are decreasing functions of

AC

nanoparticle volume fraction. Moreover the temperature rises when heat generation parameter and Brinkman number are enhanced. Key words: Copper-water nanofluid; Curved channel; First order velocity slip condi-

tions; Heat generation; Mixed convection. 1

Tel.: + 92 51 90642172.

e-mail address: [email protected]

2

ACCEPTED MANUSCRIPT

1

Introduction

Mixed convection flow has numerous applications in engineering for instance cooling of electronic devices, food processing, solidification, float glass production, nuclear reactors, microelectronic devices, coating and solar power. The communication between the shear driven flow and natural convection has important effects on the enhancement in the flow mixing

CR IP T

and heat transfer process. Volumetric internal heat generation is an important issue in engineering such as nuclear power plants, concrete or inside of earth. Basically in the presence of buoyancy forces, these flows are highly considerable as means to ease the thermal capability in numerous industrial procedures. Convection in channel is also experienced in advanced

AN US

cooling/heating systems, ejection of nuclear waste and in heat exchangers. Keeping in mind the above mentioned key features in engineering and industry the researchers focused towards the topic of mixed convection (see some studies [1-5]). It is an indensible challenge for scientists and researchers to improve the heat transfer ability of conventional liquids including water, engine oil and ethylene glycol having low thermal conductivity. Nanofluid is

M

the mixture of base fluid and nanoparticles (copper, iron, gold, titanium etc.) of nanosized (10-100nm) which is used to improve the heat transfer rate of microelectronics, computer

ED

microchips, transportation, fuel cells, food processing, biomedicine, manufacturing and solid state lightening. Mostly used nanoparticles are of spherical and tubular shaped. Choi [6] was

PT

the first who used the nanomaterial (nanoparticles) to improve thermal conductivity of fluids. Buongiorno [7] suggests that Brownian motion and thermophoresis are very important in nanofluids dynamics. Some modelers used the Buongiorno’s model to study the nanoflu-

CE

ids characteristics given in the refs. [8-11]. Xuan and Li [12] computed the expressions (relations) for the effective viscosity and density of the nanofluids. The effective viscosity

AC

for the two phase flow introduced by Brinkman’s [13] seemed to be in excellent agreement with that of Xuan and Li [14]. The analysis of Xuan and Li [12] and Brinkman [13] was extended by Tiwari and Das [15] which was later utilized by several investigators to explore the characteristics of nanofluids (for detail see refs. [16-26]). The achieved results are found in an excellent agreement with the available experimental research. Flows generated by the oscillatory movement of the channel walls are usually known as peristaltic flows. Such phenomenon is comprised due to sinusoidal motion of the channel walls which propel the fluid filled within the channel/tube. Basically the peristaltic motion

3

ACCEPTED MANUSCRIPT

is the continuous process of wave-like muscular contractions which is useful in transporting physiological materials like chyme through intestine, food bolus through oesophagus, urine transport from kidney to gallbladder, locomotion in earthworm, ovum in the fallopian tube, blood in the small blood vessels transportation of water in trees and in several others. Peristalsis is responsible in moving physiological fluids along the physiological ducts by contrac-

CR IP T

tion/expansion of circular smooth muscle behind the fluids and relaxation of circular smooth muscle ahead of it. Medical devices such as dialysis machine, open heart bypass machine, stethoscope, heart lung machine and ventilator machine obey the principle of peristalsis. The principle of peristalsis is further useful in several industrial processes like transport of sanitary and toxic materials, finger and roller pumps, hose pumps and peristaltic pumps.

AN US

The interesting experimental and theoretical studies was initially introduced by Latham [27] and Shapiro et al. [28]. These two pioneering studies were further extended for both Newtonian and non-Newtonian materials with various flow assumptions as elaborated in the refs. [29-36]. Afterwards the peristaltic flows in straight or planer channels have been considered extensively. However such consideration is not satisfactory when the flows mostly through

M

physiological conduits, arteries and glandular ducts are considered. In such situations and the industry, the flow configurations are of curved shapes. To fill this gap Sato et al. [37]

ED

firstly explored the peristalsis in curved channel. Ali et al. [38] extended the work mentioned in ref. [37] under large wavelength assumption. Heat transfer analysis on peristalsis

PT

of viscous liquid in a curved configuration is explored by Ali et al. [39]. Recently Hayat et al. [40] and [41] examined the impact of radial magnetic field on the peristaltic flow of

CE

Carreau-Yasuda liquid and Carreau liquid with wall properties and convective conditions in a curved channel respectively. Ali et al. [42] numerically discussed the heat transfer effects

AC

on peristaltic flow of an Oldroyed 8-constant liquid in a curved channel. Further Hayat et al. [43] extended the study presented in ref. [42] by considering convective heat and mass transfer effects in a curved channel. Hina et al. [44] considered the peristalsis of nanofluid in a curved channel. Magnetohydrodynamic pressure driven peristaltic flow of nanofluid in a curved geometry is analyzed by Noreen et al. [45]. Nadeem et al. [46] presented the peristaltic flow of Williamson nanomaterial in a curved channel with wall properties. Nadeem et al. [47] also studied the peristalsis of two phase nanofluid model in a curved channel. Narla et al. [48] formulated the peristaltic flow of Jeffrey nanofluid in a curved channel. Peristalsis of nanofluid has a significant role in the drug delivery systems and in can4

ACCEPTED MANUSCRIPT

cer therapy as means to destroy the tissues. Literature survey indicates that attempts on the peristalsis of two phase nanofluid model in a curved channel are rare. The intention of the present communication is to explore such fluid model describing peristalsis in a curved channel. Thus our main purpose here is to investigate mixed convection peristalsis of copper-water nanofluid in a curved channel. The channel walls verify the velocity conditions.

CR IP T

Impacts of viscous dissipation and heat generation/absorption are also taken into account. The curved channel is filled with an incompressible copper-water nanoliquid. Relevant modelling is made in view of Tiwari and Das model [15]. Solution of the dimensionless problem is utilized in terms of perturbation series. Physical explanation of the obtained results is

2

Problem formulation

AN US

discussed through plots for various flow quantities of interest.

Here we consider peristalsis of copper-water nanoliquid filled in a two-dimensional curved channel (of width 2a1 ) twisted in a circle with radius R∗ and centre at O (see Fig. 1). Velocity

M

¯ and axial X ¯ directions are dented by V¯1 and V¯2 respectively. Moreover components in radial R both the peristaltic walls are considered at different temperatures T1 and T0 respectively.

ED

Here flow is due to propagation of sinusoidal waves along the axial direction. Aspects of heat generation/absorption and mixed convection are also taken into account. Further the

PT

first order velocity and thermal slip conditions are considered at channel boundaries The shape of waves are

CE

  

2π ¯ ¯ ¯ ¯ ¯ H X, t = ± a1 + b1 sin X − ct . λ∗ Where c is the wave speed λ∗ the wavelength t¯ the time and b1 the amplitude.

(1) The

AC

ρnf denotes the nanofluid density, µef f the effective dynamic viscosity of nanofluid, Cnf the specific heat, κef f the effective thermal conductivity of nanofluid, Φ0 the heat genera¯ the body force, P¯ the pressure and T the fluid ¯ the velocity, ρb tion absorption quantity, V temperature. The ρnf , µef f , (ρβ)nf , (ρC)nf , and κef f for copper-water nanoparticle are given in the form [6,9,16,49]:

ρnf = ρbf (1 − φ) + ρnf φ, (ρβ)nf = (1 − φ) (ρβ)bf + φ (ρβ)nf , µef f = (ρC)nf = (1 − φ) (Cρ)bf + φ (Cρ)nf ,

κef f κnf = κbf κnf 5

µbf

(1 − φ)2.5 + 2κbf − 2φ (κbf − κnf ) . (2) + 2κbf + 2φ (κbf − κnf )

ACCEPTED MANUSCRIPT

Here Maxwell-Gamett’s model (MG-model) is considered to approximate the effective thermal conductivity κef f and effective dynamic viscosity µef f of the nanofluid. In the above equation φ denotes the nanoparticle volume fraction and ρbf the density of base fluid (water), ρnf the nanoparticle (copper) density. The velocity is given by ¯ R, ¯ t¯),V¯2 (X, ¯ R, ¯ t¯),0). ¯ = (V¯1 (X, V

CR IP T

(3)

The relevant equations are [9,16,21,47,49]:

¯ ∂  ∗ ¯
¯ ∗ ∂ V2 R + R V + R 1 ¯ ¯ = 0, ∂R ∂X

(4)

AN US

  ¯ ∂ P¯ ∂ V1 ¯ ∂ V¯1 R∗ V¯2 ∂ V¯1 V¯22 + V1 ¯ + ∗ ¯ ¯ − ∗ ¯ = − ¯ ρnf ∂ t¯ ∂R R + R ∂X R +R ∂R # "   2 2¯ 1 ∂ V¯1 R∗ ∂ V1 V¯1 2R∗ ∂ 2 V¯1 ∂ V¯2 +µef f
−
¯ ,(5) ¯2 + R ¯ + R∗ ∂ R ¯ + R ¯ + R∗ ¯2 − R ¯ + R∗ 2 ¯ + R∗ 2 ∂ X ∂R ∂X R

+g (ρβ)nf (T − T0 ) ,

M

 ¯    ¯ ∂ V2 ¯ ∂ V¯2 R∗ V¯2 ∂ V¯2 V¯1 V¯2 R∗ ∂P ρnf + V1 ¯ + ∗ ¯ ¯ − ∗ ¯ = − ¯ ∂X ¯ ∂ t¯ ∂R R + R ∂X R +R R∗ + R " #   2 2¯ ∂ 2 V¯2 1 ∂ V¯2 R∗ V¯2 2R∗ ∂ V¯1 ∂ V2 +µef f
+
¯ ¯2 + R ¯ + R∗ ∂ R ¯ + R ¯ + R∗ ¯2 − R ¯ + R∗ 2 ¯ + R∗ 2 ∂ X ∂R ∂X R

(6)

ED

# " ∗2 2 2 ¯2 R∗ ∂T  ∂T ∂T V 1 ∂T R ∂ T ∂ T (ρC)nf + V¯1 ¯ + ∗ ¯ ¯ = κef f
¯2 ¯ 2 + R∗ + R ¯ ∂R ¯ + R∗ + R ¯ 2 ∂X ∂ t¯ ∂R R + R ∂X ∂R "  2  2 !  ¯ 2 # ∂ V¯1 R∗ ∂ V¯2 V¯1 ∂ V2 R∗ ∂ V¯1 V¯2 µef f 2 + + ¯ ¯ ∂X ¯ + R∗ + R ¯ ¯ + R∗ + R ¯ ∂X ¯ −R ¯ + R∗ ∂R R∗ + R ∂R (7)

CE

+Φ0 .

PT



AC

The velocity and thermal slip conditions are  ¯ ¯2 
α n ˜ ∂ V V 2 1 ¯ = −H ¯ X, ¯ t¯ . V¯2 − + = 0, T = T at R (8) 1 2.5 ¯ ¯ + R∗ ∂R R (1 − φ)  ¯ 
∂ V2 V¯2 α1 n ˜ ¯ ¯=H ¯ X, ¯ t¯ . + = 0, T = T at R (9) V2 + 0 ¯ ¯ + R∗ R (1 − φ)2.5 ∂ R
¯ X ¯ whereas the flow in the channel The flow is considered unsteady in stationary frame R, can be treated as steady in moving frame (¯ r, x¯). The two frames are related by



¯ − ct¯, r¯ = R, ¯ v¯1 (¯ ¯ R, ¯ t¯ , v¯2 (¯ ¯ R, ¯ t¯ − c, x¯ = X x, r¯) = V¯1 X, x, r¯) = V¯2 X,

¯ R, ¯ t¯ , p¯ (¯ ¯ R, ¯ t¯ , T (¯ x, r¯) = T X, x, r¯) = P¯ X, 6

(10)

ACCEPTED MANUSCRIPT

and now equations. (4 − 7) yield ∂ ∂¯ v2 {(R∗ + r¯) v¯1 } + R∗ = 0, ∂¯ r ∂ x¯

(11)

"

CR IP T

# ∂¯ v1 ∂¯ v1 R∗ (¯ v2 + c) ∂¯ v1 (¯ v2 + c)2 ∂ p¯ ρnf −c + v¯1 + − ∗ =− ∗ ∂ x¯ ∂¯ r R + r¯ ∂ x¯ R + r¯ ∂¯ r " #   2 2 ∂ 2 v¯1 v1 R∗ ∂ v¯1 1 ∂¯ v¯1 2R∗ ∂¯ v2 + , (12) +µef f + − − ∂¯ r2 r¯ + R∗ ∂¯ r r¯ + R∗ ∂ x¯2 (¯ r + R∗ )2 (¯ r + R∗ )2 ∂ x¯    ∂¯ v2 ∂¯ v2 R∗ (¯ v2 + c) ∂¯ v2 v¯1 (¯ v2 + c) R∗ ∂ p¯ ρnf −c + v¯1 + − =− ∗ ∗ ∗ ∂ x¯ ∂¯ r R + r¯ ∂ x¯ R + r¯ R + r¯ ∂ x¯ " #   2 2 2R∗ ∂¯ ∂ 2 v¯2 1 ∂¯ R∗ v¯2 v1 v2 ∂ v¯2 + +µef f + + − ∂¯ r2 r¯ + R∗ ∂¯ r r¯ + R∗ ∂ x¯2 (¯ r + R∗ )2 (¯ r + R∗ )2 ∂ x¯

AN US



+g (ρβ)nf (T − T0 ) ,

(13)

M

   2  ∂T (¯ v2 + c) R∗ ∂T ∂ T 1 ∂T R∗2 ∂ 2 T ∂T (ρC)nf −c + v¯1 + + ∗ + = κef f ∂ x¯ ∂¯ r R∗ + r¯ ∂ x¯ ∂¯ r2 R + r¯ ∂¯ r (R∗ + r¯)2 ∂ x¯2 "  2  2 !  2 # R∗ ∂¯ ∂¯ v1 v2 v¯1 ∂¯ v2 R∗ ∂¯ v1 (¯ v2 + c) + +µef f 2 + ∗ + + ∗ − ∗ ∂¯ r R∗ + r¯ ∂ x¯ R + r¯ ∂¯ r R + r¯ ∂ x¯ R + r¯

ED

+Φ0 . Defining

AC

CE

PT

¯ 2π¯ x r¯ v¯1 v¯2 2πa1 H , v1 = , v2 = , δ = ∗ , η = ± , ∗ , r = λ a1 c c λ a1 µbf Cnf ρca1 R∗ ct¯ 2πa21 p¯ p= , Re = , k= , Pr = ,t = ∗, cµbf λ∗ µbf a1 κbf λ µbf Cp a21 Φ0 a21 T − T0 , β0 = , Pr = , Ec = , θ= T1 − T0 (T1 − T0 ) κbf κbf Cnf (T1 − T0 ) a21 g (ρβ)nf (T1 − T0 ) Br = Pr Ec, Gr = , cµbf x=

(14)

v1 = δ

k ∂ψ ∂ψ , v2 = − , r + k ∂x ∂r

(15)

(16)

equation (11) is satisfied identically and other expressions for long wavelength and small

7

ACCEPTED MANUSCRIPT

Reynolds number yield

CR IP T

dp = 0, dr ( !) k dp ∂ 1 ∂ 2 ψ 1 − ∂ψ 2 ∂r = (r + k) − 2 − r + k dx ∂r r+k (1 − φ)2.5 (r + k)2 ∂r !) ( (ρβ)nf −1 θ, +Gr 1 + φ (ρβ)bf   κnf + 2κbf − 2φ (κbf − κnf ) ∂ 2 θ 1 ∂θ + κnf + 2κbf + 2φ (κbf − κnf ) ∂r2 r + k ∂r  2  2 Br ∂ ψ ∂ψ 1 + 1− + + β 0 = 0, r+k ∂r (1 − φ)2.5 ∂r2

(17)

(18)

(19)

where θ depicts the dimensionless temperature, δ the wave number, k the curvature param-

AN US

eter, Re the Reynolds number, Pr the Prandtl number, Br the Brinkman number, Gr the

(21)

η(x) = 1 +  sin [x] , (22)  α1 n ˜ indicates the velocity slip parameter. The dimensionless expression for a1

ED

 where α =

(20)

M

Grashof number and β 0 the source/sink parameter. The conditions now are ! α ∂ 2 ψ 1 − ∂ψ F ∂ψ ∂r − + = 1, θ = 1, at r = −η, ψ= , 2.5 2 2 ∂r ∂r r + k (1 − φ) ! α −F ∂ψ ∂ 2 ψ 1 − ∂ψ ∂r , + + = 1, θ = 0, at r = +η, ψ= 2 ∂r r+k (1 − φ)2.5 ∂r2

PT

pressure rise per wavelength ∆pλ∗ is

∆pλ∗ =

Z2π

dp dx. dx

(23)

0

CE

It is worth mentioning that results of ref. [39] can be achieved in the absence of copper water nanoparticles, thermal Grashof number Gr, velocity slip parameter α and heat generation

AC

parameter β 0 .

3

Solution methodology

Writing ψ = ψ 0 + Brψ 1 ..., dp dp0 dp1 = + Br ..., dx dx dx ∆pλ∗ = ∆p0λ∗ + Br∆p1λ∗ ..., θ = θ0 + Brθ1 ..., 8

(24)

ACCEPTED MANUSCRIPT

one has the following systems at the zeroth and first orders:

3.1

Systems for zeroth order

∂ψ ∂ 2 ψ 0 1 − ∂r0 + ∂r2 r+k

−F0 ∂ψ 0 α ψ0 = , + 2 ∂r (1 − φ)2.5

Systems for first order

(26)

∂ψ ∂ 2 ψ 0 1 − ∂r0 + ∂r2 r+k

= 1, θ0 = 1, at r = −η,

!

= 1, θ0 = 0, at r = +η. (27)

M

3.2

!

(25)

AN US

F0 ∂ψ 0 α ψ0 = , − 2 ∂r (1 − φ)2.5

CR IP T

( !) ∂ψ 0 2 1 − 1 ∂ ∂ ψ k dp0 ∂r = (r + k)2 − 20 − − 2.5 2 r + k dx ∂r ∂r r + k (1 − φ) (r + k) !) ( (ρβ)nf −1 θ0 , +Gr 1 + φ (ρβ)bf   κnf + 2κbf − 2φ (κbf − κnf ) ∂ 2 θ0 1 ∂θ0 + β 0 = 0, + κnf + 2κbf + 2φ (κbf − κnf ) ∂r2 r + k ∂r

(28)

AC

CE

PT

ED

( !) ∂ψ 1 k dp1 1 ∂ ∂ 2ψ1 2 − = (r + k) − 2 − ∂r r + k dx ∂r r+k (1 − φ)2.5 (r + k)2 ∂r ( !) (ρβ)nf +Gr 1 + φ −1 θ1 , (ρβ)bf   1 ∂θ1 κnf + 2κf − 2φ (κbf − κnf ) ∂ 2 θ1 + κnf + 2κf + 2φ (κbf − κnf ) ∂r2 r + k ∂r  2  2 1 ∂ ψ0 1 ∂ψ 0 + + 1− = 0, r+k ∂r (1 − φ)2.5 ∂r2

F1 ∂ψ 1 α ψ1 = , − 2 ∂r (1 − φ)2.5 −F1 ∂ψ 1 α ψ1 = , + 2 ∂r (1 − φ)2.5

∂ψ

1 ∂ 2ψ1 ∂r − ∂r2 r+k

∂ψ

!

1 ∂ 2ψ1 ∂r − ∂r2 r+k

The solutions for zeroth and first order systems are

9

(29)

= 0, θ1 = 0, at r = −η,

!

= 0, θ1 = 0, at r = +η.

(30)

ACCEPTED MANUSCRIPT


1 r ( (B −24k 2 + 3kr + 4r2 β 0 − 36A(4 + 4B (C1 − C2 ) 24A 6 dp0 dp0 + B1 )r + 2k( + B1 + B2 ) + rB2 ) + 2(2Bk 2 (4k + 3r) β 0 +( dx dx dp0 dp0 2 dp0 +3A(2k 2 + 4BC1 r + r + 2k(1 + 3BC1 − BC2 + r) dx dx dx −2 (B1 − B2 ))) log [k + r])

CR IP T

ψ 0 = B3 −

(31)

1 dp0 2 + Bkβ 0 ) (− B 3 Lr5 β 02 − 2B 2 Lr4 β 0 (10A 2 69120A 5 dx
4BLr3 r2 1 L1 − L2 + (L3 − L4 ) + r L5 + 720A2 (L6 + L7 ) + k+r 3 2 dp1 dp0 dp0 +40(−216A2 (L8 + (2k + r)(2 − 2B 2 L C2 ) + 8B 2 L C1 (3k + r)) dx dx dx
+8AB 2 L L9 + L10 r + BC1 r2 + (6k + r) (9B2 − 9B1 ) β 0 + B 3 k 2 L(−2619k 2
+36kr + 4r2 )β 02 ) log [k + r] − 8 L11 + 720A2 (L12 − L13 ) log [k + r] dp0 r (2C2 − 2C1 ) +720BL(12A2 ((k + r)) L14 + B(L15 + 2k dx dp0 2 +C1 r ) + 2B 2 (2C22 r − C1 C2 (9k + r) + C12 (17k + r))) + 2ABk 2 (L16 dx dp0 dp0 +(144BC1 − 36BC2 − 2k +3 r)r)β 0 + B 2 k 4 (79k + 63r)β 02 ) log [k + r]2 dx dx
2 2 0 −960B L 2AC1 + k β (3A (L17 + 4Br (2C1 − C2 )) + Bk 2 (7k + r) β 0 ) log [k + r]3
2 +1440B 3 L (k + r) 2AC1 + k 2 β 0 log [k + r]4 ), (32)

ED

M

AN US

ψ 1 = B6 +

PT

(36A1 + A2 ) log [η + k] dp0 = , dx 36A (−η (η + 2k) + (η 2 + 2ηk + 2k 2 ) log [η + k])

(33)

AC

CE

dp1 4 = (A3 + A4 − A5 + A6 − A7 + A8 2 dx η(η + 2k) − (η + 2ηk + 2k 2 ) log [η + k] B4 B5 −A9 + A10 + (− + + B 3 ηL (−3C1 + 3C2 )2 + A11 + A12 log [η + k] 2 2
B 3 kL + 43C12 − 31C1 C2 + 8C22 ) log [η + k] − A13 log [η + k] 4 0 0 Bη(8C3 − L dp (−8B1 + 8B2 + k(8 + 3k dp ))) dx dx + log [η + k] 8 B 2 ηLβ 0 + A14 log [η + k] + A15 log [η + k] + A16 216A BL(12A2 (2B 2 A17 + A18 + A19 )) log [η + k]2 − 96A2 2 2 0 B L(2AC1 + k β ) A21 log [η + k]4 3 + A log [η + k] − ), (34) 20 72A2 48A2 θ 0 = C2 −

r (2k + r) β 0 − 2 (k 2 β 0 + 2AC1 ) log [k + r] 4A 10

(35)

ACCEPTED MANUSCRIPT

θ 1 = C4 −

1 dp0 2 4 02 2 3 0 + 3Bkβ 0 ) (3B Lr β + 4B Lr β (16A 6912A2 dx

576BL (4A (C1 + C2 ) + 3k 2 β 0 )(A(6 (B2 − B1 ) + 3k(2 + 2B (C1 − C2 ) k+r dp0 dp0 +k ) + 2Bk 3 β 0 ) + (48L(A(6 (B2 − B1 ) + 3k(2 + 2B (C1 − C2 ) + k ) dx dx dp0 2 dp0 0 +2Bk 3 β 0 )))/ (k + r)2 ) + 6Lr2 (72A2 )β + 16AB(6BC1 − 3BC2 + 2k dx dx dp0 dp0 (24BC1 − 8BC2 + k ) + 16AB(6 (B1 − B2 ) +39B 2 k 2 β 02 ) + 12Lr(72A2 dx dx dp0 144BL +k(−6 + 3BC2 + 43k ))β 0 + 5B 2 k 3 β 02 ) + (2AC1 + k 2 β 0 ) dx k+r dp0 dp0 dp0 (−24A(2 (B1 − B2 ) − 2k + 2Bk (C2 − C1 ) − k 2 +k r + r2 ) dx dx dx −B(−16k 3 + 2k 2 r + 3kr2 + r3 )β 0 ) log [k + r] − 12(16ABkL(6 (B1 − B2 ) dp0 dp0 +k(−6 + 3BC2 + 43k ))β 0 + 5B 2 k 4 Lβ 02 + 72A2 (8B(3C1 − C2 )kL dx dx 2 dp 0 + 8C3 )) log [k + r] + 72L(24A2 (2B 2 (C1 − C1 )2 +k 2 L dx dp0 dp0 dp0 +2B (C2 − C1 ) k − (2 (B2 − B1 ) + k(2 + k )) + 8ABk 2 (3B (C1 − C2 ) dx dx dx dp0 0 −2k )β ) + 3B 2 k 4 β 02 ) log [k + r]2 − 288B 2 L(2AC1 + k 2 β 0 )(4A (C1 − C2 ) dx +k 2 β 0 ) log [k + r]3 + 144B 2 L(2AC1 + k 2 β 0 )4 ), (36)

ED

M

AN US

CR IP T

+

where the involved values are given in Appendix.

Discussion

PT

4

CE

The prime goal here is to examine the influence of first order velocity slip parameter α, nanoparticle volume fraction φ, Grashof number Gr and curvature parameter on pressure

AC

rise ∆pλ∗ , pressure gradient dp/dx and stream function ψ. Plots for temperature field are prepared for various values of thermal slip parameter γ, nanoparticle volume fraction φ, heat generation/absorption parameter β 0 and Brinkman number Br. Variation in heat transfer

rate −

κef f 0 θ κf

(η) is shown through bar charts for thermal slip parameter γ, nanoparticle vol-

ume fraction φ and Brinkman number Br.

4.1

Pumping and trapping

To discuss the impact of different pertinent parameters on the basic quantities of peristaltic motion such as pumping and trapping, we have prepared Figs. 2-12. Influence of first 11

ACCEPTED MANUSCRIPT

order velocity slip parameter α, nanoparticle volume fraction φ, Grashof number Gr and curvature parameter k on pressure rise ∆pλ∗ is exhibited in the Figs. 2-5. The discussion for pressure are made in three different regions which are known as backward , peristaltic and retrograde pumping regions. It is revealed from Fig. 2 that pressure rise ∆pλ∗ decreases in the backward (∆pλ∗ > 0, Θ < 0) and peristaltic (∆pλ∗ , Θ > 0) pumping region and it

CR IP T

rises in the retrograde (∆pλ∗ < 0, Θ > 0) pumping region for larger first order velocity slip parameter α. From Fig. 3 it is noted that impact of nanoparticle volume fraction φ on pressure rise ∆pλ∗ is similar to that of α. It is noted here that ∆pλ∗ is larger for base fluid (water) and small for nanoparticles (copper) in backward and peristaltic pumping regions where it is reverse in retrograde pumping region. Fig. 4 illustrated that pressure

AN US

rise ∆pλ∗ enhances for larger Grashof number Gr. Fig. 5 discloses the effect of curvature parameter k on pressure rise ∆pλ∗ . This Fig. describe that for larger curvature the pressure rise enhances in backward (∆pλ∗ > 0, Θ < 0) and peristaltic (∆pλ∗ , Θ > 0) pumping regions where it decreases in retrograde pumping region (∆pλ∗ < 0, Θ > 0). Figs. 6-9 are made to see the behavior of pressure gradient dp/dx for different values of

M

first order velocity slip parameter α, nanoparticle volume fraction φ, Grashof number Gr and curvature parameter k. Fig. 6 elucidates that pressure gradient enhances for larger first order

ED

velocity slip parameter α. Fig. 7 shows the impact of nanoparticle volume fraction φ on pressure gradient. It depicts that dp/dx has a sinusoidal behavior and it attains maximum

PT

value when the maximum occlusion is provided by the peristaltic walls. In occlude portion of the channel the pressure gradient enhances when nanoparticle volume fraction is increased.

CE

It is due to the fact that resistance in the flow field provided by the insertion of nanoparticles becomes large when the channel is occlude. Impact of Grashof number on pressure gradient

AC

is similar as reported for α and φ (see Fig. 8). Fig. 9 describe that the pressure gradient decreases when we shift from curved to planer channel (i.e. larger k). Influence of first order velocity parameter α, nanoparticle volume fraction φ and Grashof

number Gr on trapping are disclosed in Figs. 10-12. Impact of velocity slip parameter α on streamlines is depicted in Fig. 10. It is seen that when the effects of velocity slip parameter enhances the size of trapped bolus reduces and bolus disappeared when (α = 0.3).

Fig. 11 illustrates that size of trapped bolus decreases with the addition of nanoparticle volume fraction φ. It is important to note that nanoparticles (copper) generates resistance for streamlines. Fig. 12 depicts that the size of trapped bolus enhances for larger Grashof 12

ACCEPTED MANUSCRIPT

number Gr.

4.2

Velocity distribution

Characteristics of axial velocity for different values of first order velocity parameter α, nanoparticle volume fraction φ, Grashof number Gr and curvature parameter k are shown

CR IP T

in Figs. 13-16. Fig. 13 portrays the effect of α on axial velocity. This Fig. elaborates that axial velocity is minimum at center of the channel and maximum in the vicinity of the channel walls. Fig. 14 exhibits that nanoliquids with low concentration of nanoparticles possesses smaller magnitude of the axial velocity at center. Hence larger nanoparticle volume fraction provides more resistance to the flow field. However in the vicinity of the channel walls such

AN US

observation remains invalid which is basically due to velocity slip effects. Figs 15 delineates that axial velocity enhances at the lower wall of the channel and it decreases near the upper wall of the channel for different values of Grashof number Gr. Fig. 16 shows that axial velocity is maximum at the center of the channel whereas it decreases near the channel walls

Temperature distribution

ED

4.3

M

when curvature parameter k rises.

Effects of nanoparticle volume fraction φ, heat generation/absorption parameter β 0 , Brinkman

PT

number Br and curvature parameter on temperature are shown in Figs. 17-20. Fig. 17 depicts the variation in temperature θ for increasing values of φ. It is illustrated from this plot

CE

that temperature of the nanofluid decreases with an addition of the nanoparticle volume fraction. This is a clear evidence that heat produced by the heat source is quickly transferred to the walls and consequently the temperature of the fluid decreases. High thermal

AC

conductivity of nanoparticles plays an important role in fast dissipation of the fluid temperature. This confirms the use of copper nanoparticles in different types of coolants. Fig. 18 shows that temperature is an increasing function of heat generation/absorption parameter β 0 . Fig. 19 portrays the impact of Brinkman number Br on temperature. It illustrates that temperature of the nanofluid enhances due to the loss of kinetic energy when fluid particles collide with each other. Temperature of the nanofluid decreases for larger values of curvature parameter k (see Fig. 20).

13

ACCEPTED MANUSCRIPT

5

Final outcomes

Here we investigated the mixed convection effects in peristalsis of copper-water nanofluid with first order velocity and thermal slip condition in curved configuration. The important achievements of the present study are summarized as follows:.

CR IP T

• Impact of velocity slip parameter α and nanoparticle volume fraction on pressure is qualitatively similar.

• It is observed that in the occluded part of the channel the pressure gradient attains its maximum value in the presence of nanoparticle volume fraction φ.

AN US

• Addition of nanoparticles (copper) trapping is less likely when compared with basefluid (water).

• Velocity slip parameter α have dominating effects on axial velocity.

M

• Larger viscous dissipative heat cause an increase in the temperature. • It is seen that magnitude of the axial velocity for basefluid is maximum and it is

ED

minimum when φ is enhanced.

Table 1: Numerical values of the thermophysical properties. Basefluid(water) Nanoparticles(Copper)

PT

Property

997.1

8933

Thermal conductivity (W/mK)

0.613

401

Specific heat (J/kgK)

4179

385

Thermal expansion coefficient (1/k) × 10−6

210

16.65

AC

CE

Density (kg/m3 )

References [1] I. Muhaimin, R. Kandasamy and I. Hashim, Thermophoresis and chemical reaction effects on non-Darcy MHD mixed convection heat and mass transfer past a porous wedge in the presence of variable stream condition, Chem. Eng. Res. Des., 87 (2009) 1527-1535.

14

ACCEPTED MANUSCRIPT

[2] T. Hayat, F.M. Abbasi, M. Al-Yami and S. Monaquel, Slip and Joule heating effects in mixed convection peristaltic transport of nanofluid with Soret and Dufour effects, J. Mol. Liq., 194 (2014) 93-99. [3] M. Mustafa, S. Abbasbandy, S. Hina and T. Hayat, Numerical investigation on mixed convective peristaltic flow of fourth grade fluid with Dufour and Soret effects, J. Taiwan

CR IP T

Inst. Chem. Eng., 45 (2014) 308-316.

[4] T. Hayat, M. Farooq, Z. Iqbal and A. Alsaedi, Mixed Convection Falkner-Skan Flow of a Maxwell Fluid, J. Heat Transf., 134 (2012) 114501 (6 pages).

[5] S. Srinivas, R. Gayathri and M. Kothandapani, Mixed convective heat and mass transfer

AN US

in an asymmetric channel with peristalsis, Communi. Nonlinear Sci. Numer. Simul., 16 (2011) 1845-1862.

[6] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles. In: D.A. Siginer, H.P. Wang (Eds.). Developments and Applications of Non-Newtonian Flows,

M

FED-Vol. 231/MD-vol. 66, ASME J. Heat Transfer, New York, (1995) 99-105.

ED

[7] J. Buongiorno, Convective transport in nanofluids. ASME J. Heat Transf., 128 (2005) 240-250.

PT

[8] C. Zhang, L. Zheng, X. Zhang and G. Chen, MHD flow and radiation heat transfer of nanofluids in porous media with variable surface heat flux and chemical reaction, Appl.

CE

Math. Model., 39 (2015) 165-181. [9] T. Hayat, S. Farooq, A. Alsaedi and B. Ahmad, Influence of variable viscosity and radial

AC

magnetic field on peristalsis of copper-water nanomaterial in a non-uniform porous medium, Int. J. Heat Mass Transf., 103 (2016) 1133-1143.

[10] Y. Lin, L. Zheng, X. Zhang, L. Ma and G. Chen, MHD pseudo-plastic nanofluid unsteady flow and heat transfer in a finite thin film over stretching surface with internal heat generation, Int. J. Heat Mass Transf., 84 (2015) 903-911. [11] T. Hayat, M. Shafique, A. Tanveer and A Alsaedi, Radiative peristaltic flow of Jeffrey nanofluid with slip conditions and joule heating, PloS one, 11 (2016) e0148002.

15

ACCEPTED MANUSCRIPT

[12] Y. Xuan and Q. Li, Investigation on convective heat transfer and flow features of nanofluids, ASME J. Heat Transfer 125 (2003) 151-155. [13] H.C. Brinkman, The viscosity of concentrated suspensions and solutions, J. Chem. Phys., 20 (1952) 571-581.

CR IP T

[14] Y. Xuan and Q. Li, Report of Nanjing University of Sciences and Technology (in Chinese) (1999).

[15] R.K. Tiwari and M.K. Das Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int. J. Heat Mass Transf. 50 (2007)

AN US

2002-2018.

[16] F.M. Abbasi, T. Hayat, B. Ahmad and G.Q. Chen, Slip effects on mixed convective peristaltic transport of copper-water nanofluid in an inclined channel, PloS One 9 (2014) e105440.

M

[17] M. Sheikkoleslami, M.G. Bandpy, D.D. Ganji and S. Soleimani, heat flux boundary condition for nanofluid filled enclosure in presence of magnetic field, J. Mol. liq., 193

ED

(2014) 174-184.

[18] K. Khanafer, K. Vafai and M. Lighstone, Buoyancy-driven heat transfer enhancement

3639-3653.

PT

in a two-dimensional enclosure utilizing nanofluids, Int. J. Heat Mass Transf. 46 (2003)

CE

[19] T. Hayat, Z. Hussain, A. Alsaedi and M. Mustafa, Nanofluid flow through a porous space with convective conditions and heterogeneous-homogeneous reactions, J. Taiwan

AC

Inst. Chem. Eng., In Press (2016) DOI:org/10.1016/j.jtice.2016.11.002.

[20] Y. Lin, L. Zheng and X. Zhang, Radiation effects on Marangoni convection flow and heat transfer in pseudo-plastic non-Newtonian nanofluids with variable thermal conductivity, Int. J. Heat Mass Transf., 77 (2014) 708-716. [21] S.A. Shehzad, F.M. Abbasi, T. Hayat and F. Alsaadi, Model and comparative study for peristaltic transport of water based nanofluids, J. Mol. Liq., 209 (2015) 723-728.

16

ACCEPTED MANUSCRIPT

[22] L. Zheng, C. Zhang, X. Zhang and J. Zhang, Flow and radiation heat transfer of a Nanofluid over a stretching sheet with velocity slip and temperature jump in porous medium, J. Franklin Inst., 350 (2013) 990-1007. [23] G.K. Ramesh, A.J. Chamkha and B.J. Gireesha, MHD mixed convection flow of a viscoelastic fluid over an inclined surface with a nonuniform heat source/sink, Can. J.

CR IP T

Phy., 91 (2013) 1074-1080.

[24] T. Hayat, G. Bashir, M. Waqas and A. Alsaedi, MHD 2D flow of Williamson nanofluid over a nonlinear variable thicked surface with melting heat transfer, J. Mol. Liq., 223 (2016) 836-844.

AN US

[25] M. Farooq, M. Ijaz Khan, M. Waqas, T. Hayat, A. Alsaedi, M. Imran Khan, MHD stagnation point flow of viscoelastic nanofluid with non-linear radiation effects, J. Mol. Liq., 221 (2016) 1097-1103.

[26] B.J. Gireesha, G.K. Ramesh, M. Subhas Abel and C.S. Bagewadi, Boundary layer flow

M

and heat transfer of a dusty fluid flow over a stretching sheet with non-uniform heat

ED

source/sink, Int. J. Multiphas. Flow., 37 (2011) 977-982. [27] T.W. Latham, “Fluid Motion in a Peristaltic Pump,” MS Thesis, MIT Cambridge MA,

PT

(1966).

[28] H. Shapiro, M.Y. Jaffrin and S. L. Weinberg, Peristaltic pumping with long wavelength

CE

at low Reynolds number, J. Fluid Mech., 37 (1969) 799-825. [29] Y. Wang, T. Hayat, N. Ali and M. Oberlack, Magnetohydrodynamic peristaltic motion

AC

of a Sisko fluid in a symmetric or asymmetric channel, Physica A: Statistical Mechanics and its Applications, 387 (2008) 347-362.

[30] A. Ebaid, Effects of magnetic field and wall slip conditions on the peristaltic transport of a Newtonian fluid in an asymmetric channel, Phy. Lett. A., 372 (2008) 4493-4499. [31] M. Lachiheb, Effect of coupled radial and axial variability of viscosity on the peristaltic transport of Newtonian fluid, Appli. Math. Comput., 244 (2014) 761-771. [32] S. Maiti and J.C. Misra, Non-Newtonian characteristics of peristaltic flow of blood in micro-vessels, Communi. Nonlinear Sci. Numer. Simul., 18 (2013) 1970-1988. 17

ACCEPTED MANUSCRIPT

[33] T. Hayat and N. Ali, Effect of variable viscosity on the peristaltic transport of a Newtonian fluid in an asymmetric channel, Appli. Math. Mod., 32 (2008) 761-774. [34] M. Awais, S. Farooq, H. Yasmin, T. Hayat and A. Alsaedi, Convective heat transfer analysis for MHD peristaltic flow of Jeffrey fluid in an asymmetric channel, Int. J.

CR IP T

Biomath., 7 (2014) 1450023. [35] T. Hayat, A. Bibi, H. Yasmin and B. Ahmad, Simultaneous effects of Hall current and homogeneous/heterogeneous reactions on peristalsis, J. Taiwan Inst. Chem. Eng., 58 (2016) 28-38.

[36] T. Hayat, A. Tanveer and A. Alsaedi, Mixed convective peristaltic flow of Carreau-

AN US

Yasuda fluid with thermal deposition and chemical reaction, Int. J. Heat Mass Transf., 96 (2016) 474-481.

[37] H. Sato, T. Kawai, T. Fujita and M. Okabe, Two dimensional peristaltic flow in curved

M

channels, Trans. Jpn. Soc. Mech. Eng., B 66 (2000) 679-685.

[38] N. Ali, M. Sajid and T. Hayat, Long wavelength flow analysis on curved channel, Z.

ED

Naturforsch., A 65 (2010) 191-196.

[39] N. Ali, M. Sajid, T. Javed and Z. Abbas, Heat transfer analysis of peristaltic flow in a

PT

curved channel, Int. J. Heat Mass Transf., 53 (2010) 3319-3325. [40] T. Hayat, S. Farooq, A. Alsaedi and B. Ahmad, Hall and radial magnetic field effects

CE

on radiative peristaltic flow of Carreau-Yasuda fluid in a channel with convective heat and mass transfer, J. Mag. Mag. Mater., 412 (2016) 207-216.

AC

[41] T. Hayat, S. Farooq, B. Ahmad and A. Alsaedi, Characteristics of convective heat transfer in the MHD peristalsis of Carreau fluid with Joule heating, AIP Advances., 6 (2016) 045302.

[42] N. Ali, K. Javid, M. Sajid, A. Zaman and T. Hayat, Numerical simulations of Oldroyd 8-constant fluid flow and heat transfer in a curved channel, Int. J. Heat Mass Transf., 94 (2016) 500-508.

18

ACCEPTED MANUSCRIPT

[43] T. Hayat, S. Farooq, A. Alsaedi and B. Ahmad, Numerical study for Soret and Dufour effects on mixed convective peristalsis of Oldroyd 8-constants fluid, Int. J. Thermal Sci., 112 (2017) 68-81. [44] S. Hina, M. Mustafa, S. Abbasbandy, T. Hayat and A. Alsaedi, Peristaltic motion of

CR IP T

nanofluid in a curved channel, J. Heat Transfer 136 (2014) 052001 (7 pages). [45] S. Noreen, M. Qasim and Z.H. Khan, MHD pressure driven flow of nanofluid in curved channel, J. Mag. Mag. Mater., 393 (2015) 490-497.

[46] S. Nadeem, E.N. Maraj and N.S. Akbar, Investigation of peristaltic flow of Williamson

AN US

nanofluid in a curved channel with compliant walls, Appl. Nanosci., 4 (2014) 511-521. [47] S. Nadeem and I. Shahzadi, Mathematical analysis for peristaltic flow of two phase nanofluid in a curved channel, Commun. Theory. Phys., 64 (2015) 547-554. [48] V.K. Narla, K.M. Prasad and J.V. Ramanamurthy, Peristaltic transport of Jeffrey

M

nanofluid in curved channels, Procedia Eng., 127 (2015) 869-876. [49] F.M. Abbasi, T. Hayat and B. Ahmad, Peristaltic transport of copper-water nanofluid

AC

CE

PT

ED

saturating porous medium, Physica E., 67 (2015) 47-53.

19

ACCEPTED MANUSCRIPT

6

Appendix

We here include: L1 = 160BL(3A(−2B1 + 2B2 + k(2 + 2BC1 − 2BC2 + k

dp0 )) + 2Bk 3 β 0 )2 , dx

dp0 2 dp0 0 )β + 383B 2 k 2 β 02 ), + 20AB(44BC1 − 12BC2 + 3k dx dx dp0 L3 = −80AB 2 L(−72B1 + 72B2 + k(72 + 106BC1 − 78BC2 − 665k ))β 0 , dx dp0 dp0 2 dp1 L4 = 3163B 3 k 3 Lβ 02 + 720A2 (12B 2 (15C1 − 4C2 )L − BkL + 48( + B4 + B5 ))), dx dx dx dp0 ))β 0 + 146053B 3 k 4 Lβ 02 , L5 = 80AB 2 kL(72B1 − 72B2 − k(72 − 8614BC1 + 2562BC2 + 325k dx dp0 L6 = 192B 3 (6C12 − 4C1 C2 + C22 )L − 36B 2 (3C1 − 4C2 )kL , dx dp0 dp0 dp1 L7 = B(96C3 − 96C4 − L (−192B1 + 192B2 + k(192 + 89k ))) + 48k( + B4 + B5 ), dx dx dx dp0 dp0 (−8B1 + 8B2 + k(8 + 3k ))), L8 = 8B 3 (−3C1 + C2)2 L + B(8C3 − L dx dx dp0 dp0 L9 = 54k 2 − 1590BC1 k 2 + 504BC2 k 2 − 258k 3 , L10 = 9k + 12BC1 k − 9BC2 k − 90k 2 , dx dx dp0 ))β 0 + 15311B 3 k 5 Lβ 02 , L11 = 10AB 2 k 2 L(648B1 − 648B2 + k(−648 + 7480BC1 − 2400BC2 + 411k dx L12 = 3B 3 (43C12 − 31C1 C2 + 8C22 )kL + 3B 2 L(B1 (3C1 + 2C2 ) − B2 (3C1 + 2C2 ) dp0 dp0 −k(C1 (3 − 6k ) + C2 (2 + k ))), dx dx dp0 dp0 dp1 (−18B1 + 18B2 + k(18 + 7k ))) + 6k 2 − 6B4 + 6B5 , L13 = Bk(−18C3 + 6C4 + L dx dx dx dp0 dp0 L14 = −2 (−2B1 + 2B2 + k(2 + k )), dx dx dp0 L15 = −6B1 C1 + 6B2 C1 − 4B1 C2 + 4B2 C2 + 6C1 k + 4C2 k + 6C2 k 2 , dx dp0 , L16 = −24B1 + 24B2 + 24k + 180BC1 k − 52BC2 k + 7k 2 dx dp0 L17 = 2B1 − 2B2 − 2k + 6BC1 k − 2BC2 k − k 2 , dx

AC

CE

PT

ED

M

AN US

CR IP T

L2 = (360A2

20

ACCEPTED MANUSCRIPT

A1 = (4B3 + 2F0 + η(4 + 4BC1 − 4BC2 + B1 η + B2 η + 2B1 k + 2B2 k)), A2 = Bη(−4η 2 − 3ηk + 42k 2 )β 0 + 24(3A(B1 − B2 − 2BC1 η − k

AN US

CR IP T

−3BC1 k + BC2 k) − Bk 2 (3η + 4k)β 0 , B4 ηk B5 ηk F1 B4 η 2 1 − − B5 η 2 − − − 2B 3 (6C12 − 4C1 C2 + C22 )ηL, A3 = −B6 − 2 4 4 2 2 2 0 0 0 3B 2 (3C1 − 4C2 )ηkL dp Bη 2 kL dp B 2 (15C1 − 4C2 )η 2 L dp dx dx dx A4 = − + + , 16 8 192 0 0 Bη(96C3 − 96C4 − L dp (−192B1 + 192B2 + k(192 + 89k dp ))) dx dx A5 = , 96 0 (B 2 η 2 L(−72B1 + 72B2 + k(72 + 106BC1 − 78BC2 − 665k dp ))β 0 ) dx A6 = , 1728A 0 (B 2 ηkL(72B1 − 72B2 − k(72 − 8614BC1 + 2562BC2 + 325k dp ))β 0 ) dx A7 = , 864A 0 + Bkβ 0 ) B 3 η 5 Lβ 02 3163B 3 η 2 k 3 Lβ 02 146053B 3 ηk 4 Lβ 02 B 2 η 4 Lβ 0 (10A dp dx + − + , A8 = 172800A2 138240A2 69120A2 34560A2 0 BL(3A(−2B1 + 2B2 + k(2 + 2BC1 − 2BC2 + k dp )) + 2Bk 3 β 0 )2 dx A9 = , 432A2 (η + k) 2

AC

CE

PT

ED

M

0 0 Bη 3 L(360A2 dp + 20AB(44BC1 − 12BC2 + 3k dp )β 0 + 383B 2 k 2 β 02 ) dx dx A10 = , 51840A2 0 L(−C2 (η + 2k) + 4C1 (η + 3k)) dp log [η + k] dx A11 = , 2 4B η 0 0 ) + C2 (2 + k dp ))) B 2 L(B1 (3C1 + 2C2 ) − B2 (3C1 + 2C2 ) − k(C1 (3 − 6k dp dx dx , A12 = 4 0 0 Bk(−18C3 + 6C4 + L dp (−18B1 + 18B2 + k(18 + 7k dp ))) dx dx A13 = , 12 A14 = (BC1 η 2 + 9ηk + 12BC1 ηk − 9BC2 ηk + 54k 2 − 1590BC1 k 2 + 504BC2 k 2 dp0 dp0 − 258k 3 ), −9B1 (η + 6k) + 9B2 (η + 6k) − 90ηk 2 dx dx 0 (B 2 k 2 L(648B1 − 648B2 + k(−648 + 7480BC1 − 2400BC2 + 411k dp ))β 0 dx A15 = , 864A 15311B 3 k 5 Lβ 02 log [η + k] B 3 ηk 2 (4η 2 + 36ηk − 2619k 2 )Lβ 02 log [η + k] A16 = − , 8640A2 1728A2 A17 = (2C22 η − C1 C2 (8η + 9k) + C12 (14η + 17k)), dp0 A18 = B(−6B1 C1 + 6B2 C1 − 4B1 C2 + 4B2 C2 + 6C1 k + 4C2 k + C1 η 2 dx dp dp0 dp0 dp dp0 0 0 −2C1 ηk + 4C2 ηk + 6C2 k 2 ) − 2(η + k) (−2B1 + 2B2 + k(2 + k )), dx dx dx dx dx A19 = 2ABk 2 (−24B1 + 24B2 + 144BC1 η − 36BC2 η + 24k + 180BC1 k − 52BC2 k
dp0 0 + 3η 2 − 2ηk + 7k 2 )β + B 2 k 4 (63η + 79k)β 02 , dx dp0 ) A20 = (3A(2B1 − 2B2 + 8BC1 η − 4BC2 η − 2k + 6BC1 k − 2BC2 k − k 2 dx
2 +Bk 2 (9η + 7k)β 0 ), A21 = (B 3 (η + k)L 2AC1 + k 2 β 0 21

ACCEPTED MANUSCRIPT

AN US

CR IP T

where B1 − B6 and C1 − C4 can be calculated through algebraic computations.

AC

CE

PT

ED

M

Fig. 1. Schematic diagram.

Fig. 2. α variation for ∆pλ∗ when  = 0.5, Br = 0.1, k = 2.5, γ = 0.1, φ = 0.1, Gr = 5, β 0 = 3.

22

AN US

CR IP T

ACCEPTED MANUSCRIPT

Fig. 3. φ variation for ∆pλ∗ when

AC

CE

PT

ED

M

 = 0.5, Br = 0.1, k = 2.5, γ = 0.1, α = 0.1, Gr = 5, β 0 = 3.

Fig. 4. Gr variation for ∆pλ∗ when  = 0.5, Br = 0.1, k = 2.5, γ = 0.1, φ = 0.1, α = 0.1, β 0 = 3.

23

AN US

CR IP T

ACCEPTED MANUSCRIPT

Fig. 5. k variation for ∆pλ∗ when

AC

CE

PT

ED

M

 = 0.5, Br = 0.1, α = 0.1, γ = 0.1, φ = 0.1, Gr = 5, β 0 = 3.

Fig. 6. α variation for

dp dx

when  = 0.5, Br = 0.1, k = 2.5, γ = 0.1, φ = 0.1, Gr = 5, β 0 = 3.

24

when  = 0.5, Br = 0.1, k = 2.5, γ = 0.1, α = 0.1, Gr = 5, β 0 = 3.

dp dx

AC

CE

PT

ED

M

Fig. 7. φ variation for

AN US

CR IP T

ACCEPTED MANUSCRIPT

Fig. 8. Gr variation for

dp dx

when  = 0.5, Br = 0.1, k = 2.5, γ = 0.1, α = 0.1, φ = 0.1, β 0 = 3.

25

dp dx

when  = 0.5, Br = 0.1, α = 0.1, γ = 0.1, φ = 0.1, Gr = 5, β 0 = 3.

AC

CE

PT

ED

M

Fig. 9. k variation for

AN US

CR IP T

ACCEPTED MANUSCRIPT

Fig. 10. Influence of α on streamlines when  = 0.5, Br = 0.1, Θ = 1.5, φ = 0.1, γ = 0.1, k = 2.5, Gr = 5, β 0 = 3.

26

AN US

CR IP T

ACCEPTED MANUSCRIPT

Fig. 11. Influence of φ on streamlines when

AC

CE

PT

ED

M

 = 0.5, Br = 0.1, Θ = 1.5, α = 0.1, γ = 0.1, k = 2.5, Gr = 5, β 0 = 3.

Fig. 12. Influence of Gr on streamlines when  = 0.5, Br = 0.1, Θ = 1.5, α = 0.1, γ = 0.1, φ = 0.1, k = 2.5, β 0 = 3. 27

AN US

CR IP T

ACCEPTED MANUSCRIPT

Fig. 13. Influence of α on axial velocity v2 when

AC

CE

PT

ED

M

 = 0.5, x = 0.0, Br = 0.1, Θ = 1.5, k = 2.5, γ = 0.1, φ = 0.1, Gr = 5, β 0 = 3.

Fig. 14. Influence of φ on axial velocity v2 when

 = 0.5, x = 0.0, Br = 0.1, Θ = 1.5, k = 2.5, γ = 0.1, α = 0.1, Gr = 5, β 0 = 3.

28

AN US

CR IP T

ACCEPTED MANUSCRIPT

Fig. 15. Influence of Gr on axial velocity v2 when

AC

CE

PT

ED

M

 = 0.5, x = 0.0, Br = 0.1, Θ = 1.5, k = 2.5, γ = 0.1, φ = 0.1, α = 0.1, β 0 = 3.

Fig. 16. Influence of k on axial velocity v2 when

 = 0.5, x = 0.0, Br = 0.1, Θ = 1.5, φ = 0.1, γ = 0.1, α = 0.1, Gr = 5, β 0 = 3.

29

AN US

CR IP T

ACCEPTED MANUSCRIPT

Fig. 17. φ variation for θ when

AC

CE

PT

ED

M

 = 0.5, x = 0.0, Br = 0.1, Θ = 1.5, k = 2.5, γ = 0.1, α = 0.1, Gr = 5, β 0 = 3.

Fig. 18. β 0 variation for θ when

 = 0.5, x = 0.0, Br = 0.1, Θ = 1.5, k = 2.5, Gr = 5, φ = 0.1, α = 0.1, γ = 0.1.

30

AN US

CR IP T

ACCEPTED MANUSCRIPT

Fig. 19. Br variation for θ when

AC

CE

PT

ED

M

 = 0.5, x = 0.0, Θ = 1.5, k = 2.5, γ = 0.1, φ = 0.1, α = 0.1, Gr = 5, β 0 = 3.

Fig. 20. k variation for θ when

 = 0.5, x = 0.0, Θ = 1.5, Br = 0.1, γ = 0.1, φ = 0.1, α = 0.1, Gr = 5, β 0 = 3.

31