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A model for an application to biomedical engineering through nanoparticles
International Journal of Heat and Mass Transfer 101 (2016) 112–120
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A model for an application to biomedical engineering through nanoparticles T. Hayat a, Sadaf Nawaz a, F. Alsaadi b, M. Rafiq a, M. Mustafa c,⇑ a
Department of Mathematics, Quaid-I-Azam University, 45320, Islamabad 44000, Pakistan Department of Electrical and Computer Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia c School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Islamabad 44000, Pakistan b
a r t i c l e
i n f o
Article history: Received 7 November 2015 Received in revised form 4 May 2016 Accepted 6 May 2016
Keywords: Slip condition Peristalsis Mixed convection Nanofluid Maxwell model Hamilton–Crosser model
a b s t r a c t Recent advancements in nanoscience and technology has made the nanofluid an important research topic. Various models have been put forward to estimate the effective thermal conductivity of nanofluids. Present article addresses the comparative study of Maxwell’s and Hamilton–Crosser’s model for mixed convection peristaltic flow of incompressible nanofluid in an asymmetric channel. Viscous dissipation and heat generation/absorption effects are retained. Analysis is performed for five different types of nanoparticles namely titanium oxide or titania (TiO2), aluminum oxide or alumina (Al2O3), copper oxide (CuO), copper (Cu) and silver (Ag) with water as base fluid. Velocity and thermal slip conditions are employed. Lubrication approach is adopted for problem formulation. The developed non-linear problems are solved numerically. Plots for axial velocity, temperature and heat transfer rate at the wall are obtained and analyzed. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Heat transfer enhancement determines the need for innovative coolants with improved performance. The novel concept of nanofluid has been introduced to enhance the heat transfer capability of conventional coolants such as water, ethylene glycol and oils. The terminology of nanofluid was first used by Choi [1] who demonstrated an anomalous increase in thermal conductivity of water and other liquids through dispersion of copper and aluminum nanoparticles. This enhanced feature of nanofluid has led to the plethora of diverse applications such as cooling of microelectronics, engine cooling/vehicle thermal management, magnetic drug targeting, space cooling, in grinding machining and many others. Utilization of nanofluids in water cooled nuclear reactor can produce safety margins and provides economic gains. Nanofluids are also employed in solar collectors for their tunable optical properties. Nanofluids can improve the performance efficiency of heat exchangers by reducing total heat resistance. In biomedicine, nanofluids serve as carriers for delivering drugs and radiation in cancer patients. Nanofluids can be considered as single phase liquids and hence classical theory of single phase fluids can be applied in which physical properties of nanofluids are expressed ⇑ Corresponding author. Tel.: +92 51 9085 5596. E-mail address: [email protected] (M. Mustafa). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.05.033 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
as function of both constituents and their volume fractions. Keeping this in view Tiwari and Das [2] proposed a mathematical model to estimate heat transfer behavior by varying nanoparticle volume fraction. Buongiorno [3] also suggested a mathematical model for nanofluids convective transport considering novel aspects of Brownian motion and thermophoresis. Both the above mentioned models have been applied to investigate heat transfer problems involving nanofluids [4–15]. Peristaltic flow refers to the transportation of fluid inside a channel or tube by the action of flexible walls. It is a major mechanism for fluid flow in many biological and industrial systems. Within human body it is involved in swallowing of food through esophagus, movement of chyme in the gastro-intestinal track, in the ductus efferentes of the male reproductive system, vasomotion of small blood vessels such as arterioles and capillaries. Peristaltic pumps are used to transport corrosive materials in order to avoid direct contact of the fluid with the pump’s internal surface. Many biomedical devices as dialysis machines, open heart bypass pump machines, infusion pumps, etc. are engineered on the mechanism of peristaltic. In robotic industry, the concept of peristalsis has been used for the production of robots. The pioneering work on peristalsis was performed to investigate the urine transport. Shapiro [16] in 1967 analyzed the peristaltic pumping in a two dimensional flexible tube. Later, the theoretical results determined in [16] were experimentally confirmed by Weinberg [17]. Some
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Nomenclature H 1 ; H2 X; Y Kf ðU; VÞ Kp K eff a1 ; b1
c F
g Q q h1 ; h2
U T0; T1
qeff leff ðqCÞeff ðqbÞeff e n n
qf ; qp bf bp a; b
right and left walls space coordinates in fixed frame thermal conductivity of base fluid velocity components in fixed frame thermal conductivity of nanoparticle effective thermal conductivity dimensional wave amplitude phase difference dimensionless flow rate in wave frame dimensionless flow rate in fixed frame volume flow rate in fixed frame volume flow rate in wave frame dimensionless right/left walls heat generation/absorption temperature of right/left wall effective density of nanofluid effective viscosity of nanofluid effective heat capacity of nanofluid effective thermal expansion of nanofluid dimensionless heat generation/absorption dimensionless velocity slip parameter shape factor of nanoparticles density of fluid and nanoparticle fluid thermal expansion coefficient nanoparticle thermal expansion coefficient dimensionless wave amplitudes
recent researches dealing with the peristaltic motion are mentioned in Refs. [18–27]. The effect of heat transfer cannot be ignored in peristalsis especially when dealing with the blood flow simulation related to tumors and muscles, drug transport, production of osteoinductive material, nutrients to brain cell, etc. Peristalsis is also seen in the processes of oxygenation and hemodialysis. Some references pertaining to the heat transfer phenomenon in peristalsis are mentioned in [28–37]. Peristaltic transport of nanofluids are utilized in modern drug delivery systems and in cancer therapy to destroy undesirable tissues. Despite the aforementioned applications, very little attention has been given to the studies dealing with the peristaltic transport of nanofluids [38–42]. Current study aims to address the effects of velocity and thermal slip on the peristaltic transport of nanofluids through an asymmetric channel. Two different models for effective thermal conductivity of nanofluids are utilized. Long wavelength and low Reynolds number approximation are adopted for problem formation. Arising system comprising of coupled equations has been solved numerically through NDSolve of MATHEMATICA. Graphs are sketched to see the effects of embedded parameters on velocity and temperature distributions.
t d1 þ d2 / k ðx; yÞ ðu; v Þ p ðx; yÞ P Tm c T g w Pr Br Re d Ec Gr f Cf
lf Cp d
time in fixed frame width of channel nanoparticle volume fraction wavelength space coordinates in wave frame velocity components in wave frame pressure in wave frame dimensionless space coordinates pressure in fixed frame mean temperature wave speed fluid temperature gravity stream function Prandtl number Brinkman number Reynolds number wave number Eckert number Grashoff number thermal slip parameter specific heat of fluid dynamic viscosity of fluid specific heat of nanoparticle ratio of d2 and d1
K eff K p þ 2K f 2/ðK f K p Þ ¼ : Kf K p þ 2K f þ /ðK f K p Þ
Hamilton–Crosser model the thermal conductivity of nanofluid is given by [44]:
K eff K p þ ðn 1ÞK f ðn 1Þ/ðK f K p Þ ¼ : Kf K p þ ðn 1ÞK f þ /ðK f K p Þ
ð2Þ
Here K eff is the effective thermal conductivity of the nanofluid, K f is the thermal conductivity of water, K p is the thermal conductivity of nanoparticles and / is the nanoparticle volume fraction. n in
2. Problem formulation We consider the peristaltic motion of an incompressible nanofluid flowing through a two dimensional vertical asymmetric channel of width d1 þ d2 (see Fig. 1). Nanofluid comprises of five different types of nanoparticles namely titanium oxide or titania (TiO2), aluminum oxide or alumina (Al2O3), copper oxide (CuO), copper (Cu) and silver (Ag). Water is considered as the base fluid. We take into account two different models for effective conductivity of nanofluids. Maxwell–Garnelt model for effective thermal conductivity is expressed as [43]:
ð1Þ
Fig. 1. Geometry of the problem.
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Table 1 Thermo-physical properties of water and nanoparticles.
H2O TiO2 Al2O3 CuO Cu Ag
q (kg m3)
C p (J kg1 K1)
K W m1 K1
b (l/k) 106
997.1 4250 3970 6320 8933 10,500
4179 686.2 765 531.8 385 235
0.613 8.9538 40 76.5 401 429
210 9.0 8.5 18.0 16.7 18.9
Hamilton–Crosser’s model denotes the shape factor of nanoparticles. The value of n ¼ 3 and n ¼ 6 correspond to the spherical and cylindrical nanoparticles respectively. For n ¼ 3, Hamilton–Cross er’s model reduces to the Maxwell’s model. We select rectangular coordinate system in such a way that the X axis lies along the walls of the channel and Y axis lies perpendicular to it. The flow is due to an infinite sinusoidal wave of wavelength k traveling with speed c along the walls of channel. Mathematically, wave shape are expressed as:
2p H1 X; t ¼ d1 þ a1 cos X ct ; k 2p H2 X; t ¼ d2 b1 cos X ct þ c ; k
" # @ @ @ @P @2U @2U þ gðqbÞeff ðT T m Þ; þU þV U ¼ þ leff þ @t @X @Y @X @X 2 @Y 2
qeff
ð5Þ
in the region Y < 0; a1 is the wave amplitude traveling along H1 X; t ; b1 is the wave amplitude traveling along H2 X; t ; c represents the phase difference which varies in the range 0 6 c 6 p and t denotes the time. Moreover a1 ; b1 ; d1 ; d2 and c satisfy the following condition
" # @ @ @ @P @2V @2V ; þU V ¼ þ þV þ leff @t @X @Y @Y @X 2 @Y 2
ð3Þ
where H1 X; t is the wall in the region Y > 0 and H2 X; t is the wall
2
Fig. 3. Variation of axial velocity for TiO2 for change in / when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1, f ¼ 0:1; Gr ¼ 3:0, ¼ 2:5.
qeff
ðqCÞeff
" # @ @ @ @2 T @2 T þV þU þ þU T ¼ K eff @t @X @Y @X 2 @Y 2 2 0 !2 !2 1 !2 3 @U @V @U @V Aþ 5: þ þ leff 42@ þ @X @Y @Y @X ð7Þ
2
a21 þ b1 þ 2a1 b1 cos c 6 ðd1 þ d2 Þ : The wall H1 X; t is maintained at constant temperature T 0 and temperature of wall H2 X; t is T 1 ð> T 0 Þ. The velocity field for the two dimensional flow is given by ½U X; Y; t ; V X; Y; t ; 0. The conservation laws of mass, momentum, and energy are governed by the following equations:
@U @V þ ¼ 0; @X @Y
ð6Þ
ð4Þ
In the above equations P X; Y; t represents the pressure, 1 T m ¼ T 0 þT is the mean temperature, g is the acceleration due to 2 gravity and U denotes the heat generation/absorption term. For the two-phase flow model we can take qeff denotes the effective density, leff the effective viscosity, ðqCÞeff the effective heat capacity and ðqbÞeff the effective thermal expansion of the nanofluid. These are expressed as [8,38,39]:
Fig. 2. Effective thermal conductivity for Maxwell’s and H–C model.
T. Hayat et al. / International Journal of Heat and Mass Transfer 101 (2016) 112–120
Fig. 4. Variation of axial velocity for Al2O3 for change in / when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1, f ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
Fig. 5. Variation of axial velocity for CuO for change in / when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1, f ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
Fig. 6. Variation of axial velocity for Cu for change in / when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1, f ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
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Fig. 7. Variation of axial velocity for Ag for change in / when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1, f ¼ 0:1; Gr ¼ 3:0, ¼ 2:5.
Fig. 8. Variation of temperature for TiO2 for change in / when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1, f ¼ 0:1; Gr ¼ 3:0, ¼ 2:5.
Fig. 9. Variation of temperature for Al2O3 for change in / when a ¼ 0:7, b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1, f ¼ 0:1; Gr ¼ 3:0, ¼ 2:5.
qeff ¼ ð1 /Þqf þ /qp ; ðqCÞeff ¼ ð1 /ÞðqCÞf þ /ðqCÞp ðqbÞeff ¼ ð1 /Þqf bf þ /qp bp ;
leff ¼
lf ð1 /Þ2:5
;
ð8Þ
Here q is the density, C the specific heat, b the thermal expansion coefficient and l the dynamic viscosity. The subscripts p and f correspond to the solid and fluid phases respectively.
Numerical values of the thermo-physical properties of water and nanoparticles are listed in Table 1. The transformations between laboratory and wave frames are defined as:
x ¼ X ct;
v ðx; yÞ ¼ V
y ¼ Y;
X; Y; t ;
uðx; yÞ ¼ U X; Y; t c;
pðx; yÞ ¼ P X; Y; t ;
ð9Þ
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Fig. 10. Variation of temperature for CuO for change in / when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1, f ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
Fig. 11. Variation of temperature for Cu for change in / when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1, f ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
Fig. 12. Variation of temperature for Ag for change in / when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1, f ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
in which uðx; yÞ and v ðx; yÞ are the velocity components and pðx; yÞ denotes the pressure. Eqs. (4)–(7) in view of Eq. (9) become
@u @ v þ ¼ 0; @x @y
ð10Þ
Fig. 13. Variation of velocity for TiO2 for change in velocity slip when a ¼ 0:7; b ¼ 0:6, c ¼ p=2, d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; f ¼ 0:1, / ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
Fig. 14. Variation of velocity for Al2O3 for change in velocity slip when a ¼ 0:7; b ¼ 0:6; c ¼ p=2; d ¼ 0:8; x ¼ 0; g ¼ 0:7; Br ¼ 0:3; f ¼ 0:1; / ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
Fig. 15. Variation of velocity for CuO for change in velocity when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; f ¼ 0:1, / ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
@ @ @p ðu þ cÞ ¼ ð1 /Þqf þ /qp ðu þ cÞ þ v @x @y @x " # 2 2 lf @ u @ u þ 2 þ g ð1 /Þqf bf þ /qp bp ðT T m Þ; þ 2:5 @x2 @y ð1 /Þ ð11Þ
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"
# @ @ @2T @2T ð1 /ÞðqCÞf þ /ðqCÞp ðu þ cÞ þ v þ T ¼ Kf K1 @x @y @x2 @y2 " ! # 2 2 2 lf @u @v @u @ v 2 þ þUþ þ þ : ð13Þ @x @y @x @y ð1 /Þ2:5 Introducing the following non-dimensional quantities:
x x¼ ; k
d¼
Fig. 16. Variation of velocity for Cu for change in velocity slip when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; f ¼ 0:1, / ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
y¼
d2 ; d1
Pr ¼
lf C f Kf
y ; d1
a¼
a1 ; d1
;
Ec ¼
u u¼ ; c
b¼
v¼
b1 ; d1
c2 ; C f ðT 1 T 0 Þ
g qf bf ðT 1 T 0 Þd1 ; clf
;
d¼
d1 ; k
h1 ¼
h¼
T Tm ; T1 T0
2
p¼
2
Gr ¼
v cd
u¼
d1 p ; cklf
H1 ; d1
h2 ¼
Re ¼
H2 ; d1
qf cd1 ; lf
2
Br ¼ PrEc;
@w ; @y
v ¼
e¼
d1 U ; ðT 1 T 0 ÞK f
@w : @x
ð14Þ
Eq. (10) is identically satisfied and Eqs. (11)–(13) after invoking the long wavelength and low Reynolds number approximations become:
Fig. 17. Variation of velocity for Ag for change in velocity slip when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; f ¼ 0:1, / ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
@p @3w ¼ A1 3 þ A2 Grh; @x @y
ð15Þ
@p ¼ 0; @y
ð16Þ
!2 @2h Br @2w þ e ¼ 0: K1 2 þ @y ð1 /Þ2:5 @y2
ð17Þ
Here d represents the wave number, a and b denote the dimensionless amplitudes at walls h1 and h2 respectively, p is the dimensionless pressure, while Re; Pr; Ec and Br represent the Reynolds, Prandtl, Eckert and Brinkman numbers respectively. e denotes the heat generation or absorption term and w is the stream function. The coefficients A1 ; A2 and K 1 in Eqs. (15) and (17) are:
A1 ¼
1 ð1 /Þ2:5
K p þ 2K f 2/ðK f K p Þ K p þ 2K f þ /ðK f K p Þ
K1 ¼ K1 ¼
;
! ðqbÞp A2 ¼ 1 / þ / ; ðqbÞf
0
used for Maxwell s model and
K p þ ðn 1ÞK f ðn 1Þ/ðK f K p Þ used for Hamilton—Crosser0 s model: K p þ ðn 1ÞK f þ /ðK f K p Þ
Dimensionless flow rates in the laboratory g ¼ cdQ1 and wave frames F ¼ cdq1 are related by equation
g ¼ F þ 1 þ d; Fig. 18. Variation of temperature for TiO2 for change in thermal slip when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1, / ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
@ @ ð1 /Þqf þ /qp ðu þ cÞ þ v v @x @y " # lf @p @2v @2v ; ¼ þ þ 2:5 @y ð1 /Þ @x2 @y2
ð18Þ
where Q and q represent the dimensional flow rates in the laboratory and wave frames respectively and
Z F¼
h1
h2
@w dy: @y
ð19Þ
The boundary conditions in the dimensionless form are defined as
ð12Þ
w¼
F ; 2
@w n @2w þ ¼ 1; 2:5 @y ð1 /Þ @y2
hþf
@h 1 ¼ ; @y 2
at y ¼ h1 ;
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Fig. 19. Variation of temperature for Al2O3 for change in thermal slip when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1, / ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
Fig. 22. Variation of temperature for Ag for change in thermal slip when a ¼ 0:7; b ¼ 0:6; c ¼ p=2, d ¼ 0:8; x ¼ 0, g ¼ 0:7; Br ¼ 0:3; n ¼ 0:1, / ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
Fig. 20. Variation of temperature for CuO for change in thermal slip when a ¼ 0:7; b ¼ 0:6; c ¼ p=2, d ¼ 0:8; x ¼ 0, g ¼ 0:7; Br ¼ 0:3, n ¼ 0:1; / ¼ 0:1, Gr ¼ 3:0; ¼ 2:5. K Fig. 23. Variation of heat transfer rate at the wall Keff h0 ðh1 Þ for TiO2–water f nanofluid when nanoparticle volume fraction is varied for when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8; x ¼ 0, g ¼ 0:7; Br ¼ 0:3; n ¼ 0:1, f ¼ 0:1; Gr ¼ 3:0, ¼ 2:5.
Fig. 21. Variation of temperature for Cu for change in thermal slip when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1; / ¼ 0:1, Gr ¼ 3:0; ¼ 2:5.
F w¼ ; 2
@w n @2w ¼ 1; @y ð1 /Þ2:5 @y2
hf
@h 1 ¼ ; @y 2
at y ¼ h2 : ð20Þ
K Fig. 24. Variation of heat transfer rate at the wall Keff h0 ðh1 Þ for Al2 O3–water f nanofluid when nanoparticle volume fraction is varied for when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1; f ¼ 0:1, Gr ¼ 3:0; ¼ 2:5.
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119
Wall shape in dimensional form are defined as:
h1 ðxÞ ¼ 1 þ a cosð2pxÞ h2 ðxÞ ¼ d b cosð2px þ cÞ:
ð21Þ
The system of Eqs. (15)–(17) subject to the boundary condition (20) is solved numerically by using NDSolve of MATHEMATICA. This software employs shooting method with fourth-order Runge–Kutta method for numerical computations. 3. Results and discussions Physical interpretation to the obtained numerical solutions is assigned in this section. Present study aims to analyze the behavior of velocity and temperature by varying volume fraction / for all the considered nanoparticles. We also focus in comparing the results through both Maxwell and Hamilton–Crosser (H–C) models. Fig. 2 shows the comparison of the effective thermal conductivity of the nanofluids predicted by both Maxwell and H–C models. It can be seen that the effective thermal conductivity anticipated by H–C model is greater than that determined by the Maxwell model. The difference between the effective thermal conductivity becomes substantial as / enlarges. For metallic particles such difference is greater than the metallic-oxides nanoparticles. We also observe that thermal conductivity of Ag–water nanofluid appear to be maximum at any value of /. Figs. 3–7 represent the profiles of axial velocity as a function of y for nanoparticles TiO2, Al2O3, CuO, Cu and Ag using both Maxwell and H–C models. It can be depicted from these Figs. that the axial velocity is maximum near the center of the channel. It can also be observed that magnitude of axial velocity decreases near the walls when nanoparticles volume fraction is enhanced. However it appears to increase with increase in / in central part of the channel. The results are qualitatively similar in both considered models but the effects are prominent in H–C model when compared with Maxwell model. The difference between the results of the two models grows as volume fraction / increases. Figs. 8–12 elucidate the behavior of temperature profile h as the nanoparticle volume fraction varies for all five nanoparticles. The results are discussed for Hamilton–Crosser’s and Maxwell’s model. As / enlarges from 0 to 0.2, temperature h appears to decrease. Temperature h is maximum at the center of the channel and its
K Fig. 25. Variation of heat transfer rate at the wall Keff h0 ðh1 Þ for CuO–water f nanofluid when nanoparticle volume fraction is varied for when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1; f ¼ 0:1, Gr ¼ 3:0; ¼ 2:5.
K Fig. 26. Variation of heat transfer rate at the wall Keff h0 ðh1 Þ for Cu–water f nanofluid when nanoparticle volume fraction is varied for when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8; x ¼ 0, g ¼ 0:7; Br ¼ 0:3, n ¼ 0:1; f ¼ 0:1, Gr ¼ 3:0; ¼ 2:5.
K Fig. 27. Variation of heat transfer rate at the wall Keff h0 ðh1 Þ for Ag–water f nanofluid when nanoparticle volume fraction is varied for when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1, f ¼ 0:1; Gr ¼ 3:0, ¼ 2:5.
values in Maxwell’s model is bigger than that in the H–C model. Furthermore, the difference between the results from two models increase when larger volume fraction is accounted. Figs. 13–17 give the velocity profiles for different values of velocity slip parameter in all five types of nanofluids. Magnitude of axial velocity increases with an increase in the slip effects only in the vicinity of the right wall. Figs. 18–22 give the effects of thermal slip parameter f on the temperature distribution. It can be seen that temperature of the fluid increases uniformly throughout the channel with an increase in thermal slip parameter. Here the variation in h with the slip parameter f appears to be are qualitatively similar in both Maxwell and H–C models. Figs. 23–27 provide the numerical values of the heat transfer rate at the wall for different volume fraction / in all nanoparticles used. The results are shown for both Maxwell and H–C models of effective thermal conductivity. The front row of the bars represent the values of heat transfer rate at the wall for Maxwell model, whereas the back row represents the values of heat transfer rate at the wall through the H–C model. It can be seen from the figures
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that the heat transfer rate at the wall augments when nanoparticles volume fraction is varied. Furthermore, it is observed that the values of H–C model are greater than the values of Maxwell model. Maximum heat transfer rate is observed for Ag–water nanofluids while heat transfer rate appears to be minimum for TiO2–water nanofluids. There is almost 20% growth in heat transfer rate of Ag–water nanofluids when / is increased from / ¼ 0 to / ¼ 0:1.
4. Conclusions Peristaltic flow of water based nanofluids with five different types of nanoparticles are addressed with the consideration of velocity and thermal slip conditions. Numerical solutions are derived. The key points of this study are outlined below: Velocity slip parameter accelerates the flow near the channel walls where as it restricts the fluid motion in the central part. The behaviors of thermal slip parameter f and nanoparticle volume fraction is opposite. The effects of parameters on velocity and temperature profiles are qualitatively similar in both Maxwell and H–C models. The heat transfer rate at the walls is enhanced when the nanoparticles with high thermal conductivity are used. Heat transfer rate is maximum/minimum for Ag–water/TiO2– water nanofluids. There is about 20% growth in heat transfer rate of Ag–water nanofluid when nanoparticle volume fraction varies from 0 to 0.1.
References [1] S.U.S. Choi, Enhancing thermal conductivity of the fluids with nanoparticles, ASME Fluids Eng. Div. 231 (1995) 99–105. [2] R.K. Tiwari, M.K. Das, Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids, Int. J. Heat Mass Transfer 50 (2007) 2002–2018. [3] J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transfer 128 (2006) 240–250. [4] J. Buongiorno, D.C. Venerus, N. Prabhat, et al., A benchmark study on the thermal conductivity of nanofluids, J. Appl. Phys. 106 (2009) 094312. [5] K. Khanafer, K. Vafai, M. Lightstone, Buoyancy-driven heat transfer enhancement in a two dimensional enclosure utilizing nanofluids, Int J. Heat Mass Transfer 46 (2003) 3639–3653. [6] A. Ghadimi, R. Saidur, H.S.C. Metselaar, A review of nanofluids stability properties and characterization in stationary conditions, Int. J. Heat Mass Transfer 54 (2011) 4051–4068. [7] I.M. Mahbubul, R. Saidur, M.A. Amalina, Hatest developments on the viscosity of nanofluids, Int. J. Heat Mass Transfer 55 (2012) 874–885. [8] K. Khanafer, K. Vafai, A critical synthesis of thermophysical characteristics of nanofluids, Int. J. Heat Mass Transfer 54 (2011) 4410–4428. [9] Z. Alloui, P. Vasseur, M. Reggio, Natural convection of nanofluids in a shallow cavity heated from below, Int. J. Therm. Sci. 50 (2011) 385–393. [10] M. Turkyilmazoglu, The analytical solution of mixed convection heat transfer and fluid flow of a MHD viscoelastic fluid over a permeable stretching surface, Int. J. Mech. Sci. 77 (2013) 263–268. [11] M. Turkyilmazoglu, Nanofluid flow and heat transfer due to a rotating disk, Comput. Fluids 94 (2014) 139–146. [12] M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, S. Soleimani, Heat flux boundary condition for nanofluid filled enclosure in presence of magnetic field, J. Mol. Liq. 193 (2014) 174–184. [13] M. Sheikholeslami, D.D. Ganji, Nanofluid flow and heat transfer between parallel plates considering Brownian motion using DTM, Comput. Methods Appl. Mech. Eng. 283 (2015) 651–663. [14] M. Sheikholeslami, D.D. Ganji, M.Y. Javed, R. Ellahi, Effect of thermal radiation on magnetohydrodynamics nanofluid flow and heat transfer by means of two phase model, J. Magn. Magn. Mater. 374 (2015) 36–43.
[15] M. Lomascolo, G. Colangelo, M. Milanese, A. Risi, Review of heat transfer in nanofluids: conductive, convective and radiative experimental results, Renew. Sustain. Energy Rev. 43 (2015) 1182–1198. [16] A.H. Shapiro, Pumping and retrograde diffusion in peristaltic waves, in: Proc. Workshop Ureteral Reftm Children, Nat. Acad. Sci., Washington, DC, vol. 1, 1967, pp. 109–126. [17] S.L. Weinberg, Theoretical and Experimental Treatment of Peristaltic Pumping and Its Relation to Ureteral Function, Ph.D. Thesis, MIT, Cambridge, MA, 1970. [18] S. Asghar, Q. Hussain, T. Hayat, Peristaltic motion of reactive viscous fluid with temperature dependent viscosity, Math. Comput. Appl. 18 (2013) 198–220. [19] Kh.S. Mekheimer, Y. Abd Elmaboud, A.I. Abdellateef, Particulate suspension flow induced by sinusoidal peristaltic waves through eccentric cylinders: thread annular, Int. J. Biomath. 06 (2013) 1350026. [20] S. Hina, M. Mustafa, T. Hayat, F.E. Alsaadi, Peristaltic motion of third grade fluid in curved channel, Appl. Math. Mech – Engl. Ed. 35 (2014) 73–84. [21] T. Hayat, F.M. Abbasi, B. Ahmad, A. Alsaedi, Peristaltic transport of Carreau– Yasuda fluid in a curved channel with slip effects, PLoS ONE 9 (2014) e95070. [22] M. Mustafa, S. Hina, T. Hayat, B. Ahmad, Influence of induced magnetic field on the peristaltic flow of nanofluid, Meccan 49 (2014) 521–534. [23] S. Noreen, M. Qasim, Influence of Hall current and viscous dissipation on pressure driven flow of psedoplastic fluid with heat generation: a mathematical study, PLoS ONE 10 (2015) e0129588. [24] F.M. Abbasi, T. Hayat, A. Alsaedi, Numerical analysis for MHD peristaltic transport of Carreau–Yasuda fluid in a curved channel with Hall effects, J. Magn. Magn. Mater. 382 (2015) 104–110. [25] M. Kothandapani, J. Prakash, Effect of thermal radiation parameter and magnetic field on the peristaltic flow of Williamson nanofluids in a tapered asymmetric channel, Int. J. Heat Mass Transfer 51 (2015) 234–245. [26] N.N. Jyothi, P. Devaki, S. Sreenadh, Analysis of magnetic field on the peristaltic transport of Johnson–Segalman fluid in an inclined channel bounded by flexible walls, Int. J. Curr. Res. 8 (2016) 26617–26634. [27] V.P. Rathod, L. Devindrappa, Peristaltic transport of a conducting Bingham fluid in an inclined channel with permeable walls by Adomian decomposition method, Int. J. Math. Trends Tech. 29 (2016) 70–79. [28] T. Hayat, M. Rafiq, B. Ahmad, Combined effects of rotation and thermal radiation on peristaltic transport of Jeffrey fluid, Int. J. Biomath. 8 (2015) 21. [29] M. Javed, T. Hayat, M. Mustafa, B. Ahmad, Velocity and thermal slip effects on peristaltic motion of Walters-B fluid, Int. J. Heat Mass Transfer 96 (2016) 210– 217. [30] T. Hayat, Sadaf Nawaz, A. Alsaedi, M. Rafiq, Mixed convective peristaltic flow of water based nanofluids with Joule heating and convective boundary conditions, PLoS One 11 (2016) e0153537. [31] T. Hayat, M. Rafiq, B. Ahmad, Radiative and Joule heating effects on peristaltic transport of dusty fluid in a channel with wall properties, Eur. Phys. J. Plus 129 (2015) 225. [32] S. Noreen, T. Hayat, A. Alsaedi, M. Qasim, Mixed convection heat and mass transfer in the peristaltic flow with chemical reaction and inclined magnetic field, Indian J. Phys. 87 (2013) 889–896. [33] O.U. Mehmood, N. Mustapha, S. Shafie, Heat transfer on peristaltic flow of fourth grade fluid in inclined asymmetric channel with partial slip, Appl. Math. Mech. – Engl. Ed. 33 (2012) 1313–1328. [34] S. Hina, M. Mustafa, T. Hayat, D. Naif Alotaibi, On the peristaltic motion of pseudoplastic fluid in a curved channel with heat/mass transfer and wall properties, Appl. Math. Comput. 263 (2015) 378–391. [35] T. Hayat, S. Noreen, M. Alhothuali, S. Asghar, A. Alhomaidan, Peristaltic flow under the effects of an induced magnetic field and heat and mass transfer, Int. J. Heat Mass Transfer 55 (2012) 443–452. [36] Kh.S. Mekheimer, S.Z.A. Husseny, Y. Abd Elmaboud, Effects of heat transfer and space porosity on peristaltic flow in a vertical asymmetric channel, Numer. Methods Part. Diff. Eqs. 26 (2010) 747–770. [37] T. Hayat, S. Noreen, M. Qasim, Influence of heat and mass transfer on the peristaltic transport of Phan–Thien–Tanner fluid, ZNA 68a (2013) 751–758. [38] F.M. Abbasi, T. Hayat, B. Ahmad, Impact of magnetic field on mixed convective peristaltic flow of water based nanofluids with Joule heating, Z. Naturforsch. 70 (2a) (2015) 125–132. [39] F.M. Abbasi, T. Hayat, B. Ahmad, Peristaltic transport of copper–water nanofluid saturating porous medium, Physica E 67 (2015) 47–53. [40] F.M. Abbasi, T. Hayat, A. Alsaedi, Peristaltic transport of magnetonanoparticles submerged in water: model for drug delivery system, Physica E 68 (2015) 123–132. [41] F.M. Abbasi, T. Hayat, B. Ahmad, Peristaltic flow in an asymmetric channel with convective boundary conditions and Joule heating, J. Cent. South Univ. 21 (2014) 1411–1416. [42] S.A. Shehzad, F.M. Abbasi, T. Hayat, F. Alsaadi, Model and comparative study for the peristaltic transport of water based nanofluids, J. Mol. Liq. 209 (2015) 723–728. [43] J.C. Maxwell, A Treatise on Electricity and Magnetism, second ed., Oxford University Press, Cambridge, 1904. 435–441. [44] R.L. Hamilton, O.K. Crosser, Thermal conductivity of heterogeneous two component systems, EC Fundam. 1 (1962) 187–191.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
A model for an application to biomedical engineering through nanoparticles T. Hayat a, Sadaf Nawaz a, F. Alsaadi b, M. Rafiq a, M. Mustafa c,⇑ a
Department of Mathematics, Quaid-I-Azam University, 45320, Islamabad 44000, Pakistan Department of Electrical and Computer Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia c School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Islamabad 44000, Pakistan b
a r t i c l e
i n f o
Article history: Received 7 November 2015 Received in revised form 4 May 2016 Accepted 6 May 2016
Keywords: Slip condition Peristalsis Mixed convection Nanofluid Maxwell model Hamilton–Crosser model
a b s t r a c t Recent advancements in nanoscience and technology has made the nanofluid an important research topic. Various models have been put forward to estimate the effective thermal conductivity of nanofluids. Present article addresses the comparative study of Maxwell’s and Hamilton–Crosser’s model for mixed convection peristaltic flow of incompressible nanofluid in an asymmetric channel. Viscous dissipation and heat generation/absorption effects are retained. Analysis is performed for five different types of nanoparticles namely titanium oxide or titania (TiO2), aluminum oxide or alumina (Al2O3), copper oxide (CuO), copper (Cu) and silver (Ag) with water as base fluid. Velocity and thermal slip conditions are employed. Lubrication approach is adopted for problem formulation. The developed non-linear problems are solved numerically. Plots for axial velocity, temperature and heat transfer rate at the wall are obtained and analyzed. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Heat transfer enhancement determines the need for innovative coolants with improved performance. The novel concept of nanofluid has been introduced to enhance the heat transfer capability of conventional coolants such as water, ethylene glycol and oils. The terminology of nanofluid was first used by Choi [1] who demonstrated an anomalous increase in thermal conductivity of water and other liquids through dispersion of copper and aluminum nanoparticles. This enhanced feature of nanofluid has led to the plethora of diverse applications such as cooling of microelectronics, engine cooling/vehicle thermal management, magnetic drug targeting, space cooling, in grinding machining and many others. Utilization of nanofluids in water cooled nuclear reactor can produce safety margins and provides economic gains. Nanofluids are also employed in solar collectors for their tunable optical properties. Nanofluids can improve the performance efficiency of heat exchangers by reducing total heat resistance. In biomedicine, nanofluids serve as carriers for delivering drugs and radiation in cancer patients. Nanofluids can be considered as single phase liquids and hence classical theory of single phase fluids can be applied in which physical properties of nanofluids are expressed ⇑ Corresponding author. Tel.: +92 51 9085 5596. E-mail address: [email protected] (M. Mustafa). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.05.033 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
as function of both constituents and their volume fractions. Keeping this in view Tiwari and Das [2] proposed a mathematical model to estimate heat transfer behavior by varying nanoparticle volume fraction. Buongiorno [3] also suggested a mathematical model for nanofluids convective transport considering novel aspects of Brownian motion and thermophoresis. Both the above mentioned models have been applied to investigate heat transfer problems involving nanofluids [4–15]. Peristaltic flow refers to the transportation of fluid inside a channel or tube by the action of flexible walls. It is a major mechanism for fluid flow in many biological and industrial systems. Within human body it is involved in swallowing of food through esophagus, movement of chyme in the gastro-intestinal track, in the ductus efferentes of the male reproductive system, vasomotion of small blood vessels such as arterioles and capillaries. Peristaltic pumps are used to transport corrosive materials in order to avoid direct contact of the fluid with the pump’s internal surface. Many biomedical devices as dialysis machines, open heart bypass pump machines, infusion pumps, etc. are engineered on the mechanism of peristaltic. In robotic industry, the concept of peristalsis has been used for the production of robots. The pioneering work on peristalsis was performed to investigate the urine transport. Shapiro [16] in 1967 analyzed the peristaltic pumping in a two dimensional flexible tube. Later, the theoretical results determined in [16] were experimentally confirmed by Weinberg [17]. Some
T. Hayat et al. / International Journal of Heat and Mass Transfer 101 (2016) 112–120
113
Nomenclature H 1 ; H2 X; Y Kf ðU; VÞ Kp K eff a1 ; b1
c F
g Q q h1 ; h2
U T0; T1
qeff leff ðqCÞeff ðqbÞeff e n n
qf ; qp bf bp a; b
right and left walls space coordinates in fixed frame thermal conductivity of base fluid velocity components in fixed frame thermal conductivity of nanoparticle effective thermal conductivity dimensional wave amplitude phase difference dimensionless flow rate in wave frame dimensionless flow rate in fixed frame volume flow rate in fixed frame volume flow rate in wave frame dimensionless right/left walls heat generation/absorption temperature of right/left wall effective density of nanofluid effective viscosity of nanofluid effective heat capacity of nanofluid effective thermal expansion of nanofluid dimensionless heat generation/absorption dimensionless velocity slip parameter shape factor of nanoparticles density of fluid and nanoparticle fluid thermal expansion coefficient nanoparticle thermal expansion coefficient dimensionless wave amplitudes
recent researches dealing with the peristaltic motion are mentioned in Refs. [18–27]. The effect of heat transfer cannot be ignored in peristalsis especially when dealing with the blood flow simulation related to tumors and muscles, drug transport, production of osteoinductive material, nutrients to brain cell, etc. Peristalsis is also seen in the processes of oxygenation and hemodialysis. Some references pertaining to the heat transfer phenomenon in peristalsis are mentioned in [28–37]. Peristaltic transport of nanofluids are utilized in modern drug delivery systems and in cancer therapy to destroy undesirable tissues. Despite the aforementioned applications, very little attention has been given to the studies dealing with the peristaltic transport of nanofluids [38–42]. Current study aims to address the effects of velocity and thermal slip on the peristaltic transport of nanofluids through an asymmetric channel. Two different models for effective thermal conductivity of nanofluids are utilized. Long wavelength and low Reynolds number approximation are adopted for problem formation. Arising system comprising of coupled equations has been solved numerically through NDSolve of MATHEMATICA. Graphs are sketched to see the effects of embedded parameters on velocity and temperature distributions.
t d1 þ d2 / k ðx; yÞ ðu; v Þ p ðx; yÞ P Tm c T g w Pr Br Re d Ec Gr f Cf
lf Cp d
time in fixed frame width of channel nanoparticle volume fraction wavelength space coordinates in wave frame velocity components in wave frame pressure in wave frame dimensionless space coordinates pressure in fixed frame mean temperature wave speed fluid temperature gravity stream function Prandtl number Brinkman number Reynolds number wave number Eckert number Grashoff number thermal slip parameter specific heat of fluid dynamic viscosity of fluid specific heat of nanoparticle ratio of d2 and d1
K eff K p þ 2K f 2/ðK f K p Þ ¼ : Kf K p þ 2K f þ /ðK f K p Þ
Hamilton–Crosser model the thermal conductivity of nanofluid is given by [44]:
K eff K p þ ðn 1ÞK f ðn 1Þ/ðK f K p Þ ¼ : Kf K p þ ðn 1ÞK f þ /ðK f K p Þ
ð2Þ
Here K eff is the effective thermal conductivity of the nanofluid, K f is the thermal conductivity of water, K p is the thermal conductivity of nanoparticles and / is the nanoparticle volume fraction. n in
2. Problem formulation We consider the peristaltic motion of an incompressible nanofluid flowing through a two dimensional vertical asymmetric channel of width d1 þ d2 (see Fig. 1). Nanofluid comprises of five different types of nanoparticles namely titanium oxide or titania (TiO2), aluminum oxide or alumina (Al2O3), copper oxide (CuO), copper (Cu) and silver (Ag). Water is considered as the base fluid. We take into account two different models for effective conductivity of nanofluids. Maxwell–Garnelt model for effective thermal conductivity is expressed as [43]:
ð1Þ
Fig. 1. Geometry of the problem.
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T. Hayat et al. / International Journal of Heat and Mass Transfer 101 (2016) 112–120
Table 1 Thermo-physical properties of water and nanoparticles.
H2O TiO2 Al2O3 CuO Cu Ag
q (kg m3)
C p (J kg1 K1)
K W m1 K1
b (l/k) 106
997.1 4250 3970 6320 8933 10,500
4179 686.2 765 531.8 385 235
0.613 8.9538 40 76.5 401 429
210 9.0 8.5 18.0 16.7 18.9
Hamilton–Crosser’s model denotes the shape factor of nanoparticles. The value of n ¼ 3 and n ¼ 6 correspond to the spherical and cylindrical nanoparticles respectively. For n ¼ 3, Hamilton–Cross er’s model reduces to the Maxwell’s model. We select rectangular coordinate system in such a way that the X axis lies along the walls of the channel and Y axis lies perpendicular to it. The flow is due to an infinite sinusoidal wave of wavelength k traveling with speed c along the walls of channel. Mathematically, wave shape are expressed as:
2p H1 X; t ¼ d1 þ a1 cos X ct ; k 2p H2 X; t ¼ d2 b1 cos X ct þ c ; k
" # @ @ @ @P @2U @2U þ gðqbÞeff ðT T m Þ; þU þV U ¼ þ leff þ @t @X @Y @X @X 2 @Y 2
qeff
ð5Þ
in the region Y < 0; a1 is the wave amplitude traveling along H1 X; t ; b1 is the wave amplitude traveling along H2 X; t ; c represents the phase difference which varies in the range 0 6 c 6 p and t denotes the time. Moreover a1 ; b1 ; d1 ; d2 and c satisfy the following condition
" # @ @ @ @P @2V @2V ; þU V ¼ þ þV þ leff @t @X @Y @Y @X 2 @Y 2
ð3Þ
where H1 X; t is the wall in the region Y > 0 and H2 X; t is the wall
2
Fig. 3. Variation of axial velocity for TiO2 for change in / when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1, f ¼ 0:1; Gr ¼ 3:0, ¼ 2:5.
qeff
ðqCÞeff
" # @ @ @ @2 T @2 T þV þU þ þU T ¼ K eff @t @X @Y @X 2 @Y 2 2 0 !2 !2 1 !2 3 @U @V @U @V Aþ 5: þ þ leff 42@ þ @X @Y @Y @X ð7Þ
2
a21 þ b1 þ 2a1 b1 cos c 6 ðd1 þ d2 Þ : The wall H1 X; t is maintained at constant temperature T 0 and temperature of wall H2 X; t is T 1 ð> T 0 Þ. The velocity field for the two dimensional flow is given by ½U X; Y; t ; V X; Y; t ; 0. The conservation laws of mass, momentum, and energy are governed by the following equations:
@U @V þ ¼ 0; @X @Y
ð6Þ
ð4Þ
In the above equations P X; Y; t represents the pressure, 1 T m ¼ T 0 þT is the mean temperature, g is the acceleration due to 2 gravity and U denotes the heat generation/absorption term. For the two-phase flow model we can take qeff denotes the effective density, leff the effective viscosity, ðqCÞeff the effective heat capacity and ðqbÞeff the effective thermal expansion of the nanofluid. These are expressed as [8,38,39]:
Fig. 2. Effective thermal conductivity for Maxwell’s and H–C model.
T. Hayat et al. / International Journal of Heat and Mass Transfer 101 (2016) 112–120
Fig. 4. Variation of axial velocity for Al2O3 for change in / when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1, f ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
Fig. 5. Variation of axial velocity for CuO for change in / when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1, f ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
Fig. 6. Variation of axial velocity for Cu for change in / when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1, f ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
115
Fig. 7. Variation of axial velocity for Ag for change in / when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1, f ¼ 0:1; Gr ¼ 3:0, ¼ 2:5.
Fig. 8. Variation of temperature for TiO2 for change in / when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1, f ¼ 0:1; Gr ¼ 3:0, ¼ 2:5.
Fig. 9. Variation of temperature for Al2O3 for change in / when a ¼ 0:7, b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1, f ¼ 0:1; Gr ¼ 3:0, ¼ 2:5.
qeff ¼ ð1 /Þqf þ /qp ; ðqCÞeff ¼ ð1 /ÞðqCÞf þ /ðqCÞp ðqbÞeff ¼ ð1 /Þqf bf þ /qp bp ;
leff ¼
lf ð1 /Þ2:5
;
ð8Þ
Here q is the density, C the specific heat, b the thermal expansion coefficient and l the dynamic viscosity. The subscripts p and f correspond to the solid and fluid phases respectively.
Numerical values of the thermo-physical properties of water and nanoparticles are listed in Table 1. The transformations between laboratory and wave frames are defined as:
x ¼ X ct;
v ðx; yÞ ¼ V
y ¼ Y;
X; Y; t ;
uðx; yÞ ¼ U X; Y; t c;
pðx; yÞ ¼ P X; Y; t ;
ð9Þ
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Fig. 10. Variation of temperature for CuO for change in / when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1, f ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
Fig. 11. Variation of temperature for Cu for change in / when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1, f ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
Fig. 12. Variation of temperature for Ag for change in / when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1, f ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
in which uðx; yÞ and v ðx; yÞ are the velocity components and pðx; yÞ denotes the pressure. Eqs. (4)–(7) in view of Eq. (9) become
@u @ v þ ¼ 0; @x @y
ð10Þ
Fig. 13. Variation of velocity for TiO2 for change in velocity slip when a ¼ 0:7; b ¼ 0:6, c ¼ p=2, d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; f ¼ 0:1, / ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
Fig. 14. Variation of velocity for Al2O3 for change in velocity slip when a ¼ 0:7; b ¼ 0:6; c ¼ p=2; d ¼ 0:8; x ¼ 0; g ¼ 0:7; Br ¼ 0:3; f ¼ 0:1; / ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
Fig. 15. Variation of velocity for CuO for change in velocity when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; f ¼ 0:1, / ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
@ @ @p ðu þ cÞ ¼ ð1 /Þqf þ /qp ðu þ cÞ þ v @x @y @x " # 2 2 lf @ u @ u þ 2 þ g ð1 /Þqf bf þ /qp bp ðT T m Þ; þ 2:5 @x2 @y ð1 /Þ ð11Þ
117
T. Hayat et al. / International Journal of Heat and Mass Transfer 101 (2016) 112–120
"
# @ @ @2T @2T ð1 /ÞðqCÞf þ /ðqCÞp ðu þ cÞ þ v þ T ¼ Kf K1 @x @y @x2 @y2 " ! # 2 2 2 lf @u @v @u @ v 2 þ þUþ þ þ : ð13Þ @x @y @x @y ð1 /Þ2:5 Introducing the following non-dimensional quantities:
x x¼ ; k
d¼
Fig. 16. Variation of velocity for Cu for change in velocity slip when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; f ¼ 0:1, / ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
y¼
d2 ; d1
Pr ¼
lf C f Kf
y ; d1
a¼
a1 ; d1
;
Ec ¼
u u¼ ; c
b¼
v¼
b1 ; d1
c2 ; C f ðT 1 T 0 Þ
g qf bf ðT 1 T 0 Þd1 ; clf
;
d¼
d1 ; k
h1 ¼
h¼
T Tm ; T1 T0
2
p¼
2
Gr ¼
v cd
u¼
d1 p ; cklf
H1 ; d1
h2 ¼
Re ¼
H2 ; d1
qf cd1 ; lf
2
Br ¼ PrEc;
@w ; @y
v ¼
e¼
d1 U ; ðT 1 T 0 ÞK f
@w : @x
ð14Þ
Eq. (10) is identically satisfied and Eqs. (11)–(13) after invoking the long wavelength and low Reynolds number approximations become:
Fig. 17. Variation of velocity for Ag for change in velocity slip when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; f ¼ 0:1, / ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
@p @3w ¼ A1 3 þ A2 Grh; @x @y
ð15Þ
@p ¼ 0; @y
ð16Þ
!2 @2h Br @2w þ e ¼ 0: K1 2 þ @y ð1 /Þ2:5 @y2
ð17Þ
Here d represents the wave number, a and b denote the dimensionless amplitudes at walls h1 and h2 respectively, p is the dimensionless pressure, while Re; Pr; Ec and Br represent the Reynolds, Prandtl, Eckert and Brinkman numbers respectively. e denotes the heat generation or absorption term and w is the stream function. The coefficients A1 ; A2 and K 1 in Eqs. (15) and (17) are:
A1 ¼
1 ð1 /Þ2:5
K p þ 2K f 2/ðK f K p Þ K p þ 2K f þ /ðK f K p Þ
K1 ¼ K1 ¼
;
! ðqbÞp A2 ¼ 1 / þ / ; ðqbÞf
0
used for Maxwell s model and
K p þ ðn 1ÞK f ðn 1Þ/ðK f K p Þ used for Hamilton—Crosser0 s model: K p þ ðn 1ÞK f þ /ðK f K p Þ
Dimensionless flow rates in the laboratory g ¼ cdQ1 and wave frames F ¼ cdq1 are related by equation
g ¼ F þ 1 þ d; Fig. 18. Variation of temperature for TiO2 for change in thermal slip when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1, / ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
@ @ ð1 /Þqf þ /qp ðu þ cÞ þ v v @x @y " # lf @p @2v @2v ; ¼ þ þ 2:5 @y ð1 /Þ @x2 @y2
ð18Þ
where Q and q represent the dimensional flow rates in the laboratory and wave frames respectively and
Z F¼
h1
h2
@w dy: @y
ð19Þ
The boundary conditions in the dimensionless form are defined as
ð12Þ
w¼
F ; 2
@w n @2w þ ¼ 1; 2:5 @y ð1 /Þ @y2
hþf
@h 1 ¼ ; @y 2
at y ¼ h1 ;
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Fig. 19. Variation of temperature for Al2O3 for change in thermal slip when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1, / ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
Fig. 22. Variation of temperature for Ag for change in thermal slip when a ¼ 0:7; b ¼ 0:6; c ¼ p=2, d ¼ 0:8; x ¼ 0, g ¼ 0:7; Br ¼ 0:3; n ¼ 0:1, / ¼ 0:1; Gr ¼ 3:0; ¼ 2:5.
Fig. 20. Variation of temperature for CuO for change in thermal slip when a ¼ 0:7; b ¼ 0:6; c ¼ p=2, d ¼ 0:8; x ¼ 0, g ¼ 0:7; Br ¼ 0:3, n ¼ 0:1; / ¼ 0:1, Gr ¼ 3:0; ¼ 2:5. K Fig. 23. Variation of heat transfer rate at the wall Keff h0 ðh1 Þ for TiO2–water f nanofluid when nanoparticle volume fraction is varied for when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8; x ¼ 0, g ¼ 0:7; Br ¼ 0:3; n ¼ 0:1, f ¼ 0:1; Gr ¼ 3:0, ¼ 2:5.
Fig. 21. Variation of temperature for Cu for change in thermal slip when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1; / ¼ 0:1, Gr ¼ 3:0; ¼ 2:5.
F w¼ ; 2
@w n @2w ¼ 1; @y ð1 /Þ2:5 @y2
hf
@h 1 ¼ ; @y 2
at y ¼ h2 : ð20Þ
K Fig. 24. Variation of heat transfer rate at the wall Keff h0 ðh1 Þ for Al2 O3–water f nanofluid when nanoparticle volume fraction is varied for when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1; f ¼ 0:1, Gr ¼ 3:0; ¼ 2:5.
T. Hayat et al. / International Journal of Heat and Mass Transfer 101 (2016) 112–120
119
Wall shape in dimensional form are defined as:
h1 ðxÞ ¼ 1 þ a cosð2pxÞ h2 ðxÞ ¼ d b cosð2px þ cÞ:
ð21Þ
The system of Eqs. (15)–(17) subject to the boundary condition (20) is solved numerically by using NDSolve of MATHEMATICA. This software employs shooting method with fourth-order Runge–Kutta method for numerical computations. 3. Results and discussions Physical interpretation to the obtained numerical solutions is assigned in this section. Present study aims to analyze the behavior of velocity and temperature by varying volume fraction / for all the considered nanoparticles. We also focus in comparing the results through both Maxwell and Hamilton–Crosser (H–C) models. Fig. 2 shows the comparison of the effective thermal conductivity of the nanofluids predicted by both Maxwell and H–C models. It can be seen that the effective thermal conductivity anticipated by H–C model is greater than that determined by the Maxwell model. The difference between the effective thermal conductivity becomes substantial as / enlarges. For metallic particles such difference is greater than the metallic-oxides nanoparticles. We also observe that thermal conductivity of Ag–water nanofluid appear to be maximum at any value of /. Figs. 3–7 represent the profiles of axial velocity as a function of y for nanoparticles TiO2, Al2O3, CuO, Cu and Ag using both Maxwell and H–C models. It can be depicted from these Figs. that the axial velocity is maximum near the center of the channel. It can also be observed that magnitude of axial velocity decreases near the walls when nanoparticles volume fraction is enhanced. However it appears to increase with increase in / in central part of the channel. The results are qualitatively similar in both considered models but the effects are prominent in H–C model when compared with Maxwell model. The difference between the results of the two models grows as volume fraction / increases. Figs. 8–12 elucidate the behavior of temperature profile h as the nanoparticle volume fraction varies for all five nanoparticles. The results are discussed for Hamilton–Crosser’s and Maxwell’s model. As / enlarges from 0 to 0.2, temperature h appears to decrease. Temperature h is maximum at the center of the channel and its
K Fig. 25. Variation of heat transfer rate at the wall Keff h0 ðh1 Þ for CuO–water f nanofluid when nanoparticle volume fraction is varied for when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1; f ¼ 0:1, Gr ¼ 3:0; ¼ 2:5.
K Fig. 26. Variation of heat transfer rate at the wall Keff h0 ðh1 Þ for Cu–water f nanofluid when nanoparticle volume fraction is varied for when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8; x ¼ 0, g ¼ 0:7; Br ¼ 0:3, n ¼ 0:1; f ¼ 0:1, Gr ¼ 3:0; ¼ 2:5.
K Fig. 27. Variation of heat transfer rate at the wall Keff h0 ðh1 Þ for Ag–water f nanofluid when nanoparticle volume fraction is varied for when a ¼ 0:7; b ¼ 0:6, c ¼ p=2; d ¼ 0:8, x ¼ 0; g ¼ 0:7, Br ¼ 0:3; n ¼ 0:1, f ¼ 0:1; Gr ¼ 3:0, ¼ 2:5.
values in Maxwell’s model is bigger than that in the H–C model. Furthermore, the difference between the results from two models increase when larger volume fraction is accounted. Figs. 13–17 give the velocity profiles for different values of velocity slip parameter in all five types of nanofluids. Magnitude of axial velocity increases with an increase in the slip effects only in the vicinity of the right wall. Figs. 18–22 give the effects of thermal slip parameter f on the temperature distribution. It can be seen that temperature of the fluid increases uniformly throughout the channel with an increase in thermal slip parameter. Here the variation in h with the slip parameter f appears to be are qualitatively similar in both Maxwell and H–C models. Figs. 23–27 provide the numerical values of the heat transfer rate at the wall for different volume fraction / in all nanoparticles used. The results are shown for both Maxwell and H–C models of effective thermal conductivity. The front row of the bars represent the values of heat transfer rate at the wall for Maxwell model, whereas the back row represents the values of heat transfer rate at the wall through the H–C model. It can be seen from the figures
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that the heat transfer rate at the wall augments when nanoparticles volume fraction is varied. Furthermore, it is observed that the values of H–C model are greater than the values of Maxwell model. Maximum heat transfer rate is observed for Ag–water nanofluids while heat transfer rate appears to be minimum for TiO2–water nanofluids. There is almost 20% growth in heat transfer rate of Ag–water nanofluids when / is increased from / ¼ 0 to / ¼ 0:1.
4. Conclusions Peristaltic flow of water based nanofluids with five different types of nanoparticles are addressed with the consideration of velocity and thermal slip conditions. Numerical solutions are derived. The key points of this study are outlined below: Velocity slip parameter accelerates the flow near the channel walls where as it restricts the fluid motion in the central part. The behaviors of thermal slip parameter f and nanoparticle volume fraction is opposite. The effects of parameters on velocity and temperature profiles are qualitatively similar in both Maxwell and H–C models. The heat transfer rate at the walls is enhanced when the nanoparticles with high thermal conductivity are used. Heat transfer rate is maximum/minimum for Ag–water/TiO2– water nanofluids. There is about 20% growth in heat transfer rate of Ag–water nanofluid when nanoparticle volume fraction varies from 0 to 0.1.
References [1] S.U.S. Choi, Enhancing thermal conductivity of the fluids with nanoparticles, ASME Fluids Eng. Div. 231 (1995) 99–105. [2] R.K. Tiwari, M.K. Das, Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids, Int. J. Heat Mass Transfer 50 (2007) 2002–2018. [3] J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transfer 128 (2006) 240–250. [4] J. Buongiorno, D.C. Venerus, N. Prabhat, et al., A benchmark study on the thermal conductivity of nanofluids, J. Appl. Phys. 106 (2009) 094312. [5] K. Khanafer, K. Vafai, M. Lightstone, Buoyancy-driven heat transfer enhancement in a two dimensional enclosure utilizing nanofluids, Int J. Heat Mass Transfer 46 (2003) 3639–3653. [6] A. Ghadimi, R. Saidur, H.S.C. Metselaar, A review of nanofluids stability properties and characterization in stationary conditions, Int. J. Heat Mass Transfer 54 (2011) 4051–4068. [7] I.M. Mahbubul, R. Saidur, M.A. Amalina, Hatest developments on the viscosity of nanofluids, Int. J. Heat Mass Transfer 55 (2012) 874–885. [8] K. Khanafer, K. Vafai, A critical synthesis of thermophysical characteristics of nanofluids, Int. J. Heat Mass Transfer 54 (2011) 4410–4428. [9] Z. Alloui, P. Vasseur, M. Reggio, Natural convection of nanofluids in a shallow cavity heated from below, Int. J. Therm. Sci. 50 (2011) 385–393. [10] M. Turkyilmazoglu, The analytical solution of mixed convection heat transfer and fluid flow of a MHD viscoelastic fluid over a permeable stretching surface, Int. J. Mech. Sci. 77 (2013) 263–268. [11] M. Turkyilmazoglu, Nanofluid flow and heat transfer due to a rotating disk, Comput. Fluids 94 (2014) 139–146. [12] M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, S. Soleimani, Heat flux boundary condition for nanofluid filled enclosure in presence of magnetic field, J. Mol. Liq. 193 (2014) 174–184. [13] M. Sheikholeslami, D.D. Ganji, Nanofluid flow and heat transfer between parallel plates considering Brownian motion using DTM, Comput. Methods Appl. Mech. Eng. 283 (2015) 651–663. [14] M. Sheikholeslami, D.D. Ganji, M.Y. Javed, R. Ellahi, Effect of thermal radiation on magnetohydrodynamics nanofluid flow and heat transfer by means of two phase model, J. Magn. Magn. Mater. 374 (2015) 36–43.
[15] M. Lomascolo, G. Colangelo, M. Milanese, A. Risi, Review of heat transfer in nanofluids: conductive, convective and radiative experimental results, Renew. Sustain. Energy Rev. 43 (2015) 1182–1198. [16] A.H. Shapiro, Pumping and retrograde diffusion in peristaltic waves, in: Proc. Workshop Ureteral Reftm Children, Nat. Acad. Sci., Washington, DC, vol. 1, 1967, pp. 109–126. [17] S.L. Weinberg, Theoretical and Experimental Treatment of Peristaltic Pumping and Its Relation to Ureteral Function, Ph.D. Thesis, MIT, Cambridge, MA, 1970. [18] S. Asghar, Q. Hussain, T. Hayat, Peristaltic motion of reactive viscous fluid with temperature dependent viscosity, Math. Comput. Appl. 18 (2013) 198–220. [19] Kh.S. Mekheimer, Y. Abd Elmaboud, A.I. Abdellateef, Particulate suspension flow induced by sinusoidal peristaltic waves through eccentric cylinders: thread annular, Int. J. Biomath. 06 (2013) 1350026. [20] S. Hina, M. Mustafa, T. Hayat, F.E. Alsaadi, Peristaltic motion of third grade fluid in curved channel, Appl. Math. Mech – Engl. Ed. 35 (2014) 73–84. [21] T. Hayat, F.M. Abbasi, B. Ahmad, A. Alsaedi, Peristaltic transport of Carreau– Yasuda fluid in a curved channel with slip effects, PLoS ONE 9 (2014) e95070. [22] M. Mustafa, S. Hina, T. Hayat, B. Ahmad, Influence of induced magnetic field on the peristaltic flow of nanofluid, Meccan 49 (2014) 521–534. [23] S. Noreen, M. Qasim, Influence of Hall current and viscous dissipation on pressure driven flow of psedoplastic fluid with heat generation: a mathematical study, PLoS ONE 10 (2015) e0129588. [24] F.M. Abbasi, T. Hayat, A. Alsaedi, Numerical analysis for MHD peristaltic transport of Carreau–Yasuda fluid in a curved channel with Hall effects, J. Magn. Magn. Mater. 382 (2015) 104–110. [25] M. Kothandapani, J. Prakash, Effect of thermal radiation parameter and magnetic field on the peristaltic flow of Williamson nanofluids in a tapered asymmetric channel, Int. J. Heat Mass Transfer 51 (2015) 234–245. [26] N.N. Jyothi, P. Devaki, S. Sreenadh, Analysis of magnetic field on the peristaltic transport of Johnson–Segalman fluid in an inclined channel bounded by flexible walls, Int. J. Curr. Res. 8 (2016) 26617–26634. [27] V.P. Rathod, L. Devindrappa, Peristaltic transport of a conducting Bingham fluid in an inclined channel with permeable walls by Adomian decomposition method, Int. J. Math. Trends Tech. 29 (2016) 70–79. [28] T. Hayat, M. Rafiq, B. Ahmad, Combined effects of rotation and thermal radiation on peristaltic transport of Jeffrey fluid, Int. J. Biomath. 8 (2015) 21. [29] M. Javed, T. Hayat, M. Mustafa, B. Ahmad, Velocity and thermal slip effects on peristaltic motion of Walters-B fluid, Int. J. Heat Mass Transfer 96 (2016) 210– 217. [30] T. Hayat, Sadaf Nawaz, A. Alsaedi, M. Rafiq, Mixed convective peristaltic flow of water based nanofluids with Joule heating and convective boundary conditions, PLoS One 11 (2016) e0153537. [31] T. Hayat, M. Rafiq, B. Ahmad, Radiative and Joule heating effects on peristaltic transport of dusty fluid in a channel with wall properties, Eur. Phys. J. Plus 129 (2015) 225. [32] S. Noreen, T. Hayat, A. Alsaedi, M. Qasim, Mixed convection heat and mass transfer in the peristaltic flow with chemical reaction and inclined magnetic field, Indian J. Phys. 87 (2013) 889–896. [33] O.U. Mehmood, N. Mustapha, S. Shafie, Heat transfer on peristaltic flow of fourth grade fluid in inclined asymmetric channel with partial slip, Appl. Math. Mech. – Engl. Ed. 33 (2012) 1313–1328. [34] S. Hina, M. Mustafa, T. Hayat, D. Naif Alotaibi, On the peristaltic motion of pseudoplastic fluid in a curved channel with heat/mass transfer and wall properties, Appl. Math. Comput. 263 (2015) 378–391. [35] T. Hayat, S. Noreen, M. Alhothuali, S. Asghar, A. Alhomaidan, Peristaltic flow under the effects of an induced magnetic field and heat and mass transfer, Int. J. Heat Mass Transfer 55 (2012) 443–452. [36] Kh.S. Mekheimer, S.Z.A. Husseny, Y. Abd Elmaboud, Effects of heat transfer and space porosity on peristaltic flow in a vertical asymmetric channel, Numer. Methods Part. Diff. Eqs. 26 (2010) 747–770. [37] T. Hayat, S. Noreen, M. Qasim, Influence of heat and mass transfer on the peristaltic transport of Phan–Thien–Tanner fluid, ZNA 68a (2013) 751–758. [38] F.M. Abbasi, T. Hayat, B. Ahmad, Impact of magnetic field on mixed convective peristaltic flow of water based nanofluids with Joule heating, Z. Naturforsch. 70 (2a) (2015) 125–132. [39] F.M. Abbasi, T. Hayat, B. Ahmad, Peristaltic transport of copper–water nanofluid saturating porous medium, Physica E 67 (2015) 47–53. [40] F.M. Abbasi, T. Hayat, A. Alsaedi, Peristaltic transport of magnetonanoparticles submerged in water: model for drug delivery system, Physica E 68 (2015) 123–132. [41] F.M. Abbasi, T. Hayat, B. Ahmad, Peristaltic flow in an asymmetric channel with convective boundary conditions and Joule heating, J. Cent. South Univ. 21 (2014) 1411–1416. [42] S.A. Shehzad, F.M. Abbasi, T. Hayat, F. Alsaadi, Model and comparative study for the peristaltic transport of water based nanofluids, J. Mol. Liq. 209 (2015) 723–728. [43] J.C. Maxwell, A Treatise on Electricity and Magnetism, second ed., Oxford University Press, Cambridge, 1904. 435–441. [44] R.L. Hamilton, O.K. Crosser, Thermal conductivity of heterogeneous two component systems, EC Fundam. 1 (1962) 187–191.