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Nanoparticle shapes effects on unsteady physiological transport of nanofluids through a finite length non-uniform channel
Accepted Manuscript Nanoparticle Shapes Effects on Unsteady Physiological Transport of Nanofluids through a Finite Length Non-Uniform Channel Noreen Sher Akbar, Adil Wahid Butt, Dharmendra Tripathi PII: DOI: Reference:
S2211-3797(17)30944-0 http://dx.doi.org/10.1016/j.rinp.2017.07.019 RINP 792
To appear in:
Results in Physics
Received Date: Revised Date: Accepted Date:
30 May 2017 7 July 2017 7 July 2017
Please cite this article as: Akbar, N.S., Butt, A.W., Tripathi, D., Nanoparticle Shapes Effects on Unsteady Physiological Transport of Nanofluids through a Finite Length Non-Uniform Channel, Results in Physics (2017), doi: http://dx.doi.org/10.1016/j.rinp.2017.07.019
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1 Nanoparticle Shapes Effects on Unsteady Physiological Transport of Nanofluids through a Finite Length Non-Uniform Channel Noreen Sher Akbar1*, Adil Wahid Butt2, Dharmendra Tripathi3 1
DBS&H, CEME, National University of Sciences and Technology, Islamabad, Pakistan
2
DBS&H, MCE, National University of Sciences and Technology, Islamabad, Pakistan
3
Department of Mechanical Engineering, Manipal University Jaipur, Rajasthan-303007, India
Abstract An analytical investigation is presented to study the unsteady peristaltic transport of nanofluids. Three different geometries of nanoparticle viz bricks, cylinder and platelets are considered in our analysis. The flow geometry is taken as nonuniform channel of finite length to explore our model for wide range of biomedical applications. Exact solutions are obtained for the non-dimensional governing equations subject to physically realistic boundary conditions. The effects of nanoparticle shapes on effective thermal conductivity, axial velocity, transverse velocity, temperature, and pressure difference distributions along the length of non-uniform channel with variation of different flow parameters are discussed with the help of graphical illustrations. It is observed that platelet shaped nanoparticles carry maximum velocity whereas brick shaped nanoparticles are the best to enhance the thermal conductivity. An inherent property of peristaltic transport i.e. trapping is also discussed. This model is applicable in drugs delivery system where different geometries of drugs are delivered and it is also applicable to design a microperistaltic pump for transportation of nanofluids.
Keywords: Unsteady flow; Peristalsis; Nanofluids; Nanoparticle Shapes; Non-uniform Channel. 2
Corresponding author: E-mail: [email protected] (Noreen Sher Akbar)
2 1. INTRODUCTION The study of nanofluids flow have received a lot of attention in last few decades mainly because of significant enhancement of their properties specially thermal conductivity at modest nanoparticle concentrations. It is prepared by dispersing nanometer-sized materials (nanoparticles, nanofibers, nanotubes, nanowires, nanorods, nanosheets or droplets) in base fluids. Motivated from wide engineering and industrial applications of nanofluids, many investigations on nanofluids and its applications are being reported in literature in which some summary and comparative studies of those in the form of review reports [1-10] are written. Wang and Mujumdar [1] have presented a review report on study of heat transfer characteristics of nanofluids; Yu et al. [2] reviewed a comparative study of thermal conductivity and heat transfer enhancements on nanofluid; Kakac et al. [3] reported a review of convective heat transfer enhancement with nanofluids; Özerinç et al. [4] wroted a state-of-the-art review on nanofluids; Ghadimi et al. [5] discussed the stability properties and characterization of nanofluids in stationary conditions; Yu and Xie [6] presented a review on preparation, stability mechanisms, and applications of nanofluids; Mahian et al. [7] reviewed applications of nanofluids in solar energy; Solangi et al. [8] investigated a comprehensive review of thermo-physical properties and convective heat transfer to nanofluids; Kakaç and Pramuanjaroenkij [9] reported a stateof-the-art review single-phase and two-phase treatments of convective heat transfer enhancement with nanofluids. The real world problems are increasing exponentially day to day life. There is a need of working on their solutions. Diseases in living organs are also a great challenge for the biomedical engineers and doctors. There is a requirement to develop mathematical models and design the biomedical devices. Peristalsis is an inherent property of physiological systems and is found in the blood circulation, bypass surgery, ureter eggs in the fallopian tube, chyme movement and gastrointestinal tract. It is defined as physiological transport by relaxation of muscles and contraction of an extensible channel or tube. Peristalsis has a huge number of biomedical applications such as hemolysis, bypass surgery, artificial esophagus, artificial lymph, artificial small intestine, peristaltic pump, heart-lung machine and industrial applications such as finger and roller pumps.
3 Inspired from the application of this natural mechanism, many mathematical models on peristaltic transport of Newtonian fluids are discussed in [10-19]. Recently, the combined study of peristaltic transport and nanofluid flow has received much attention from researchers of biomedical science and engineering specially in drug delivery systems [20]. Many combined papers [21-25] are recently published to discuss physical mechanism and influence of thermal conductivity during the peristaltic movement. Tripathi and Beg [26] discussed their model in application in drug delivery systems; Akbar and Nadeem [27] investigated Phan‐Thien‐Tanner nanofluid flow through peristaltic diverging tube; Akbar [28] reported Bioconvection peristaltic flow of nanofluids; Reddy and Makinde [29] studied the MHD peristaltic transport of Jeffrey nanofluid in an asymmetric channel. All above studies are focused on steady peristaltic transport which is theoretically valid however most of the physical flow problems are unsteady. Recent advancements in peristaltic study of nanofluids have attracted the researchers across the globe to study the effects of different shaped nanoparticles [30-33] and their applications. Motivated from the above studies, and direct applications of unsteady peristaltic transport which was first modelled by Li and Brasseur [34] for Newtonian fluids, we present an analytical approach to study the non-steady peristaltic transport of nanofluids with three different shape of nanoparticles through non-uniform channel. This model is extension of Li and Brasseur [34]. The effects of pertinent parameters and shapes of nanoparticles on peristaltic flow characteristics are discussed with help of computational illustrations. For further detail see Refs. [35-42].
2. MATHEMATICAL FORMULATION The geometric model for the peristaltic transport of nanofluid with different nanoparticles via a non-uniform finite length channel, as depicted in Fig.1 is taken as: [26,37]
π h (ζ , t ) = a(ζ ) − b cos 2 (ζ − ct ) , λ
(1)
4 where a(ζ ) = a0 + α ζ , is the half width of the channel at any axial distance ζ from inlet
and a 0 , b , λ , ζ , c , ~ t are the half width at the inlet, amplitude, wavelength, axial coordinate, wave velocity and time. α is non-uniformity constant, when α → 0 , the
non-uniform channel reduces to a uniform channel.
Fig. 1: Schematic representation of unsteady peristaltic transport through finite length non-uniform channel
The peristaltic flow geometry is approximated to a finite length non-uniform channel with sinusoidal waves propagating along the flow direction. The channel walls are assumed to be distensible and identical in constitution. Damping characteristics are ignored. Flow equations will be modified with low-Reynolds number regime (laminar flows) under long wavelength assumption. The magnitudes for temperature T at the wall of the channel ( η = h ) are denoted as T1 . Under the usual Boussinesq approximation, with an appropriate reference pressure, the transport equations for the regime are respectively with the following assumptions: (a) laminar incompressible flow, (b) no chemical reactions, (c) negligible external forces, (d) negligible viscous dissipation, (e) negligible radioactive heat transfer, (f) nanoparticles and base fluid locally in thermal equilibrium. Law of conservation of mass in component form:
5 ∂u ∂v + = 0, ∂ζ ∂η
(2)
Axial momentum equation:
∂
∂
∂
∂~ p
∂ 2 u~
∂ 2 u~
(~
~
)
ρ nf ~ + u~ ~ + v~ ~ u~ = − ~ + µ nf ~ 2 + ~ 2 + g (ργ )nf T − T0 , ∂η ∂η ∂ζ ∂ζ ∂t ∂ζ
(3)
Transverse momentum equation: ∂ ~ ∂ ~ ∂ ~ ∂ 2 v~ ∂ 2 v~ ∂~ p ρ nf ~ + u ~ + v ~ v = − ~ + µ nf ~ 2 + ~ 2 , ∂η ∂η ∂η ∂ζ ∂t ∂ζ
(4)
Energy equation: ∂ 2T~ ∂ 2T~ ~ ∂ ~ ∂ ~ ∂ ~ (ρc p )nf ~ + u ~ + v ~ T = k nf ~ 2 + ~ 2 + Q0 , ∂η ∂η ∂ζ ∂t ∂ζ
(5)
~ In above equations u~ axial velocity, v~ transverse velocity, η~ transverse coordinate, T
~ is the temperature, Q0 constants heat absorption parameter, ρ nf is the nanofluid density, kf is the thermal conductivity of the fluid, g is the acceleration due to gravity, ~p is
~ pressure, T0 is wall temperature, γ
nf
is the thermal expansion coefficient and (ρc p )nf is
the heat capacitance. To linearize the boundary value problem, a set of dimensionless parameters are given below: ~ ~ 2 ~ ~ pa 0 ζ η~ ct u~ v~ h αξ ξ = ,η = , t = , u = , v = ,p= ,h = =1+ − ε cos 2 π (ξ − t ), λ a0 λ c cδ µcλ a0 δ (6) 3~ ~ ~ ~ ρ f ca0 gγ f ρ f a 0 T0 a0 T − T0 Q0 a 0 b δ = ,φ = ,θ = ~ , Re = , GrT = ,β = ~ . λ a0 µf T0 k f T0 νf2 where δ , ε ,ν ,θ , φ , Re, GrT , β are non-dimensional wave number, amplitude ratio,
kinematic viscosity, dimensionless temperature, rescaled nanoparticle volume fraction, Reynolds number, thermal Grashof number and heat absorption parameter respectively. Under the long wavelength approximation (i.e. peristaltic wavelength is much greater than channel width, viz, λ > a ), it follows that δ → 0 and also the Reynolds number vanishes ( Re → 0 ). Prescribing δ → 0 negates conduit curvature effects and Re → 0
6 negates convective inertial forces relative to viscous hydrodynamic forces. Implementing these approximations, it follows that Equations (2) − (6) become: ∂u ∂v + = 0, ∂ξ ∂η
(7)
( ργ )nf ∂p µ nf ∂ 2u θ. = + Gr T ∂ξ µ f ∂η 2 ( ργ ) f
(8)
∂p = 0. ∂η
(9)
kf ∂ 2θ +β = 0. 2 ∂η knf
(10)
The relevant boundary conditions are specified as follows:
θ (ξ ,η , t ) η =0 = 0, θ (ξ ,η , t ) η = h = 0 ,
(11)
∂u (ξ ,η , t ) = 0 , u (ξ ,η , t ) η =h = 0, ∂η η =0
(12)
v(ξ ,η , t ) η = 0 = 0 , v(ξ ,η , t ) η = h = | = , | = ,
∂h , ∂t
(13) (14)
The thermo physical properties of the nanofluids [30,35] are defined as follows: () = (1 − )() + () .
= ( ) . , ! = ! "
#$ %(& )# (& )'# #$ ( #$ %(& )# % '# #$ (
(15a) ).
(15b)
In above equations; ρ f density of the base fluid, ρ s density of the nanoparticles, k f thermal conductivity of the base fluid, ks thermal conductivity of the nanoparticles, γ nf is the thermal expansion coefficient, γ f is the thermal expansion coefficient of base fluid is the nanoparticle volume fraction, and γ s is the thermal expansion coefficient of the
nanoparticles and m is the shape factor as formulated by the Hamilton-Crosser Model. The value of those shape factors as given by Timofeeva et al. [30] are as follows:
7 Ser
Nanoparticles Type
Shape
Shape Factor (m)
1.
Bricks
3.7
2.
Cylinders
4.9
3.
Platelets
5.7
3. ANALYTICAL SOLUTIONS
The analytical solutions of governing equations (8) − 10) are obtained as: #
0
, (-, ., /) = 12 "2 + # 45(5 − .)),
6(-, ., /) =
(16)
3
(;<) 3 >
(;<)3
DE
2789: (;<) = ?'0@ 10A 2%2 @ (%B2 C%1'0 2 ( G B89: 0A (;<) DF >3 H . 1B2( )
.
(17)
To find the transverse velocity, we use the axial velocity equation (17) in the mass
conservation equation (7) along with boundary condition (13). The expression for transverse velocity takes the form: (;<)3
>
>
DP
089: (;<) 7 10A %2 = ?0A L2M1% ?2 NCG >3 >3 DF R E DE DP %BL02 A DF DF RF BL2 ( )H .
%L02 'Q2 0 (
K (-, ., /) =
.
(18)
Using the transverse vibration boundary condition leads to: (1 − ) 1.S
T2 TU
=
2 A V W Q V
TW T2
+ 51 T T − XYZ
([\)3 ([\)
_ #
Q
T2
]^ 51 + 45B ` T a. L BL # 3
(19)
Re-arranging and integrating above equation w.r.t - gives pressure gradient:
TW T
Q
= 2A bc(/) +
d∗fgh 1i( U) 1( ) .
j + XYZ
([\)3 Q ([\)
_ #
^L + L # 451 `. 3
(20)
where c(/) is an arbitrary function of time /, yet to be determined. Integration of
8 equation (20) w.r.t. - provides the pressure difference Δl across the length of the channel:
Q
Δl(-, /) = m ]2A nc(/) +
d∗fgh 1i( U) 1( ) .
o + XYZ
([\)3 Q ([\)
_ #
^L + L # 451 `a pq. 3
(21)
Upon substituting Δl(-, /) = (-, /) − (0, /) and - = r , and using the finite length condition, we obtain:
Q
− = m ]2A nc(/) +
d∗fgh 1i( U) 1( ) .
o + XYZ
([\)3 Q ([\)
_ #
^L + L # 451 `a p- . 3
(22)
where c(/) is evaluated by re-arranging the above integral and using appropriate integration techniques in Mathematica 10.0 software: c (/ ) =
(;<) 3 A t Av wxy z(FH{) > t )V 89: (;<) = % ? mu 2 V C PA (|H}) . ~ ~u>3
(st su ) mu"
.
tA
mu A V
P
(23)
4. EVALUATION OF RESULTS AND INTERPRETATION
The effective thermal conductivity of the nanofluid
#3 #
for different shape particles is
shown in Fig. 2. The noticeable deduction is that the thermal conductivity increases as the nanoparticle volume fraction increases. The brick shaped nanoparticles have least effective thermal conductivity and the platelets shaped nanoparticles have higher effective thermal conductivity. The temperature profile of the fluid is represented in Fig.
3() − 3(). Graphical representation depicts that maximum temperature exists when
platelets shaped nanoparticles are used, in comparison to the cylinder and bricks shaped
nanoparticles. Heat transfer is considerably lesser when bricks shaped nanoparticles are used. Also it is prominent that temperature is slightly lower in case of uniform channel
( = 0) as compared to the non-uniform channel ( = 0.1). In both cases, temperature
remains directly proportional to the heat absorption parameter 4 . The axial velocity profile 6(-, ., /) is investigated in Fig. 4() − 4(p). From the pictorial presentation, it is perceived that the axial velocity of platelets shaped nanoparticles is maximum and
minimum for bricks shaped nanoparticles. Axial velocity is directly proportional to both the thermal Grashof number XYZ and heat absorption parameter 4 . Maximum velocity is
9 attained at the center of the channel and it decreases as we move towards the boundaries
of the channel. It is clearly seen that velocity is greater when the uniform channel = 0 is considered as equated to the non-uniform channel = 0.1, because of larger area in non-uniform channel.
The transverse velocity profile K(-, ., /) against the transverse coordinate . is displayed
in Fig. 5() − 5(p ). Just like the axial velocity, transverse velocity is also maximum for platelets shaped nanoparticles and minimum for the bricks shaped nanoparticles.
Transverse velocity takes a rise as with an increment in thermal Grashof number XYZ and
heat absorption parameter 4 . However, in case of transverse velocity, the non-uniform channel = 0.1 portrays greater velocity as that of uniform channel = 0. Fig. 6() − 6(p ) demonstrate the consequences of variations in the thermal Grashof number XYZ and
heat absorption parameter 4 on the pressure gradient of the fluid flow. The pressure
gradient in non-uniform channel = 0.1 takes a significantly different rise as in case of
the uniform channel = 0. Uniform behaviour of pressure gradient is seen for uniform channel, however the rise in pressure gradient in case of non-uniform channel is much greater as the length of the channel is increased. The behaviour remains the same as we vary the physical constraints XYZ and 4 . For non-uniform channel, pressure gradient
increases as we move across the length of the channel. Fig. 7 present the streamlines for bricks, cylinder and platelets shapes particles from the graphs it is seen that the shape of the bolus is same as we consider the shape of the particles. The use of different shaped nanoparticles not only enhances the velocity of the base fluid, but also enhances the heat transfer through the fluid medium. For instance, the increase and decrease in the velocity of the fluid for drug delivery system can be controlled by the use of platelets or bricks shaped nanoparticles. Similarly, in industries the heat transfer through nanofluids is required to be controlled, which can be done with the help of bricks shaped nanoparticles.
10
Fig. 2
2.5
Bricks Cylinder Platelets
knf / kf
2
1.5
1
0
0.05
0.1
φ
Fig. 2 Effective thermal conductivity of the nanofluid
0.15
#3 #
0.2
for different shape particles.
11
Figs. 3(a,b) Temperature profile ( ,(-, ., /) Kq . ) for various value of heat absorption
parameter 4 in uniform channel = 0, and non-uniform channel = 0.1.
12
Fig. 4(a)
20000
Bricks β = 3, 5 Cylinders Platelets
u ( ξ, η, t )
15000
10000
5000
ξ = 1, t = 0.5, φ = 0.1, GrT = 1 ε = 0.1, δ = 0.1, α = 0 0
0
0.2
0.4
η
0.6
0.8
1
Fig. 4(b)
10000
Bricks β = 3, 5
Cylinders Platelets
u ( ξ, η, t )
7500
5000
2500
ξ = 1, t = 0.5, φ = 0.1, GrT = 1 ε = 0.1, δ = 0.1, α = 0.1 0
0
0.5
1
η
1.5
2
13
Fig. 4(c)
24000
GrT = 3, 5
Bricks Cylinders Platelets
u ( ξ, η, t )
18000
12000
6000
ξ = 1, t = 0.5, φ = 0.1, β = 1 ε = 0.1, δ = 0.1, α = 0 0
0
0.2
0.4
η
0.6
0.8
1
Fig. 4(d)
12000
Bricks GrT = 3, 5
Platelets
9000
u ( ξ, η, t )
Cylinders
6000
3000
ξ = 1, t = 0.5, φ = 0.1, β = 1 ε = 0.1, δ = 0.1, α = 0.1 0
0
0.5
1
η
1.5
2
Fig. 4(a-d) Axial velocity profile ( 6(-, ., /) Kq . ) for various value of (a) Heat
absorption parameter 4 . (b) Grashof number XYZ in uniform channel = 0, and nonuniform channel = 0.1.
14
Fig. 5(a) 3E-13
ξ = 1, t = 0.5, φ = 0.1, GrT = 1 ε = 0.1, δ = 0.1, α = 0
β = 3, 5
v ( ξ, η, t )
2.4E-13
1.8E-13
1.2E-13
Bricks 6E-14
Cylinders Platelets
0
0
0.2
0.4
η
0.6
0.8
1
Fig. 5(b) 4000
ξ = 1, t = 0.5, φ = 0.1, GrT = 1 ε = 0.1, δ = 0.1, α = 0.1 β = 3, 5
v ( ξ, η, t )
3000
2000
Bricks
1000
Cylinders Platelets 0
0
0.5
1
η
1.5
2
15
Fig. 5(c) 4E-13
ξ = 1, t = 0.5, φ = 0.1, β = 1 ε = 0.1, δ = 0.1, α = 0
GrT = 3, 5
v ( ξ, η, t )
3E-13
2E-13
Bricks
1E-13
Cylinders Platelets 0
0
0.2
0.4
η
0.6
0.8
1
Fig. 5(d)
6000
ξ = 1, t = 0.5, φ = 0.1, β = 1 ε = 0.1, δ = 0.1, α = 0.1
GrT = 3, 5
v ( ξ, η, t )
4500
3000
Bricks
1500
Cylinders Platelets 0
0
0.4
0.8
η
1.2
1.6
2
Figs. 5(a-d) Transverse velocity profile ( K (-, ., /) Kq . ) for various value of (a) Heat
absorption parameter 4 . (b) Grashof number XYZ in uniform channel = 0, and nonuniform channel = 0.1.
16
Fig. 6(a)
2100
Bricks
ε = 0.4, t = 0.5, GrT = 1 φ = 0.1, δ = 0.1, α = 0
Cylinders
dP / dξ
1800
Platelets
1500
β = 3, 5
1200
900
600
0
0.5
1
1.5
2
2.5
3
ξ Fig. 6(b)
20000
Bricks Cylinders Platelets
15000
dP / dξ
ε = 0.4, t = 0.5, GrT = 1 φ = 0.1, δ = 0.1, α = 0.1
β = 3, 5 10000
5000
0
0.5
1
1.5
ξ
2
2.5
3
17
Fig. 6(c) Bricks
3600
ε = 0.4, t = 0.5, β = 1 φ = 0.1, δ = 0.1, α = 0
Cylinders Platelets
dP / dξ
3200
GrT = 3, 4 2800
2400
2000 0
0.5
1
1.5
2
2.5
3
ξ Fig. 6(d)
16000
Bricks Cylinders
ε = 0.4, t = 0.5, β = 1 φ = 0.1, δ = 0.1, α = 0.1
Platelets
dP / dξ
12000
GrT = 3, 4 8000
4000
0
0.5
1
1.5
2
2.5
3
ξ
Figs. 6(a-d) Pressure difference vs. axial distance for various values of (a) Heat
absorption parameter 4 . (b) Grashof number XYZ in uniform channel = 0, and nonuniform channel = 0.1.
18
4
2
0
2
4 0
1
2
3
4
0
1
2
3
4
4
2
0
2
4
19
4
2
0
2
4 0
1
2
3
4
Figs. 7(a-c). Streamlines for shapes (a) Brick (b) Cylinder and (c) Platelet for α = 0.2, ε
= 0.1, φ = 0.2, δ = 0.1, XYZ = 0.5 , t = 0.1, β = 0.4. 5. CONCLUSIONS
The effects of three different shape nanoparticles (bricks, cylinder and platelets) on unsteady peristaltic transport of nanofluids through non-uniform channel are computed and discussed. On the basis of illustrative discussion, the concluding remarks are pointed out as: •
The thermal conductivity of the nanofluids for bricks, cylinder and platelets is sequenced as : bricks < cylinder < platelets.
•
Temperature profile enhances with increasing the magnitude of absorption parameter and the temperature sequence of temperature profile for bricks, cylinder and platelets is bricks < cylinder < platelets.
•
Axial velocity profile enlarges with increasing the magnitude of absorption parameter and Grashof number. The velocity sequence for bricks, cylinder and platelets is bricks < cylinder < platelets.
20 •
The effects of absorption parameter, Grashof number and shape nanoparticles on transverse velocity profile and also pressure difference are similar to that of axial velocity profile.
•
The size of trapped bolus is maximum in the case of
bricks nanoparticles
however it is minimum for platelets nanoparticles. This model is applicable to drug delivery system and also applicable to the physiological transport where nanoparticles play important role. REFERENCES
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S2211-3797(17)30944-0 http://dx.doi.org/10.1016/j.rinp.2017.07.019 RINP 792
To appear in:
Results in Physics
Received Date: Revised Date: Accepted Date:
30 May 2017 7 July 2017 7 July 2017
Please cite this article as: Akbar, N.S., Butt, A.W., Tripathi, D., Nanoparticle Shapes Effects on Unsteady Physiological Transport of Nanofluids through a Finite Length Non-Uniform Channel, Results in Physics (2017), doi: http://dx.doi.org/10.1016/j.rinp.2017.07.019
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1 Nanoparticle Shapes Effects on Unsteady Physiological Transport of Nanofluids through a Finite Length Non-Uniform Channel Noreen Sher Akbar1*, Adil Wahid Butt2, Dharmendra Tripathi3 1
DBS&H, CEME, National University of Sciences and Technology, Islamabad, Pakistan
2
DBS&H, MCE, National University of Sciences and Technology, Islamabad, Pakistan
3
Department of Mechanical Engineering, Manipal University Jaipur, Rajasthan-303007, India
Abstract An analytical investigation is presented to study the unsteady peristaltic transport of nanofluids. Three different geometries of nanoparticle viz bricks, cylinder and platelets are considered in our analysis. The flow geometry is taken as nonuniform channel of finite length to explore our model for wide range of biomedical applications. Exact solutions are obtained for the non-dimensional governing equations subject to physically realistic boundary conditions. The effects of nanoparticle shapes on effective thermal conductivity, axial velocity, transverse velocity, temperature, and pressure difference distributions along the length of non-uniform channel with variation of different flow parameters are discussed with the help of graphical illustrations. It is observed that platelet shaped nanoparticles carry maximum velocity whereas brick shaped nanoparticles are the best to enhance the thermal conductivity. An inherent property of peristaltic transport i.e. trapping is also discussed. This model is applicable in drugs delivery system where different geometries of drugs are delivered and it is also applicable to design a microperistaltic pump for transportation of nanofluids.
Keywords: Unsteady flow; Peristalsis; Nanofluids; Nanoparticle Shapes; Non-uniform Channel. 2
Corresponding author: E-mail: [email protected] (Noreen Sher Akbar)
2 1. INTRODUCTION The study of nanofluids flow have received a lot of attention in last few decades mainly because of significant enhancement of their properties specially thermal conductivity at modest nanoparticle concentrations. It is prepared by dispersing nanometer-sized materials (nanoparticles, nanofibers, nanotubes, nanowires, nanorods, nanosheets or droplets) in base fluids. Motivated from wide engineering and industrial applications of nanofluids, many investigations on nanofluids and its applications are being reported in literature in which some summary and comparative studies of those in the form of review reports [1-10] are written. Wang and Mujumdar [1] have presented a review report on study of heat transfer characteristics of nanofluids; Yu et al. [2] reviewed a comparative study of thermal conductivity and heat transfer enhancements on nanofluid; Kakac et al. [3] reported a review of convective heat transfer enhancement with nanofluids; Özerinç et al. [4] wroted a state-of-the-art review on nanofluids; Ghadimi et al. [5] discussed the stability properties and characterization of nanofluids in stationary conditions; Yu and Xie [6] presented a review on preparation, stability mechanisms, and applications of nanofluids; Mahian et al. [7] reviewed applications of nanofluids in solar energy; Solangi et al. [8] investigated a comprehensive review of thermo-physical properties and convective heat transfer to nanofluids; Kakaç and Pramuanjaroenkij [9] reported a stateof-the-art review single-phase and two-phase treatments of convective heat transfer enhancement with nanofluids. The real world problems are increasing exponentially day to day life. There is a need of working on their solutions. Diseases in living organs are also a great challenge for the biomedical engineers and doctors. There is a requirement to develop mathematical models and design the biomedical devices. Peristalsis is an inherent property of physiological systems and is found in the blood circulation, bypass surgery, ureter eggs in the fallopian tube, chyme movement and gastrointestinal tract. It is defined as physiological transport by relaxation of muscles and contraction of an extensible channel or tube. Peristalsis has a huge number of biomedical applications such as hemolysis, bypass surgery, artificial esophagus, artificial lymph, artificial small intestine, peristaltic pump, heart-lung machine and industrial applications such as finger and roller pumps.
3 Inspired from the application of this natural mechanism, many mathematical models on peristaltic transport of Newtonian fluids are discussed in [10-19]. Recently, the combined study of peristaltic transport and nanofluid flow has received much attention from researchers of biomedical science and engineering specially in drug delivery systems [20]. Many combined papers [21-25] are recently published to discuss physical mechanism and influence of thermal conductivity during the peristaltic movement. Tripathi and Beg [26] discussed their model in application in drug delivery systems; Akbar and Nadeem [27] investigated Phan‐Thien‐Tanner nanofluid flow through peristaltic diverging tube; Akbar [28] reported Bioconvection peristaltic flow of nanofluids; Reddy and Makinde [29] studied the MHD peristaltic transport of Jeffrey nanofluid in an asymmetric channel. All above studies are focused on steady peristaltic transport which is theoretically valid however most of the physical flow problems are unsteady. Recent advancements in peristaltic study of nanofluids have attracted the researchers across the globe to study the effects of different shaped nanoparticles [30-33] and their applications. Motivated from the above studies, and direct applications of unsteady peristaltic transport which was first modelled by Li and Brasseur [34] for Newtonian fluids, we present an analytical approach to study the non-steady peristaltic transport of nanofluids with three different shape of nanoparticles through non-uniform channel. This model is extension of Li and Brasseur [34]. The effects of pertinent parameters and shapes of nanoparticles on peristaltic flow characteristics are discussed with help of computational illustrations. For further detail see Refs. [35-42].
2. MATHEMATICAL FORMULATION The geometric model for the peristaltic transport of nanofluid with different nanoparticles via a non-uniform finite length channel, as depicted in Fig.1 is taken as: [26,37]
π h (ζ , t ) = a(ζ ) − b cos 2 (ζ − ct ) , λ
(1)
4 where a(ζ ) = a0 + α ζ , is the half width of the channel at any axial distance ζ from inlet
and a 0 , b , λ , ζ , c , ~ t are the half width at the inlet, amplitude, wavelength, axial coordinate, wave velocity and time. α is non-uniformity constant, when α → 0 , the
non-uniform channel reduces to a uniform channel.
Fig. 1: Schematic representation of unsteady peristaltic transport through finite length non-uniform channel
The peristaltic flow geometry is approximated to a finite length non-uniform channel with sinusoidal waves propagating along the flow direction. The channel walls are assumed to be distensible and identical in constitution. Damping characteristics are ignored. Flow equations will be modified with low-Reynolds number regime (laminar flows) under long wavelength assumption. The magnitudes for temperature T at the wall of the channel ( η = h ) are denoted as T1 . Under the usual Boussinesq approximation, with an appropriate reference pressure, the transport equations for the regime are respectively with the following assumptions: (a) laminar incompressible flow, (b) no chemical reactions, (c) negligible external forces, (d) negligible viscous dissipation, (e) negligible radioactive heat transfer, (f) nanoparticles and base fluid locally in thermal equilibrium. Law of conservation of mass in component form:
5 ∂u ∂v + = 0, ∂ζ ∂η
(2)
Axial momentum equation:
∂
∂
∂
∂~ p
∂ 2 u~
∂ 2 u~
(~
~
)
ρ nf ~ + u~ ~ + v~ ~ u~ = − ~ + µ nf ~ 2 + ~ 2 + g (ργ )nf T − T0 , ∂η ∂η ∂ζ ∂ζ ∂t ∂ζ
(3)
Transverse momentum equation: ∂ ~ ∂ ~ ∂ ~ ∂ 2 v~ ∂ 2 v~ ∂~ p ρ nf ~ + u ~ + v ~ v = − ~ + µ nf ~ 2 + ~ 2 , ∂η ∂η ∂η ∂ζ ∂t ∂ζ
(4)
Energy equation: ∂ 2T~ ∂ 2T~ ~ ∂ ~ ∂ ~ ∂ ~ (ρc p )nf ~ + u ~ + v ~ T = k nf ~ 2 + ~ 2 + Q0 , ∂η ∂η ∂ζ ∂t ∂ζ
(5)
~ In above equations u~ axial velocity, v~ transverse velocity, η~ transverse coordinate, T
~ is the temperature, Q0 constants heat absorption parameter, ρ nf is the nanofluid density, kf is the thermal conductivity of the fluid, g is the acceleration due to gravity, ~p is
~ pressure, T0 is wall temperature, γ
nf
is the thermal expansion coefficient and (ρc p )nf is
the heat capacitance. To linearize the boundary value problem, a set of dimensionless parameters are given below: ~ ~ 2 ~ ~ pa 0 ζ η~ ct u~ v~ h αξ ξ = ,η = , t = , u = , v = ,p= ,h = =1+ − ε cos 2 π (ξ − t ), λ a0 λ c cδ µcλ a0 δ (6) 3~ ~ ~ ~ ρ f ca0 gγ f ρ f a 0 T0 a0 T − T0 Q0 a 0 b δ = ,φ = ,θ = ~ , Re = , GrT = ,β = ~ . λ a0 µf T0 k f T0 νf2 where δ , ε ,ν ,θ , φ , Re, GrT , β are non-dimensional wave number, amplitude ratio,
kinematic viscosity, dimensionless temperature, rescaled nanoparticle volume fraction, Reynolds number, thermal Grashof number and heat absorption parameter respectively. Under the long wavelength approximation (i.e. peristaltic wavelength is much greater than channel width, viz, λ > a ), it follows that δ → 0 and also the Reynolds number vanishes ( Re → 0 ). Prescribing δ → 0 negates conduit curvature effects and Re → 0
6 negates convective inertial forces relative to viscous hydrodynamic forces. Implementing these approximations, it follows that Equations (2) − (6) become: ∂u ∂v + = 0, ∂ξ ∂η
(7)
( ργ )nf ∂p µ nf ∂ 2u θ. = + Gr T ∂ξ µ f ∂η 2 ( ργ ) f
(8)
∂p = 0. ∂η
(9)
kf ∂ 2θ +β = 0. 2 ∂η knf
(10)
The relevant boundary conditions are specified as follows:
θ (ξ ,η , t ) η =0 = 0, θ (ξ ,η , t ) η = h = 0 ,
(11)
∂u (ξ ,η , t ) = 0 , u (ξ ,η , t ) η =h = 0, ∂η η =0
(12)
v(ξ ,η , t ) η = 0 = 0 , v(ξ ,η , t ) η = h = | = , | = ,
∂h , ∂t
(13) (14)
The thermo physical properties of the nanofluids [30,35] are defined as follows: () = (1 − )() + () .
= ( ) . , ! = ! "
#$ %(& )# (& )'# #$ ( #$ %(& )# % '# #$ (
(15a) ).
(15b)
In above equations; ρ f density of the base fluid, ρ s density of the nanoparticles, k f thermal conductivity of the base fluid, ks thermal conductivity of the nanoparticles, γ nf is the thermal expansion coefficient, γ f is the thermal expansion coefficient of base fluid is the nanoparticle volume fraction, and γ s is the thermal expansion coefficient of the
nanoparticles and m is the shape factor as formulated by the Hamilton-Crosser Model. The value of those shape factors as given by Timofeeva et al. [30] are as follows:
7 Ser
Nanoparticles Type
Shape
Shape Factor (m)
1.
Bricks
3.7
2.
Cylinders
4.9
3.
Platelets
5.7
3. ANALYTICAL SOLUTIONS
The analytical solutions of governing equations (8) − 10) are obtained as: #
0
, (-, ., /) = 12 "2 + # 45(5 − .)),
6(-, ., /) =
(16)
3
(;<) 3 >
(;<)3
DE
2789: (;<) = ?'0@ 10A 2%2 @ (%B2 C%1'0 2 ( G B89: 0A (;<) DF >3 H . 1B2( )
.
(17)
To find the transverse velocity, we use the axial velocity equation (17) in the mass
conservation equation (7) along with boundary condition (13). The expression for transverse velocity takes the form: (;<)3
>
>
DP
089: (;<) 7 10A %2 = ?0A L2M1% ?2 NCG >3 >3 DF R E DE DP %BL02 A DF DF RF BL2 ( )H .
%L02 'Q2 0 (
K (-, ., /) =
.
(18)
Using the transverse vibration boundary condition leads to: (1 − ) 1.S
T2 TU
=
2 A V W Q V
TW T2
+ 51 T T − XYZ
([\)3 ([\)
_ #
Q
T2
]^ 51 + 45B ` T a. L BL # 3
(19)
Re-arranging and integrating above equation w.r.t - gives pressure gradient:
TW T
Q
= 2A bc(/) +
d∗fgh 1i( U) 1( ) .
j + XYZ
([\)3 Q ([\)
_ #
^L + L # 451 `. 3
(20)
where c(/) is an arbitrary function of time /, yet to be determined. Integration of
8 equation (20) w.r.t. - provides the pressure difference Δl across the length of the channel:
Q
Δl(-, /) = m ]2A nc(/) +
d∗fgh 1i( U) 1( ) .
o + XYZ
([\)3 Q ([\)
_ #
^L + L # 451 `a pq. 3
(21)
Upon substituting Δl(-, /) = (-, /) − (0, /) and - = r , and using the finite length condition, we obtain:
Q
− = m ]2A nc(/) +
d∗fgh 1i( U) 1( ) .
o + XYZ
([\)3 Q ([\)
_ #
^L + L # 451 `a p- . 3
(22)
where c(/) is evaluated by re-arranging the above integral and using appropriate integration techniques in Mathematica 10.0 software: c (/ ) =
(;<) 3 A t Av wxy z(FH{) > t )V 89: (;<) = % ? mu 2 V C PA (|H}) . ~ ~u>3
(st su ) mu"
.
tA
mu A V
P
(23)
4. EVALUATION OF RESULTS AND INTERPRETATION
The effective thermal conductivity of the nanofluid
#3 #
for different shape particles is
shown in Fig. 2. The noticeable deduction is that the thermal conductivity increases as the nanoparticle volume fraction increases. The brick shaped nanoparticles have least effective thermal conductivity and the platelets shaped nanoparticles have higher effective thermal conductivity. The temperature profile of the fluid is represented in Fig.
3() − 3(). Graphical representation depicts that maximum temperature exists when
platelets shaped nanoparticles are used, in comparison to the cylinder and bricks shaped
nanoparticles. Heat transfer is considerably lesser when bricks shaped nanoparticles are used. Also it is prominent that temperature is slightly lower in case of uniform channel
( = 0) as compared to the non-uniform channel ( = 0.1). In both cases, temperature
remains directly proportional to the heat absorption parameter 4 . The axial velocity profile 6(-, ., /) is investigated in Fig. 4() − 4(p). From the pictorial presentation, it is perceived that the axial velocity of platelets shaped nanoparticles is maximum and
minimum for bricks shaped nanoparticles. Axial velocity is directly proportional to both the thermal Grashof number XYZ and heat absorption parameter 4 . Maximum velocity is
9 attained at the center of the channel and it decreases as we move towards the boundaries
of the channel. It is clearly seen that velocity is greater when the uniform channel = 0 is considered as equated to the non-uniform channel = 0.1, because of larger area in non-uniform channel.
The transverse velocity profile K(-, ., /) against the transverse coordinate . is displayed
in Fig. 5() − 5(p ). Just like the axial velocity, transverse velocity is also maximum for platelets shaped nanoparticles and minimum for the bricks shaped nanoparticles.
Transverse velocity takes a rise as with an increment in thermal Grashof number XYZ and
heat absorption parameter 4 . However, in case of transverse velocity, the non-uniform channel = 0.1 portrays greater velocity as that of uniform channel = 0. Fig. 6() − 6(p ) demonstrate the consequences of variations in the thermal Grashof number XYZ and
heat absorption parameter 4 on the pressure gradient of the fluid flow. The pressure
gradient in non-uniform channel = 0.1 takes a significantly different rise as in case of
the uniform channel = 0. Uniform behaviour of pressure gradient is seen for uniform channel, however the rise in pressure gradient in case of non-uniform channel is much greater as the length of the channel is increased. The behaviour remains the same as we vary the physical constraints XYZ and 4 . For non-uniform channel, pressure gradient
increases as we move across the length of the channel. Fig. 7 present the streamlines for bricks, cylinder and platelets shapes particles from the graphs it is seen that the shape of the bolus is same as we consider the shape of the particles. The use of different shaped nanoparticles not only enhances the velocity of the base fluid, but also enhances the heat transfer through the fluid medium. For instance, the increase and decrease in the velocity of the fluid for drug delivery system can be controlled by the use of platelets or bricks shaped nanoparticles. Similarly, in industries the heat transfer through nanofluids is required to be controlled, which can be done with the help of bricks shaped nanoparticles.
10
Fig. 2
2.5
Bricks Cylinder Platelets
knf / kf
2
1.5
1
0
0.05
0.1
φ
Fig. 2 Effective thermal conductivity of the nanofluid
0.15
#3 #
0.2
for different shape particles.
11
Figs. 3(a,b) Temperature profile ( ,(-, ., /) Kq . ) for various value of heat absorption
parameter 4 in uniform channel = 0, and non-uniform channel = 0.1.
12
Fig. 4(a)
20000
Bricks β = 3, 5 Cylinders Platelets
u ( ξ, η, t )
15000
10000
5000
ξ = 1, t = 0.5, φ = 0.1, GrT = 1 ε = 0.1, δ = 0.1, α = 0 0
0
0.2
0.4
η
0.6
0.8
1
Fig. 4(b)
10000
Bricks β = 3, 5
Cylinders Platelets
u ( ξ, η, t )
7500
5000
2500
ξ = 1, t = 0.5, φ = 0.1, GrT = 1 ε = 0.1, δ = 0.1, α = 0.1 0
0
0.5
1
η
1.5
2
13
Fig. 4(c)
24000
GrT = 3, 5
Bricks Cylinders Platelets
u ( ξ, η, t )
18000
12000
6000
ξ = 1, t = 0.5, φ = 0.1, β = 1 ε = 0.1, δ = 0.1, α = 0 0
0
0.2
0.4
η
0.6
0.8
1
Fig. 4(d)
12000
Bricks GrT = 3, 5
Platelets
9000
u ( ξ, η, t )
Cylinders
6000
3000
ξ = 1, t = 0.5, φ = 0.1, β = 1 ε = 0.1, δ = 0.1, α = 0.1 0
0
0.5
1
η
1.5
2
Fig. 4(a-d) Axial velocity profile ( 6(-, ., /) Kq . ) for various value of (a) Heat
absorption parameter 4 . (b) Grashof number XYZ in uniform channel = 0, and nonuniform channel = 0.1.
14
Fig. 5(a) 3E-13
ξ = 1, t = 0.5, φ = 0.1, GrT = 1 ε = 0.1, δ = 0.1, α = 0
β = 3, 5
v ( ξ, η, t )
2.4E-13
1.8E-13
1.2E-13
Bricks 6E-14
Cylinders Platelets
0
0
0.2
0.4
η
0.6
0.8
1
Fig. 5(b) 4000
ξ = 1, t = 0.5, φ = 0.1, GrT = 1 ε = 0.1, δ = 0.1, α = 0.1 β = 3, 5
v ( ξ, η, t )
3000
2000
Bricks
1000
Cylinders Platelets 0
0
0.5
1
η
1.5
2
15
Fig. 5(c) 4E-13
ξ = 1, t = 0.5, φ = 0.1, β = 1 ε = 0.1, δ = 0.1, α = 0
GrT = 3, 5
v ( ξ, η, t )
3E-13
2E-13
Bricks
1E-13
Cylinders Platelets 0
0
0.2
0.4
η
0.6
0.8
1
Fig. 5(d)
6000
ξ = 1, t = 0.5, φ = 0.1, β = 1 ε = 0.1, δ = 0.1, α = 0.1
GrT = 3, 5
v ( ξ, η, t )
4500
3000
Bricks
1500
Cylinders Platelets 0
0
0.4
0.8
η
1.2
1.6
2
Figs. 5(a-d) Transverse velocity profile ( K (-, ., /) Kq . ) for various value of (a) Heat
absorption parameter 4 . (b) Grashof number XYZ in uniform channel = 0, and nonuniform channel = 0.1.
16
Fig. 6(a)
2100
Bricks
ε = 0.4, t = 0.5, GrT = 1 φ = 0.1, δ = 0.1, α = 0
Cylinders
dP / dξ
1800
Platelets
1500
β = 3, 5
1200
900
600
0
0.5
1
1.5
2
2.5
3
ξ Fig. 6(b)
20000
Bricks Cylinders Platelets
15000
dP / dξ
ε = 0.4, t = 0.5, GrT = 1 φ = 0.1, δ = 0.1, α = 0.1
β = 3, 5 10000
5000
0
0.5
1
1.5
ξ
2
2.5
3
17
Fig. 6(c) Bricks
3600
ε = 0.4, t = 0.5, β = 1 φ = 0.1, δ = 0.1, α = 0
Cylinders Platelets
dP / dξ
3200
GrT = 3, 4 2800
2400
2000 0
0.5
1
1.5
2
2.5
3
ξ Fig. 6(d)
16000
Bricks Cylinders
ε = 0.4, t = 0.5, β = 1 φ = 0.1, δ = 0.1, α = 0.1
Platelets
dP / dξ
12000
GrT = 3, 4 8000
4000
0
0.5
1
1.5
2
2.5
3
ξ
Figs. 6(a-d) Pressure difference vs. axial distance for various values of (a) Heat
absorption parameter 4 . (b) Grashof number XYZ in uniform channel = 0, and nonuniform channel = 0.1.
18
4
2
0
2
4 0
1
2
3
4
0
1
2
3
4
4
2
0
2
4
19
4
2
0
2
4 0
1
2
3
4
Figs. 7(a-c). Streamlines for shapes (a) Brick (b) Cylinder and (c) Platelet for α = 0.2, ε
= 0.1, φ = 0.2, δ = 0.1, XYZ = 0.5 , t = 0.1, β = 0.4. 5. CONCLUSIONS
The effects of three different shape nanoparticles (bricks, cylinder and platelets) on unsteady peristaltic transport of nanofluids through non-uniform channel are computed and discussed. On the basis of illustrative discussion, the concluding remarks are pointed out as: •
The thermal conductivity of the nanofluids for bricks, cylinder and platelets is sequenced as : bricks < cylinder < platelets.
•
Temperature profile enhances with increasing the magnitude of absorption parameter and the temperature sequence of temperature profile for bricks, cylinder and platelets is bricks < cylinder < platelets.
•
Axial velocity profile enlarges with increasing the magnitude of absorption parameter and Grashof number. The velocity sequence for bricks, cylinder and platelets is bricks < cylinder < platelets.
20 •
The effects of absorption parameter, Grashof number and shape nanoparticles on transverse velocity profile and also pressure difference are similar to that of axial velocity profile.
•
The size of trapped bolus is maximum in the case of
bricks nanoparticles
however it is minimum for platelets nanoparticles. This model is applicable to drug delivery system and also applicable to the physiological transport where nanoparticles play important role. REFERENCES
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