Heat transfer analysis with temperature-dependent viscosity for the peristaltic flow of nano fluid with shape factor over heated tube

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 4

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Heat transfer analysis with temperaturedependent viscosity for the peristaltic flow of nano fluid with shape factor over heated tube A. Bintul Huda a,*, Noreen Sher Akbar b a b

Mathematics & Statistics Department, Riphah International University I-14, Islamabad, Pakistan DBS&H, CEME, National University of Sciences and Technology, Islamabad, Pakistan

article info

abstract

Article history:

The present article address the nanofluid flow with the interaction of shape factor and heat

Received 3 July 2017

transfer in a vertical tube with temperature-dependent viscosity. Flow study has been done

Received in revised form

in a flexible tube with low Reynolds number (Re<<0 i.e and long wavelength (d<<0 i.e

7 August 2017

assumption. Mathematica software is employed to evaluate the exact solutions of velocity

Accepted 8 August 2017

profile, temperature profile, axial velocity profile, pressure gradient and stream function.

Available online xxx

The influence of heat source/sink parameter (b), Grashof number (Gr) and the viscosity parameter (a) and nanoparticle volume fraction (f) on velocity, temperature, pressure

Keywords:

gradient, pressure rise and wall shear stress distributions is presented graphically. Three

Biophysics

types of shape factor i.e cylinder platelets and bricks are discussed. Streamline plots are

Heat transfer

also computed to illustrate bolus dynamics and trapping phenomena which characterize

Flexible heated tube

peristaltic propulsion. It is seen that with an increment in Grash of number, Gr, nanofluid

Temperature-dependent viscosity

velocity is significantly increases i.e. flow acceleration is induced across the tube diameter.

Nanoparticles

Once again the copper-methanol nanofluid in shape of platelets achieves the best

Shape factor

acceleration. © 2017 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.

Introduction Research and advancements in the area of nano-science have received a lot of attention since last two decades. Nanofluid dynamics is a part of nano-science that deals with the study of concentration, volume fraction and energy transport of nanoparticle with base fluids. In last few decades, the work on nanofluiddynamics is much attraction for the reader and industry people. Das et al. [1] reported a review report on heat transfer in nanofluids where they discussed the effects in thermal conductivity by suspension of small particles of

micrometer sized and suggested the directions for future developments. In continuation of review report on nanofluids, Wang and Mujumdar [2] presented the heat transfer characteristics of nanofluids; Trisaksri, and Wongwises [3] further elaborated the report on nanofluids; Murshed et al. [4] studied the thermo physical and electro kinetic properties of nanofluids; Wang and Mujumdar [5] extend his review report for theoretical and numerical studies on nanofluids; Kakac and Pramuanjaroenkij [6] reviewed the convective heat transfer enhancement with nanofluids; Wen et al. [7] focused his review for nanofluids applications; Yu and Xie [8]presented preparation, stability mechanisms, and applications of

* Corresponding author. E-mail address: [email protected] (A.B. Huda). http://dx.doi.org/10.1016/j.ijhydene.2017.08.054 0360-3199/© 2017 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved. Please cite this article in press as: Huda AB, Akbar NS, Heat transfer analysis with temperature-dependent viscosity for the peristaltic flow of nano fluid with shape factor over heated tube, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.08.054

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nanofluids in their review report; Mahian et al. [9] discussed the applications of nanofluids for solar energy; Taylor et al. [10] extended the application of nanofluids for Small particles, big impacts; Younes et al. [11] studied the thermal conductivity of nanofluids; Sarkar et al. [12] presented another review report on hybrid nanofluids and its applications; Kakac and Pramuanjaroenkij [13] most recently presented a new review report where they studied single-phase and two-phase treatments of convective heat transfer enhancement with nanofluids. They have compared Nusselt number and friction factor correlations of nanofluids, nanofluid characters in common phenomena's such as boiling, radiation, entropy generation and jet flows, experimental consistency, chemical nature, mass diffusion coefficients and properties of novel nanofluids and noted that in most of the works the results shown common agreement. Most recently some interesting works [14e24] on nanofluids and its application in different domain have been incorporated. Peristaltic transport is a biological mechanism which entails the conveyance of material induced by a progressive wave of contraction or expansion along the length of a distensible vessel (tube). This effectively mixes and propels the fluid in the direction of the wave propagation. Peristaltic flows of non-Newtonian viscous fluids are encountered in many complex physiological systems including urine transport from the kidney to the bladder, chyme motion in the gastrointestinal tract, movement of ovum in the female fallopian tube, vasomotion of small blood vessels, transport of spermatozoa, and swallowing food through the esophagus (and other biomedical applications which are summarized in Fung [25]) and also phloem trans-location in plants as described by Thaine [26] and Thompson [27]. In a mathematical context, peristaltic flows fall in the category of moving boundary value problems. They have as a result mobilized considerable interest in recent years. Many analytical investigations of peristaltic propulsion have therefore been communicated and these have addressed a diverse range of geometries under various assumptions such as large wavelength, small amplitude ratio, small wave number, small Deborah number, low Reynolds number and creeping flow, etc. Representative works in this regard include Ellahi [28]; Hameed and Nadeem, [29]; Tan and Masuoka, [30e32]; Mahomed and Hayat, [33]; Fetecau and Fetecau, [34]; Malik et al., [35]; Dehghan and Shakeri, [36]. Some relevant studies on the topic can be found from the list of references (Nadeem and Akbar, [37,38]) and several references therein. Non-Newtonian models are extremely diverse and include viscoelastic, viscoplastic, micro-continuum and other formulations. Variable-viscosity models are also an important sub-section of rheological liquids. They have therefore also been investigated in the context of peristaltic flows, since viscosity variation (e.g. with temperature) is an important characteristic of certain physiological (and industrial) materials. Peristaltic transport of a power-law fluid with variable consistency was examined by Shukla and Gupta [39]. They observed that for zero pressure drops, flow rate flux is elevated for greater amplitude of the peristaltic wave whereas it is suppressed with increasing pseudo-plastic nature of the fluid. They further noted that wall friction is reduced as the consistency decreases. Srivastava et al. [40] studied the

peristaltic transport of a fluid with variable viscosity through a non-uniform tube. They showed that the pressure rise is markedly lowered as the fluid viscosity decreases at zero flow rates but is infect independent of viscosity variation at a certain value of flow rate. However beyond this critical flow rate, the pressure rise is enhanced with greater viscosity. Abd El Hakeem et al. [41] studied using perturbation expansions, the influence of variable viscosity and an inserted endoscope on peristaltic viscous flow. They employed an exponential decay model for viscosity and observed that pressure rise is decreased with increasing viscosity ratio whereas it is enhanced with increasing wave number, amplitude ratio and radius ratio. Further studies of variable-viscosity peristaltic flow include Abd El Hakeem et al. [42] for magneto hydrodynamic fluids, Khan et al. [43] for inclined pumping of nonNewtonian fluids and Akbar [44] present nanoparticle volume fraction for phase model. Another significant development in medical engineering in recent years has been the emergence of nanofluids, a subcategory of nanoscale materials. Nanofluids comprise base fluids (water, air, ethylene glycol etc) with nano-size solid particles suspended in them. Nanofluids have gained much attention from investigators due to their high thermal conductivity and pioneering work in developing such fluids was first performed by Choi [45] Nanoparticle are generally synthesized from metals, oxides, carbides, or carbon nanotubes owing to high thermal conductivities associated with these materials. In a medical engineering context as refer in Ref. [45], nanoparticle have been found to achieve exceptional performance in enhancing thermal and mass diffusion properties of, for example, drugs injected into the blood stream. Biocompatibility of the selected metallic oxides is crucial for safe deployment of nanofluids in medicine. New potential applications for nanoparticle in nanoparticle blood diagnostic systems, asthma sensors, carbon nanotubes in catheters and stents and anti-bacterial treatment for wounds via peristaltic pump delivery was identified by Harris and Graffagnini [46]. Nanoparticles possess many unique attributes which make them particularly attractive for clinical applications. These include a surface to mass ratio which is much greater than that of other particles, site-specific targeting features which can be achieved by attaching targeting ligands to surface of particles (or via magnetic guidance), quantum properties, enhanced ability to absorb and carry other compounds, excellent large functional surface which can bind, adsorb and convey secondary compounds (drugs, probes and proteins). Further advantages encompass controllable deployment of particle degradation characteristics which can be successfully modulated by judicious selection of matrix constituents, and flexibility in administration methods (nasal, parenteral, intraocular). In neuro-pharmacological hemodynamic as refer in Ref. [47], it has been clinically verified that nanoparticle can easily penetrate the blood brain barrier (BBB) facilitating the introduction of therapeutic agents into the brain. Fullstone et al. [47] have also recently described the exceptional characteristics of nanoparticle (size, shape and surface chemistry) in assisting effective delivery of drugs within cells or tissue (achieved via modulation of immune system interactions, blood clearance profile and interaction with target cells). They have further shown that erythrocytes aid in effective

Please cite this article in press as: Huda AB, Akbar NS, Heat transfer analysis with temperature-dependent viscosity for the peristaltic flow of nano fluid with shape factor over heated tube, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.08.054

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nanoparticle distribution within capillaries. Further studies include Tan et al. [48]. Simulation of peristaltic flows of nanoparticle is therefore extremely relevant to improve administration of nanofluids in medicine. Representative  g [49] who studies in this regard include Tripathi and Be considered analytically the thermal and nano-species buoyancy effects on heat, mass and momentum transfer in peristaltic propulsion of nanofluids in peristaltic pumping devices, employing the Buongiornio formulation which incorporates Brownian motion and thermophoresis. Ebaid and Aly [50] studied magnetic field effects on electricallyconducting nanofluid propulsion by peristaltic waves with applications in cancer therapy. Akbar et al. [51] investigated peristaltic slip nanofluid hydrodynamics in an asymmetric channel, obtaining series solutions for temperature, nanoparticle concentration, stream function and pressure gradient. Akbar and Nadeem [52] considered peristaltic flow of Phan-Thien-Tanner nanofluid in a diverging conduit with the g homotopy perturbation method. Further analyses include Be and Tripathi [53] who considered double-diffusive convection of nanofluids in finite length pumping systems. These studies did not consider variable viscosity effects. Further literature can be viewed through Refs. [54e64]. In the present article, we therefore consider the peristaltic propulsion of nanofluid in a vertical conduit with temperature-dependent viscosity. Basic formulation is employed for the nanofluid with a viscosity modification. Heat transfer is also considered and heat source/sink and thermal buoyancy effects featured. Various nano-particles are considered i.e. Titanium oxide-water, Copper oxide-water and Silver-water. Analytical solutions are derived to examine the effects of heat generation/absorption parameter, Grashof number, viscosity parameter and nanoparticle volume fraction on velocity, temperature, pressure gradient, pressure rise and wall shear stress variables. Streamline visualization is also computed to assess trapping hydrodynamics. The mathematical model is of potential importance in better understanding medical peristaltic pump nano-pharmacological delivery systems.

Fig. 1 e Geometry of problem. becomes steady in a wave frame ðr; zÞ moving with the same speed as the wave moves in the Z direction. The transformations between the two frames (i.e. laboratory and wave frame) are:

 r ¼ R; z ¼ Z  ct; v ¼ V; w ¼ W  c; p z; r; t ¼ p Z; R; t ;

(2)

The governing equations for conservation of mass, momentum and heat (energy) of an incompressible nanofluid, assuming the nanofluid to be a dilute suspension in thermal equilibrium, may be written as defined in Ref. [61]: 1 vðruÞ vw þ ¼ 0; vz r vr

(3)

     
 vu
 2 vu u vu vu vp v ¼ þ 2mnf T þ mnf T  rnf u þ w vr vr vr vr vr r vr r   
 vu vw v m T þ ; þ vz nf vr vz (4)

Mathematical formulation Consider axisymmetric flow of a variable-viscosity nanofluid in a circular tube of finite length L. The tube walls are flexible and a sinusoidal wave propagates along the walls of the tube. Isothermal conditions are enforced at the walls which are maintained at a temperature, T0. At the center of the tube, a symmetric temperature condition is imposed. The geometric model is illustrated in Fig. 1 with respect to a cylindrical coordinate system ðR; ZÞ . The geometry of the wall surface is simulated via the following relation: h¼ aþb sin

 2p
Z  ct ; l

(1)

where h denotes the height of the tube wall, a denotes the radius of the tube, b is the wave amplitude, l is the wave length and c is the peristaltic wave speed. In the fixed coordinates system ðR; ZÞ, the hydrodynamics is unsteady. It

    
 vu vw vw vw vp 1 v þw ¼ þ mnf T r þ rnf u vr vz vz r vr vz vr  

 vw  v þ 2mnf T þ ðrgÞnf g T  T0 ; vz vz

knf vT vT vT ¼ þu þw vr vz vt r cp

nf

 2  v T 1 vT v2 T Q0 þ 2 þ : þ 2 r vr vz vr rcp

(5)

(6)

nf

where r and z are the co-ordinates in the wave frame. z is taken as the center line of the tube and r is orientated transverse to it, u and w are the velocity components in the r and z directions respectively, T is the local temperature of the fluid. Further mnf is the effective dynamic viscosity of nanofluid, anf is the effective thermal diffusivity of nanofluid, rnf is the effective density of nanofluid, (rcp)nf is the heat capacitance of the nanofluid and knf is the effective thermal conductivity of the nanofluid, and the subscript s designates solid. These are defined, respectively, as follows:

Please cite this article in press as: Huda AB, Akbar NS, Heat transfer analysis with temperature-dependent viscosity for the peristaltic flow of nano fluid with shape factor over heated tube, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.08.054

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m0 eaq

knf  ; rnf ¼ ð1  fÞrf þ frs ; rcp nf


 rcp nf ¼ ð1  fÞ rcp f þ f rcp s; ðrgÞnf ¼ ð1  fÞðrgÞf þ fðrgÞS ;   ks þ ðm þ 1Þkf  ðm þ 1Þðkf  ksÞf knf ¼ kf ks þ ðm þ 1Þkf þ ðkf  ksÞf mnf ¼

ð1  fÞ2:5

; anf ¼


(7) In above equation rf density of the base fluid, rs density of the nanoparticle, kf thermal conductivity of the base fluid, ks thermal conductivity of the nanoparticle, gnf is the thermal expansion coefficient, gf is the thermal expansion coefficient of base fluid f is the nanoparticle volume fraction, and gs is the thermal expansion coefficient of the nanoparticle and m is the shape factor as formulated by the Hamilton-Crosser Model. The value of those shape factors is given as in Table below:

The Reynolds model for nanofluid viscosity can be defined as follows: mnf eaq ¼ ; and eaq ¼ 1  aq; m0 ð1  4Þ2:5

a < < 1:

(12)

where a is the viscosity parameter and m0 constant viscosity of the fluid. The non-dimensional boundary conditions are prescribed as: vw vq ¼ 0; ¼ 0 at r ¼ 0; vr vr w ¼ 1; q ¼ 0;

at r ¼ hðzÞ

(13) Where hðzÞ ¼ 1 þ ε cosð2pzÞ

(14)

Analytical solutions of the boundary value problem Ser

Nanoparticle Type

Shape

Shape Factor (m)

1.

Bricks

3.7

2.

Cylinders

4.9

Closed-form solutions are feasible for the transformed, nondimensional boundary value problem. Solving Eqs. 9e11 together with boundary conditions (13) and (14) therefore generates the following expressions for temperature and axial velocity: qðr; zÞ ¼

   1 ks þ ðm þ 1Þkf  ðm þ 1Þðkf  ksÞf
2 b h  r2 : 4 ks þ ðm þ 1Þkf þ ðkf  ksÞf

wðr; zÞ ¼ 3.

5.7

Platelets

  dP ðr2  h2 Þ ðr4  h4 Þ ð1  fÞ2:5 L3  L4 dz 4 8 ! ðrgÞs þ ð1  fÞ þ f Gr ð1  fÞ2:5 ðrgÞf   ðr2  h2 Þ ðr4  h4 Þ ðr6  h6 Þ þ L6 þ L7 : L5 2 4 6

(15)

(16)

where: We introduce the following non-dimensional variables:


T  T0 r z w lu a2 p ct ; q¼ ; t¼ ; r¼ ; z¼ ; w¼ ; u¼ ; p¼ a l c ac clmf l T0 gaa2 T0 rnf b Q0 a2 ε ¼ ; Gr ¼ ; b¼ : a cmf kf T0

K¼



2

(8)

These represent respectively the r dimensionless radial coordinate, z dimensionless axial coordinate, w, u dimensionless radial and axial velocity components, p dimensionless pressure, q dimensionless temperature function, t dimensionless time, ε radius ratio, Gr Grashof number and b heat source/sink parameter. Implementing these variables in Eqs. (2)e(5) and invoking the assumptions of low Reynolds number and long wavelength, the non-dimensional governing equations after dropping the dashes can be written as the following steady-state equations in the wave frame. vp ¼ 0; vr

   kf ; L ¼ ð1  fÞ rf gf þ fðrs gsÞ = rf gf ; knf

(9)

   ðrgÞnf dp 1 v mnf vw ¼ þ Gr q; r dz r vr m0 vr ðrgÞf

(10)

kf v q 1 vq þb þ ¼ 0; vr2 r vr knf

(11)

Gr bK ð1  fÞ2:5 L1 h ; L2 ¼ ; L3 ¼ ð1  fÞ2:5 ; 4 2 2 L1 abKh2 abK ; ; L6 ¼ L2 L4 ¼  ð1  fÞ2:5 ; L5 ¼ L2 þ L2 4 4 4 2 2 abKh abK abKh abK  L4 : ; L8 ¼ L4 þ L3 ; L9 ¼ L4 L7 ¼ L3  L3 4 4 4 4 (17) L1 ¼ L

The volumetric flow rate of nanofluid in the tube is given by: ZhðzÞ F¼

rwdr;

(18)

0

Axial pressure gradient emerges as:  2:5 4 s ð1  fÞ þ f ðrgÞ ðrgÞf dp F  h L9 ð1  fÞ ¼ ; dz h4 L8 ð1  fÞ2:5

(19)

It follows that the mean flow rate is given by: F ¼ 2Q 

2

ε2  1; 2

(20)

Please cite this article in press as: Huda AB, Akbar NS, Heat transfer analysis with temperature-dependent viscosity for the peristaltic flow of nano fluid with shape factor over heated tube, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.08.054

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Integrating Eq. (19) over the interval [0,1] leads to an expression for the pressure rise: DP ¼

Z1   dp dz: dz

(21)

0

Expression of wall shear stress is evaluated using Srz ¼

vw

; vr r¼h

(22)

Results and interpretation Numerical computations, based on the exact solutions derived in section 3, have been conducted to assess the influence of flow parameters on the peristaltic flow characteristics (see Table 1). These are depicted in Figs. 2e12. These computations are based on nanofluid properties documented in Table 2. Further solutions are given for the velocity and temperature fields in Tables. Fig. 2(a)-(b) Present the radial temperature distributions, q(r) with variation in heat source parameter (b) and nanoparticle volume fraction (f). Significant elevation in temperature is sustained as the increases in heat absorption parameter. Maximum temperatures are attained at r ¼ 0 and Platelets achieves greater temperatures than Cylinder and Bricks. Copper methanol nanofluid in the form of cylinder generally demonstrates the best cooling performance compared

Fig. 3 e Velocity profile for different values of the (a)b ¼ 5.0, 7.0. (b) Gr ¼ 4.0, 5.0 and (c) f ¼ 0.02, 0.03.(Blue line Bricks Red line Cylinder Purple line Platelets). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 2 e Temperature profile against the radial axis r for different values of (a)b ¼ 5.0, 7.0 (b) f ¼ 0.2, 0.4. (Blue line Red line Cylinder Purple line Platelets ). Bricks (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

with bricks and platelets form as it successfully minimizes temperature increase in the flow. The same pattern of behavior is confirmed by increasing the nanoparticle volume fraction f (Fig. 2b). With increasing nano-particle volume fraction f, Maximum temperatures are attained at r ¼ 0 and Platelets achieves greater temperatures where again copper methanol nanofluid in bricks is found to consistently attain the lowest temperatures. In all cases symmetric parabolic distributions are observed across the tube cross-section, and the maximum temperature always arises at the tube centerline. Fig. 3(a)-(c)Depict the velocity evolution in the tube for different values ofb,Gr, andf. It is observed that when we increase the value of heat absorption parameterb, the velocity of nanoparticle in all shapes increases (Fig. 2a). Heat introduction into the fluid therefore enhances momentum transfer also and accelerates the axial flow. Parabolic distributions are observed across the tube diameter, with the core flow accelerated substantially due to greater heat source (absorption).

Please cite this article in press as: Huda AB, Akbar NS, Heat transfer analysis with temperature-dependent viscosity for the peristaltic flow of nano fluid with shape factor over heated tube, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.08.054

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Fig. 4 e Pressure gradient dp/dz against the axial distance z for different values of (a)b ¼ 5.0, 8.0. (b) Gr ¼ 2.0, 3.0 and (c) f ¼ 0.1, 0.15. (Blue line Bricks Red line Cylinder Purple line Platelets). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Note the solid colored lines correspond to b ¼ 5.0 whereas the dotted color lines are associated with b ¼ 7.0. Axial velocity is maximized with copper-methanol nanofluid in shape of platelets, whereas it is minimized in bricks shape. It is also apparent in Fig. 3b, that with an increment in Grashof number, Gr, nanofluid velocity is significantly increases i.e. flow acceleration is induced across the tube diameter. Once again the copper-methanol nanofluid in shape of platelets achieves the best acceleration. The Grashof number is a representation of thermal buoyancy effect relative to viscous hydrodynamic force. When Gr ¼ 1 these forces are the same order of magnitude in the tube. For Gr > 1 thermal buoyancy force exceeds viscous force and vice versa for Gr < 1. Thermal buoyancy is known to decelerate viscous flows, since it opposes momentum development and inhibits propulsion in the tube. Fig. 3c shows with the increasing nano-particle volume fractionf, increases the axial velocity, again especially in the

Fig. 5 e Pressure rise DP against the flow rate Q for different values of (a)b ¼ 5.0,8.0. (b) Gr ¼ 2.0, 3.0 and (c) f ¼ 0.1, 0.15. (Blue line Bricks Red line Cylinder Purple line Platelets). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

core region.As solid lines are for small nano-particle volume fraction f and dotted lines are for large values of nano-particle volume fractionf so raising the nano-particle volume fractionf there is an increase in the velocity. Again the coppermethanol nanofluid in shape of platelets achieves the best acceleration in the core flow. Fig. 4(a)-(c)Present the evolution of axial pressure gradient along the tube (i.e. with axial coordinate). The magnitudes of pressure gradient increase with greater heat absorption parameter (b) as shown in Fig. 3a. Values are highest for copper methanol nanofluid in platelets shape and lowest for bricks shape. These are generally maximized at the central zone of the tube length but are clearly non-zero both at the entry of the tube (z ¼ 1.0) and at the exit (z ¼ 1.0). A pressure gradient is therefore maintained throughout the tube irrespective of whether copper methanol nanofluid is the transported material. Fig. 4b shows that with increasing Grashof number, Gr, an adverse response in pressure gradient is

Please cite this article in press as: Huda AB, Akbar NS, Heat transfer analysis with temperature-dependent viscosity for the peristaltic flow of nano fluid with shape factor over heated tube, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.08.054

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Fig. 6 e Variation of wall shear stress Srz against axial distance z for different values of (a)b ¼ 2.0, 2.5. (b) Gr ¼ 2.0, 2.2 and (c) f ¼ 0.1, 0.15. (Blue line Bricks Red line Cylinders Purple line Platelets). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Please cite this article in press as: Huda AB, Akbar NS, Heat transfer analysis with temperature-dependent viscosity for the peristaltic flow of nano fluid with shape factor over heated tube, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.08.054

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Fig. 7 e Stream lines for Copper methanol nanofluid in bricks shape for different values of b.

Fig. 8 e Stream lines of Copper methanol nanofluid in cylinders shape for different values of b.

Fig. 9 e Stream lines of Copper methanol nanofluid in platelets shape for different values of b. Please cite this article in press as: Huda AB, Akbar NS, Heat transfer analysis with temperature-dependent viscosity for the peristaltic flow of nano fluid with shape factor over heated tube, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.08.054

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Fig. 10 e Stream lines of Copper methanol nanofluid in bricks shape for different values of Gr.

Fig. 11 e Stream lines of Copper methanol nanofluid in cylinders shape for different values of Gr.

Fig. 12 e Stream lines of Copper methanol nanofluid in platelets shape for different values of Gr. Please cite this article in press as: Huda AB, Akbar NS, Heat transfer analysis with temperature-dependent viscosity for the peristaltic flow of nano fluid with shape factor over heated tube, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.08.054

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Table 1 e Thermo physical properties of base fluid and Copper. Physical properties r(kg/m3) cp(J/kg-K) k(W/m-K) g105(1/K)

Base fluid

Nanoparticle

Methanol

Copper

792 2545 0.2035 6.2

8933 385 401 1.67

computed i.e. it decrease significantly throughout the length of the tube. Increasing thermal buoyancy force in the peristaltic flow regime therefore depresses pressure gradient consistently. Maximum magnitudes are noted copper methanol nanofluid in shape of platelets for whereas the lowest magnitudes correspond to the bricks shape copper methanol nanofluid. This has implications in clinical applications, since for sustaining greater pressure gradients, which affect the efficiency of delivery of drugs. Fig. 4c illustrates that with increasing nano-particle volume fractionf, pressure gradient is substantially elevated. Copper methanol nanofluid in shape of platelets is capable to achieve the peak values of pressure gradient and the bricks shape copper methanol nanofluid again produces the lowest magnitudes.

Fig. 5(a)-(c)Provide the response in pressure rise (Dp) with a variation in heat absorption, Grashof number and nanoparticle volume fraction, respectively for Copper methanol nanofluid in shape of bricks, cylinder and platelets are studied. Pressure rise generally increases with an increase in heat absorption parameter (b), Grashof number (Gr) and also nanoparticle volume fraction (f) in the peristaltic pumping region which corresponds to the volumetric flow rate range, 2.0  Q  0.9. However in the augmented pumping region, which approximately corresponds to the range, 0.9 < Q  2. in which is 0.91Q2 pressure rise decreases with an increase in Grashof number (Gr), heat absorption parameter (b), and nanoparticle volume fraction (f), which is the reverse of the response in the peristaltic pumping region. In all the plots, the pressure rise-volumetric flow rate (Dp-Q) demonstrates a linear relationship. Minimum values are computed always at the lowest volumetric flow rate and are negative; maximum values arise at the maximum volumetric flow rate and are positive. In the peristaltic pumping region, Copper methanol nanofluid in shape of bricks consistently attains the minimal values of pressure rise, whereas in the augmented pumping region, it achieves the maximum pressure rise values. Conversely Copper methanol nanofluid in shape of platelets attains the maximum pressure rise in the peristaltic pumping

Table 2 e Variation of the velocity profile with different values of sink parameter (b) for Copper Methanol nanofluid cases. Copper Methanol nanofluid (f¼0.04) w(r, z) R 0.9 0.7 0.5 0.3 0.1 0 0.2 0.4 0.6 0.8

Bricks

Cylinders

Platelets

b¼5.0

b¼7.0

b¼5.0

b¼7.0

b¼5.0

b¼7.0

0.0000 0.1659 0.0709 0.0611 0.1482 0.1711 0.1279 0.0231 0.1111 0.1586

0.0000 0.3664 0.2927 0.1367 0.0294 0.0012 0.0545 0.1829 0.3351 0.3259

0.0000 0.1841 0.0908 0.0435 0.1326 0.1559 0.1118 0.0047 0.1312 0.1737

0.0000 0.3983 0.3283 0.1689 0.0586 0.0296 0.0845 0.2163 0.3708 0.3524

0.0000 0.1964 0.1044 0.0316 0.1219 0.1456 0.1008 0.0077 0.1449 0.1841

0.0000 0.4201 0.3526 0.1909 0.0787 0.0491 0.1051 0.2391 0.3953 0.3705

Fig. 13 e Stream lines of Copper methanol nanofluid in bricks shape for different values of f. Please cite this article in press as: Huda AB, Akbar NS, Heat transfer analysis with temperature-dependent viscosity for the peristaltic flow of nano fluid with shape factor over heated tube, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.08.054

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 4

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Fig. 14 e Stream lines of Copper methanol nanofluid in cylinders shape for different values of f.

Fig. 15 e Stream lines of Copper methanol nanofluid in platelets shape for different values of f. region, whereas it achieves the minimum magnitudes in the augmented pumping region. Fig. 6(a)-(c)illustrate the axial response in wall tube shear stress (Srz) for Copper methanol nanofluid in shape of bricks, cylinder and platelets, for variation with different thermophysical parameters. The oscillatory nature of peristaltic propulsion is clearly captured in the profiles, owing to the sinusoidal wave propagation along the flexible tube wall. It is evident that Copper methanol nanofluid in shape of platelets invariably achieves the highest shear stress magnitudes as compared with Copper methanol nanofluid in shape of bricks and cylinder. The lowest shear stress is observed for the Copper methanol nanofluid in shape of bricks. Therefore it is established again that the presence of nano-particles serves to increase velocity and thereby increase in shear stresses at the wall. Close inspection of the profiles reveals that in the first region, magnitudes of shear stress start decreasing at the entry point of the tube until a maximum constriction is reached and thereafter the shear stress magnitudes increase to the end of the contracting section in the range(0z1). This

characteristic is also exhibited for subsequent regions i.e. (1z2 and 2z3) i.e. it is repeated until the termination of the tube i.e. exit. Similar patterns are observed in all of Fig. 6(a)-(c). The figures respectively show that shear stresses are elevated with increasing heat source (b), Grashof number (Gr) and nano-particle volume fraction (f), i.e. the flow is accelerated at the inner wall of the tube with greater heat absorption, thermal buoyancy and nano-particle volume fraction. Figs. 7e14 depict streamline visualizations for the influence of different parameters in the peristaltic flow. These permit a better appraisal of the influence of trapping of boluses of nanofluid which is a characteristic phenomenon associated with creeping-type peristaltic dynamics. The effects of heat source parameter (b) on trapping phenomena for Copper methanol nanofluid in of bricks, cylinders and platelets cases are presented respectively in Figs. 7(a)-9(b). The number of trapped bolus increases with an increase in the value of heat source or sink parameter b for Copper methanol nanofluid in all cases. The modification in trapped

Please cite this article in press as: Huda AB, Akbar NS, Heat transfer analysis with temperature-dependent viscosity for the peristaltic flow of nano fluid with shape factor over heated tube, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.08.054

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i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 4

Table 3 e Variation of then Pressure gradient dp/dz with different values of Grashof number, (Gr) for Copper Methanol nanofluid cases. Copper Methanol nanofluid (f¼0.04) dp/dz

Bricks

z

Cylinders Gr¼3.0

Gr¼2.0

Gr¼3.0

Gr¼2.0

Gr¼3.0

0.7354 0.7685 0.8378 0.8378 0.7685 0.7354 0.7685 0.8378 0.8378 0.7685 0.7354

0.7184 0.7459 0.8005 0.8005 0.7459 0.7184 0.7459 0.8005 0.8005 0.7459 0.7184

0.7463 0.7791 0.8471 0.8471 0.7791 0.7463 0.7791 0.8471 0.8471 0.7791 0.7463

0.7302 0.7577 0.8119 0.8119 0.7577 0.7302 0.7577 0.8119 0.8119 0.7577 0.7302

0.7534 0.7861 0.8532 0.8532 0.7861 0.7534 0.7861 0.8532 0.8532 0.7861 0.7534

0.7379 0.7655 0.8194 0.8194 0.7655 0.7379 0.7655 0.8194 0.8194 0.7655 0.7379

0.1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1.0

Table 4 e Variation of the temperature profile for different values of nanoparticle volume fraction (f) for Methanol and Copper nanofluid cases. CopperMethanol nanofluid q(r, z)

Platelets

Gr¼2.0

Bricks

Cylinders

Platelets

r

f¼0.2

f¼0.4

f¼0.2

f¼0.4

f¼0.2

f¼0.4

0.9 0.7 0.5 0.3 0.1 0 0.2 0.4 0.6 0.8

0.0000 1.0515 1.8611 2.4286 2.7542 2.8378 2.6795 2.2791 1.6367 0.7524

0.0000 2.0779 3.6776 4.7991 5.4425 5.6077 5.2948 4.5036 3.2343 1.4868

0.0000 1.1808 2.0901 2.7273 3.0931 3.1869 3.0091 2.5594 1.8381 0.8449

0.0000 2.4218 4.2864 5.5936 6.3435 6.5361 6.1712 5.2491 3.7697 1.7329

0.0000 1.2669 2.2424 2.9262 3.3185 3.4193 3.2284 2.7461 1.9721 0.9065

0.0000 2.6507 4.6915 6.1223 6.9431 7.1538 6.7546 5.7453 4.1261 1.8967

phenomena again for Copper methanol nanofluid in of bricks, cylinders and platelets with variation in Grashof number (Gr) are illustrated in Figs. 10(a)e12(b) respectively. The number of trapped boluses and also the size of the bolus both increase with an increase in Grashof number (Gr).in Figs. 13(a)-14(b) illustrated respectively the number of trapped boluses and also the size of the bolus both decrease with an increase in nano-particle volume fraction (f). Fig. 15(a) (b) present the effects of in nano-particle volume fraction (f) in platelets shape for Copper methanol nanofluid in the tube. The trapped bolus quantity the magnitude of the bolus markedly increases for the Copper methanol nanofluid (f) in platelets case, with an increase in nano-particle volume fraction (f).Therefore different responses are observed for the different nanoparticle cases. Tables 2e4 provide solutions for velocity, Pressure gradient and temperature functions, based on numerical evaluation of the closed-form solutions.

Conclusions A mathematical study has been conducted of peristaltic propulsion and heat transfer in a temperature-dependent variable-viscosity nanofluid propagating through a flexible tube

under thermal buoyancy and heat generation effect. The transformed boundary value problem has been linearized via appropriate creeping flow and long wavelength approximations and solved exactly. Numerical evaluation of the closedform solutions has been conducted in symbolic software to evaluate the influence of heat source, Grashof number and viscosity parameter on axial velocity, axial pressure gradient, temperature, pressure rise, wall shear stress and also streamline plots. Copper methanol nanofluid in shape of bricks, cylinders and platelets has been examined in detail. The computations have shown that. 1) Increasing heat absorption parameter (b) generally accelerates the axial flow i.e. increases velocity and is greatest for Copper methanol nanofluid in shape of platelets and lowest for Copper methanol nanofluid in shape of bricks. The presence of nano-particles therefore aids the flow. 2) Axial flow is also accelerated with increasing Grashof number, (Gr) i.e. with greater thermal buoyancy force, with highest velocity magnitudes achieved by Copper methanol nanofluid in shape of platelets. 3) Axial velocity is significantly increased with an increase in nano-particle volume fraction (f), again with Copper methanol nanofluid in shape of platelets achieving the highest acceleration. 4) Temperature is elevated with heat absorption parameter (b) and is highest for Copper methanol nanofluid in shape of platelets lowest for Copper methanol nanofluid in shape of bricks. 5) Temperature also generally increase significantly with increase in nano-particle volume fraction (f). Copper methanol nanofluid in the form of cylinder generally demonstrates the best cooling performance. 6) Axial pressure gradient is enhanced with increasing nano-particle volume fraction (f) i.e. decreasing viscosity. Maximum magnitudes are associated with Copper methanol nanofluid in shape of platelets whereas the lowest values are computed for Copper methanol nanofluid in shape of bricks. 7) Axial pressure gradient increases with greater heat absorption parameter (b) with highest values achieved in

Please cite this article in press as: Huda AB, Akbar NS, Heat transfer analysis with temperature-dependent viscosity for the peristaltic flow of nano fluid with shape factor over heated tube, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.08.054

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platelets shape Copper methanol nanofluid and lowest values corresponding to bricks shape Copper methanol nanofluid. 8) Axial pressure gradient is also increases substantially with increasing Grashof number, Gr, throughout the length of the tube. 9) Pressure rise increases with an increase in heat absorption parameter (b), Grashof number (Gr) and also nanoparticle volume fraction (f) in the peristaltic pumping region i.e. the range 2.0
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