Results in Physics xxx (1017) xxx-xxx

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Results in Physics

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Dynamics of variable-viscosity nanofluid flow with heat transfer in a flexible vertical tube under propagating waves

A. Bintul Hudaa'*, Noreen Sher Akbarb, O. Anwar Begc, M. Yaqub Khana

a Mathematics & Statistics Department Riphah International University 1-14, Islamabad, Pakistan b DBS&H, CEME, National University of Sciences and Technology, Islamabad, Pakistan

c Fluid Mechanics, Spray Research Group, School of Computing, Science and Engineering, Newton Bldg, G77, University ofSalford, Manchester M54WT, UK

ARTICLE INFO ABSTRACT

Background and objectives: The present investigation addresses nanofluid flow and heat transfer in a vertical tube with temperature-dependent viscosity. A Tiwari-Das type formulation is employed for the nanofluid with a viscosity modification. As geometry of the problem is flexible tube so flow equations are modeled considering cylindrical coordinates. Governing partial differential equations are simplified and converted into differential equations using non-dimensionless variables with low Reynolds number (Re < 0 i.e. inertial forces are small as compared to the viscous forces) and long wavelength (5 < 0 i.e. physiologically valid that length of tube is very large as compared to width of the tube) approximations. Methods results conclusions: Mathematica software is employed to evaluate the exact solutions of velocity profile, temperature profile, axial velocity profile, pressure gradient and stream function. The influence of heat source/sink parameter (b), Grashof number (Gr) and the viscosity parameter (a) and nanoparticle volume fraction (/) on velocity, temperature, pressure gradient, pressure rise and wall shear stress distributions is presented graphically. Three different nanofluid suspensions are investigated- Titanium oxide-water, Copper oxide-water and Silver-water. Streamline plots are also computed to illustrate bolus dynamics and trapping phenomena which characterize peristaltic propulsion. The computations show that wall shear stress is maximum for the Silver-water nanofluid case. Furthermore the pressure rise is reduced with increasing Grashof number, heat absorption parameter and viscosity parameter in the augmented pumping region whereas the contrary response is observed in the peristaltic pumping region. Significant modification in the quantity of trapped boluses is found with different nanofluids and the size of the trapped bolus decreased in the Titanium oxide-water nanofluid case with either greater heat source or sink parameter. The study is relevant to drug delivery systems exploiting nano-particles.

© 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND

license (http://creativecommons.org/licenses/by-nc-nd/4XI/).

Article history:

Received 10 November 2016 Accepted 24 December 2016 Available online xxxx

Keywords: Biophysics Heat transfer Flexible tube

Temperature-dependent viscosity

Nanoparticles

Drug delivery

Introduction

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 [1])

* Corresponding author. E-mail address: abhripha@gmail.com (A.B. Huda).

and also phloem trans-location in plants as described by Thaine [2] and Thompson [3]. 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 [4]; Hameed and Nadeem, [5]; Tan and Masuoka, [6-8]; Mahomed and Hayat, [9]; Fetecau and Fetecau, [10]; Malik et al., [11]; Dehghan and Shakeri, [12]. Some relevant studies on the topic can be found from the list of references (Nadeem and Akbar, [13,14]) and several references therein. Simulations of peristalsis, which is derived from the Greek word peri-stalitikos, which means clasping and compressing, are also of

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A.B. Huda et al./Results in Physics xxx (2017) xxx-xxx

great relevance to advanced medical and biochemical engineering pumps. Roller and Finger pumps using viscous fluid operate on this principle. Peristaltic pumps offer considerable advantages over conventional pumping mechanisms including corrosion mitigation and leak-free designs. Furthermore such pumps can avoid internal backflow and thereby can consistently deliver accurate dosing without slip and exhibit excellent repeatability and metering capabilities. The tube components in peristaltic pumps also achieve longer service times and minimize maintenance. They have therefore been deployed in many diverse areas of biomedicine including diabetic treatment [15] and drug delivery in cancer therapy [16]. They have also been integrated into medical recirculating cooling pumps with applications in cooling electro-tips of RF catheters in order to mitigate blood coagulation on the tip during ablation therapy. The mechanism of peristaltic transport has also been exploited in biohazard management including sanitary fluid transport and safe conveyance of corrosive fluid where the contact of the fluid with the machinery parts is prohibited. Further details are provided in Diniz et al. [17].

Non-Newtonian models are extremely diverse and include vis-coelastic, viscoplastic, micro-continuum and other formulations. Variable-viscosity models are also an important sub-section of rhe-ological 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 [18]. They observed that for zero pressure drop, 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. [19] studied the peristaltic transport 4 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 rate but is infact 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. [20] 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. [21] for magnetohydrodynamic fluids, Khan et al. [22] for inclined pumping of non-Newtonian fluids and Akbar

[23] present nanoparticle volume fraction for phase model.

Another significant development in medical engineering in

recent years has been the emergence of nanofluids, a sub-category 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

[24] Nanoparticles 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 [24], nanoparticles have been found to achieve exceptional performance in enhancing thermal and mass diffusion properties of, for example, drugs injected into the blood stream. Biocompatability of the selected metallic oxides is crucial for safe deployment of nanofluids in medicine. New potential applications for nanoparticles in nanoparticle blood diagnostic systems, asthma sensors, carbon nanotubes in catheters and stents and antibacterial treatment for wounds via peristaltic pump delivery was identified by Harris and Graffagnini [25]. 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 adsorb 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, intra-ocular). In neuro-pharmacological hemodynamics as refer in [26], it has been clinically verified that nanoparticles can easily penetrate the blood brain barrier (BBB) facilitating the introduction of therapeutic agents into the brain. Fullstone et al. [26] have also recently described the exceptional characteristics of nanoparticles (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 nanoparticle distribution within capillaries. Further studies include Tan et al. [27]. Simulation of peristaltic flows of nanoparticles is therefore extremely relevant to improve administration of nanofluids in medicine. Representative studies in this regard include Tripathi and Beg [28] who 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 [29] studied magnetic field effects on electrically-conducting nanofluid propulsion by peristaltic waves with applications in cancer therapy. Akbar et al. [30] investigated peristaltic slip nanofluid hydrodynamics in an asymmetric channel, obtaining series solutions for temperature, nano-particle concentration, stream function and pressure gradient. Akbar and Nadeem [31] considered peristaltic flow of Phan-Thien-Tanner nanofluid in a diverging conduit with the homotopy perturbation method. Further analyses include Beg and Tripathi [32] 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. [33-44].

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 nano-fluid 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.

Nanofluid peristaltic transport model

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 tempera-

A.B. Huda et al./Results in Physics xxx (2017) xxx-xxx

Fig. 1. Geometry of problem.

ture condition is imposed. The geometric model is illustrated in Fig. 1 with respect to a cylindrical coordinate system (R , Z).

The geometry of the wall surface is simulated via the following relation:

h = a + b sin2^ (Z - ct)

where h denotes the height of the tube wall, a denotes the radius of the tube, b is the wave amplitude, k is the wave length and c is the peristaltic wave speed. In the fixed coordinates system (R , Z) , the hydrodynamics is unsteady. It becomes steady in a wave frame (h, h) 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:

h = R,h = Z - ch, h = h w = W - c, h(h h h) = h(Z, R, t),

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:

1 d(zh) dw_ 0

dp dz d_ dh

» « du

2 duh uh

-Mnf(h) gh - T

duh dwh

N (h)( dh+âh

dwh dwh

dz d_ " dh

21nf (h) dW

dww ~dh

Inf (h) n dh

(PC)nfg(h - T0)'

dh _ dh _ dh k, dh + u dh+ w dl = (PkP)

(Pcp)nf'

where r and h are the co-ordinates in the wave frame. h is taken as the center line of the tube and hr is orientated transverse to it, uh and w are the velocity components in the h and h directions respectively, h is the local temperature of the fluid. Further inf is the effective dynamic viscosity of nanofluid, anf is the effective thermal diffusiv-ity of nanofluid, pnf is the effective density of nanofluid, (pcp)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:

_ l0e-

-Pnf = (1 - /) Pf + «

(1-/)25 ' -n' (Pcp)nf '

(Pcp)nf = (1 - /) (Pcp)/ + /(PcP)s' (PC)nf = (1 - /) (PC)f + /(PJ)s'

k _ k fh+2kf-2/(kf-fe)/ ^nf f ks+2kf+2/(kf -ks) y

In above equation Pf density of the base fluid, Ps density of the nanoparticles, kf thermal conductivity of the base fluid, ks thermal conductivity of the nanoparticles, yn{ is the thermal expansion coefficient, j is the thermal expansion coefficient of basefluid / is the nanoparticle volume fraction, and cs is the thermal expansion coefficient of the nanoparticles.

We introduce the following non-dimensional variables:

r = t z = t w = w u = ku v = g =

' a ' k ' vv c ' " ac' f ckif ' h

|_iT-Tal t = Çt

£ = b' Gr =

_ gaa2TpPnf b _ Q0a2

f> _ Q0a

These represent respectively the r dimensionless radial coordinate, z dimensionless axial coordinate, w,u dimensionless radial and axial velocity components, p dimensionless pressure, Q dimen-

A.B. Huda et al./Results in Physics xxx (2017) xxx-xxx

The Reynolds model for nanofluid viscosity can be defined as follows:

Fig. 2. Velocity profile for different values of the (a) b = 2.0, 2.2. (b) Gr = 2.0,2.2 and (c) a = 0.10,0.15. (Red line Pure water Purple line Cuo + H2O Blue line

Ag + H2O Dark Purple line TiO2 + H2O ). (For interpretation of the

references to color in this figure legend, the reader is referred to the web version of this article.)

sionless temperature function, t dimensionless time, e radius ratio, Gr Grashof number and p heat source/sink parameter. Implementing these variables in Eqs. (2)-(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.

= 0, dr '

1 @ /ftf r dr V lo

dw' д

(qc)n/r в

1 о kf n

+1 @r+b kkf —0'

(9) (10)

Inf _ e-ah

l0 (1 - Ф)2

and e-ah — 1 - ah, a < 1.

where a is the viscosity parameter and i0 constant viscosity of the fluid.

The non-dimensional boundary conditions are prescribed as:

@w n дв n „

~7t- — 0, — = 0 at r — 0,

w —-1, в — 0, at r — h(z) where h(z) = 1 + e cos(2pz)

Analytical solutions of the boundary value problem

Closed-form solutions are feasible for the transformed, non-dimensional boundary value problem. Solving Eqs. (9)-(11) together with boundary conditions (13) and (14) therefore generates the following expressions for temperature and axial velocity:

1 ffks + 2kf + 2/(kf - ks)\\b(h2 _ j2) .15.

h(r;Z) H\ks + 2kf - 2/(kf - ks)JJb(h r ): (15)

w(r,z)=f (1 - /)25(L3^_L4^) + ((1 _ /) + /(f Gr(1 - /)25 (L5 + L6^ + L7^ ■

where:

K — kf, L =((1 - /)(PfCf) + /(PsCs))/(PfCf)L = LG-t-, L2 — ■^

L3 = ^(1 - /)2'5, L4 = -L-(1 - /)", ¿5 = L2 + L2^; ¿6 = -L2af;

r r r abKh2 r r r abK r abKh2 r r abK L7 = —¿3 — L3 -^4 , Lg = —¿4 + L3-4--L4 -^4 , L9 = L4-4-

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

Axial pressure gradient emerges as:

dp F - h4L9(1 - /)25((1 - /) + /

r h(z)

F — / rwdr 0

dz h4L8(1 - /)25

It follows that the mean flow rate is given by:

F = 2Q - "2 - 1'

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

DP - i'(D dz (21)

Expression of wall shear stress is evaluated using

Sz(22)

Results and discussion

Numerical computations, based on the exact solutions derived in Section ''Analytical solutions of the boundary value problem",

282 283

288 289

A.B. Huda et al./Results in Physics xxx (2017) xxx-xxx

Fig. 3. Temperature profile against the radial axis r for different values of (a) b = 2.0, 2.2. (b) / = 0.01, 0.04 and (c) r = 2.0,4.0. (Red line Pure water Purple line Cuo

+ H2O Blue line Ag + H2O Dark Purple line TiO2 + H2O ). (For interpretation of the references to color in this figure legend, the reader is referred to the

web version of this article.)

have been conducted to assess the influence of flow parameters on the peristaltic flow characteristics. These are depicted in Figs. 2-12. These computations are based on nanofluid properties documented in Table 1. Further solutions are given for the velocity and temperature fields in Tables 2-4.

Fig. 2(a)-(c) depict the velocity evolution in the tube for different values of b, Gr and a. It is observed that when we increase the value of heat absorption parameter b, the velocity of the pure water and other nanoparticles also 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). Note the solid colored lines correspond to b = 2.0 whereas the dotted color lines are associated with b = 2.2. Axial velocity is maximized with silver-water (Ag + H2o) nanofluid, whereas it is minimized with pure water. Copper-water nanofluid also achieves slightly greater acceleration in the core flow than does Tianium oxide nanofluid, however both under-perform compared with silver-nanofluid. It is also apparent in Fig. 2b, that with an increment in Grashof number, Gr, nanofluid velocity is significantly decreased i.e. flow deceleration is induced across the tube diameter. Once again the silver-water nanofluid achieves the best acceleration, and out-performs the Copper water nanofluid, titaniums oxide nanofluid and achieves dramatically greater velocities than pure water. The Grashof number is a representation of thermal buoyancy effect relative

to viscous hydrodynamic force. When Gr =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. This observation is consistent with many other studies on peristaltic nanofluids, including [33-40]. Fig. 2c shows that an increase in viscosity parameter (a), decreases the axial velocity, again especially in the core region. Because increasing in viscosity parameter there will be more resistance in the fluid that will reduce the velocity. As solid lines are for small viscosity parameters dotted lines are for large values of viscosity parameter so raising the viscosity parameter there is decline in the velocity. Based on the Reynolds model (Reynolds 1886), originally for non-Newtonian tribological flows, the nanofluid viscosity will decrease with increasing values of a; and higher temperature (Q). This will reduce the viscous forces in the fluid and will lead to acceleration, as observed in Fig. 2c. pure water is found to sustain significant deceleration as compare with the nanofluid cases. Again Silver water nanofluid achieves the best acceleration in the core flow.

Fig. 3(a)-(b) present the radial temperature distributions, Q(r) with variation in heat source parameter (b) and nano-particle volume fraction (/). Significant elevation in temperature is sustained as the increases in heat absorption parameter. Maximum temperatures are attained for pure water and the minimum temperatures are associated with silver-water nanofluid. Copper water nanofluid

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+ H2O Blue line Ag + H2O

web version of this article.)

Dark Purple line TiO2 + H2O

: (a) b = 2.0, 2.2. (b) Gr = 2.0,2.2 and (c) a = 0.10,0.15. (Red line Pure water Purple line Cuo ). (For interpretation of the references to color in this figure legend, the reader is referred to the

achieves greater temperatures than silver water nanofluid but distinctly lower temperatures than Titanium oxide nanofluid. Silver water nanofluid therefore generally demonstrates the best cooling performance compared with pure water and nanofluids as it successfully minimizes temperature increase in the flow. This pattern of behavior is also confirmed by increasing the nanoparticle volume fraction u (Fig. 3b) where again silver water nanofluid is found to consistently attain the lowest temperatures. With increasing nano-particle volume fraction, the desired effect of cooling the pumping fluid is achieved. Titanium water nanofluid is observed to produce the highest temperatures, then Copper water nanofluid and finally silver water nanofluid. Therefore for temperature regulation in peristaltic nanofluid pumping, the optimized cooling performance is associated with Silver water nanofluids. In all cases symmetric parabolic distributions are observed across the tube cross-section, and the maximum temperature always arises at the tube centerline.

Fig. 4(a)-(c) present the evolution of axial pressure gradient along the tube (i.e. with axial coordinate). The magnitudes of pressure gradient decrease with greater heat absorption parameter ( b ) as shown in Fig. 3a. Values are highest for Titanium oxide and lowest for Silver water nanofluid. Values are also generally maximized at the central zone of the tube length but are clearly non-zero both at the entry of the tube (z = 0) and at the exit (z = 1.0). A pressure gradient is therefore maintained throughout the tube irrespective

of whether pure water or nanofluid is the transported material. Copper water nanofluid achieves greater pressure gradient magnitudes than silver water nanofluid but substantially lower values than Titanium oxide nanofluid. Fig. 4b shows that with increasing Grashof number, Gr, a similar response in pressure gradient is computed i.e. it decreases significantly throughout the length of the tube. Increasing thermal buoyancy force in the peristaltic flow regime therefore depresses pressure gradient consistently. Maximum magnitudes are again noted for Titanium oxide nanofluid whereas the lowest magnitudes correspond to Silver water nano-fluid. This has implications in clinical applications, since for sustaining greater pressure gradients, which affect the efficiency of delivery of drugs, Titanium oxide nano-particles would potentially perform better than the other nano-particles. Fig. 4c illustrates that with increasing viscosity parameter (a) which is associated with decreasing nanofluid dynamic viscosity (if based on Reynolds exponential decay model, Eq. (12)), pressure gradient is substantially elevated. Titanium oxide nanofluid is observed to achieve the peak values of pressure gradient and Silver water nanofluid again produces the lowest magnitudes. Overall therefore irrespective of the parameter being varied, Titanium oxide nanofluid prevails as the best choice for sustaining high axial pressure gradient values in peristaltic nanofluid propulsion.

Fig. 5(a)-(c) provide an insight into the response in pressure rise (Dp) with a variation in heat absorption, Grashof number

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Fig. 5. Pressure rise DP against the flow rate Q for different values of(a) b = 2.0, 2.2. (b) Gr = 2.0, 2.2 and (c) a = 0.10, 0.15. (Red line Pure water Purple line Cuo + H2O

Blue line Ag + H2O Dark Purple line TiO2 + H2O ). (For interpretation of the references to color in this figure legend, the reader is referred to the web

version of this article.)

and viscosity parameter, respectively. Again pure water and three nano-particle suspension cases are considered. Pressure rise generally increases with an increase in heat absorption parameter (b), Grashof number (Gr) and also viscosity parameter a in the peristaltic pumping region which corresponds to the volumetric flow rate range, -0.5 6 Q 6 0.5. However in the augmented pumping region, which approximately corresponds to the range, 0.5 < Q6 2. in which is 0.51 6 Q 6 2 pressure rise decreases with an increase in Grashof number (Gr), heat absorption parameter (b), and viscosity parameter (a), 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, Titanium oxide nanofluid consistently attains the minimal values of pressure rise, whereas in the augmented pumping region, it achieves the maximum pressure rise values. Conversely silver water nanofluid attains the maximum pressure rise in the peristaltic pumping region, whereas it achieves the minimum magnitudes in the augmented pumping region. The implication is that no single nanofluid achieves a maximum pressure rise in both pumping regions. Therefore designers may judiciously select different nanofluids to optimize pressure rise in different regions of the peristaltic flow.

Fig. 6(a)-(c) depicts the axial response in wall tube shear stress (Srz) for pure water and three different nanofluids, for variation with different thermo-physical parameters. The oscillatory nature of peristaltic propulsion is clearly captured in the profiles, owing to the sinousoidal wave propagation along the flexible tube wall. It is evident that silver water nanofluid invariably achieves the highest shear stress magnitudes as compared with pure water and other nanofluids (copper oxide water, Titanium oxide water). This is consistent with the acceleration computed earlier in connection with Silver water nanofluid. The lowest shear stress is observed for the pure water case. Therefore it is established again that the presence of nano-particles serves to increase velocity and there by 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 (0 6 z 6 1). This characteristic is also exhibited for subsequent regions i.e. (1 6 z 6 2 and 2 6 z 6 3) 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 viscosity parameter (a) i.e. the flow is accelerated at the inner wall of the tube with greater heat absorption, thermal buoyancy and decreasing viscosity of the nanofluid. These trends are indeed

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Fig. 6. Variation of wall shear stress Srz against axial distance z for different values of (a) b = 2.0, 2.2. (b) Gr = 2.0, 2.2 and (c) a = 0.10, 0.15. (Red line Pure water Purple

line Cuo + H2O Blue line Ag + H2O Dark Purple line TiO2 + H2O ). (For interpretation of the references to color in this figure legend, the reader is referred

to the web version of this article.)

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(a) P = 2 (b) p = 2.2

Fig. 8. Stream lines for CuO + H2O for different values of b

(a) P = 2 (b) P = 2.2

Fig. 9. Stream lines of Ag + H2O for different values of b.

486 consistent with the earlier computations presented for axial veloc- Figs. 7-14 depict streamline visualizations for the influence 489

487 ity and also concur with other studies of nanofluid transport under of different parameters in the peristaltic flow. These permit a 490

488 peristaltic waves e.g. Akbar [14] and Nadeem, and Ijaz [36]. better appraisal of the influence of trapping of boluses of nano- 491

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1.5 1.0 0.5 0.0 -0.5

1.5 1.0 0.5 0.0 -0.5

(b) p= 2.2

(a) p = 2

Fig. 10. Stream lines of Tio2 + H2O for different values of b.

(a) « = 0.15 (b) « = 2

Fig. 11. Stream lines of Pure Water for different values of a.

-10 12 -10 12

(a) a = 0.15 (b) a = 2

Fig. 12. Stream lines of CuO + H2O for different values of a.

492 fluid which is a characteristic phenomenon associated with

493 creeping-type peristaltic dynamics. The effects of heat source

494 parameter (b) on trapping phenomena for pure water CuO

495 + H2O nanofluid and also Ag + H2O nanofluid cases are presented

496 respectively in Figs. 7a-9b. The number of trapped bolus

increases with an increase in the value of heat source or sink 497

parameter b in all pure water, copper water and silver water 498

nanofluids cases. Fig. 10(a)-(b) demonstrate that the quantity 499

of trapped boluses decreases for the Titanium oxide water nano- 500

fluid (Tio2 + H2O) case with the increase of heat source parame- 501

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Table 1

Thermo-physical properties of pure water and other metallic nano-particles.

Physical properties H2O CuO Ag TiO2

Cp (J/kgK) 4179.0 540.0 235 686.2

q (kg/m3) 997.1 6500.0 10500.0 4250.0

k (W/mk) 0.613 18.0 429.0 401.0

y x 10-5(1/K) 21.0 0.85 1.89 1.67

502 ter b.The modification in trapped phenomena again for pure

503 water, CuO + H2O and Ag + H2O nanofluids with variation in vis-

504 cosity parameter (a) are illustrated in Figs. 11a-13b respectively.

505 The number of trapped boluses and also the size of the bolus

506 both increase with an increase in viscosity parameter (a) i.e.

507 with decreasing viscosity of the nanofluid. Fig. 14a and b present

508 the effects of viscosity parameter (a) for Tio2 + H2O nanofluid in

Physics xxx (2017) xxx-xxx 11

the tube. The trapped bolus quantity decreases for the Tio2 + H2- 509

O nanofluid case, with an increase in viscosity parameter a 510

whereas the magnitude of the bolus markedly increases. There- 511

fore different responses are observed for the different nano- 512

particle cases. Tables 2-4 provide solutions for velocity and 513

temperature functions, based on numerical evaluation of the 514

closed-form solutions. They show that silver water nanofluid 515

generally attains higher axial velocities than the pure water or 516

other nanofluid cases, provided the nano-particle volume frac- 517

tion is the same i.e. / = 0.004. Maximum velocity and tempera- 518

ture, as shown and elaborated in earlier graphs, consistently 519

arise at the centerline of the channel i.e. at r = 0. The tempera- 520

ture and velocity profiles always increase at the center of the 521

tube with an increase in heat source parameter (b), viscosity 522

parameter (a) and also with the nanoparticle volume fraction 523

(u). Conversely the temperature and velocity magnitudes are 524

found to fall in the near-wall regions of the tube. 525

Table 2

Variation of the velocity profile for different values of sink parameter P for pure water, CuO, Ag, and TiO2 case.

w(r,z) r Pure water ( / = 0.00) CuO (/ = 0.004) Ag (/ = 0.004) TiO2 (/ = 0.004)

b = 2 b = 2.2 b = 2 b = 2.2 b = 2 b = 2.2 b = 2 b = 2.2

-0.9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.7 -0.1026 -0.0928 -0.0721 -0.0629 -0.0499 -0.0382 -0.0763 -0.0677

-0.5 -0.2270 -0.2146 -0.1702 -0.1585 -0.1438 -0.1288 -0.1755 -0.1644

-0.3 -0.3271 -0.3142 -0.2511 -0.2389 -0.2254 -0.2098 -0.2563 -0.2448

-0.1 -0.3865 -0.3736 -0.2992 -0.2871 -0.2747 -0.2591 -0.3044 -0.2928

0 -0.4018 -0.3889 -0.3116 -0.2995 -0.2874 -0.2718 -0.3167 -0.3052

0.2 -0.3729 -0.3599 -0.2881 -0.2759 -0.2633 -0.2477 -0.2933 -0.2817

0.4 -0.3002 -0.2873 -0.2292 -0.2171 -0.2032 -0.1876 -0.2345 -0.2231

0.6 -0.1898 -0.1778 -0.1404 -0.1292 -0.1144 -0.1001 -0.1456 -0.1349

0.8 -0.0652 -0.0574 -0.0438 -0.0365 -0.0257 -0.0163 -0.0473 -0.0404

Table 3

Variation of the velocity profile with different values of viscosity parameter (a) for pure water, CuO, Ag and TiO2 nanofluid cases.

w(r,z) r Pure water ( / = 0.00) CuO (/ = 0.004) Ag (/ = 0.004) TiO2 (/ = 0.004)

a = 0.15 a = 2 a = 0.15 a = 2 a = 0.15 a = 2 a = 0.15 a = 2

-0.9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.7 -0.1008 0.0089 -0.0689 0.0698 -0.0454 0.1833 -0.0734 0.0504

-0.5 -0.2329 -0.1569 -0.1744 -0.0588 -0.1501 0.0631 -0.1793 -0.0794

-0.3 -0.3414 -0.3321 -0.2638 -0.2104 -0.2439 -0.1076 -0.2681 -0.2273

-0.1 -0.4065 -0.4462 -0.3176 -0.3115 -0.3013 -0.2249 -0.3213 -0.3254

0 -0.4233 -0.4766 -0.3316 -0.3385 -0.3162 -0.2564 -0.3351 -0.3516

0.2 -0.3915 -0.4193 -0.3052 -0.2876 -0.2881 -0.1971 -0.3091 -0.3022

0.4 -0.3121 -0.2824 -0.2395 -0.1669 -0.2182 -0.0577 -0.2439 -0.1851

0.6 -0.1929 -0.0994 -0.1419 -0.0111 -0.1167 0.1135 -0.1469 -0.0322

0.8 -0.0624 0.0362 -0.0399 0.0816 -0.0201 0.1767 -0.0437 0.0653

Table 4

Variation of the Velocity profile for different values of nanoparticle volume fraction 0 for CuO, Ag, and TiO2 case.

w(r,z) r CuO Ag TiO2

/ = 0.01 / = 0.04. / = 0.01 / = 0.04 / = 0.01 / = 0.04

-0.9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.7 0.1647 0.1419 0.1554 0.1168 0.1669 0.1486

-0.5 0.2916 0.2511 0.2751 0.2067 0.2954 0.2631

-0.3 0.3805 0.3277 0.3589 0.2698 0.3855 0.3433

-0.1 0.4316 0.3716 0.4071 0.3059 0.4371 0.3894

0 0.4447 0.3829 0.4194 0.3152 0.4504 0.4012

0.2 0.4198 0.3616 0.3961 0.2976 0.4253 0.3788

0.4 0.3571 0.3075 0.3368 0.2531 0.3617 0.3222

0.6 0.2564 0.2208 0.2419 0.1818 0.2598 0.2314

0.8 0.1179 0.1015 0.1112 0.0835 0.1194 0.1063

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A.B. Huda et al./Results in Physics xxx (2017) xxx-xxx

(a) a = 0.15

1.5 1.0 0.5 0.0 -0.5

(b) a = 2

Fig. 13. Stream lines of Ag + H2O for different values of a.

(a) a = 0.15 (b) a = 2

Fig. 14. Stream lines of Tio2 + H2O for different values of a.

Conclusions (4) Temperature is elevated with heat absorption parameter (b)

and is highest for pure water and lowest for Silver-water

A mathematical study has been conducted of peristaltic nanofluid.

propulsion and heat transfer in a temperature-dependent variable- (5) Temperature generally decreases significantly with increase

viscosity nanofluid propagating through a flexible tube under thermal in nanoparticle volume fraction (u) i.e. nano-particles cool

buoyancy and heat generation effect. The transformed boundary the regime and Silver water nanofluid demonstrates supe-

value problem has been linearized via appropriate creeping flow rior cooling to copper oxide nanofluid and Titanium oxide

and long wavelength approximations and solved exactly. Numerical nanofluid.

evaluation of the closed-form solutions has been conducted in sym- (6) Axial pressure gradient is enhanced with increasing viscosity

bolic software to evaluate the influence of heat source, Grashofnum- parameter (a) i.e. decreasing viscosity. Maximum magni-

ber and viscosity parameter on axial velocity, axial pressure gradient, tudes are associated with Titanium oxide nanofluid whereas

temperature, pressure rise, wall shear stress and also streamline the lowest values are computed for Silver water nanofluid.

plots. Pure water and also three nanofluid cases (Copper oxide- Pressure gradient is lower for pure water as compared to

water, Silver-water and Titanium oxide water nanofluids) have been Titanium oxide and Copper water nanofluids but higher than

examined in detail. The computations have shown that. Silver water nanofluid.

(7) Axial pressure gradient decreases with greater heat absorp-

(1) Increasing heat absorption parameter (b) generally acceler- tion parameter (b) with highest values achieved by Titanium ates the axial flow i.e. increases velocity and is greatest for oxide and lowest values corresponding to Silver water silver-water (Ag + H2o) nanofluid and lowest for pure water. nanofluid.

The presence of nano-particles therefore aids the flow. (8) Axial pressure gradient decreases substantially with increas-

(2) Axial flow is decelerated with increasing Grashof number, ing Grashof number, Gr , throughout the length of the tube. (Gr) i.e. with greater thermal buoyancy force, with highest (9) Pressure rise increases with an increase in heat absorption velocity magnitudes achieved by Silver-water nanofluid. parameter (b), Grashof number (Gr) and also viscosity

(3) Axial velocity is significantly increased with an increase in parameter (a) in the peristaltic pumping region i.e. the range viscosity parameter (a) i.e. with decreasing viscosity of the -0.5 6 Q 6 0.5. However the opposite trend is observed in nanofluid, again with Silver water nanofluid achieving the the augmented pumping region i.e. in the range 0.5 < Q 6 2, highest acceleration.

RINP 503 ARTICLE IN PRESS No. of Pages 13, Model 5G

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A.B. Huda et al./Results in Physics xxx (2017) xxx-xxx 13

where pressure rise decreases with an increase in Grashof number (Gr), heat absorption parameter (b) and viscosity parameter (a)

(10) Wall shear stress is a maximum for Silver water nanofluid and is minimized for the pure water case.

(11) The quantity of trapped boluses is elevated with an increase in the value of heat source parameter (b) for CuO + H2O nanofluid and alsoAg + H2O nanofluid cases whereas it decreases for the Titanium oxide water nanofluid (Tio2 + H2O) case.

(12) The number of trapped boluses and also the size of the bolus both increase with an increase in viscosity parameter (a) i.e. with decreasing viscosity of the nanofluid, for CuO + H2O and Ag + H2O nanofluids whereas the quantity of trapped boluses is reduced whereas the magnitude of the bolus markedly increases, for the Tio2 + H2O nanofluid case, with an increase in viscosity parameter (a).

The present study has overall provided a reasonable assessment

of nano-particle influence on peristaltic nanofluid dynamics. It has

however ignored inclination of the tube [39] and also species diffusion [40] which may be simulated via the Buonjiornio model [38].

These aspects will be considered in future investigations.

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