Peristaltic flow of Eyring-Powell nanofluid under the action of an electromagnetic field

Engineering Science and Technology, an International Journal 22 (2019) 266–281

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Peristaltic flow of Eyring-Powell nanofluid under the action of an electromagnetic field B. Mallick, J.C. Misra ⇑,1,2,3 Centre for Healthcare Science and Technology, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, India

a r t i c l e

i n f o

Article history: Received 5 May 2018 Revised 26 November 2018 Accepted 1 December 2018 Available online 17 December 2018 Keywords: EMHD peristaltic transport Eyring-Powell nanofluid Debye-Hückel approximation Heat transfer Joule heating

a b s t r a c t Electro-kinetic peristaltic flow of Eyring-Powell nanofluid has been investigated in this paper. The flow is considered to take place in an asymmetric wavy micro-channel. The system is supposed to be acted on by an external magnetic field and to be exposed to Joule heating. Velocity and temperature of the fluid have been calculated by using Debye-Hückel approximation. Flow of the nanofluid is considered to take place under the combined action of an external transverse magnetic field and a horizontal static electric field. Considering low Reynolds number and using long wavelength approximation, variations of axial velocity, pressure gradient, wall shear stress and temperature profiles have been investigated both analytically and numerically by using efficient mathematical softwares. The peristaltic pumping characteristics and the trapping of fluid bolus have also been thoroughly examined in the light of electroosmosis. The study shows that even mild electroosmosis can cause higher pressure gradient in the axial direction. It bears the promise of important application to weak peristaltic transport modulation for EMHD nanofluid flows. The study further reveals that the occurrence of trapping can be controlled by suitably adjusting the magnitude of the electric field. The study also reports the thermal transport mechanism in the fluid, under the action of Joule heating and viscous energy dissipation. It shows that an increase in the volume fraction of the nanoparticles in the fluid can considerably enhance the momentum transport in the core region of the microchannel. Ó 2018 Karabuk University. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction Nanofluids are fluids that contain particles of nanometer size. These particles are termed as nanoparticles. According to Buongimo [1], nanofluids are colloidal suspensions of nanoparticles in some base fluid. Water, oil and ethylene glycol are common examples of base fluids. The nanoparticles used in preparation of nanofluids are usually made of metals, metallic oxides and carbides. Since nanofluids exhibit some distinctive properties, they have found a wide range of applications, especially in the heat transfer process [2]. Kakak and Pramuanjaroenkij [3] made an observation that the convective heat transfer coefficient and thermal conductivity of nanofluids are much higher than those of the corresponding base fluid. As the volume concentration of nanofluids rises, they exhibit some novel acoustical properties. Consider⇑ Corresponding author. E-mail address: [email protected] (J.C. Misra). 1 2 3

Peer review under responsibility of Karabuk University. Formerly, Pro Vice-Chancellor, SOA University, Bhubaneswar, India. Former Professor and Head, Department of Mathematics, IIT Kharagpur, India. Ex-President, Mathematical Sciences Section, Indian Science Congress, India.

ing Brownian motion of nanofluid buoyancy flow Doogonchi and Ganji [4] examined the thermoradiative heat transfer over a stretching sheet. Effect of electromagnetic field on heat and mass transfer for peristaltic flow has been examined by Bhatti et al. [5]. They considered a two phase fluid model and conclude that presence of a magnetic field provide resistance to flow. A detailed discussion on the effects of magnetic/electromagnetic fields on the flow of electrically conducting fluids has been made in a recent communication by Misra and Adhikary [6]. Recently Tripathi et al. [7] discussed the flow dynamics of a biological fluid through an asymmetric microchannel under the action of a static electric field. They made an observation that formation of trapping bolus is very sensitive to an external electric field. However, most of the existing scientific literature in this area dealt with flow and heat transfer in the absence of Joule heating phenomena. Peristaltic transport of fluids is a particular type of fluid transport that takes place due to the propagation of progressive waves of area contraction/expansion. This is a natural process for the transport of various physiological fluids. This phenomenon has been explained in detail by different researchers (Fung and Yih [8], Yin and Fung [9], Misra et al. [10,11]. Misra and Maiti [12]

https://doi.org/10.1016/j.jestch.2018.12.001 2215-0986/Ó 2018 Karabuk University. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

B. Mallick, J.C. Misra / Engineering Science and Technology, an International Journal 22 (2019) 266–281

267

Nomenclature Symbols Definitions u ¼ ðu0 ; t0 Þ Velocity vector ðx0 ; y0 Þ Cartesian coordinate of a point J Electric current density B Magnetic field vector E Electric field vector I Identity tensor S Cauchy stress tensor T Temperature of fluid T av Average temperature Constant height of the channel walls d1 ; d2 a1 ; a2 Amplitudes of the walls c Speed of wave trains t0 Time 0 0 Geometry of the channel walls h1 ; h2 p Pressure c0 A material parameter of the fluid Electrolyte concentration n0 e Charge of an electron z Valance of ions kB Boltzmann constant Radiative heat flux qr cp Specific heat k Thermal conductivity H k Mean absorption coefficient of nanofluid Temperature at lower wall T0 T1 Temperature at upper wall Re; Re Reynolds number of the base fluid, nanofluid Hartmann number of the base fluid, nanofluid Ha; Ha Nr; Nr Radiation parameter of the base fluid, nanofluid Pr; Pr Prandtl number of the base fluid, nanofluid Ec; Ec Eckert number of the base fluid, nanofluid Joule heating parameter of the base fluid, nanofluid Sp ; Sp

investigated the pumping characteristics during blood flow via peristaltic mechanism through vessels of the micro-circulatory system, having varying cross-section. However, in all the studies mentioned above, neither electroosmosis, nor mixture of nanoparticles and base fluids have been considered. When a fluid contains particles of micrometer/nanometre sizes, electro-kinetic phenomena are often observed, when the fluid is in motion. The occurrence of an electrical double layer (also simply called as double layer) of charges on the interface is considered to be the cause of several electrokinetic phenomena, including electroosmosis, or electroosmotic flow (cf. Hunter [13] and Lykelema [14]). The motion of a fluid takes place in the tangential direction, whenever a force, originated by an external electric field, or by the concentration gradient, or by the pressure gradient. Several researchers have dwelt upon various studies pertaining to the electrokinetic transport of different types of fluids (cf. [15,16] and Misra et al. [17,18]). In these communications, in addition to extensive discussions on various specific problems of electroosmosis, references to other previous studies on electro-osmotic flows are available. But in none of these papers, electroosmosis of nanofluids has been discussed. Neither there is any consideration of peristalsis. During the last several decades, researchers have studied various fluid mechanical problems, where the fluids do not obey Newton’s law of viscosity. These studies constitute a separate area of study called non-Newtonian fluid mechanics. However, it has been the experience of researchers that a single non-Newtonian model cannot be applied to deal with all sorts of non-Newtonian behaviour. In view of this, various authors have formulated different

A; B Eyring-Powell fluid parameters U HS Helmholtz-Smoluchowski velocity F Flow rate Greek symbols Definitions a Angle shown in Fig. 1 k Wave length q Density l Dynamic viscosity u Electrical potential U Viscous dissipation factor f Zeta potential b0 Material parameter s0 Extra stress tensor r Electrical conductivity  Dielectric constant 0 Permittivity of vacuum / Volume fraction of the nanoparticles j Debye-Hückel parameter b Mobility of the medium w Stream function d Wave number rH Stefan–Boltzmann constant Subscripts nf Properties of nanofluid f Properties of base fluid np Properties of nanoparticles 0 Terms associated with zeroth order system 1 Terms associated with first order system Superscript 0 Symbols representing dimensional quantities

non-Newtonian models to represent different types of nonNewtonian fluid flow behaviour. Different constitutive relations have been proposed, some of which have been derived empirically (e.g. power-law fluid model), while some others have been proposed on the basis of fundamental theories of the physics of fluids. For example, Eyring-Powell fluid model is a non-Newtonian model that has been derived from the kinetic theory of liquids. This model is quite suitable to describe the shear thinning properties of human blood and some other materials, like toothpaste and ketchup. Depending on the objectives of various studies, different nonNewtonian models have been used in the past by different researchers [19,20] to solve various problems related to flows of physiological fluids. By using Eyring-Powell fluid model, some studies have also been performed by several authors. Nadeem and Saleem [21] analysed mathematically the rotating flow of Eyring-Powell fluid. Malik et al. [22] analysed the problem of mixed convection magnetohydrodynamic flow of Eyring-Powell nanofluid on a sheet in stretching motion, while effect of thermal radiation on the flow of Eyring-Powell fluid was studied by Hayat et al. [23]. Eyring-Powell nanofluid flow on an exponentially stretching sheet was analysed by Hayat et al. [24]. Effect of external magnetic field on non-Newtonian blood flow in an axisymmetric cylindrical vessel was discussed by Ali et al. [25]. Since human blood possesses shear thinning properties, and since our final goal in the present study is to explore the electromagneto–hydrodynamic flow of blood, we have used the EyringPowell fluid model, because this model, as mentioned above, bears the potential to take care of the shear thinning properties of the

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fluid. It may be mentioned that in many situations, the system is subjected to electro-magnetic fields (rather than only electric or magnetic fields). For example, during MRI tests, the human body is exposed to electro-magnetic radiation. In such a situation, blood flow is governed by the principles of electro-magneto–hydrody namics (EMHD). Moreover, the fluid flow is supposed to be exposed to Joule heating. This consideration is in agreement with the fundamental law of electrodynamics that when electric current is passed through a conductor, heat is produced. This type of heating was first observed by James Joule and is referred to as Joule hating. Blood being an electrically conducting fluid, consideration of Joule heating is quite relevant to the presented study, particularly because electroosmosis/electroosmotic flow is one of the main concerns of the study. The study shows very clearly that for electroosmotically actuated peristaltic flow of Eyring-Powell fluids, mixed homogeneously with nano-particles of gold, the physical variables are highly influenced by an external electromagnetic field. Moreover, the volume fraction of the nanofluid has an important bearing on the flow physics of the fluid considered here. The study reveals that trapping phenomenon observed in the peristaltic flow can be avoided by adjusting the intensity of the electromagnetic field applied externally. The present study is expected to find important applications in a variety of physical problems involving flows of non-Newtonian fluids, both in the industrial sector and in the field of biomedical engineering and technology. 2. Mathematical modelling 2.1. The Eyring-Powell nanofluid model

0


 2p 0 0 h1 ¼ d1 þ ðx0  ct 0 Þtana þ a1 cos ðx  ct0 Þ k
 2p 0 0 ðx  ct 0 Þ þ /0 ; and h2 ¼ d2  ðx0  ct0 Þtana  a2 cos k

ð1Þ ð2Þ

in which a1 ; a2 are the wave amplitudes, k is the wave length, d1 ; d2 are the heights (assumed constants) of the upper and lower wall of the channel, c refers to the speed of the wave trains, /0 ð0 6 /0 6 pÞ denotes the phase difference between the two moving walls and a represents the angle as indicated in Fig. 1. Although two-phase nanofluid models are expected to yield more accurate results, it was observed by Hadad et al. [27] that two-phase and single-phase models are closely identical, so far as the hydrodynamical fields are concerned. Moreover, the governing equations for the study of nanofluid flows, based on a singlephase model are easy to understand and the analysis is less complicated. While reviewing the state-of-the art of single-phase and two-phase nanofluid flows and heat transfer, Kakak and Pramuanjaroenkij [28] suggested that some terms of single-phase governing equations can be adjusted suitably in order to improve the accuracy of the results. In view of these remarks, the present formulation presented below is based on the consideration of a singlephase model. For an incompressible electrically conducting Erying-Powell nanofluid [29,30], the governing equations may be written as

r  u ¼ 0;

We denote by ðx0 ; y0 Þ cartesian coordinates of a representative particle of the fluid mass, with O as the origin of the coordinate system, x0 -axis being along the central line and y0 -axis along the transverse direction (cf. Fig. 1. We consider here the peristaltic flow of a viscous, electrically conducting Eyring-Powell nanofluid, considering long wavelength approximation and low Reynolds number, through a non-uniform channel having height d1 þ d2 . The flow geometry is sketched in Fig. 1. The sinusoidal waves are supposed to propagate in the form of wave trains travelling with a constant

0

velocity c along the channel walls, viz. y0 ¼ h1 and y0 ¼ h2 . Geometrical configurations of these two walls can be put mathematically as (cf. [26])

Du qnf 0 ¼ r  S þ F B ; Dt   DT qcp nf 0 ¼ knf r2 T  r  qr þ lnf U þ Sf ; Dt

ð3Þ ð4Þ ð5Þ

  where the definitions of qnf ; lnf ; cp nf and knf are given below in

Eq. (9). DtD0  @t@ 0 þ u  r represents the material derivative, u is the velocity vector, q the density, cp the specific heat capacity, k the thermal conductivity, U the viscous dissipation factor, T is the

Fig. 1. Physical sketch of the problem.

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temperature and p is the pressure. Using Ohm’s law, the rate of external volumetric heat generation due to Joule heating can be expressed as Sf ¼ ðJ  JÞ=rnf , where rnf is the effective electrical conductivity of the nanofluid [31]. If S denotes the Cauchy stress tensor for the Eyring-Powell nanofluid (cf. [29]), we can write

S ¼ pI þ s0 ;

with s0 ¼




lnf

 _ 0  1 1 c P; þ 0 0 sinh c0 b c_

net external body force is the combined interaction between the axial electrical force together with the superimposed transverse magnetic field, so that F B can be expressed as [32]

F B ¼ qe E þ J  B;

ð7Þ

where the electric current density J ¼ rnf ðu  BÞ of the nanofluid is given by Ohm’s law. Here E ¼ ðEx0 ; 0; 0Þ is the intensity of the alter  nating current in the x-direction, B ¼ 0; By0 ; 0 the magnetic field 0 having strength By0 in y -direction and rnf is the electrical conductivity of the nanofluid. The influence of electromagnetic induction for the present problem is considered to be ignorably small. Considering Rosseland approximation (c.f [33]), the heat flux qr may be H

rH is the Stefan–Boltz-

mann constant and k is the absorption coefficient of the nanofluid. Following Raptis [34], fluid-phase temperature differences during the flow are taken to be very small, so that we may express T 4 as a linear combination of temperature. Let T 1 and T 0 denote the temperature at the upper and lower walls of the microchannel respectively. Using Taylor’s series expansion about ðT 1  T 0 Þ and 4

neglecting second and higher order terms, we can express T as T 4  4ðT 1  T 0 Þ3 T  3ðT 1  T 0 Þ4 . Therefore, divergence of the radiative heat flux vector qr is given by

$  qr ¼ 

16rH ðT 1  T 0 Þ3 H

3k

r2 T:

ð8Þ

Influence of the nanoparticles present in the fluid is accounted for by considering the following expressions of the involved material parameters (cf. [35]):

lnf ¼ ð1/Þ2:5 ;

qnf ¼ ð1  /Þqf þ /qnp ;       cp nf ¼ ð1  /Þ cp f þ / cp np ;

¼1þ

rs 3 r 1 / f



;

rs rs rf þ2  rf 1 /

and

knf ¼

ð9Þ knp þ2kf 2/ðkf knp Þ knp þ2kf þ/ðkf knp Þ



kf :

In Eq. (9), / represents the volume fraction of the nanoparticles of the fluid. The subscript nf ; f and np denote quantities referring to the nanofluid, base fluid and nanoparticles respectively. In the present study, human blood has been considered as the base fluid and very small spherical gold ð AuÞ particles as the nanoparticles. Values of the physical constants of the base fluid and the nanoparticles are presented in Table 1. 2.2. Charge distribution In terms of the net charge density qe , dielectric constant  and permittivity of the vacuum 0 Poission-Boltzmann equation for the electrical potential distribution may be expressed (cf. [37]) as

Base fluid f (Human Blood)

Nanoparticlenp Gold (Au)

ð1:08  0:02Þ  103

19:4  103 129.1

3:5  0:8Þ  10 0:53  0:11 0:60  0:1

kðW=mKÞ rðS=mÞ

3

qe ¼ 0 r2 u0 :

317 4:10  107

ð10Þ

In the case of flow over a microchannel, this equation governs the distribution of ions, when the axial gradient of ionic concentration is negligibly small. In the case of a symmetric electrolyte, for the binary fluid consisting of two different kinds of equal and opposite charges, say nþ and n , the net charge may be expressed as



qe ¼ 2n0 ezsinh

 ezu0 ; kB T av

ð11Þ

which follows from Boltzmann distribution [37]. It may be noted that when the electrical potential is small in comparison to the temperature field potential, i:e:jezu0 j < jkB T av j, one may apply DebyeHückel approximation

sinhðezu0 =kB T av Þ  ðezu0 =kB T av Þ which transforms the net charge densities of ion in a unit volume of fluid as qe ¼ 2n0 e2 z2 u0 =kB T av . The approximation can be applied,

when the concentration of the electrolyte is low, less than 103 mol/l. Further discussion on this is available in [37]. Inserting the above expression of qe in Eq. (10), we obtain 2

d u0

0

02

dy

¼

2n0 e2 z2 0 u; k B T av

ð12Þ

z being the valence of ions, e the charge of an electron, n0 the average concentration of the electrolyte, kB the Boltzmann constant and T av stands for the average temperature. Solving Eq. (12), we can determine the potential u0 for the electrical double layer (EDL). Zeta potential of the EDL is taken to be constant here. Debye-Hückel approximation are known to hold good for dilute solutions

n0 < 103 , when the applied voltage across the EDL is much less than the temperature voltage, kB Te av . Let us now introduce the normalized variables

u¼

lf

rnf rf



q kg=m3

ð6Þ

the nanofluid, b0 and c0 denote the material parameters. Assuming that the fifth and higher order terms in the Taylor’s series expansion



_0 1 c_0 1 c_0 c_03 are very small, we may write sinh  cc0  6c of sinh 03 . The c0 c0

H

Physical constants

cp ðJ=kgKÞ

in which P ¼ ru þ ðruÞT is the Rivlin Erickson tensor, I is the idenqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tity tensor and c_0 ¼ 12 tr ðPÞ2 . Here, lnf is the dynamic viscosity of

expressed as qr ¼ 4rH rT 4 =3k , in which

Table 1 Values of parameters for base fluid and nano-particle (Refs. [35,36])

u0 f

;

y¼

y0 ; d1

0

h1 ¼

h1 ; d1

0

h2 ¼

h2 : d1

ð13Þ

Considering these new variables, Eq. (12) assumes the form 2

d u dy

2

¼ j2 u:

ð14Þ

This equation is satisfied by the non-dimensional electroosmotic potential function, defined in terms of the zeta potential f of the medium. In Eq. (14), jð¼ d1 =kD Þ denotes the non-dimensional Debye-Hückel parameter, which represents the ratio between half the height of the channel and the Debye length kD , defined by  1=2 kD ¼ 0 kB T av =2n0 e2 z2 . It may mentioned here that although the term ‘zeta potential’ is usually used for the electrokinetic potential in studies pertaining to colloidal dispersions, it has applications in many other studies that involve dispersive systems, including those of micro-pharmaceuticals. Solving Eq. (14) with the assumption of constant zeta potential ðu0 ¼ fÞ at both the walls of the micro-channel, we obtain

u¼

cosh j2 ½2y  ðh1 þ h2 Þ : cosh j2 ðh1  h2 Þ

ð15Þ

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2.3. Governing equations In the two-dimensional rectangular Cartesian coordinate system, the Eqs. (3)–(5) may be written as: 0

0

@u @ t þ ¼ 0; @x0 @y0  0  0 @ s0xy @u @u0 @p0 @ s0 0 @u qnf þ t0 0 ¼  0 þ xx0 þ 0 þ qe Ex0  rnf B2y0 u0 ; 0 þu 0 @t @x @y @x @x @y  0  0 0 0 0 @ s @ s0yy @t @ t @ t @p yx 0 qnf þ t0 0 ¼  0 þ 0 þ 0 ; 0 þu 0 @t @x @y @y @x @y !  0 0 2   @T 0 @T @T @ T @2T 0 0 qcp nf þ u þ t þ ¼ k nf @x02 @y02 @t0 @x0 @y0    0  2

@qr @qr @u @ t0  þ þ þ rnf E2x þ B20 u02 ; þ lnf @x0 @y0 @y0 @x0

ð16Þ

h

_0

ð17Þ ð18Þ

ð19Þ

i

0 c_0 3 @u0  6c þ @@xt0 ; 03 @y0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð20Þ h _0

i 2  0 2 @u0 @ t0 2 _03 0 s0yy ¼ 2 lnf þ b01c_0 cc0  6cc 03 @@yt00 ; c_0 ¼ 2 @u þ @y0 þ @x0 þ 2 @@yt0 ; @x0 c

c0

_03

 6cc 03

i

@u0 ; @x0

h

s0xy ¼ s0yx ¼ lnf þ b01c_0

Ec ¼

_0 c

c0

x¼ p¼

x0 ct0 u0 ; t¼ ; u¼ ; k k c 2 d1 p0

c lf k

; c_ ¼

t0 d d s0 t¼ ; d¼ 1; s¼ 1 ; c k c lf

d1 c_0 T  T0 and h ¼ c T1  T0 0

ð21Þ

Eqs. (16)–(20) are now transformed to:

@u @ t þ ¼ 0; @x @y
 @u @u @t @p @ sxx @ sxy þu þt Re d ¼ þd þ þ bj2 u  Ha2 u; @t @x @x @y @x @y
 @t @t @t @p @ syx @ syy Re d2 þu þt ¼  þ d2 þd @y @t @x @y @x @y ! 
  2 1 þ Nr @h @h @h @ h @2h Re d þu þt ¼ d2 2 þ 2 @t @x @y @x @y Pr  2 2 @u @t Sp þ Hau þd þ þ Ec @y @x Pr

ð22Þ ð23Þ ð24Þ

h

i

@u ; @x

h

sxy ¼ syx ¼ 1 þ B  A3 c_ 2

i

ð25Þ



@u þ d @@xt @y

; rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i

2ffi 2  2 @u @t @t syy ¼ 2 1 þ B  A3 c_ 2 @@yt ; and c_ ¼ 2d2 @u þ þ d þ 2 : @x @y @x @y and Re ¼

qnf qf

cd q

 l Re=l in which Re ¼ l1 f is the Reynolds number of nf f f

the base fluid, d ¼ dk1 is the wave number, B ¼ l B=l and A ¼ l A=l , nf f nf f 2

with B ¼ b0 c0 dc1 l and A ¼ 2cBc02 d2 represent the Eyring-Powell fluid f

1

(base fluid) parameters. qffiffiffiffiffiffiffiffiffiffi qffiffiffiffi r =r r Ha ¼ Ha lnf =lf in which Ha ¼ lf d1 By0 is the Hartmann numnf

f

f

ber corresponding to the base fluid,

j j ¼ pffiffiffiffiffiffiffiffiffiffi and b ¼ UcHS reprel =l



 fE sents the mobility of the medium, where U HS ¼  0l x0 nf

f

f

corresponds to the Helmholtz-Smoluchowski velocity, Nr ¼ k Nr=k in which Nr ¼ nf

f

the base fluid,

16 ðDT Þ3 3k kf

r

@w @y

is the radiation parameter of

where Sp ¼

t ¼ d

and

rf E2x d21 kf DT

c2

ðcp Þf DT

is the Eckert number of the

is a non-dimensional quantity rep-

@w : @x

ð26Þ

Consideration of long wavelength approximation and low effective Reynolds number transformations and use of Eqns. (23)–(25) yield the following set of partial differential equations:

20 3 !2 1 @p @ 4@ A @ 2 w A @ 2 w5 @w þ bj2 u  Ha2 ¼ 1þB ; @x @y 3 @y2 @y2 @y @p ¼0 @y 

and

ð27Þ ð28Þ

1 þ Nr Pr



!

!2  2 @2h @2w Sp Ha @w þ Ec þ ¼0 þ @y2 @y2 Pr Pr @y

ð29Þ

Combining Eqs. (27) and (28), we have

2 !3 3   @4w A @2 @2w 5 1þB  4 @y4 3 @y2 @y2  Ha2

where

sxx ¼ 2d 1 þ B  A3 c_ 2

Sp rnf =rf knf =kf

in which Ec ¼

resenting Joule heating in case of the base fluid. DT ¼ T 1  T 0 denotes the temperature difference, T 1 and T 0 being respectively the temperatures of the upper and lower walls of the micro-channel. The present analysis is restricted to long wavelength approximation ðd 1Þ, where the effective Reynolds number is small. In case of physiological fluid flows, particularly for blood flows in arteries, consideration of long wave length is meaningful. The justification of this assertion is that in order to avoid bending rigidity of the arterial wall and effects of non-linear convective acceleration terms, the wave length has to be long, while the wave amplitude should be comparatively small. Similar is the situation for the flow of urine through the ureter. Now on introducing nondimensional stream function w, we write

u¼ in which u0 ; t0 represent the velocity components in the longitude and transverse directions. In terms of the dimensionless

Ec

ðcp Þnf =ðcp Þf base fluid, Sp ¼

in which

s0xx ¼ 2 lnf þ b10 c_0

lf ðcp Þ ðcp Þ ðlÞ Pr ¼ ðlÞnf  c nf  k Pr=k , with Pr ¼ k f is the Prandtl number of f f ð p Þf nf f the base fluid,

sinh j2 ½2y  ðh1 þ h2 Þ @2w þ bjj2 ¼0 2 @y cosh j2 ðh1  h2 Þ

ð30Þ

Writing the boundary conditions of the problem in terms of w, we have

F @w H ¼ 0 at y ¼ h1 ¼ 1 þ k ðx  t Þ þ acos½2pðx  t Þ; w¼ ; 2 @y ð31Þ F @w H w¼ ; ¼ 0 at y ¼ h2 ¼ d  k ðx  tÞ  bcos½2pðx  t Þ þ /0 ; 2 @y along with the thermal boundary conditions : h ¼ 0 at y ¼ h1 ;

h ¼ 1 at y ¼ h2 : H

a is the parameter that In Eq. (31), a ¼ da11 ; b ¼ bd11 ; d ¼ dd12 ; and k ¼ ktan d1

depicts the non-uniform geometry of the channel. For hemodynamical studies, time-dependent exponential decay of the dimensionless flow rate was experimentally observed by Kikuchi [38], who also analysed the structural independence of the micro-channel on the flow rate. Let us consider the dimensionless flow rate as

F ¼ exp At . Expanding F in Taylor’s series and considering that the Eyring-Powell fluid parameter A is small, we write

F ¼ 1  At:

ð32Þ

Thus we can take F 0 ¼ 1 and F 1 ¼ t in the perturbation expansion.

B. Mallick, J.C. Misra / Engineering Science and Technology, an International Journal 22 (2019) 266–281

2

3. Method of solution

w1 ¼ A5 þ A6 y þ

Due to non-linearity of Eq. (30), a perturbation technique has been used to obtain the solution, by considering Eyring-Powell fluid parameter A as the perturbation parameter (details are available in [39,40]), which is taken to be small.

The zeroth order system is given by



and

1 þ Nr



@ 2 h0 @y2

Pr

!

@ 2 w0 þ Ec @y2

!2 þ

ð33Þ

 2 Sp Ha @w0 þ ¼ 0; ð34Þ Pr Pr @y

with the boundary conditions:

w0 ¼ F20 ;

@w0 @y

¼ 0 at y ¼ h1 ;

h0 ¼ 0 at y ¼ h1 ;

w0 ¼  F20 ;

@w0 @y

ð35Þ

The solutions of Eqs. (33) and (34) subject to the respective boundary conditions given in (35) are found in the form

2

dydy and G 1 ð yÞ ¼

ð42Þ

ffiffiffiffiffiffiffiffiffiffi y þ N 9 exp þN 6 exp pHay ð1þBÞ " ! # "

Ha pffiffiffiffiffiffiffiffiffiffi þ 2j y ð1þBÞ ! #

"

Ha þN 12 exp  pffiffiffiffiffiffiffiffiffiffi þ 2j y þ N 10 exp ð1þBÞ " ! # "

þN 13 exp

! #

p2Ha ffiffiffiffiffiffiffiffiffiffi þ j y þ N 10 exp  p2Ha ffiffiffiffiffiffiffiffiffiffi þ j y ð1þBÞ ð1þBÞ

ffiffiffiffiffiffiffiffiffiffi y þ N 14 exp j  p2Ha ð1þBÞ

"

! # ffiffiffiffiffiffiffiffiffiffi  j y p2Ha ð1þBÞ ! #

Ha 2j  pffiffiffiffiffiffiffiffiffiffi y ð1þBÞ ! #

þN 15 exp

RR

@ 2 w0 @ 2 w1 @y2 @y2



Ha pffiffiffiffiffiffiffiffiffiffi  2j y ; ð1þBÞ

RR @w0 @w1 dydy and G 3 ð yÞ ¼ dydy: @y @y

4. Validation of the present model

ð37Þ @ 2 w0 @y2

Pr

#

G 3 ð yÞ

Expressions of the coefficients have been included in the Appendix.

3

bj2 sinh j2 ½2y  ðh1 þ h2 Þ þ  ð36Þ   j Ha2  j2 1 þ B cosh j2 ðh1  h2 Þ " # Pr Sp 2 Ha A3 þ A4 y  y  EcG1 ð yÞ  G1 ð yÞ ; and h0 ¼ 1 þ Nr 2Pr Pr

RR

2Ha

! ! h

i 2 ffiffiffiffiffiffiffiffiffiffi þ N 2 exp p3Hay ffiffiffiffiffiffiffiffiffiffi þ N 3 Sinh 3j y  h1 þh G2 ð yÞ ¼ N1 exp p3Hay 2 ð1þBÞ ð1þBÞ ! h

i Hay 2 ffiffiffiffiffiffiffiffiffiffi p þ N exp þN4 Sinh j y  h1 þh y 5 2 ð1þBÞ ! " ! # "

G3 ð yÞ ¼

ðL1 L6  L4 L3 Þ Ha 6 7 exp4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y5 ðL1 L5  L4 L2 Þ 1þB

where G1 ð yÞ ¼

A7 þ A8 y  2EcG3 ð yÞ 

3

L3 L5  L2 L6 6 Ha 7 w0 ¼ A1 þ A2 y þ exp4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y5 L1 L5  L2 L4 1þB þ

1 þ Nr

"

þN 11 exp

¼ 0 at y ¼ h2 ;

h0 ¼ 1 at y ¼ h2 :

2

Pr

ð41Þ

in which

 @ 4 w0 sinh j2 ½2y  ðH1 þ H2 Þ @ 2 w0 1þB  Ha2 þ bjj2 ¼0 4 2 @y @y cosh j2 ðH1  H2 Þ 

  L5 L6  L2 L7 6 Ha 7 exp4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   y5 L1 L5  L2 L4 1þB

2 3   L4 L6  L1 L7 Ha 6 7 1 exp4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y5 þ G2 ð yÞ; þ 3 L2 L4  L1 L5 1þB h1 ¼

3.1. Zeroth order system and its solution

271

3

RR @w0 2 @y

dydy. The derived

expressions of all the coefficients of Eqs. (36) and (37) are presented in the Appendix.

The numerical results for the different physical variables involved in the present study have been computed for different sets of values of the involved parameters. The results have been presented graphically in the succeeding section and an elaborate discussion has been made on the basis of the computed results. In this section, we make an attempt to validate the results computed for the velocity of the nanofluid on the basis of the theoretical analysis performed in the preceding section, by comparing them with the results reported earlier by Hina [20]. Both the sets of results have been presented graphically in Fig. 2, after bringing both the studies to a common platform. For this purpose, the

3.2. First order system and its solution



2 !3 3 2  @ 4 w1 1 @ 2 @ 2 w0 5  Ha2 @ w1 ¼ 0 1þB  4 2 4 2 3 @y @y @y @y2

! ! !   1 þ Nr @ 2 h1 @ 2 w0 @ 2 w1 and þ 2Ec @y2 @y2 @y2 Pr    2Ha @w0 @w1 ¼ 0; þ @y @y Pr

ð38Þ

ð39Þ

with boundary conditions:

w1 ¼ F21 ;

@w1 @y

¼ 0 at y ¼ h1 ;

h1 ¼ 0 at y ¼ h1 ;

w1 ¼  F21 ;

h1 ¼ 0 at y ¼ h2 :

@w1 @y

¼ 0 at y ¼ h2 ;

ð40Þ

The solution of the first order system subject to the conditions (40) has been obtained in the form

Fig. 2. Comparison of results (velocity) between present study and those reported earlier by Hina [25].

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results of the present study presented in Fig. 2 have been computed by considering a very small value of the Debye-Hückel

parameter j ¼ 1  105 , where by the electroosmotic effect is

problem. The results presented have also been adequately discussed in the sub-sections that follow.

negligibly small. Moreover, the angle a (cf. Fig. 1) and the phase difference between the walls were set equal to zero. These considerations make the channel symmetric. It may be mentioned that the results reported by Hina [20] are valid for the peristaltic flow of Eyring-Powell fluid on a symmetric channel. Fig. 2 exhibits a good conformity of the results of the present model with those reported by Hina [20]. This observation may be considered as a satisfactory validation of the present study.

5.1. Velocity distribution

5. Results and discussion Table 2 gives the values of various flow variables associated with the present study on the electro-magneto–hydrodynamic peristaltic flow of Eyring-Powell nanofluid exposed to Joule heating. The computational results presented here correspond to the values of the dimensional variables and the values of the physical constants presented in Table 1. As mentioned earlier, for the purpose of numerical simulations, computational work has been carried out along with the following range of parametric values (estimated on the basis of dimensional values mentioned in Table 1 and Table 2). The computational work has been performed in a PC with Intel CORE i5 processor and 4 GB RAM. For the numerical simulations, we used the software MATHEMATICA in WINDOWS 10 operating system with 3 min per computation CPU time.

0 6 Ha 6 6;

0 6 j 6 20;

0:01 6 A 6 0:04;

0 6 / 6 0:5;

0 6 Nr 6 3:0;

0 6 B 6 1:5;

0 6 Sp 6 0:6;

0 6 Ec 6 0:4:

Further, for the mobility of the medium, we have taken b ¼ 2 throughout the computational procedure. Considering the dynamic viscosity of blood as lf ¼ 3:2  103 Pa s and using it in the for  mula Pr ¼ lf cp f =kf , we found that the Prandtl number for the blood is 21. Variations in velocity distribution, pumping characteristics, trapping phenomena and isotherm patterns for the peristaltic transport of the fluid, as well as the wall shear stress and temperature distribution were thoroughly investigated. The results have been presented graphically in the sequel. Variations in streamlines and isotherm patterns due to changes in the (a) strength of the applied magnetic field, (b) nanoparticle volume fraction and (c) Debye-Hückel parameter have been presented in an elegant manner. Moreover, change that takes place in the streamline pattern at different points of time has also been illustrated. All the figures depict very clearly the physics of the

Table 2 Typical values of physical parameters [20,26,41] Parameter

Symbol

Value

Unit

Half channel height Amplitude of the channel Angle of inclination Charge of electron

d1 ; d2 a1 ; a2

10–50 1–5 0  p2

lm lm

e

1:6  1019

radian C

Boltzmann constant

kB

Ionic concentration Average absolute temperature Ionic valency Electric potential Strength of characteristics electric field Magnetic field strength Stefan–Boltzmann constant

n0 T av z f Ex0 B0

1:38  1023 1 300 1 0.025 0–20 0–8

mol=m3 K – V KV =m TðTeslaÞ

rH

5:67  108

Absorption coefficient of nanofluid

k

a

H

101

J/K

W=m2 K4 m1

Variation in the distribution of velocity in the axial direction with change in magnetic field strength in the case of electro-mag neto–hydrodynamic flow is illustrated in Fig. 3(a). It is observed that the velocity distribution is considerably influenced by the applied magnetic field. This may be attributed to the fact that due to the existence of magnetic field, a Lorentz force of magnitude jJ  Bj per unit volume is generated in the flowing mass of the fluid. The plots presented in Fig. 3(a) show that as the strength of the magnetic field increase, the velocity profile becomes more and more flat at the centre of the micro-channel. This observation may be physically interpreted through the assertion that as the magnetic field strength increases, there will be a greater suppressive effect on the flow field, whereby the velocity profile becomes more and more flat in the neighbourhood of the centre of the micro-channel, because magnetic field intensity is proportional to the axial velocity ðu0 Þ. The effect is more pronounced at three locations, viz. (i) Stern layer, (ii) Diffuse layer and (iii) the central position of the micro-channel. Moreover, Fig. 3(a) reveals that the fluid velocity enhances with higher actuation of the magnetic field in the Stern layer and in the neighbourhood of the centre of the micro-channel, while in the diffuse layer, the velocity of the nanofluid can be reduced by increasing the intensity of the magnetic field. Fig. 3(b) depicts the effect of the Debye-Hückel constant ðjÞ on the velocity distribution. The plots of this figure show that for higher values of j, the velocity profile is much different from the case of lower values of j. For values of j P 10, the velocity attains its maximum near the boundary of the channel and minimum in the central region. It may be argued that such a behaviour of fluid flow owes its origin to the complex interaction of the electromagnetic body forces. Electrical force acting on the fluid mass is usually stronger within the electrical double layer (EDL). Thus for a given magnetic field, the fluid velocity is expected to be much higher in the EDL than in the centre of the channel. Since the DebyeHückel parameter is inversely proportional to the Debye length, for a given height ðd1 Þ of the micro-channel, a double layer with less density (characterized by smaller value of kD ), corresponds to a higher value of j. Thus when j is increased, the EDL is gradually shifted towards the boundary of the microchannel. As a consequence, mobile charges travel along the vicinity of the channel walls. One may observe from Fig. 3(c) that with an increase in nanoparticle volume fraction, there is a reduction in the velocity of the nanofluid in the EDL, but in the vicinity of the centre of the micro-channel, it is enhanced. It may be further noted from Fig. 3(d) that the fluid parameter B has a similar influence on the flow of the nanofluid. 5.2. Pumping characteristics of peristaltic flow Fig. 4(a) and (b) give us an idea of the change in axial pressure gradient along the length of the micro-channel for different values of the magnetic field strength and different values of the DebyeHückel parameter j. Fig. 4(a) shows that when the magnetic field intensity is enhanced, the pressure gradient increases and that it is maximum at some location in the narrower portion of the microchannel, while the pressure gradient is less in the wider portion. We may further observe that with gradual increase in the magnetic field strength, greater pressure is required by a given volume of the fluid to pass through the tapered portion of the micro-channel fx : ð0:5 6 x 6 0:7Þ [ ð1:5 6 x 6 1:7Þg. However, when the inten-

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273



Fig. 3. Velocity distribution (a) for different Ha, when x = 0.1, t = 0.2, k ¼ 0:6, a = 0.8, b = 0.6, /0 ¼ p=3, d = 0.8, j ¼ 10:0, b ¼ 2:0, B = 0.3, A = 0.02, / ¼ 0:02, (b) for different j, when x = 0.1, t = 0.2, k ¼ 0:6, a = 0.8, b = 0.6, /0 ¼ p=3, d = 0.8, Ha = 4.0, b ¼ 2:0, B = 0.3, A = 0.02, / ¼ 0:02, (c) For different /, when x = 0.1, t = 0.2, k ¼ 0:6, a = 0.8, b = 0.6, /0 ¼ p=3, d = 0.8, Ha = 4.0, j ¼ 10:0; b ¼ 2:0, B = 0.3, A = 0.02, (d) for different B, when x = 0.1, t = 0.2, k ¼ 0:6, a = 0.8, b = 0.6, /0 ¼ p=3, d = 0.8, Ha = 4.0, j ¼ 10:0; b ¼ 2:0, A = 0.02, / ¼ 0:02.

sity of the magnetic field is lower ðHa 6 1Þ, the axial pressure gradient vanishes at two different locations on the axis of the channel. It may be noted that the trend of change in pressure gradient with variation in j is altogether different. These observations imply that when the intensity of the magnetic field is low, a small amount of pressure is required to transport a given volume of the liquid and that the charged particles move in the regions adjacent to the channel walls. This phenomenon may be considered responsible for maintaining a nanoscopic electrical double layer (EDL).

Fig. 4(c) shows the nature of change in pressure gradient distribution, when the nanoparticle volume fraction changes. From the plots presented in this figure, one may observe that the pressure gradient can be controlled by adjusting the nanoparticle volume fraction. One can have an idea of the nature of the distribution of the pressure gradient for different values of the fluid parameter B. It may be observed that as the value of B increases, the pressure gradient is enhanced and that it attains its maximum in the region fx : x 2 ð0:5; 0:7Þg.

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Fig. 4. Variation of dp (a) for different Ha, when y = 0.0, t = 0.2, k ¼ 0:6, a = 0.8, b = 0.6, /0 ¼ p=3; d ¼ 0:8; j ¼ 10:0, b ¼ 1:0, B = 0.3, A = 0.02, / ¼ 0:01, (b) for different j, when dx y = 0.0, t = 0.2, k ¼ 0:6, a = 0.8, b = 0.6, /0 ¼ p=3, d = 0.8, Ha = 4.0, b ¼ 1:0, B = 0.1, A = 0.02, / ¼ 0:01, (c) for different /, when y = 0.0, t = 0.2, k ¼ 0:6, a = 0.8, b = 0.6, /0 ¼ p=3, d = 0.8, j ¼ 10:0, Ha = 4.0, b ¼ 1:0, B = 0.1, A = 0.02, and (d) for different B, when y = 0.0, t = 0.2, k ¼ 0:6, a = 0.8, b = 0.6, /0 ¼ p=3, d = 0.8, j ¼ 10:0, Ha = 4.0, b ¼ 1:0, A = 0.02, / ¼ 0:01.

5.3. Distribution of shear stresses We know that a real fluid, while flowing on a channel, develops shear stress on the walls of the chanel. It has been observed that oscillating shear stress is directly involved in the impairment of atherosclerotic lesions in pathological states of arteries [42]. Keeping this in view, we have investigated here the variation of the amplitude of shear stress at the upper wall of the nonuniform micro-channel under the present consideration. Distribution of shear stress has been examined for different values of the physical constants. The results have been presented in Fig. 5(a)–(d). The impact of Eyring-Powell fluid parameters is depicted in Fig. 5(a)–(b). From these figures, it may be observed that the shear stress/skin friction increases with an increase in the value of the fluid parameter A. Interpreted physically, we can say that the higher values of the parameter A provide more resistance to the flow dynamics. On the other hand, for the parameter B, a reverse trend is noted. Thus the parameter B shows refusal like behaviour with the fluid motion throughout the microchannel. The physical implication of this observation is that since B is inversely

proportional to the base fluid viscosity

lf for fixed values of the

material parameters b0 and c0 , as the viscosity of the base fluid diminishes the shear stress/skin friction during the fluid motion is reduced. Fig. 5(c) shows that with a rise in Debye-Hückel parameter j, the amplitude of the wall shear stress increases. It is evident from Fig. 5(d) that for a considerably strong magnetic field ðHa > 1Þ, the magnitude of the shear stress increases with an increase in the intensity of the externally applied magnetic field. Fig. 5 also reveals that the shear stress is minimum at x ¼ 0:5, irrespective of the magnetic field intensity as well as the values of A; B and j. Further, it is important to observe that the wall shear stress is inversely proportional to the width of the channel. 5.4. Distribution of temperature in the nanofluid Fig. 6(a)–(d) depict the variation of temperature distribution in the nanofluid for various values of the parameters /; Nr; Sp (Joule heating parameter) and Ec (Eckert number), when nanoparticles of gold are mixed with blood. Fig. 6(a) represents the influence of nanoparticles volume fraction, when there is equilibrium and there

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275



Fig. 5. Shear stress distribution at upper wall of the channel (a) for different A, when x = 0.1, t = 0.2, k ¼ 0:6, a = 0.8, b = 0.6, /0 ¼ p=3, d = 0.8, Ha ¼ 4:0; j ¼ 10:0, b ¼ 2:0, B = 0.3, A = 0.02, / ¼ 0:02, (b) for different B, when x = 0.1, t = 0.2, k ¼ 0:6, a = 0.8, b = 0.6, /0 ¼ p=3, d = 0.8, Ha = 4.0, j ¼ 10:0; b ¼ 2:0, A = 0.02, / ¼ 0:02, (c) for different j, when x = 0.1, t = 0.2, k ¼ 0:6, a = 0.8, b = 0.6, /0 ¼ p=3, d = 0.8, Ha = 4.0, b ¼ 2:0, B = 0.3, A = 0.02, / ¼ 0:01, and (d) for different Ha, when x = 0.1, t = 0.2, k ¼ 0:6, a = 0.8, b = 0.6, /0 ¼ p=3, d = 0.8, j ¼ 10:0; b ¼ 2:0, A = 0.02, B = 0.3, / ¼ 0:02.

is no slip between the nanoparticles and blood (base fluid). From different plots of Fig. 6(a), it is evident that a rise in the volume fraction of the nanofluid is accompanied by an expansion of the temperature profile. One can have an idea of the influence of thermal radiation on temperature distribution in the nanofluid from Fig. 6(b). Different plots of this figure indicate that with change in thermal radiation, there occurs reduction in the influence of temperature. It is worthwhile to note that with a rise in thermal radiation, the thermal boundary layer thickness reduces. Joule heating is known to be an inexorable phenomenon, when a conducting medium is subject to the influence of an external electric field. Fig. 6(c) illustrates the impact of Joule heating on temperature distribution. The different plots of this figure show that temperature of the nano-fluidic mass rises, as the Joule heating parameter increases. The influence of Eckert number on the temperature distribution in the nanofluid mass under consideration is depicted in Fig. 6(d). One may note that the temperature of the nanofluid is increased significantly owing to viscous dissipation and also that as the viscous dissipation effect increases, there occurs a rise in the boundary layer thickness. Table 3 exhibits the variations that take place in Nusselt number and heat transfer coefficient at both the walls of the channel, due to alternation of different parameters of the present study, viz. Pr; Sp ; Ec and /. The table shows that when thermal radiation increases at both the walls of the microchannel, the Nusselt

number is reduced and the heat transfer diminishes. However, when the Prandtl number/Joule heating parameter/Eckert number/ nano-fluid volume fraction is increased, the tabulated values show that both the Nusselt number and the magnitude of heat transfer increase. It may further be noted that the influence of Eckert number on both the physical quantities is greater. 5.5. Trapping phenomenon in the peristaltic motion of the nanofluid The phenomenon of trapping is very important in studies related to peristaltic transport of a fluid. It manifests in the form of a bolous (a fluid mass) that circulates internally, which is surrounded by streamlines that are of closed form (cf. [10–15]). The propagating peristaltic waves push the bolus ahead and it moves with the same velocity as that of wave propagation of peristalsis. Due to the importance of the trapping phenomenon in peristaltic motion of a fluid, we have made an endeavour to depict in Figs. 7–9 the nature of the streamlines, corresponding to different values of the various parameters involved in the present theoretical study. Figs. 7(a)–(b) give us an idea of the effect of the magnetic environment on bolus formation. These figures show that the bolus size increases, when the magnetic field intensity is enhanced. Fig. 8 presents the variation in the streamline patterns, with change in the value of j. A comparison between Fig. 8(a) and (b) reveals that by changing j, it is possible to bring about a major

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Fig. 6. Thermal distribution (a) for different /, when x = 0.0, t = 0.2, k ¼ 0:6, a = 0.8, b = 0.6, /0 ¼ p=3, d = 0.3, Ha = 4.0, j ¼ 10:0, b ¼ 2:0, A = 0.02, B = 0.2, Nr = 1.0, Pr = 21.0, Sp ¼ 0:1, Ec = 0.2, (b) for different Nr, when x = 0.0, t = 0.2, k ¼ 0:6, a = 0.8, b = 0.6, /0 ¼ p=3, d = 0.8, Ha = 4.0, b ¼ 2:0, B = 0.3, A = 0.02, / ¼ 0:02, Pr = 21.0, Sp ¼ 0:1, Ec = 0.2, (c) for different Sp , when x = 0.0, t = 0.2, k ¼ 0:6, a = 0.8, b = 0.6, /0 ¼ p=3, d = 0.8, Ha = 4.0, j ¼ 10:0; b ¼ 2:0, B = 0.3, A = 0.02, Nr = 1.0, Pr = 21.0, Sp ¼ 0:1, Ec = 0.2, and (d) for different Ec, when x = 0.0, t = 0.2, k ¼ 0:6, a = 0.8, b = 0.6, /0 ¼ p=3, d = 0.8, Ha = 4.0, j ¼ 10:0; b ¼ 2:0, A = 0.02, / ¼ 0:02, Nr = 1.0, Pr = 21.0, Sp ¼ 0:1, Ec = 0.2.

Table 3 Nusselt number and heat transfer coefficient at the two walls when x ¼ 0:3; t ¼ 0:2; j ¼ 10:0; b ¼ 2:0; B ¼ 0:2; A ¼ 0:02; Ha ¼ 4:0. Nr

Pr

Sp

Ec

/

Nuh1

Z h1

Nuh2

Z h2

1.0 2.0

21.0

0.1

0.2

0.02

1.3358 1.0295 1.4193 1.3967 2.2124 1.3469

3.1038 2.3921 3.2978 3.2454 5.1407 3.1297

0.8860 0.4664 1.0025 0.9469 2.1009 0.9233

2.0587 1.0838 2.3294 2.2003 4.9014 2.1453

23.0 0.2 0.4 0.04

variation in the streamline pattern. These two figures clearly show that the bolus size breaks down as the Debye-Hückel parameter is enhanced. These figures indicate that as j gradually increases, the dimension of the bolus trapped inside, gets reduced and subsequently for considerably large values of j, the bolus is likely to

vanish altogether. Finally, Fig. 9(a)–(b) show that as time progresses, the occurrence of trapping phenomenon is greatly affected. The size of the bolus that is formed at the initial stage is gradually reduced with the passage of time and the bolus vanishes altogether at some point of time.

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277



Fig. 7. Streamline pattern for different Hartmann number (a) Ha ¼ 4:0, (b) Ha ¼ 6:0 for fixed values t ¼ 0:1; k ¼ 0:6; a ¼ 0:8; b ¼ 0:6; /0 ¼ p3 ; d ¼ 0:8; j ¼ 10:0; b ¼ 2:0; B ¼ 0:3; A ¼ 0:02; / ¼ 0:02.

Fig. 8. Streamline pattern for different Debye-Hückel parameter (a) b ¼ 2:0; B ¼ 0:3; A ¼ 0:02; / ¼ 0:02.

j ¼ 9, (b) j ¼ 11 for fixed values t ¼ 0:1; k ¼ 0:6; a ¼ 0:8; b ¼ 0:6; /0 ¼ p3 ; d ¼ 0:8; Ha ¼ 4:0;

5.6. Isotherm contours Fig. 10 illustrates the change in isotherm patterns when volume fraction of nano-particles is changed. Fig. 10(a) and (b) indicates that the isotherm patterns become more distorted with a rise in the volume fraction of the nanoparticles in the fluid. Physically this fact may be attributed to the increasing fluid temperature due to rise in nanoparticle volume fraction. Further from Fig. 11(a)–(b) one may observe the isotherm contour at different Joule heating parameter. It may be noted that temperature contours increase due to rise in thermal energy generation in the system due to joule

heating phenomenon. In due course, the temperature attains a steady state. 6. Summary and conclusions The study explored several important information on electrokinetically modulated peristaltic flow of Eyring-Powell fluid mixed with metallic nanoparticles. The investigation corresponds to a situation, where the system is exposed to Joule heating in the presence of a magnetic field. The governing equations used here have been derived on the basis of Debye-Hückel approximation. The

278

Fig. 9. Streamline pattern B ¼ 0:3; A ¼ 0:02; / ¼ 0:02.

B. Mallick, J.C. Misra / Engineering Science and Technology, an International Journal 22 (2019) 266–281

at

different

time

(a)

t ¼ 0:2,

(b)

t ¼ 0:4

for

fixed

values



k ¼ 0:6; a ¼ 0:8; b ¼ 0:6; /0 ¼ p3 ; d ¼ 0:8; Ha ¼ 4:0; j ¼ 10:0; b ¼ 2:0;



Fig. 10. Isotherm patterns for different nanoparticle volume fraction (a) / ¼ 0, (b) / ¼ 0:04 for fixed values k ¼ 0:6; a ¼ 0:8; b ¼ 0:6; /0 ¼ p3 ; d ¼ 0:8; Ha ¼ 4:0; j ¼ 10:0; b ¼ 2:0; B ¼ 0:3; A ¼ 0:02; Nr ¼ 1; Pr ¼ 21; Sp ¼ 0:1; Ec ¼ 0:2.

theoretical analysis makes use of a perturbation technique. The application of the analysis has been illustrated by computing the derived analytical expressions for an appropriate set of values of the involved parameters. Based on the study, we can conclude the following:  The axial velocity of the nanofluid and the wall shear stress in the micro-channel are considerably enhanced due to an increase in the Debye-Hückel parameter, but there is an opposite trend in the case of the axial pressure gradient.  For the nanofluid under consideration, both the heat transfer and Nusselt number are greater than those for the base fluid.

 The volume fraction increase of the nanoparticles of the fluid bears the potential to enhance momentum transport in the core region of the micro-channel. Acknowledgements The authors wish to express their deep sense of gratitude to Science and Engineering Research Board, Department of Science and Technology, Government of India, New Delhi for the financial support of this investigation through Grant No. SB/S4/MS: 864/14. The authors are thankful to the esteemed reviewers for their valuable comments based upon which the manuscript has been revised.

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Fig. 11. Isotherm patterns for different Joule heating parameter (a) Sp ¼ 0:2, (b) Sp ¼ 0:2 for fixed values k ¼ 0:6; a ¼ 0:8; b ¼ 0:6; /0 ¼ p3 ; d ¼ 0:8; Ha ¼ 4:0; j ¼ 10:0; b ¼ 2:0; B ¼ 0:3; A ¼ 0:02; / ¼ 0:02; Nr ¼ 1; Pr ¼ 21; Ec ¼ 0:2.

Appendix A ! exp

M1 ¼

1 ffi pHah ffiffiffiffiffiffiffiffi ð1þBÞ

! exp

2 ffi pHah ffiffiffiffiffiffiffiffi ð1þBÞ

! exp

1 ffi pffiffiffiffiffiffiffiffi Hah

ð1þBÞ

; M2 ¼

ðh1 h2 Þ

! !

2 ffi pffiffiffiffiffiffiffiffi Hah

exp

ð1þBÞ

; M3 ¼

ðh1 h2 Þ

!

2bj2 tanhj2 ðh1 h2 Þ

Ha 1 ffiffiffiffiffiffiffiffiffiffi exp pHah ; L ¼ pffiffiffiffiffiffiffiffiffiffi j½Ha2 j2 ð1þBÞ 1 ð1þBÞ ð1þBÞ !

bj2

Ha 1 ffiffiffiffiffiffiffiffiffiffi exp  pHah  M2 ; L2 ¼  pffiffiffiffiffiffiffiffiffiffi ð1þBÞ ð1þBÞ !

0 L3 ¼ L6 ¼ M 3  ðh1Fh  2Þ

Ha 2 ffiffiffiffiffiffiffiffiffiffi exp  pHah  M2 , L5 ¼  pffiffiffiffiffiffiffiffiffiffi ð1þBÞ ð1þBÞ " #

1 M 3 L5 L3 L5 M 1 þL2 L6 M 1 þL3 L4 M 2 L1 L6 M 2 Þ A2 ¼ F 0 ðL2 L4 L1 L5 Þðh1 h2 ÞðM3 L2ðLh41L , h2 ÞðL2 L4 L1 L5 Þ

"

½Ha2 j2 ð1þBÞ

Ha 2 ffiffiffiffiffiffiffiffiffiffi exp pHah L4 ¼ pffiffiffiffiffiffiffiffiffiffi ð1þBÞ ð1þBÞ

;

j

2

Pr 1þNr

2S



pffiffiffiffiffiffiffiffiffi ð1þBÞ 8ð1þBÞ

N1 ¼

Pr

;

N2 ¼

þ 12 A4 ¼  h1þNr 1 h2

Sp Pr

ÞGðh2 ÞÞ ðh1 þ h2 Þ þ EcðGðhh11h . 2

!4

3

ðL1 L6 L4 L3 Þ ðL1 L5 L4 L2 Þ

2 ðh2 Þ M 4 ¼ G2 ð3hð1hÞG , 1 h2 Þ

Pr

ÞGðh2 ÞÞ ðh1 þ h2 Þ  EcðGðhh11h ; 2



Ha

pffiffiffiffiffiffiffiffiffi ð1þBÞ 8ð1þBÞ Ha

;



N3 ¼

bjj ; 3 ½Ha2 j2 ð1þBÞ cosh3 j2ðh1 h2 Þ 3

4ð1þBÞ

2

ðL1 L6 L4 L3 Þ L3 L5 L2 L6 N 5 ¼ 3 L1 L5 L2 L4 ðL1 L5 L4 L2 Þ

2 9j2 Ha 1þB

5

6

Ha8 1

; L6 ¼  h1Fh þ M 4  13 G0 ðh1 Þ; 4 ; 3 2 ð1þBÞ Ha2 j2 ð1þBÞ cosh3 j2 ðh1 h2 Þ ½ 1þB !5 !4 " !2 #

2

2 L3 L5 L2 L6 L3 L5 L2 L6 2 2 Ha 2 Ha 2 Ha pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 12j bj L L L L 3bj j L L L L j þ4 pffiffiffiffiffiffiffiffiffi 1 5 2 4 1 5 2 4 ð1þBÞ ð1þBÞ ð1þBÞ N5 ; N ; N8 ¼ N6 ¼ !3 ; 7 ¼ j j 2 2 2 2 ½Ha j ð1þBÞcosh2ðh1 h2 Þ ½Ha j ð1þBÞcosh2ðh1 h2 Þ Ha ffi 2 pffiffiffiffiffiffiffiffi ð1þBÞ 8



 ðN6 þN7 Þexpð12jðh1 þh2 ÞÞ ðN6 N7 Þexpð12jðh1 þh2 ÞÞ ðL L L L Þ L L L L Ha N9 ¼  2   # ; N 10 ¼  2   # ; N 11 ¼ 3 L31 L55 L22 L64 ðL11 L65 L44 L32 Þ pffiffiffiffiffiffiffi ; 2 "  2 " 

N4 ¼ 

2



Sp

 12

!4

3

L3 L5 L2 L6 L1 L5 L2 L4

h1 h2

 M1 ,

#

bj tanh ðh h Þ 4 L3 Þ 2 L6 1 1 ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi  ððLL11 LL65 L  j Ha2 j22 11þB2 ; exp pHah exp  pHah A1 ¼ F20  A2 h1  LL31 LL55 L L2 L4 L4 L2 Þ ð ð ÞÞ ð1þBÞ ð1þBÞ !

A3 ¼ 12 h1 Prp þ EcGðh1 Þ þ h1

 M1 ,

4ð1þBÞ

p2Ha ffiffiffiffiffiþj

3b3 j5 j6

j2 Ha2

3

1þB

N 12 ¼

þ4j

pHaffiffiffiffiffi

1þB

3bjj2

pHaffiffiffiffiffi

ðL1 L6 L4 L3 Þ ðL1 L5 L4 L2 Þ

þj2

2

1þB

4

2 

2 pHaffiffiffiffiffi j2 þ4Ha 1þB 1þB

½Ha2 j2 ð1þBÞcoshj2ðh1 h2 Þ

p2Ha ffiffiffiffiffij 1þB

12bj2 j2

;

N 13 ¼ 

pHaffiffiffiffiffi

3

2 

ðL1 L6 L4 L3 Þ ðL1 L5 L4 L2 Þ

4j

1þB

5 pHaffiffiffiffiffi 1þB

½Ha2 j2 ð1þBÞcoshj2ðh1 h2 Þ

pHaffiffiffiffiffi

1þB

þj2

1þB

N 14 ¼  2ð1þBÞ

N11  3 ; pHaffiffiffiffiffi 1þB

280

B. Mallick, J.C. Misra / Engineering Science and Technology, an International Journal 22 (2019) 266–281

ðN12 þN13 Þexpð

Þ  2   ; N 10 2 2Ha 3Ha p 4jffiffiffiffiffi Ha þj2 ffiffiffiffiffij 2ð1þBÞ p 1þB 1þB 1þB  2

L3 L5 L2 L6 ffiffiffiffiffi 3bj2 j4 pHa expð2jðh1 þh2 ÞÞ L1 L5 L2 L4 1þB ¼  2 ; 2 ffiffiffiffiffi 4½Ha2 j2 ð1þBÞ cosh2 j2 ðh1 h2 Þ 2jpHa 1þB  2



L1 L6 L4 L3 2 4 Ha L L L L pffiffiffiffiffi 3bj j expð2jðh1 þh2 ÞÞ L1 L5 L2 L4 3bj2 j4 L1 L6 L4 L3 1þB 1 5 2 4 ¼ ;  2 ; N 22 ¼  2 2½Ha2 j2 ð1þBÞ cosh2 j2 ðh1 h2 Þ 2 ffiffiffiffiffi 4½Ha2 j2 ð1þBÞ cosh2 j2 ðh1 h2 Þ 2jþpHa

N 13 ¼

N 18

N 21

 2

L3 L5 L2 L6 pHaffiffiffiffiffi expð2jðh1 þh2 ÞÞ L1 L5 L2 L4 ðN13 N12 Þexpð Þ 1þB ¼  2 ;  2   ; N 17 ¼ 2 2Ha 3Ha2 þp 4jffiffiffiffiffi Ha þj2 ffiffiffiffiffiþj ffiffiffiffiffi 2ð1þBÞ p 4½Ha2 j2 ð1þBÞ cosh2 j2 ðh1 h2 Þ 2jþpHa 1þB 1þB 1þB 1þB  2



L1 L6 L4 L3 L L L L ffiffiffiffiffi 3bj2 j4 pHa expð2jðh1 þh2 ÞÞ L1 L5 L2 L4 3b2 j2 j4 L3 L5 L2 L6 1þB 1 5 2 4 N 19 ¼  ; N ¼  2 ; 20 2 2j 2 2 2½Ha j ð1þBÞ cosh 2 ðh1 h2 Þ 2 ffiffiffiffiffi 4½Ha2 j2 ð1þBÞ cosh2 j2 ðh1 h2 Þ 2jþpHa

12jðh1 þh2 Þ

3bj2 j4

1jðh þh Þ 1 2 2

1þB

N 11 ¼

N17   2 ; ffiffiffiffiffi 2jþpHaffiffiffiffiffi 4ð1þBÞ j2 þjpHa

1þB

N 15 ¼

N18

 2 ; ffiffiffiffiffi 2jpHaffiffiffiffiffi 4ð1þBÞ j2 jpHa 

1þB

N 27 ¼ 

N 24 ¼

1þB

6bj2

N

22 3 ; ffiffiffiffiffi 2ð1þBÞ pHa

N 28 ¼

1þB

N19  3 ; ffiffiffiffiffi 2ð1þBÞ pHa

L1 L6 L4 L3 L1 L5 L2 L4



1þB

L3 L5 L2 L6 L1 L5 L2 L4

N 14 ¼

1þB

4

 pHaffiffiffiffiffi

N20

1þB

1þB

1þB ; N 5 ¼ N 8 þ N 24 ;  2 ! 2 2j Ha 3 2 2 2 j ð1þBÞ j  pffiffiffiffiffi ½Ha j ð1þBÞ cosh 2ðh1 h2 Þ

1þB

1þB

N 6 ¼ N 14 þ N 27

1þB



1 L7 1  M 2 LL42 LL64 L  M 4 ; A7 ¼ ðh2Ech  M1 ðG3 ðh2 Þ  G3 ðh1 ÞÞ þ 2EcG3 ðh1 Þ L1 L5 1 h2 Þ ! !



2 L7 1 1 L7 1 ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi  LL42 LL64 L  13 Gðh1 Þ; A8 ¼ ðh12Ec ðG3 ðh1 Þ  G3 ðh2 ÞÞ. A5 ¼ F21  A6 h1  LL51 LL65 L exp pHah exp  pHah L2 L4 L1 L5 h2 Þ ð1þBÞ ð1þBÞ

L7 ¼

1  h1Fh 2

þ M4 

1 0 G ðh2 Þ; A6 3

¼

F1 h1 h2



1þB

N21  2 ; N 12 ¼   2 ; ffiffiffiffiffi 2jpHaffiffiffiffiffi ffiffiffiffiffi 2jþpHaffiffiffiffiffi 4ð1þBÞ j2 jpHa 4ð1þBÞ j2 þjpHa



L5 L6 L2 L7 L1 L5 L2 L4

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