study of transient peristaltic heat flow through a finite porous channel

Mathematical and Computer Modelling 57 (2013) 1270–1283

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Mathematical and Computer Modelling

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Study of transient peristaltic heat flow through a finite porous channelDharmendra Tripathi ∗Department of Mathematics, National Institute of Technology Delhi, Delhi-110077, India

a r t i c l e i n f o

Article history:Received 26 July 2011Received in revised form 4 October 2012Accepted 7 October 2012

Keywords:Heat and mass transferTransient flowPeristaltic pumpingFinite porous channelRefluxTrapping

a b s t r a c t

Analytical and computational studies on transient peristaltic heat flow through a finitelength porous channel are presented in this paper. Results for the temperature field, axialvelocity, transverse velocity, pressure gradient, local wall shear stress, volume flow rate,averaged volume flow, mechanical efficiency, and stream function are obtained underthe assumption of low Reynolds number (Re → 0) and long wavelength approximation(a ≪ λ → ∞). The current two-dimensional analysis is applicable in biofluid mechanics,industrial fluid mechanics, and some of the engineering fields. The impact of physicalparameters such as permeability parameter, Grashof number and thermal conductivityon the velocity field, pressure distribution, local wall shear stress, mechanical efficiencyof peristaltic pump, and two inherent phenomena (reflux and trapping) are depicted withthe help of computational results. Themain conclusions that can be drawn out of this studyis that peristaltic heat flow resists more porous medium whereas the peristaltic heat flowimproves with increasing magnitude of Grashof number, and thermal conductivity. Theresults of Tripathi (2012) [42] can be obtained by taking out the effects of porosity fromthis model.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

A variety of complex rheological fluids can easily be transported from one place to another place with a special type ofpumping known as Peristaltic pumping. This pumping principle is called peristalsis. The mechanism includes involuntaryperiodic contraction followed by relaxation or expansion of the ducts the fluids move through. This leads to the rise ofpressure gradient that eventually pushes the fluid forward. This type of pumping is first observed in physiology where thefood moves in the digestive tract, urine transports from the kidney to the bladder through ureters, semen movement inthe vas deferens, movement of lymphatic fluids in lymphatic vessels, bile flow from the gall bladder into the duodenum,spermatozoa in the ductus efferentes of the male reproductive tract and cervical canal, ovum moves in the fallopian tube,and blood circulation in small blood vessels. Historically, however, the engineering analysis of peristalsis was initiatedmuchlater than physiological studies. Applications in industrial fluid mechanics are like aggressive chemicals, high solids slurries,noxious fluid (nuclear industries) and other materials which are transported by peristaltic pumps. Roller pumps, hosepumps, tube pumps, finger pumps, heart-lung machines, blood pump machines, and dialysis machines are engineered onthe basis of peristalsis. Owing to the importance of peristaltic flow, some significant investigations [1–14] on peristaltic flowhave recently been reported. Takagi and Balmforth [1,2] discussed the peristaltic pumping with a rigid object and viscousfluids in an elastic tube. Chiu-On and Ye [3] introduced the Lagrangian approach for peristaltic pumping and Dudchenkoand Guria [4] studied the self-sustained peristaltic waves. Beg and Tripathi [5] investigated the peristaltic flow of nanofluidswith double-diffusive convection. Tripathi [6–10] reported the applications of fractional calculus in peristaltic pumping andTripathi [11,12] studied the blood flow model with couple stress fluids through the porous medium with and without slip

∗ Tel.: +91 1127785355.E-mail addresses: [email protected], [email protected].

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D. Tripathi / Mathematical and Computer Modelling 57 (2013) 1270–1283 1271

Fig. 1. Geometry of peristaltic flow pattern through finite porous channel.

effect. Tripathi [13] and Pandey et al. [14] presented the three layered peristaltic flow for viscous fluids and power law fluidsrespectively. They pointed out the effects of viscosities of core, intermediate and peripheral layers on peristaltic flowpattern.

Khaled and Vafai [15] presented an interesting review article the role of porous media in modeling flow and heat transferin biological tissues. They concluded that models for convective transport through porous media are widely applicable in theproduction of the osteoinductive material, simulation of blood flow of tumors and muscles, and in modeling blood flow,when fatty plaques of cholesterol and artery-clogging clots are formed in the lumen, transport of drugs, and nutrients tobrain cells. In the same lineNarasimhan [16]wrote another review article on the role of porousmedium in bio-thermo-fluidsmechanics. He studied the two broad categories of bio-mass and bio-heat transport of human physiology and discussed theapplication in LDL transport in arteries, drug delivery, drug eluting stents, functions of organs modeled as porous medium,and porous medium modeling of microbial transport.

A porous medium (which contains a number of small holes distributed throughout the matter) plays a key role in thestudy of transport process in bio-fluidmechanics, industrialmechanics, and engineering fields. A good example of peristalsisin a porousmedium is focused in intestinal fluid dynamics byMiyamoto et al. [17] and Jeffrey et al. [18]. Miyamoto et al. [17]studied the two-dimensional laminar flow in a circular porous tube and considered a small water absorption or secretionin the intestinal perfusion experiment whereas Jeffrey et al. [18] discussed the flow fields generated by peristaltic reflex inisolated guinea pig ileum. Elshehawey et al. [19] developed another model for peristaltic transport through an asymmetricporous channel and focused the application to intra uterine fluid motion in a sagittal cross-section of the uterus. Someauthors [20–23] studied peristaltic flow of Newtonian fluid and non-Newtonian fluids such as power law fluid, magnetofluid and Maxwell fluid through the porous medium. They discussed the effect of permeability parameter on pressure andfriction force across one wavelength through channel, asymmetric channel and tube.

The discipline of heat transfer is concerned with only two things: temperature, and the flow of heat. Temperaturerepresents the amount of thermal energy available, whereas heat flow represents the movement of thermal energy fromone place to another. A heat transfer mechanism can be grouped into three broad categories: conduction, convection, andradiation. In view of the wide range of applications of the heat transfer effect in peristaltic flow pattern, Vajravelu et al. [24]reported the peristaltic transport and heat transfer through vertical porous annulus. Srinivas and Kothandapani [25] studiedthe effect of heat transfer on peristaltic transport in an asymmetric channel. Hayat et al. [26] investigated the peristaltic flowwith heat transfer in porous space. Nadeem et al. [27] incorporated the MHD fluids with variable viscosity and discussedthe effect of heat transfer on peristaltic flow. Akbar and Nadeem [28] modeled for blood flow through a tapered artery witha stenosis by the heat transfer simulation of non-Newtonian fluids with the Reiner–Rivlin model. Tripathi and Beg [29]reported the influence of heat transfer on unsteady physiological magneto-fluid flow.

None of the above studies deals with the unsteady peristaltic flow through the finite length porous channel/tube. Whilefrom an application’s point of view, it is needed to study the time dependent flow through the bounded geometries (lengthof the channel is finite) of a flow pattern. Considering these facts, Li and Brasseur [30] presented a model for unsteadyperistaltic transport of incompressible Newtonian fluid through the finite length tube. They compared their results withexperimental results (intrabolus pressure during oesophageal peristaltic transport by using a manometer) and found goodagreement with manometric observations. Since the physical property of food bolus is not only of Newtonian character,this study does not cover non-Newtonian behavior of food bolus. Subsequently, Misra and Pandey [31], Tripathi et al. [32],Pandey and Tripathi [33–38] and Tripathi [39–41] have improved Li and Brasseur’s model for non-Newtonian fluids such as

1272 D. Tripathi / Mathematical and Computer Modelling 57 (2013) 1270–1283

Fig. 2. Time dependent velocity profiles at φ = 0.6, ξ = 1.0, ∂p∂ξ

= 1.0 for (a) t = 0.2,Gr = 1.0, β = 1.0, K = 0.1, 1.0, 3.0, 10.0 (b) t = 0.2, K =

1.0, β = 1.0,Gr = 0.0, 1.0, 2.0, 3.0 (c) t = 0.2, K = 1.0,Gr = 1.0, β = 0.0, 1.0, 2.0, 3.0 (d) K = 1.0, β = 1.0,Gr = 1.0, t = 0.0, 0.1, 0.2, 0.3.

power-law fluids, Micropolar fluids, couple stress fluids, MHD fluids, visco-plastic (Casson model), visco-elastic fluids(Maxwell and Jeffrey models), and viscous fluids with variable viscosity. The applications of these models in oesophagealswallowing have been discussed. Tripathi [42] further studied the effect of heat transfer on peristaltic flow pattern througha finite length channel and discussed the impact of physical parameters on flow behavior. According to the author’sknowledge, until now no investigation has been made to study the transient peristaltic heat flow through the finite porouschannel which may have future applications in biofluid mechanics, industrial fluid mechanics and engineering fields.Therefore, in the present paper, we further extend the peristaltic flow model [42] for a finite porous channel and comparethe present results with the results of [42]. This paper is designed as follows; section two presents the mathematical


a b

c d

Fig. 3. Pressure vs. axial distance for K = 0.1, 0.2, 0.3,Gr = 1.0, β = 2.0, φ = 0.9, l = 2.0 at various instants (a) t = 0.0, 1.0 (b) t = 0.25 (c) t = 0.5(d) t = 0.75. Dotted curves represent the position of wave and colored solid curves show pressure along the length of the channel.

model wherein a dimensionless set of governing equations subject to appropriate boundary conditions is derived. Analyticalsolutions with analysis are developed in section three. Section four provides detailed computational results with discussionand physical interpretation. The final section concludes our findings.

2. Flow regime and mathematical formulation

The constitutive equation for the deformable wall geometry (Misra and Pandey [31]) due to propagation of train waves,considered in the present investigation, takes the form:

h(ξ , t) = a − φ cos2π

λ(ξ − ct), (1)

where h, ξ , t, a, φ, λ and c represent the transverse vibration of thewall, the axial coordinate, time, halfwidth of the channel,the amplitude of the wave, the wavelength and the wave velocity respectively (see Fig. 1).

The governing equations, which describe the flow of an incompressible fluid with heat transfer through porous medium,are given as follows:

ρ

∂

∂ t+ u

∂

∂ξ+ v

∂

∂η

u = −

∂ p

∂ξ+ µ

∂2u

∂ξ 2+∂2u∂η2

− µ

u

K+ ρ gα(T − T0),

ρ

∂

∂ t+ u

∂

∂ξ+ v

∂

∂η

v = −

∂ p∂η

+ µ

∂2v

∂ξ 2+∂2v

∂η2

− µ

v

K,

∂ u

∂ξ+∂v

∂η= 0,

ρ cp

∂

∂ t+ u

∂

∂ξ+ v

∂

∂η

T = k

∂2T

∂ξ 2+∂2T∂η2

+ Φ

, (2)

where ρ, u, v, η, p, µ, K , g, α, T , cp, k and Φ indicate the fluid density, axial velocity, transverse velocity, transversecoordinate, pressure, fluid viscosity, permeability parameter, acceleration due to gravity, coefficient of linear thermal


a b

c d

Fig. 4. Pressure vs. axial distance for Gr = 1.0, 2.0, 3.0, K = 1.0, β = 2.0, φ = 0.9, l = 2.0 at various instants (a) t = 0.0, 1.0 (b) t = 0.25 (c) t = 0.5(d) t = 0.75. Dotted curves represent the position of wave and colored solid curves show pressure along the length of the channel.

expansion of fluid, temperature, specific heat at constant pressure, thermal conductivity and constant heat addition/absorption.

The non-dimensional variables are given as:

ξ =ξ

λ, η =

η

a, t =

ctλ, u =

uc, v =

v

cδ, δ =

aλ, h =

ha

= 1 − φ cos2 π(ξ − t),

l =lλ, φ =

φ

a, p =

pa2

µcλ, ψ =

ψ

ac, Q =

Qac, Re =

ρcaδµ

, K =Ka2,

Gr =gρ2αa3(T1 − T0)

µ2, θ =

T − T0T1 − T0

, β =a2Φ

k(T1 − T0), Pr =

µ cpk

(3)

where δ, l, ψ, Q , Re,Gr, θ, β and Pr are thewavenumber, length of the channel, stream function, volume flow rate, Reynoldsnumber, Grashof number, dimensionless temperature, dimensionless heat source/sink parameter and Prandtl number.

Introducing the non-dimensional parameters from Eq. (3) into Eq. (2) and applying large wavelength and low Reynoldsnumber approximations, lead to:

∂p∂ξ

=∂2u∂η2

−uK

+ Gr θ,

∂p∂η

= 0,

∂u∂ξ

+∂v

∂η= 0,

∂2θ

∂η2+ β = 0,

. (4)

The corresponding boundary conditions are given [42] as:

θ = 0 at η = 0, θ = 1 at η = h, i.e. the temperature conditions, (5)


a b

c d

Fig. 5. Pressure vs. axial distance for β = 2.0, 4.0, 6.0, K = 1.0,Gr = 1.0, φ = 0.9, l = 2.0 at various instants (a) t = 0.0, 1.0 (b) t = 0.25 (c) t = 0.5(d) t = 0.75. Dotted curves represent the position of wave and colored solid curves show pressure along the length of the channel.

∂u∂η(ξ, η, t)

η=0

= 0, i.e., the regularity condition, (6)

u(ξ , η, t)|η=h = 0, i.e., the no slip condition, (7)

v(ξ, η, t)|η=0 = 0, i.e., the absence of any transverse velocity, (8)

v(ξ, η, t)|η=h =∂h∂t, i.e., the transverse vibration of the wall, (9)

p|ξ=0 = p0 and p|ξ=l = pl, i.e. the finite length condition. (10)

3. Analytical solutions and analysis

Solving Eqs. (4) and using boundary conditions (5)–(7), temperature field and axial velocity are obtained as

θ =η

h+β

2(ηh − η2), (11)

u =1k2

cosh(kη)cosh(kh)

− 1

∂p∂ξ

−Gr2

βh +

2h

+

Grβ2k2

h2

+2k2

cosh(kη)cosh(kh)

−

η2 +

2k2

(12)

where k =

1K .

On the substitution of u, integrating continuity equation from 0 to η, and using boundary condition (8), the transversevelocity is derived as

v = −1k2

sinh(kh)k cosh(kh)

− η

∂2p∂ξ 2

−Gr2

β −

2h2

∂h∂ξ

+

∂p∂ξ

−Gr2

βh +

2h

sinh(kη) tanh(kh)

cosh(kh)∂h∂ξ

+Grβ2∂h∂ξ

2h sinh(kη)k cosh(kh)

+

h2

+2k2

sinh(kη) tanh(kh)

cosh(kh)

. (13)


a b

c d

Fig. 6. Local wall shear stress vs. axial distance for K = 0.1, 0.2, 0.3,Gr = 1.0, β = 2.0, φ = 0.9, l = 2.0 at various instants (a) t = 0.0, 1.0 (b) t = 0.25(c) t = 0.5 (d) t = 0.75. Dotted curves represent the position of wave and colored solid curves show local wall shear stress along the length of the channel.

At the wall, the transverse velocity is expressed as

∂h∂t

= −1k2

tanh(kh)− kh

k

∂2p∂ξ 2

−Gr2

β −

2h2

∂h∂ξ

+

∂p∂ξ

−Gr2

βh +

2h

tanh2(kh)

∂h∂ξ

+Grβ2∂h∂ξ

2h tanh(kh)

k+

h2

+2k2

tanh2(kh)

. (14)

Integrating equation (14) with respect to ξ , the pressure gradient is obtained as

∂p∂ξ

=k3A(t)+

φ

2 cos 2π(ξ − t)

kh − tanh(kh)+

Gr2

2h

+ β

h(1 − h)−

2k2

+2kh3

3(kh − tanh(kh))

, (15)

where A(t) is an arbitrary function of t , and on further integration from 0 to ξ , yields

p(ξ , t)− p(0, t) = k3 ξ

0

A(t)+

φ

2 cos 2π(s − t)

kh − tanh(kh)ds

+Gr2

ξ

0

2h

+ β

h(1 − h)−

2k2

+2kh3

3(kh − tanh(kh))

ds. (16)

The pressure difference between inlet and outlet of the channel is found by substituting ξ = l, and given by

p(l, t)− p(0, t) = k3 l

0

A(t)+

φ

2 cos 2π(ξ − t)

kh − tanh(kh)dξ

+Gr2

l

0

2h

+ β

h(1 − h)−

2k2

+2kh3

3(kh − tanh(kh))

dξ . (17)


a b

c d

Fig. 7. Local wall shear stress vs. axial distance for Gr = 1.0, 2.0, 3.0, K = 1.0, β = 2.0, φ = 0.9, l = 2.0 at various instants (a) t = 0.0, 1.0 (b) t = 0.25(c) t = 0.5 (d) t = 0.75. Dotted curves represent the position of wave and colored solid lines show local wall shear stress along the length of the channel.

a b

c d

Fig. 8. Local wall shear stress vs. axial distance for β = 2.0, 4.0, 6.0, K = 1.0,Gr = 1.0, φ = 0.9, l = 2.0 at various instants (a) t = 0.0, 1.0 (b) t = 0.25(c) t = 0.5 (d) t = 0.75. Dotted curves represent the position of wave and colored solid lines show local wall shear stress along the length of the channel.


a

b

c

Fig. 9. Pressure difference across one wavelength vs. averaged flow rate. Colored lines represent the pressure at fixed flow rate at φ = 0.5 for(a) K = 1.0, 2.0, 3.0,Gr = 1.0, β = 2.0, (b) Gr = 1.0, 2.0, 3.0, K = 1.0, β = 2.0, (c) β = 2.0, 4.0, 6.0, K = 1.0,Gr = 1.0.

Local wall shear stress [42] is defined as τw =∂u∂η

η=h

, in the view of Eqs. (12) and (15), it reduces to

τw = tanh(kh)

3k2

A(t)+

φ

2 cos 2π(ξ − t)+ Grβh3

3(kh − tanh(kh))

−

Grβhk2

. (18)

Volume flow rate [42] is defined as Q (ξ , t) = h0 udη, integrating, in view of Eq. (12), it yields

Q (ξ , t) =1k2

tanh(kh)− kh

k

∂p∂ξ

−Gr2

βh +

2h

+

Grβ2

h2

+2k2

tanh(kh)

k−

h3

3+

2hk2

. (19)

The averaged flow rate is expressed [42] in terms of flow rate as:

Q = Q − h + 1 −φ

2. (20)

A simple manipulation of Eq. (19), followed by application of Eq. (20), yields the pressure gradient as

∂p∂ξ

=k3Q + h − 1 +

φ

2

tanh(kh)− kh

+Gr2

2h

+ β

h(1 − h)−

2k2

+2kh3

3(kh − tanh(kh))

. (21)


a

b

c

Fig. 10. Mechanical efficiency vs. ratio of averaged flow rate andmaximum averaged flow rate. Colored curves represent the efficiency of pump at φ = 0.5for (a) K = 1.0, 2.0, 3.0,Gr = 1.0, β = 2.0 (b) Gr = 1.0, 2.0, 3.0, K = 1.0, β = 1.0 (c) β = 1.0, 1.5, 2.0, K = 1.0,Gr = 1.0.

Mechanical efficiency [42] is obtained as

E =Q1p1

φ(I1 −1p1), (22)

where

I1 =

1

0

∂p∂ξ

cos 2πξdξ,

and1p1 is the pressure difference across a wavelength, which is given by

1p1 = p(1)− p(0)

= k3 1

0

Q + h − 1 +

φ

2

tanh(kh)− kh

dξ +Gr2

1

0

2h

+ β

h(1 − h)−

2k2

+2kh3

3(kh − tanh(kh))

dξ . (23)


a b

Fig. 11. Averaged flow rate vs. amplitude. Colored curves represent reflux limit for (a) Gr = 1.0, 2.0, 3.0, K = 1.0, β = 1.0 (b) β = 1.0, 2.0, 3.0, K =

1.0,Gr = 1.0.

Moreover, the maximum flow rate, obtained by substituting1p1 = 0, is

Q0 = 1 −φ

2−

Gr2

10

2h + β

−

2k2

+ h1 − h +

2kh23(kh−tan(kh))

dξ +

10

htanh(kh)−khdξ 1

01

tanh(kh)−khdξ. (24)

The stream function [42] is obtained as:

ψ =

sinh(kη)k cosh(kh)

− η

3k2

Q + h − 1 +

φ

2

− Grβh3

3k(tanh(kh)− kh)

+

Grβ2k2

h2η −

η3

3

− η. (25)

In order to evaluate the reflux limit, the averaged reflux flow rate [42] is expanded in a power series in terms of asmall parameter ε (=ψ − Q + 1 −

φ

2 ) about the wall. The coefficients of the first two terms in the expansion of η, i.e.,η = h + a1ε + a2ε2 + · · · can be evaluated from Eq. (25), which yields:

a1 = −1, a2 =kh

2 cosh(kh)

3k2

Q + h − 1 +

φ

2

− Grβh3

3(tanh(kh)− kh)

−

Grβh2k2

. (26)

The reflux limit [42] is obtained as:

Q < 1 −φ

2−

Grβk

1 −

φ

2

+

13k

10

h4cosh(kh)(tanh(kh)−kh)dξ

+ 10

h2cosh(kh)(tanh(kh)−kh)dξ 1

0h

cosh(kh)(tanh(kh)−kh)dξ. (27)

Results obtained by Tripathi [42] are the limiting case (K → ∞) of the results found in present paper. Results of thepresent study and those reported in [42] have been compared in the next section.

4. Computational results and discussion

In this section we have presented the numerical and computational results to analyze the impact of heat transferand porosity on velocity field (u(η)), pressure distribution (p(ξ , t) − p(0, t)), local wall shear stress (τw), averaged flowrate (Q ), maximum averaged flow rate (Q0), mechanical efficiency (E), reflux limit, and trapping through the Figs. 2–12.Figures are drawn using computational programs in C-language and Mathematica software. One of the cases known asfree pumping, i.e., pressures at the two ends of channel are zero, i.e., pl = p0 = 0, is taken to discuss the characteristicof peristaltic heat flow pattern through the finite length porous channel. This condition is important in this study as itdistinguishes the finite length channel flow with infinite length channel flow.

Fig. 2(a)–(d) present the velocity profiles under the effect of permeability parameter (K ), thermal conductivity (β),Grashof number (Gr), and time (t). Figures show the relation between axial velocity and transverse displacement. It is foundthat relation is nonlinear (parabolic shape). Fig. 2(a) depicts the velocity profile region reduction with enhancingmagnitudeof permeability parameter in β ∈ (−0.6, 0.6). The effects of thermal conductivity and Grashof number on velocity profilesare shown in Fig. 2(a) and (c). It is observed that the velocity profile region increases with increasing magnitudes of thermalconductivity and Grashof number in β ∈ (−0.6, 0.6). Fig. 2(d) reveals that the velocity profile is an increasing functionof time.


Fig. 12. Streamlines in the wave frame at Q = 0.5, φ = 0.5 when (a) K = 1.0,Gr = 1.0, β = 1.0 (b) K = 1.2,Gr = 1.0, β = 1.0(c) K = 1.3,Gr = 1.0, β = 1.0 (d) K = 1.4,Gr = 1.0, β = 1.0 (e) K = 1.0,Gr = 2.0, β = 1.0 (f) K = 1.0,Gr = 3.0, β = 1.0(g) K = 1.0,Gr = 4.0, β = 1.0 (h) K = 1.0,Gr = 1.0, β = 2.0 (i) K = 1.0,Gr = 1.0, β = 4.0.

The keen examination of the pressure distribution along the length of the channel at various instances (t = 0.0–1.0) hasbeen done through the Fig. 3(a)–(d). Fig. 3(a) (t = 0) shows that the pressure first tends to rise sharply at the inlet, thenreaches a peak, thereafter falling at a lower rate to zero at the middle of the bolus, further comes down to a lower trough,


and finally rises sharply to meet the leading end of the bolus. The same distribution is repeated for the next bolus. After onefourth of the periodic cycle (Fig. 3(b)), the bolus has moved ahead and a trailing bolus is on the way to entry. The graphsfor higher values of t represent a systematic progress of the boluses in the channel. Eventually, at the time t = 1, whichrepresents the completion of one period, the pressure distribution resembles that at t = 0; this indicates that a new cycleis ready to set out (Fig. 3(a)).

The effects of permeability parameter, Grashof number and thermal conductivity on pressure distribution along thelength of the channel are presented in Figs. 3–5. It is observed that pressure along the length of the channel reduces withincreasingmagnitude of permeability parameter, Grashof number, and thermal conductivity. From the governing equationsof motion, it is clear that the value of permeability parameter increases (K → ∞), the porosity of channel reduces (movesto results of [42]); that indicates that pressure is required more when the porosity of channel is enhanced. When Gr → 0and β → 0, i.e., the peristaltic flow of viscous fluid without the effect of heat transfer, the pressure rises to a maximum. It isphysically interpreted that if heat transfer is effective then less pressure is required for the peristaltic flow of viscous fluidthrough the channel.

Figs. 6–8(a)–(d) illustrate the influence of porosity and heat transfer on local wall shear stress along the length of thechannel at various instances. It is revealed that the range between peak and trough of local wall shear stress distributionreduces with increasing magnitude of permeability parameter, Grashof number, and thermal conductivity.

A relation between pressure across one wavelength and volumetric averaged flow rate under the effect of porosity andheat transfer is presented through Fig. 9(a)–(c). It is observed that volumetric averaged flow rate enhances with increasingmagnitude of permeability parameter, Grashof number, and thermal conductivity. That implies that the volumetric flowrate reduces with the effect of porosity while it increases with the effect of heat transfer.

Mechanical efficiency is an important characteristic of a peristaltic pump. It is defined as the ratio of the average rate perwavelength at which work is done by moving fluid against a pressure head and average rate at which the walls do workon the fluid. Emphasizing the influence of porosity and heat transfer on mechanical efficiency, Fig. 10(a)–(c) are plottedbetween mechanical efficiency (E) and the ratio of averaged flow rate to maximum averaged flow rate (Q/Q0). There is anonlinear relation between them. First E rises with Q/Q0 from zero to themaximum value, it falls thereafter to zero. Figuresreveal that the mechanical efficiency increases with increasing magnitude of permeability parameter, Grashof number, andthermal conductivity.

Reflux is an inherent phenomenon and is related to the retrograde motion of the fluid particles close to the inner surfaceof the wall of the vessel in which the flow is considered. Fig. 11(a) and (b) are plotted between amplitude and averaged flowrate to discuss the influence of Grashof number and thermal conductivity on the reflux limit. It is found that the relationbetween them is a half parabolic curve. The upper portion of the curve is a no reflux region while the lower portion is areflux region. It is inferred that the area of the reflux region increases with increasing magnitude of Grashof number andthermal conductivity. The expression for reflux limit is obtained for very small value of permeability parameter. Therefore,the effect of this parameter on reflux limit is ignored in the discussion.

Trapping is another inherent physiological phenomenon which refers to closed circulating streamlines that exist at veryhigh flow rates and when occlusions are very large. The trapped region moves with the wave velocity. Streamlines in waveframe are drawn to discuss the impact of porosity and heat transfer on trapping through Fig. 12(a)–(i). Fig. 12(a)–(d) revealthat the number of trapped boluses diminishes with increasing the magnitude of permeability parameter. It indicates thattrapping reduces with increasing magnitude of permeability parameter. Fig. 12(a), (e)–(g) show the number of trappedboluses decreases with increasing themagnitude of Grashof number. Fig. 12(a), (h), (i) depict the number of trapped bolusesdecreases with increasing magnitude of thermal conductivity.

5. Conclusions

The significant features of peristaltic flowof viscous fluid under the influence of porosity andheat transfer through a finitelength channel havenumerically and graphically beenpresented in the above section. On the basis of discussion, it is revealedthatmore pressure is required for peristaltic flow of viscous fluid through a finite channel, when themedium ismore porous.The other revelation is that less pressure is required for peristaltic flow of viscous fluid when heat transfer is effective. It isalso found that the local wall shear stress is the increasing function of permeability parameter, Grashof number, and thermalconductivity. Another observation is that volumetric flow rate enhances with the increasing magnitude of said parameters.It is further inferred that the mechanical efficiency reduces under the effect of porosity while it increases under the effect ofheat transfer. The area of reflux region increases with increasing the effect of heat transfer. Finally, it is concluded that thetrapping enhances with increasing the effect of porosity, whereas it reduces with increasing the effect of heat transfer. Thismodel motivates the study of bio-thermo-fluidmechanics and industrial-thermo-fluids mechanics where a porousmediumis present.

References

[1] D. Takagi, N.J. Balmforth, Peristaltic pumping of rigid objects in an elastic tube, Journal of Fluid Mechanics 672 (2011) 219–244.[2] D. Takagi, N.J. Balmforth, Peristaltic pumping of viscous fluid in an elastic tube, Journal of Fluid Mechanics 672 (2011) 196–218.[3] Chiu-On Ng, Ye Ma, Lagrangian transport induced by peristaltic pumping in a closed channel, Physical Review E 80 (2009) 056307.


[4] O.A. Dudchenko, G.Th. Guria, Self-sustained peristaltic waves: explicit asymptotic solutions, Physical Review E 85 (2012) 020902(R).[5] O.A. Beg, D. Tripathi, Mathematica simulation of peristaltic pumping with double-diffusive convection in nanofluids: a bio-nanoengineering model,

Proceedings of the Institution of Mechanical Engineers, Part N: Journal of Nanoengineering and Nanosystems (2012)http://dx.doi.org/10.1177/1740349912437087.

[6] D. Tripathi, Numerical study on creeping flow of Burgers’ fluids through a peristaltic tube, Transactions of the ASME Journal of Fluids Engineering 133(2011) 121104-1–121104-9.

[7] D. Tripathi, A mathematical model for the peristaltic flow of chyme movement in small intestine, Mathematical Biosciences 233 (2011) 90–97.[8] D. Tripathi, Peristaltic transport of fractional Maxwell fluids in uniform tubes: application of an endoscope, Computers and Mathematics with

Applications 62 (2011) 1116–1126.[9] D. Tripathi, Numerical and analytical simulation of peristaltic flows of generalized Oldroyd-B fluids, International Journal for Numerical Methods in

Fluids 67 (2011) 1932–1943.[10] D. Tripathi, Numerical study on peristaltic flow of generalized Burgers’ fluids in uniform tubes in presence of an endoscope, International Journal of

Numerical Methods in Biomedical Engineering 27 (2011) 1812–1828.[11] D. Tripathi, Peristaltic hemodynamic flow of couple-stress fluids through a porous medium with slip effect, Transport in Porous Media 92 (2012)

559–572.[12] D. Tripathi, Peristaltic flow of couple-stress conducting fluids through a porous channel: applications to blood flow in the micro-circulatory system,

Journal of Biological Systems 19 (2011) 461–477.[13] D. Tripathi, A mathematical study on three layered oscillatory blood flow through stenosed arteries, Journal of Bionic Engineering 9 (2012) 119–131.[14] S.K. Pandey, M.K. Chaube, D. Tripathi, Peristaltic transport of multilayered power-law fluids with distinct viscosities: a mathematical model for

intestinal flows, Journal of Theoretical Biology 278 (2011) 11–19.[15] A.R.A. Khaled, K. Vafai, The role of porous media in modeling flow and heat transfer in biological tissues, International Journal of Heat and Mass

Transfer 46 (2003) 4989–5003.[16] Arunn Narasimhan, The role of porous medium modeling in biothermofluids, Journal of the Indian Institute of Science 91 (2011) 343–366.[17] Y. Miyamoto, M. Hanano, T. Iga, Concentration profile in the intestinal tract and drug absorption model: two-dimensional laminar flow in a circular

porous tube, Journal of Theoretical Biology 102 (1983) 585–601.[18] B. Jeffrey, H.S. Udaykumar, K.S. Schulze, Flow fields generated by peristaltic reflex in isolated guinea pig ileum: impact of contraction depth and

shoulders, American Journal of Physiology—Gastrointestinal and Liver Physiology 285 (2003) G907–G918.[19] E.F. Elshehawey, N.T. Eldabe, E.M. Elghazy, A. Ebaid, Peristaltic transport in an asymmetric channel through a porous medium, Applied Mathematics

and Computation 182 (2006) 140–150.[20] A.R. Rao, M. Mishra, Peristaltic transport of a power-law fluid in a porous tube, Journal of Non-Newtonian Fluid Mechanics 121 (2004) 163–174.[21] N.A.S. Afifi, N.S. Gad, Interaction of peristaltic flow with pulsatile magneto-fluid through a porous medium, Acta Mechanica 149 (2001) 229–237.[22] Kh.S. Mekheimer, Nonlinear peristaltic transport through a porous medium in an inclined planar channel, Journal of Porous Media 6 (2003) 189–201.[23] T. Hayat, N. Ali, S. Asghar, Hall effects on peristaltic flow of a Maxwell fluid in a porous medium, Physics Letters A 363 (2007) 397–403.[24] K. Vajravelu, G. Radhakrishnamacharya, V. Radhakrishnamurty, Peristaltic transport and heat transfer in a vertical porous annulus with long wave

approximation, International Journal of Non-Linear Mechanics 42 (2007) 754–759.[25] S. Srinivas, M. Kothandapani, Peristaltic transport in an asymmetric channel with heat transfer—a note, International Communications in Heat and

Mass Transfer 35 (2008) 514–522.[26] T. Hayat, M.U. Qureshi, Q. Hussain, Effect of heat transfer on the peristaltic flow of an electrically conducting fluid in porous space, Applied

Mathematical Modelling 33 (2008) 1862–1873.[27] S. Nadeem, N.S. Akbar, M. Hameed, Peristaltic transport and heat transfer of a MHD Newtonian fluid with variable viscosity, International Journal for

Numerical Methods in Fluids 63 (2010) 1375–1393.[28] N.S. Akbar, S. Nadeem, Simulation of heat and chemical reactions on Reiner–Rivlin fluidmodel for blood flow through a tapered artery with a stenosis,

Heat and Mass Transfer 46 (2010) 531–539.[29] D. Tripathi, O.A. Beg, A study of unsteady physiological magneto-fluid flow and heat transfer through a finite length channel by peristaltic pumping,

Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine 226 (2012) 631–644.[30] M. Li, J.G. Brasseur, Non-steady peristaltic transport in finite length tubes, Journal of Fluid Mechanics 248 (1993) 129–151.[31] J.C.Misra, S.K. Pandey, Amathematicalmodel for oesophageal swallowing of a food bolus,Mathematical and ComputerModelling 33 (2001) 997–1009.[32] D. Tripathi, S.K. Pandey, M.K. Chaube, Peristaltic transport of power law fluid in finite length vessels, International Journal of Mathematical Sciences

and Engineering Applications 3 (2009) 279–289.[33] S.K. Pandey, D. Tripathi, A mathematical model for peristaltic transport of micro-polar fluids, Applied Bionics and Biomechanics 8 (2011) 279–293.[34] S.K. Pandey, D. Tripathi, Effects of non-integral number of peristaltic waves transporting couple stress fluids in finite length channels, Zeitschrift fur

Naturforschung A: Journal of Physical Sciences 66a (2011) 172–180.[35] S.K. Pandey, D. Tripathi, Peristaltic transport of a Casson fluid in a finite channel: application to flows of concentrated fluids in oesophagus,

International Journal of Biomathematics 3 (2010) 473–491.[36] S.K. Pandey, D. Tripathi, A mathematical model for swallowing of concentrated fluids in oesophagus, Applied Bionics and Biomechanics 8 (2011)

309–321.[37] S.K. Pandey, D. Tripathi, Unsteady peristaltic transport of Maxwell fluid through a finite length tube: application to oesophageal swallowing, Applied

Mathematics and Mechanics. English Edition 33 (2012) 15–24.[38] S.K. Pandey, D. Tripathi, Unsteady model of transportation of Jeffrey fluid by peristalsis, International Journal of Biomathematics 3 (2010) 453–472.[39] D. Tripathi, Unsteady peristaltic transport of integral and non-integral number of viscous fluid bolus through channel, International Journal of

Mathematical Sciences and Engineering Applications 3 (2009) 223–231.[40] D. Tripathi, A mathematical model for swallowing of the viscoelastic food bolus through oesophagus, International Journal of Mathematical Sciences

and Engineering Applications 3 (2009) 255–263.[41] D. Tripathi, Amathematicalmodel for themovement of foodbolus of varying viscosities through the oesophagus, ActaAstronautica 69 (2011) 429–439.[42] D. Tripathi, A mathematical model for swallowing of food bolus through the oesophagus under the influence of heat transfer, International Journal of

Thermal Sciences 51 (2012) 91–101.

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