flow of a jeffrey fluid through a porous medium in narrow tubes

Journal of Porous Media, 18 (1): 71–78 (2015)

FLOW OF A JEFFREY FLUID THROUGH A POROUSMEDIUM IN NARROW TUBES

Nallapu Santhosh,1 G. Radhakrishnamacharya,1 & Ali J. Chamkha2,∗

1Department of Mathematics, National Institute of Technology, Warangal, India2Mechanical Engineering Department, Prince Mohammad Bin Fahd University (PMU), Al-Khobar 31952, Kingdom of Saudi Arabia

∗Address all correspondence to Ali J. Chamkha E-mail: [email protected]

Original Manuscript Submitted: 3/14/2014; Final Draft Received: 8/25/2014

In this paper, the effects of porous medium on a two-fluid model for the flow of Jeffrey fluid in tubes of small diametersare studied. It is assumed that the core region consists of Jeffrey fluid and Newtonian fluid in the peripheral region. Theanalytical solutions for velocity, flow flux, effective viscosity, core hematocrit and mean hematocrit have been derivedand the effects of various relevant parameters on these flow variables have been studied. It is noticed that the flow exhibitsthe anomalous Fahraeus-Lindqvist effect.

KEY WORDS: porous media, effective viscosity, narrow tubes, Jeffrey fluid

1. INTRODUCTION

The word microcirculation refers to the flow throughsmaller blood vessels of the circulatory system, with di-ameters ranging from 20µm (microns) to 500µm. It iswell known that the flow of blood through tubes, smallerthan 500µm in diameter, is accompanied by anomalouseffects. In particular, it has been observed that the ap-parent viscosity of blood increases with tube diameter(Fahraeus-Lindqvist effect) and the hematocrit within thetube is lower than that in the feed reservoir (Fahraeus ef-fect). These effects have been confirmed by several in-vestigators (Fahraeus and Lindqvist, 1931; Seshadri andJaffrin, 1977).

Most of the models on the blood flow deal with a one-phase model. However, it is realized that blood, being asuspension of corpuscles, behaves like a non-Newtonianfluid at lower shear rates (Haynes and Burton, 1959; Her-shey and Chow, 1966). The studies of Cokelet (1972)and Haynes (1960) indicate that blood can no longerbe treated as a single-phase homogeneous viscous fluidin smaller vessels. Bugliarello and Sevilla (1970) haveconsidered a two-fluid model with both fluids as New-

tonian fluids and with different viscosities. Sharan andPopel (2001) considered a two-phase model to discuss theflow of blood in narrow tubes. Following the theoreticalstudy of Haynes (1960) and experimentally tested modelof Bugliarello and Sevilla (1970), two-fluid modeling ofblood flow has been discussed and used by a good numberof researchers. Several non-Newtonian fluid models havebeen considered for blood flow in small-diameter tubes.Chaturani and Upadhya (1979, 1981) analyzed two-fluidmodels assuming Newtonian fluid in the peripheral regionand polar fluids in the core region. Haldar and Andersson(1996) have presented a two-layered blood flow model inwhich the core region is occupied by a Casson-type fluidand the peripheral region by a Newtonian fluid. Chat-urani and Bharatiya (2001) analyzed a two-phase mag-netic fluid model for pulsatile flow of blood. Shukla et al.(1980) considered a two-fluid model to discuss the flowof blood through a stenosed tube. Chamkha et al. (2002)considered micropolar fluid flow in a vertical parallelplate channel with asymmetric heating. Khan et al. (2013)investigated the peristaltic motion of an incompressiblenon-Newtonian fluid with variable viscosity through aporous medium in an inclined symmetric channel under

1091–028X/15/$35.00 c⃝ 2015 by Begell House, Inc. 71

72 Santhosh, Radhakrishnamacharya, & Chamkha

the effect of the slip condition. Ellahi et al. (2013) ana-lyzed a non-Newtonian nanofluid with coaxial cylindersfor constant and space-dependent viscosity.

A porous medium is a material which contains anumber of small holes distributed throughout the mat-ter. Flows through porous medium have several applica-tions present in nature: flow in sand beds, in petroleumreservoir rocks, slurries, sedimentation, etc. Many indus-trial applications involve the modeling of flow throughporous media, such as filters, catalyst beds, and packing.Porous materials are used in various engineering devicessuch as catalytic converters and fuel cells due to advan-tages in dispersion and chemical reaction by their largecontact area. The flow of non-Newtonian fluids through aporous medium under different conditions was studied byChamkha et al. (1994), Kumar et al. (2012), and Ellahi etal. (2012).

One non-Newtonian fluid model which has receivedconsiderable attention in the recent past is the Jeffreyfluid. It is one of the simplest models which accountsfor rheological effects of a visco-elastic fluid. The Jeffreyfluid model is a relatively simple linear model using timederivatives instead of convective derivatives. It is a signif-icant generalization of the Newtonian fluid model as thelatter one can be deduced as a special case of the former.It appears to be an appropriate model to describe somephysiological and industrial fluids (Hayat et al., 2006; Va-jravelu et al., 2011). Ellahi et al. (2014) has studied the ef-fects of magnetohydrodynamics on the peristaltic flow ofa Jeffrey fluid in a rectangular duct under the constraintsof low Reynolds number and long wavelength. Pandeyand Tripathi (2010) have considered the peristaltic motionof a Jeffrey fluid under the effect of magnetic field in anasymmetric channel. Khan et al. (2012) has investigatedthe peristaltic flow of a Jeffrey fluid with variable viscos-ity through a porous medium in an asymmetric channel.Jyothi et al. (2013) have considered the pulsatile flow of aJeffrey fluid in a circular tube lined internally with porousmaterial. Akbar et al. (2013) has studied the Jeffrey fluidmodel for the peristaltic flow of chyme in the small intes-tine with a magnetic field. Abd-Alla et al. (2014) haveinvestigated the peristaltic flow of a Jeffrey fluid in anasymmetric channel.

However, the flow of a Jeffrey fluid in tubes of smalldiameter has not received much attention. Recently, San-thosh and Radhakrishnamacharya (2014) studied a two-fluid model for the flow of Jeffrey fluid in the presenceof a magnetic field through a porous medium in tubes ofsmall diameters. The objective of this paper is to studythe effects of porous medium on a two-fluid model for the

flow of a Jeffrey fluid in tubes of small diameters. It is as-sumed that the core region consists of Jeffrey fluid andthe peripheral region consists of Newtonian fluid. Fol-lowing the analysis of Chaturani and Upadhya (1979),the linearized equations of motion have been solved andanalytical solution has been obtained. The analytical ex-pressions for velocity, flow rate, effective viscosity, corehematocrit, and mean hematocrit are obtained. The resultsare depicted graphically and discussed for various rele-vant parameters.

2. PROBLEM FORMULATION

Let us consider an axisymmetric flow of an incompress-ible fluid through a porous medium in a circular tube ofconstant radiusa. It is assumed that the flow in the tubeis represented by a two-fluid model consisting of a coreregion of radiusb, occupied by Jeffrey fluid and a periph-eral region of thicknessε (a − b = ε) by Newtonian fluidas shown in Fig. 1. Letµp andµc be the viscosities of thefluid in peripheral region and core region, respectively.The axisymmetric cylindrical polar coordinate (r, z) ischosen, where thez axis is taken along the axis of thetube.

The equations governing the steady two-dimensionalflow of an incompressible Jeffrey fluid for the presentproblem (Santhosh and Radhakrishnamacharya, 2014)areEquations of Motion:

ρ

[vr

∂

∂r+ vz

∂

∂z

]vr = −∂p

∂r+

1

r

∂

∂r(rSrr)

+∂

∂z(Srz)−

µc

k0vr (1)

FIG. 1: Geometry of the problem

Journal of Porous Media

Flow of a Jeffrey Fluid through a Porous Medium 73

ρ

[vr

∂

∂r+ vz

∂

∂z

]vz = −∂p

∂z+

1

r

∂

∂r(rSzr)

+∂

∂z(Szz)−

µc

k0vz (2)

Equation of continuity:

∂vr∂r

+vrr

+∂vz∂z

= 0 (3)

in which

Srr =2µc

1 + λ1

[1 + λ2

(vr

∂

∂r+ vz

∂

∂z

)](∂vr∂r

)(4)

Srz = Szr =µc

1 + λ1

[1 + λ2

(vr

∂

∂r+ vz

∂

∂z

)]×(∂vz∂r

+∂vr∂z

)(5)

Szz =2µc

1 + λ1

[1 + λ2

(vr

∂

∂r+ vz

∂

∂z

)](∂vz∂z

)(6)

are the extra stress components.Further,vr andvz are the velocity components in ther

andz directions, respectively;λ1 is the ratio of relaxationto retardation times,λ2 is the retardation time;p is thepressure;ρ is the density; andko is the permeability ofthe porous medium.

It is assumed that the flow is in thez-direction onlyand hence the velocity componentvr = 0. Consequently,the equations governing the flow of fluid (Jeffrey fluid) inthe core region (0 ≤ r ≤ b) reduce to:

∂p

∂r= 0 (7)

µc

1 + λ1

1

r

∂

∂r

(r∂vz∂r

)− µc

k0vz −

∂p

∂z= 0 (8)

Let vz(r) = v1(r) be the velocity in the peripheral re-gion andv2(r) be in the core region. The equations gov-erning the flow of fluid are:Peripheral region (Newtonian fluid):

µp1

r

∂

∂r

(r∂v1∂r

)−µp

k0v1 −

∂p

∂z= 0 for b ≤ r ≤ a (9)

Core region (Jeffrey fluid):

µc

1 + λ1

1

r

∂

∂r

(r∂v2∂r

)−µck0

v2−∂p

∂z= 0 for 0≤r≤b (10)

where(∂p)/(∂z) is the constant pressure gradient.

The boundary conditions for the problem are given by

v1 = 0 at r = a (11a)

v1 = v2, τ1 = τ2 at r = b (11b)

v2 is finite at r = 0 (11c)

Condition (11a) is the classical no-slip boundary con-dition for the velocity, (11b) denotes the continuity of ve-locities and stresses at the interface, and (11c) is the reg-ularity condition.

Solving Eqs. (9) and (10) under the conditions (11),we get

v1(ξ) =a2P

µpDa

(1− I0 (Dξ)

I0 (D)

)for d ≤ ξ ≤ 1 (12)

v2(ξ)=a2P

µpDa

[1− I0 (Dd)

I0 (D)+µ′(1− I0 (αξ)

I0 (αd)

)]for 0 ≤ ξ ≤ d (13)

where

ξ =r

a, d =

b

a, P = −∂p

∂z, µ′ =

µp

µc,

Da=k0a2

, D =

√1

Da, α =

√1 + λ1D. (14)

Here Da is the Darcy number. The flow flux in the periph-eral region, denoted byQp, is defined by

Qp = 2πa2∫ 1

d

v1(ξ)ξdξ (15)

Substituting forv1(ξ) from (12) in (15), we get

Qp =a4Pπ

µpDa

(1− d2 + 2

dI1 (Dd)− I1 (D)

DI0 (D)

)(16)

Similarly, the flow flux in the core region is given by

Qc=2πa2∫ d

0

v2(ξ)ξdξ=a4Pπ

µpDa

[d2−d2

I0 (Dd)

I0 (D)

+ µ′(d2 − 2

dI1 (αd)

αI0 (αd)

)](17)

Thus, the flow flux through the tube is given by

Q = Qp +Qc (18)

Volume 18, Number 1, 2015


Using (16) and (17) in (18), we get

Q =a4Pπ

µpDa

[1 + 2

dI1 (Dd)− I1 (D)

DI0 (D)− d2

I0 (Dd)

I0 (D)

+ µ′(d2 − 2

dI1 (αd)

αI0 (αd)

)](19)

Comparing (19) with flow flux for Poiseuille flow, we getthe effective viscosity as

µeff = µp

/{8Da

[1 + 2

dI1 (Dd)− I1 (D)

DI0 (D)

− d2I0 (Dd)

I0 (D)+ µ′

(d2 − 2

dI1 (αd)

αI0 (αd)

)]}(20)

2.1 Mean Hematocrit for Cell-free Wall Layer

The percentage volume of red blood cells is called thehematocrit and is approximately 40%–45% for adult hu-man beings. For example, a hematocrit of 20% means thatthere are 20 millilitres of red blood cells in 100 millilitresof blood; i.e., the red blood cells constitute one fifth of thetotal volume of blood.

The core hematocritHc is related to the hematocritH0

of blood leaving or entering the tube by

H0Q = HcQc (21)

Substituting forQc andQ from (17) and (19) in (21),we get (after simplification),

Hc =Hc

H0= 1

+

1− d2 + 2dI1 (Dd)− I1 (D)

DI0 (D)

d2 − d2I0 (Dd)

I0 (D)+ µ′

(d2 − 2

dI1 (αd)

αI0 (αd)

) (22)

whereHc is the normalized core hematocrit.The mean hematocrit within the tubeHm is related to

the core hematocritHc by

Hmπa2 = Hcπb

2 (23)

On simplification, we get

Hm =Hm

H0=

Hc

H0d2 (24)

whereHm is the normalized mean hematocrit.

Substituting forHc from equation (22) in (24), we get

Hm= d2

1+ 1− d2 + 2dI1 (Dd)− I1 (D)

DI0 (D)

d2−d2I0(Dd)

I0(D)+µ′(d2−2

dI1(αd)

αI0(αd)

) (25)

3. RESULTS AND DISCUSSION

In order to discuss the effects of Darcy number and var-ious other parameters on effective viscosityµeff , corehematocritHc and mean hematocritHm, they havebeen numerically computed and graphically presented inFigs. 2–13. The value ofd (nondimensional core radius)is calculated from the relation:d = 1 − (ε/a) in which

FIG. 2: Effect ofλ1 onµeff (H0 = 20% and Da = 0.5)

FIG. 3: Effect of Da onµeff (H0 = 20% andλ1 = 1.0)



FIG. 4: Effect ofλ1 onµeff (H0 = 40% and Da = 0.5)

FIG. 5: Effect of Da onµeff (H0 = 40% andλ1 = 1.0)

FIG. 6: Effect ofλ1 onHc (H0 = 20% and Da = 0.05)

FIG. 7: Effect of Da onHc (H0 = 20% andλ1 = 1)

FIG. 8: Effect ofλ1 onHc (H0 = 40% and Da = 0.05)

FIG. 9: Effect of Da onHc (H0 = 40% andλ1 = 1)



FIG. 10: Effect ofλ1 onHm (H0 = 20% and Da = 0.05)

FIG. 11: Effect of Da onHm (H0 = 20% andλ1 = 1)

FIG. 12: Effect ofλ1 onHm (H0 = 40% and Da = 0.05)

FIG. 13: Effect of Da onHm (H0 = 40% andλ1 = 1)

ε denotes the peripheral layer thickness for a given hema-tocrit. The values ofε are 3.12µ for 40% hematocrit,3.60µ for 30% and 4.67µ for 20% [see Table 1 in Haynes(1960)].

The variation of effective viscosity (µeff ) for differentvalues of Jeffrey parameter (λ1), Darcy number (Da), tubehematocrit (H0) and tube radius (a) are shown in Figs. 2–5. It is observed that the effective viscosity decreases withJeffrey parameter (λ1) and Darcy number (Da) but in-creases with tube hematocrit (H0) (Figs. 2–5). These re-sults are in agreement with the results of Haynes (1960),Chaturani and Upadhya (1979) and Srivastava (2007).Further, for given values of Jeffrey parameter (λ1), Darcynumber (Da) and tube hematocrit (H0), the effective vis-cosity (µeff ) increases with tube radius (a); i.e., the flowexhibits Fahraeus-Lindqvist. For higher values of Jeffreyparameter (λ1) and Darcy number (Da) the increase in ef-fective viscosity with tube radius is not very significantfor values of tube radius larger than 75µm.

The effects of Jeffrey parameter (λ1), Darcy number(Da), tube hematocrit (H0) and tube radius (a) on corehematrocit (Hc) and mean hematocrit (Hm) are depictedin Figs. 6–13. It is seen that the core hematocrit (Hc) de-creases with the increase in Jeffrey parameter (λ1), Darcynumber (Da), tube hematocrit (H0) and tube radius (a)(Figs. 6–9). However, the decrease in core hematocrit(Hc) with tube radius (a) is not significant for valuesof tube radius greater than 160µm. Figures 10–13 showthat the mean hematocrit (Hm) decreases with Jeffrey pa-rameter (λ1) and Darcy number (Da) but increases withtube hematocrit (H0) and tube radius (a). However, theincrease in mean hematocrit (Hm) with tube radius (a)



is not significant for values of tube radius greater than160µm.

4. CONCLUSION

A two-fluid model for the flow of a Jeffrey fluid throughporous medium in tubes of small diameters is studied.With the assumption that there is Jeffrey fluid in the coreregion and Newtonian fluid in the peripheral region, ana-lytical expressions for effective viscosity, core hematocritand mean hematocrit are obtained. It is noticed that forgiven values of Jeffrey parameter, Darcy number and tubehematocrit, the effective viscosity increases with tube ra-dius. Further, the core hematocrit decreases with tube ra-dius and the mean hematocrit increases with tube radius.

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flow of a jeffrey fluid through a porous medium in narrow tubes

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