PERISTALTIC FLOW OF AN ELLIS FLUID MODEL IN AN …parameters on pumping characteristics and...

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International Journal of Mechanical Engineering and Technology (IJMET) Volume 9, Issue 2, February 2018, pp. 15–27 Article ID: IJMET_09_02_002

Available online at http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=9&IType=2

ISSN Print: 0976-6340 and ISSN Online: 0976-6359

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PERISTALTIC FLOW OF AN ELLIS FLUID

MODEL IN AN INCLINED UNIFORM TUBE

WITH WALL PROPERTIES

K. Thanesh Kumar, A. Kavitha

Department of Mathematics, School of Advanced Sciences, VIT University,

Vellore, India

R. Saravana

Department of Mathematics, Madanapalle Institute of Technology and Science,

Madanapalle, India

ABSTRACT

The present article deals with the peristaltic transport of an Ellis fluid model in an

inclined uniform tube with the wall properties using long wavelength and low

Reynolds number approximations. The analytical expressions have been obtained for

stream function, velocity and temperature distribution. The results are plotted and

discussed in detail for the shear thinning and shear thickening fluid cases. The impact

of various parameters on the flow behavior such as rigidity parameter 1E , stiffness

parameter 2E , viscous damping force parameter 3E and Brickman number are studied.

It is found that the velocity profile is an increasing function of rigidity parameter,

stiffness parameter and viscous damping force parameter due to the less resistance

offered by the walls for shear thinning fluid, but quite opposite behavior is depicted

for shear thickening fluid. It is seen that Brickman number enhances the temperature

for all cases.

Keywords: Peristaltic flow, uniform tube, Shear thickening, shear thinning, Ellis fluid

Cite this Article: K. Thanesh Kumar, A. Kavitha and R. Saravana, Peristaltic Flow of

an Ellis Fluid Model in an Inclined Uniform Tube with Wall Properties, International

Journal of Mechanical Engineering and Technology 9(2), 2018. pp. 15–27.

http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=9&IType=2

K. Thanesh Kumar, A. Kavitha and R. Saravana


1. INTRODUCTION

The complex rheological fluids usually transported from one place to another place with a

special type of pumping known as Peristaltic pumping. The mechanism includes involuntary

periodic contraction followed by relaxation or expansion of the ducts and the mechanism is

observed in physiology situations such as the food propels through the digestive tract, urine

transport from the kidney to the bladder through ureters, lymphatic fluids propels through

lymphatic vessels, bile flows from the gall bladder into the duodenum, spermatozoa move

through the ducts efferentes of the male reproductive tract and cervical canal, ovum moves

through the fallopian tube, and blood circulates in small blood vessels. The engineering

analysis of peristalsis such as aggressive chemicals, high solid slurries, noxious fluid and

other materials are transported by peristaltic pumps. Roller pumps, hose pumps, tube pumps,

ginger pumps, heart lung machines, blood pump machines, and dialysis machines are

engineered on the basis of peristalsis.

The study of peristalsis has received considerable attention in the last few decades mainly

because of its relevance of engineering and biological systems. Several studies have been

made analyzing both theoretical and experimental aspects of the peristaltic motion of

Newtonian and non-Newtonian fluids in different situations. In the context of such

physiological and industrial applications, the dynamics of peristaltic mechanism has been

discussed in detail by various researchers [1-10]. Kavitha et al. [11] studied the peristaltic

transport of a Jeffrey fluid between porous walls with suction and injection. Eldabe et al. [12]

analyzed the MHD peristaltic flow of a couple stress fluids with heat and mass transfer in a

porous medium. Saravana et al. [13] discussed the MHD peristaltic flow of a Jeffrey fluid in a

non-uniform porous medium channel with wall properties, slip conditions, heat and mass

transfer. Saravana et al. [14] reported the peristaltic transport of a third grade fluid in an

inclined asymmetric channel. Kavitha et al. [15] studied the peristaltic transport of Jeffrey

fluid in contact with Newtonian fluid in an inclined channel.

The Ellis equation is given by Rathy [16] is as follows

rz

n

rzrzdr

dwττητη

1

10

−+=

(1)

The Ellis fluid model is one of the fluid model which exhibits the non-linear relationship

between the shear stress and strain rate. The fluid model has its significance depending on

nonlinear factor 1η , for 1η = 0 the model represents the Newtonian fluid, for 1η <0 it represents

shear thickening fluid, and for 1η >0 the model exhibits the behavior of shear thinning fluid.

Narahari et al. [17] studied the peristaltic flow of an Ellis fluid through a circular tube.

Vajravelu et al. [18] analyzed the peristaltic flow of Herschel-Bulkily fluid in an inclined tube

and the results of flow characteristics reveal many interesting behaviors. Nadeem et al. [19]

investigated the tangent hyperbolic fluid in a uniform inclined tube. Hemadri Reddy et al. [20]

investigated the effect of induced magnetic field on the peristaltic pumping of a Carreau fluid

in an inclined symmetric channel filled with porous material. The effects of various

parameters on pumping characteristics and frictional forces are discussed. Hemadri Reddy et

al. [21] made a study on the peristaltic pumping of a non-Newtonian micropolar fluid in an

inclined channel. The pressure rise over one wavelength and frictional force are obtained.

Srinivas et al. [22] discussed the effect of thickness of the porous material on the peristaltic

pumping when the inclined channel walls are provided with non-erodible porous lining. Hari

Prabakaran et al. [23] analyzed the peristaltic transport of a Bingham fluid in contact with a

Newtonian fluid in an inclined channel.

Peristaltic Flow of an Ellis Fluid Model in an Inclined Uniform Tube with Wall Properties


Keeping this in mind, we study the peristaltic flow of an Ellis fluid model in an inclined

uniform tube with the wall properties using long wavelength and low Reynolds number

approximations.

2. MATHEMATICAL FORMULATION

Consider the peristaltic flow of an Ellis fluid through a uniform inclined tube of radius a (Fig.

1) with heat transfer. The peristaltic wave is represented by

( ) ( )ctZSinbatZHR −+==λ

π2,

(1)

Where a is the radius of the tube at inlet, b is the wave amplitude of the tube, λ is the

wavelength and c is the wave speed.

Figure 1 Physical Model

Under the assumptions that the tube length is an integral multiple of the wavelength λ and

the pressure difference across the ends of the tube is a constant, the flow is inherently

unsteady in the laboratory frame ( )ZR ,,θ and becomes steady in the wave frame ( )zr ,,θ

which is moving with velocity c along the wave. The transformation between these two

frames is given by

, ,r R z Z ctθ θ= = = − ,2

2R

−Ψ=ψ and ( ) ( )zptZp =, (2)

Where ψ and Ψ are stream functions in the wave and the laboratory frame respectively.

We assume that the flow is inertia free and the wavelength is infinite. Using the non-

dimensional quantities.



1

2

1

2 2

1

2 3 3 3

1 2 33 3 2

0 1

, , , , , ,

, , , , ,

1 1, , , ,

, , , , , ,( )

n

npo

o rn

o

c e r

p

r z ct F wr z t F w

a c c c

a

crpa q b uacp r q p u

c c a c a k

T TQQ u w

a c a c r z r r T T

c a ca a ma c C aE R E E E B

c T T c c

ττ

λ λ λµµ

µφ

λ µ π λ

ψ ψ ψψ θ

π π

ηρ σδ

λ µ λ µ λ λ µ

+

= = = = = =

= = = = = =

−∂ ∂= = = − = =

∂ ∂ −

′−= = = = = = =

−

2

2

d

o

c

T a (3)

Where ,U w the radial and axial velocities in the wave frame.

The governing equations of motion and energy in simplified form can be written as

(lubrication approach)

0u u w

r r z

∂ ∂+ + =

∂ ∂ (4)

( ) ( )1sinrz zz

e

rw w pR u w g

r z z r r z

τ τδ δ ρ α

∂ ∂∂ ∂ ∂ + = − + + +

∂ ∂ ∂ ∂ ∂ (5)

3 2 ( ) ( )cosrz rr

e

ru u pR u w g

r z r z r r

τ τδδ δ ρ α

∂ ∂∂ ∂ ∂ + = − + + −

∂ ∂ ∂ ∂ ∂ (6)

2 22 2

2 2

1Pre r r r z z r z z

u w u wR u w Br

r z r r r z r r z r

θ θ θ θ θδ δ δτ τ δ τ τ δ

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + = + + + + + +

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ (7)

0w = , at ( )2

sin ( )r h z a b z tπ

λ= = + − (8)

0 0rz at rτ = = (9)

0=Ψ , at 0r = (10)

0θ = , at r h= (11)

0r

θ∂=

∂ , at 0r = (12)

Under the assumption of long wavelength and low Reynolds number Equations (5) - (7)

takes the following form

[ ]1 1

sinrz

pr

r r z Fτ α

∂ ∂= −

∂ ∂ (12)

rz

n

rzrzdr

dwττητ

1

1

−+= (13)

2

2

10rz

wBr

r r r r

θ θτ

∂ ∂ ∂ + + =

∂ ∂ ∂ (14)

Where r c rB E P= and

1

n

n

cF

ga

µ

ρ +=

The governing equation of motion of the flexible wall is given by



0( )L H p p= − (15)

Where L is the operator that is used to characterize the motion of the stretched membrane

with damping forces and 0p is the pressure on the outside surface of the wall due to tension in

the muscle, which is assumed to be zero, and L can be written as

2 2

2 2L m C

z t tσ

∂ ∂ ∂′= − + +

∂ ∂ ∂ (16)

After dimensionless it becomes

3 3 2

1 2 33 2

sin ( )p L h h h hE E E A

z F z z z t z t

α ∂ ∂ ∂ ∂ ∂− = = + + =

∂ ∂ ∂ ∂ ∂ ∂ ∂ at r h= (17)

3. SOLUTION OF THE PROBLEM

Equation (12) after simplification with boundary condition (9) can be written as

2r z

rAτ =

, (18)

Where sin

, ,p

A p k p kz F

α∂= − = =

∂

Substituting (18) in to (13) and using boundary condition (8) we get

( ) ( )2 2 1 1 1

4 2 ( 1)

nn n

n

AAw r h r h

n

η+ += − + −+

(19)

Now solving equation (14) using (11) and (12) we get

( ) ( )12

4 4 3 3 1

1 264 2 ( 3)

nn n

n

Br ABrAh r h r

n

ηθ

++ +

+= − + −

+ (20)

Velocities in terms of stream function relation can be defined as

1 1,u w at r h

r z r r

ψ ψ∂ ∂= − = =

∂ ∂ and 0 0at rψ = = (21)

where [ ]3

1 2 3

sin 2 ( )( , ) 8 cos 2 ( ).( )

2

z tA z t z t E E E

ππ φ π

π

− = − − + +

4. RESULTS AND DISCUSSIONS

Velocity Profiles

Equation (20) gives the expression for velocity as a function of w. Velocity profiles are

plotted from Figs 2 - 8. From Figs. 2, 3 and 4 are drawn to study the effect of rigidity

parameter, stiffness parameter and viscous damping force parameter 1, ,

2 3E E E on the velocity

distribution w. From the figures, It is found that the velocity profiles are parabolic and the

velocity increases with increasing 1,

2E E and

3E . Figs. 5 and 6 are drawn to study the effect of

Ellis parameter 1η in the case of shear thinning and shear thickening on velocity distribution

w. It is observed that the velocity increases for shear thinning ( 1η >0) and opposite behavior is

depicted for shear thickening case ( 1η <0). Figs. 7 and 8 are drawn to study the effect of

gravity parameter F and α (angle of inclination) on the velocity distribution w. It is observed

that the velocity increases with increasing F and α .



Figure 2 The variation of w with r for different values of 1E for fixed 2E =1, 3E =1.5, z=0.22, t=0.25,

n=2, φ =0.01, 1η =0.5, α = / 3π , F=0.5

Figure 3 The variation of w with r for different values of 2E for fixed 1E =0.5, 3E =1.5, z=0.22,

t=0.25, n=2, φ =0.01, 1η =0.5, α = / 3π , F=0.5



Figure 4 The variation of w with r for different values of 3E for fixed 1E =0.5, 2E =1, z=0.22, t=0.25,

n=2, φ =0.01, 1η =0.5, α = / 3π , F=0.5

Figure 5 The variation of w with r for different values of 321

,, EEE with 1η =0.5 and for fixed

z=0.22, t=0.25, n=2, φ =0.01, α = / 3π , F=0.5



Figure 6 The variation of w with r for different values of 321

,, EEE with 1η = -0.5 and for fixed

z=0.22, t=0.25, n=2, φ =0.01, α = / 3π , F=0.5

Figure 7 The variation of w with r for different values of F for fixed 1E =0.5, 2E =1, 3E =1.5, z=0.22,

t=0.25, n=2, φ =0.01, 1η =0.5, α = / 3π



Figure 8 The variation of w with r for different values of α for fixed 1E =0.5, 2E =1, 3E =1.5, z=0.22,

t=0.25, n=2, φ =0.01, 1η =0.5, F= 0.5

Temperature Profiles

Equation (22) gives the expression for temperature as a function of r, the temperature profiles

are almost parabolic are plotted from Figs. 9 to 14. In Figs 9- 12, we observed that

temperature is increases with increasing Br, 1E , 2E , 3E . Figs. 13 and 14 are drawn to study the

effect of temperature for shear thinning and shear thickening case. It is observed that

temperature is increases in shear thinning case and opposite behavior is depicted for shear

thickening case.

Figure 9 The variation of θ with r for different values of Br, for fixed 1E =0.5, 2E =1, 3E =1.5,

z=0.22, t=0.25, n=2, φ =0.01, 1η =0.5, α = / 3π



Figure 10 The variation of θ with r for different values of 1E , for fixed 2E =1, 3E =1.5, Br=1,

z=0.22, t=0.25, n=2, φ =0.01, 1η =0.5, α = / 3π

Figure 11 The variation of θ with r for different values of 2E , for fixed 1E =0.5, 3E =1.5, Br=1

z=0.22 , t=0.25, n=2, φ =0.01, 1η =0.5, α = / 3π



Figure 12 The variation of θ with r for different values of 3E , for fixed 1E =0.5, 2E =1, Br=1,

z=0.22 , t=0.25, n=2, φ =0.01, 1η =0.5, α = / 3π

Figure 13 The variation of θ with r for different values of 1η >0, for fixed 1E =0.5, 2E =1, 3E =1.5,

Br=1, z=0.22, t=0.25, n=2, φ =0.01,α = / 3π

Figure 14 The variation of θ with r for different values of 1η <0, for fixed 1E =0.5, 2E =1, 3E =1.5,

Br=1, z=0.22, t=0.25, n=2, φ =0.01,α = / 3π



5. CONCLUSIONS

In the current study, we have discussed the peristaltic flow of an Ellis fluid in an inclined

uniform tube with wall properties. The exact solution is calculated for velocity and

temperature profiles.

• Velocity profiles are increasing for increasing rigidity parameter, stiffness parameter.

For viscous damping force parameter, the velocity increases in small variations.

• Velocity profiles are increasing for Ellis parameter in shear thinning case and opposite

behavior depicted for shear thickening case.

• Velocity profiles are increasing for increasing gravity parameter F and angle of

inclinationα .

• Temperature profiles are increasing for increasing Brickman number Br, 1E , 2E , 3E .

• Temperature profiles are increasing in shear thinning case and opposite behavior is

observed in shear thickening case.

REFERENCES

[1] Latham T W. Fluid motions in a peristaltic pump, M.S. Thesis, MIT, Cambridge,

Massachusetts, 1966.

[2] Akbar N S, Nadeem S, Lee C, Characteristics of Jeffrey fluid model for peristaltic flow of

chimed in small intestine with magnetic field. Results in Physics 3;(2013):152-160.

[3] Akbar N S, Nadeem S, Lee C, Zafar Hayat Khan, Rizwan Ul Haq, Numerical study of

Williamson nano fluid in an asymmetric channel, Results in Physics,3;9(2013): 161-166.

[4] Shit G C, Roy M. Hydro magnetic effect on inclined peristaltic flow of a couple stress

fluid Alexandria Eng. 53, (2014) 949-958.

[5] Abd-Alla A M, Abo-Dahab S M, and EI-Shaharany HD. Influence of heat and mass

transfer initial stress and radially varying magnetic field on the peristaltic flow in an

annulus with gravity field Magn Mater 363; (2014): 166-178.

[6] Abbas M A, Bai Y Q, Rashidi M M, Bhatti M M. Application of drug delivery in magneto

hydrodynamics peristaltic blood flow of nano fluid in a non-uniform channel. J. Mech

Med Bio. (2015): 1650052.

[7] Akbar NS, Raza M, Ellahi R, Influence of induced magnetic field and hear flux the

suspension of carbon nanotubes for the peristaltic flow in a permeable channel. J.Magn

Mater 381, 2015, 405-415.

[8] Saravana R, Hemadri Reddy R, Suresh Goud J, Sreenadh S, MHD Peristaltic flow of a

Hyperbolic tangent fluid in a non-uniform channel with heat and mass transfer, IOP Conf.

Series: Materials Science and Engineering 263 (2017) 062006, 1-15. doi:10.1088/1757-

899X/263/6/062006.

[9] Ramesh k, Debakar M, Magneto hydrodynamics peristaltic transport of couple stress fluid

through porous medium in an inclined asymmetric channel with heat transfer. J.magm

Mater 394, (2015):335-348.

[10] Sinha A, Shit G C, Ranjit N K, Peristaltic transport of MHD flow and heat transfer in as

asymmetric channel: Effects of variable viscosity, velocity slip and temperature jump.

Alexandria Eng J (2015). (Add page number, volume etc.)

[11] Kavitha A, Hemadri Reddy R, Sarvana R, Sreenadh S, Srinivas ANS. Peristaltic transport

of a Jeffery fluid between porous walls with suction and injection, International Journal of

Mechanical and Material Engineering. 7(2) (2012), 152-157.



[12] Eldabe N T, Elshaboury S M, Alfaisal A H. MHD peristaltic flow of a couple stress fluids

with heat and mass transfer through porous medium. Innovative Systems Design and Eng.,

2013, 3, 51-67.

[13] Saravana R, Sreenadh S, Venkataramana S, Hemadri Reddy R, Kavitha A, Influence of

slip conditions, wall properties and heat transfer on MHD peristaltic flow of a Jeffrey fluid

in a non-uniform porous channel. Int Journal of Inovat Technol Creat Eng. 2011, 1 (11):

10-24.

[14] Saravana R, Hemadri Reddy R, Sreenadh S, Venkataramana S, Kavitha A, Influence of

slip, heat and mass transfer on Peristaltic transport of third grade fluid in an inclined

asymmetric channel, Int J Appl Math Mech 9 (11), 51-86.

[15] Kavitha A, Hemadri Reddy R, Sarvana R, Sreenadh S. Peristaltic transport of a Jeffrey

fluid in contact with a Newtonian fluid in an inclined channel. Ain Shams Engineering

Journal 8 (2017), 683–687.

[16] Rathy, R.K. (1976). An introduction to fluid dynamics, Oxford & IBH Publishing Co.,

New Delhi, Bombay, Calcutta.

[17] Narahari M, Sreenadh S, Arunachalam P.V. Peristaltic pumping flow of an-Ellis fluid

through a circular tube, Bulletun of Pure & applied Sciences Vol. 19E (No 1) 2000; P

493-503.

[18] Vajravelu K, Sreenadh S, Ramesh Babu V, Peristaltic transport of a Herschel–Bulkley

fluid in an inclined tube, International Journal of Non-Linear Mechanics, Volume 40 Issue

1, January 2005, Pages 83–90.

[19] Sohail Nadeem and Noreen Sher Akbar Series Solutions for the Peristaltic Flow of a

Tangent Hyperbolic Fluid in a Uniform Inclined Tube, Z. Naturforsch. 65a, 887 – 895

(2010).

[20] Hemadri Reddy R, Kavitha A, Sreenadh S, and Saravana R, Effect of induced magnetic

field on peristaltic transport of a carreau fluid in an inclined channel filled with porous

material. International Journal of Mechanical and Materials Engineering, Vol.6 (2011),

No.2, 240-249.

[21] Hemadri Reddy R, Kavitha A, Sreenadh S, Hariprabakran P, Peristaltic Pumping of a

Micropolar Fluid in an Inclined Channel International Journal Of Innovative Technology

& Creative Engineering (ISSN: 2045-711) Vol.1 No.6 June 2012.

[22] Srinivas A.N.S., Hemadri Reddy R, Sreenadh S, and Kavitha A, Effect of thickness of the

porous material on the peristaltic pumping when the inclined channel walls are provided

with non-erodible porous lining, Applied Mathematics Elixir Appl. Math. 48 (2012)

pp. 9510-9513

[23] Hari Prabakaran P, Hemadri,Reddy R, Sreenadh S, Saravana R and Kavitha A, Peristaltic

Pumping Of A Bingham Fluid In Contact With A Newtonian Fluid In An Inclined

Channel Under Long Wavelength Approximation, Advances and Applications in Fluid

Mechanics Volume 13, Number 2, 2013, Pages 127-139.

PERISTALTIC FLOW OF AN ELLIS FLUID MODEL IN AN …parameters on pumping characteristics and...

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