Effect Of Elasticity On Herschel - Bulkley Fluid Flow In A Tube

CH.Badari Narayana et al., International Journal of Advanced Research in Innovative Discoveries in Engineering and Applications[IJARIDEA]
Vol.1, Issue 2,27 December 2016, pg. 9-16
9
© 2016, IJARIDEA All Rights Reserved
Effect Of Elasticity On Herschel - Bulkley Fluid
Flow In A Tube
CH.Badari Narayana1
, P.Devaki2
, S.Sreenadh3
1
Department of Mathematics, Hindu Degree College, Machilipatanam, Andhra Pradesh, 521001,India
2
Department of Mathematics, Madanapalle Institute of Technology and Science, Madanapalle, Andhra Pradesh, 517 325, India.
3
Department of Mathematics, Sri Venkateswara University, Tirupati, Andhra Pradesh, 517 501, India.
Abstract— The impact of flexibility on Herschel-Bulkley liquid in a tube is researched. The issue is unraveled scientifically
for two unique sorts: one flux is computed taking the worry of the flexible tube into thought and the other flux is acquired by
considering the weight range relationship. Speed of the inelastic tube is additionally considered. The impact of various
parameters on flux and speed are talked about through charts. The outcomes acquired for the stream attributes uncover
many intriguing practices that warrant additionally contemplate on the non-Newtonian liquid stream wonders, particularly
the shear-diminishing marvels. Shear diminishing decreases the divider shear stretch.
Keywords— Elastic tube, Herschel-Bulkley Fluid, Inlet pressure, Outlet Pressure, Yield Stress.
I. INTRODUCTION
As of late, the stream of non-Newtonian liquids in a versatile tube is notable to be vital kind of stream
and finds different viable applications in building and prescription. It is outstanding that the circulatory
arrangement of a living body works as a method for transporting and dispersing fundamental substances to
the tissues and evacuating by-results of digestion system. Subsequently there must be sufficient course at
all circumstances to the imperative organs of the body. Such a vital circulatory framework is comprised of
heart and veins. Veins are characterized as corridors, vessels and veins. These veins are versatile in nature.
Demonstrating of veins assumes a crucial part in the field of pharmaceutical for planning counterfeit
organs of the body.
Many creators are intrigued on non-Newtonian liquid streams in flexible tubes due to their applications
in the realm of prescription. Rubinow and Keller [1] made a logical investigation of the streams in
versatile tubes and talked about the pertinence of their outcomes to the issue of blood stream in corridors.
Fung [2] concentrated the stream of Newtonian liquid in a versatile tube. Vajravelu et al. [3] concentrated
the stream of Herschel-Bulkley liquid in a flexible tube. The stream of Newtonian and power law liquids
in flexible tubes was explored by Sochi [4]. All the more as of late Peristaltic transport of Herschel
Bulkley liquid in a flexible tube was researched by Vajravelu et al.[5].
Biofluids are liquids which are available in channels of the living beings. Blood, Interstitial liquid,
Sweat, Mother drain, tears, Cerebrospinal Fluid, thus on are a portion of the cases of biofluids. Herschel-
Bulkley liquid is a semisolid instead of a genuine liquid. A point by point exchange of the impropriety of
the utilization of such models for liquids is talked about in the late audit paper by Krishnan and Rajagopal
[6]. While such materials won't not be liquids, there is esteem in considering them as they give some
thought of the conduct of liquids of enthusiasm under specific breaking points.
These works has spurred the creators to focus on the enduring laminar stream of a Herschel-Bulkley
liquid in a flexible tube. In this paper the impact of shear diminishing, shear thickening, versatility on
liquid stream attributes has been talked about. Diagrams are plotted for showing the conduct of various
parameters on flux and speed.

10
II. FORMULATION AND SOLUTION OF THE PROBLEM
Consider the Poiseuille flow of a Herschel-Bulkley fluid in an elastic tube of radius ( )a z . The flow
is axisymmetric. The axisymmetric geometry facilitates the choice of the cylindrical coordinate system
( , , )R Zθ to study the problem. The fluid enters the tube at the pressure p1 and leaves it at the pressure 2p ,
1( )p< . while the pressure outside the tube is 0p . Let z denote the distance along the tube from the inlet
end. The pressure of the fluid in the tube decreases from 1(0)p p= to 2( )p L p= . The pressure difference
at z inside the tube is denoted by 0(0)p p− . Due to the pressure difference between inside and the outside
the tube, the tube may expand or contract. which will be due to the elastic property of the wall . This
conductivity σ of the tube at z will be function of pressure difference.
The governing equations reduce to:
( )
1
rz
p
r
r r z
τ
∂ ∂
= −
∂ ∂
(1)
where rzτ is the shear stress and is given by
0
n
rz
u
r
τ µ τ
∂ 
= − + 
∂ 
(2)
Here 0τ represents the yield stress of the tube, n is the power – law index and µ is the viscosity.
A. Non-dimensionalization of The Flow Quantities
The following non-dimensionalized quantities are introduced to make the basic equations and the
boundary conditions dimensionless:
2
0 0
1
0
2
0 0
, , , ,
, , ,
n
n
r z u q
r z u q
a L U a U
aq a
Q a P P
a U a L U
π
π µ
+
= = = =
= = =
( ) ( )
( )
0
0
0 0
0
0
00
, ,
,
rz
rzn n
a
a n
U a U a
T r
T r
aU a
τ τ
τ τ
µ µ
µ
= =
= =
(3)
where 0a is the radius of the tube in the absence of elasticity, L is the length of the tube and U is the
average velocity of the fluid.
The non-dimensional governing equations are (dropping the bars)
( )
1
rzr P
r r
τ
∂
=
∂
(4)
where
0
n
rz
u
r
τ τ
∂ 
= − + 
∂ 
and (5)

11
p
P
z
∂
= −
∂
(6)
The non-dimensional boundary conditions are
rzτ is finite at 0r = (7)
0u = at r a= (8)
B. Exact Solution Of The Problem
Solving equations (4) and (5) subjected to conditions we obtain the velocity field as
( )
1 1
0 0
2
.
1 2 2
k k
P P
u a r
P k
τ τ
+ +
    
= − − −    
+      
(9)
where
1
k
n
= Using the boundary condition
00 at
u
r r
r
∂
= =
∂
the upper limit of the plug flow region is obtained as
0
0
2
.r
P
τ
=
Also by using the condition atyx a r aτ τ= = (Bird et. al., [13]) we obtain
2 a
P
a
τ
=
Hence
0 0
, 0 1.
a
r
a
τ
τ τ
τ
= = < < (10)
Using relation (10) and taking 0r r= in equation (9), we get the plug flow velocity as
( )
1
1
01 for 0 .
2 1
k k
k
p
P a
u r r
k
τ
+
+ 
= − ≤ ≤ 
+ 
(11)
The volume flux Q through any cross-section is given by
0
0
3
0
r a
k k
p
r
Q u r dr u r dr Fa P+
= + =∫ ∫ (12)
where
( )
( )
( )( )
( )( )
1
1
1 2 1 2
1
2 1 2 3
k
k
k
k k k
τ τ τ
+
+
 − − + +
− 
+ + + 
(13)
From equation (2.12), we find that,
1 1
3
1
k k
k
dp Q
dz F a +
   
= −   
   
` (14)
Due to elastic property of the tube wall, the tube radius ‘a’ is a function of z. Integrating equation (14), we
get
( )
3 1
0
1
( ) (0)
( )
n z
n
Q
p z p dz
F a z
+
 
= − 
 
∫ (15)

12
The integration constant is (0)p , the pressure at 0z = . The exit pressure is given by equation (2.15)
with 1z = .
Now, let us turn our attention to the calculation of the radius ( )a z Let the tube be initially straight and
uniform, with a radius 0a . Assume that the tube is thin walled and that the external pressure is zero. (If the
external pressure is not zero, we should replace p , below , by the difference of internal and external
pressures). Then a simple analysis yields the average circumferential stress in the wall:
( ) ( )p z a z
h
θθσ = (Fung, [2]) (16)
where h is the wall thickness. Let the axial tension be zero, and assume that the material obeys Hooke’s
law. Then the circumferential strain is
1
( )zz rre v v
E
θθ θθσ σ σ= − − (17)
where υ is the Poisson’s ratio and E is the Young’s modules of the wall material. But zzσ is assumed to
be zero and rrσ is, in general, much smaller than θθσ for thin-walled tubes. Hence equation (17) reduces to
e
E
θθ
θθ
σ
= (18)
The strain eθθ is equal to the change of radius divided by the original radius, 0a :
0
0 0
( ) ( )
1
a z a a z
e
a a
θθ
−
= = − (19)
Combining (16), (18) and (19) we obtain:
1
0
0( ) 1 ( )
a
a z a p z
Eh
−
 
= − 
 
(20)
Substituting (20) into (14), we may write the result as
(3 1)
0
3 1
0
1
1 ( )
n n
n
a Q
p z dp dz
Eh F a
− +
+
   
− = −  
  
(21)
Recognizing the boundary conditions (0)p p= when 0z = and (1)p p= when 1z = and integrating
Equation (20) from (0)p to (1)p on the left and 0 to 1 on the right, we obtain the pressure-flow
relationship:
3 3
0 0
3 1
10 0
1
1 (1) 1 (0)
3
n n n
n
a aEh Q
p p
na Eh Eh F a
− −
+
       
− − − =      
      
(22)
which shows that the flow is not a linear function of pressure drop (0) (1)p p− . where F is given by
equation (13)
Newtonian fluid : 0, 1nτ = =
In this case equation (21) becomes
3 3
0 0
4
10 0
1
1 (1) 1 (0) 12
3
a aEh
p p Q
a Eh Eh a
− −
     
− − − =    
      (23)
When 1n = and 0τ = eq (22) reduces to the corresponding results of Fung [2] for the flow of Newtonian
fluid in an elastic tube.
Bingham Fluid: 1n =
In this case equation (2.21) becomes

13
3 3
0 0
4
10 0
1 (1) 1 (0)
3
a aEh Q
p p
a Eh Eh Fa
− −
     
− − − =    
      (24)
where
2 3 41
6(1 ) 4(1 ) (1 )
48
F τ τ τ = − − − + −  (25)
Power-Law Fluid: 0τ =
In this case equation (2.21) become
3 3
0 0
3 1
10 0
(12 )
1 (1) 1 (0)
3
n n n
n
a aEh Q
p p
na Eh Eh a
− −
+
     
− − − =    
      (26)
III. ANOTHER SOLUTION OF THE PROBLEM
The solution obtained above is based on the assumption that the tube-wall material obeys Hooke’s law.
But blood vessel walls need not obey this law. Hence the solution may not be valid for blood vessel in
general except possibly as an approximation.
A simpler result can be obtained if we assume the pressure-radius relationship to be linear:
0 2a a ap= + (27)
Hence 0a
is the tube radius when the transmural pressure is zero. α is a compliance constant. Equation
(3.1) is a suitable representation of the pulmonary blood vessels. Using Eq (27), we have
2dp dp da da
dz da dz dzα
= =
(28)
On substituting Eq (28) into Eq(14) and arranging terms, we obtain
[ ]
3 2
3 1 1
( )
3 2 2
nn
n da da Q
a z
dz n dz F
α+
+  
= = −  
+   (29)
Since the right hand side term is a constant independent of z, we obtain at once the integrated result
[ ] ( )
3 2
( ) 3 2
2
n
n Q
a z n C
F
α+  
= − + + 
  (30)
The integration constant can be determined by the boundary condition that when 0, ( ) (0)z a z a= = . Hence
[ ]
3 2
(0)
3 2
n
a
C
n
+
=
+ Then by putting 1z = , we obtain the following result from Eq.(30)
[ ] [ ]
1
13 2 3 2
(3 2)
2
(0) (1)
n
nn n
n Q
F
a a
α
+ +
+ 
 
 
 = −
  (31)
where F is given by eq (13)
The pressure-flow relationship is obtained by substituting Eq.(27) into Eq.(31). Thus for a Newtonian
fluid (n=1) the flow varies with the difference of the fifth power of the tube radius at the entry section

14
( 0z = ) minus the exit section ( 1z = ). If the ratio ( ) (0)a z a is 1/2, then [ ]
3 2
(1)
n
a
+
is only about of
[ ]
3 2
(0)
n
a
+
, and is negligible by comparison. Hence when (1)a is one-half of (0)a or smaller, the flow
varies directly with the fifth power of the tube radius at the entry, whereas the radius (and the pressure) at
the exit section has little effect on the flow.
Deductions:
(i) Newtonian fluid : 0, 1nτ = =
[ ] [ ]
5 5
30 (0) (1)Q a aα = −
(ii) Bingham Fluid: 1n =
[ ] [ ]
5 55
(0) (1)
2
Q
a a
F
α
= −
where
2 3 41
6(1 ) 4(1 ) (1 )
48
F τ τ τ = − − − + − 
(iii) Power-Law Fluid: 0τ =
[ ] [ ]( )
1
13 2 3 2(3 2)
12 (0) (1)
2
n
nn nn
Q a a
α + ++ 
= − 
 
IV.RESULTS AND DISCUSSION
In this issue the unfaltering laminar stream of a Herschel-Bulkley liquid in a versatile tube is examined.
As the tube is versatile in nature, the stream of the liquid in the tube will be affected because of stress and
because of weight. So the flux of the flexible tube, is figured in two distinct strategies: first is by
considering worry of the versatile tube and the other by considering the weight span relationship. Speed of
the inelastic tube is likewise assessed. The impacts of various parameters like delta weight, outlet weight,
Young's modulus, consistence consistent, yield stress are talked about through charts.
Case 1: Stress is considered
Fig.2 demonstrates the variety of flux with sweep for various estimations of yield stress. At a given
span of the tube the flux diminishes with the expanding yield push. For a given yield stretch, the flux
diminishes with expanding range. From this we watch that as the liquid turns out to be thick as tooth glue
the liquid stream turns out to be moderate. It is seen from Fig.3 that as the Young's modulus expands, the
flux of the flexible tube additionally increments. As the gulf weight expands, the flux is diminishing which
is appeared in Fig.4. In any case, in the event of outlet weight the stream conduct is distinctive, that is as
the outlet weight expands the flux is expanding which is appeared in fig.5. It is genuine in light of the fact
that if the outlet weight is settled and gulf weight is all the more then the liquid stream will get turned
around once in a while or the stream turns out to be moderate, though the delta weight is settled and outlet
weight is all the more, then the liquid stream will be more.
Case 2: Pressure-Radius connection is considered

15
It is seen from Fig.6 that as the consistence steady builds the flux of the flexible tube is expanding. Fig.7
demonstrates the variety of flux with sweep for various yield stretch qualities. Here we see that the flux of
the flexible tube diminishes with the expanding yield push. For shear diminishing (n<1) and shear
thickening (n>1) liquids it is watched that as the file builds the liquid stream is diminishing. This conduct
is seen in Figs. (8&9). From both the cases it is watched that the liquid stream is more if weight is
considered than the thought of stress. One can comprehend from this that a flexible tube can make the
liquid stream easily if the tube experiences certain weight yet in the event that the versatile tube
experiences certain anxiety then the liquid stream may not be so natural to stream. Additionally from
Fig.10, we see that as the file of the liquid expands the speed is diminishing. Fig.11 demonstrates the
variety of speed with sweep for various yield push values. As the yield push builds the speed of the liquid
is observed to diminish. Christo Ananth et al. [9] proposed a system, this fully automatic vehicle is
equipped by micro controller, motor driving mechanism and battery.
Fig.1 Physical Model
0.5 1 1.5 2 2.5 3
0
0.5
1
1.5
2
2.5
3
3.5
x 10
-15
RADIUS
FLUX
τ=0.2
τ=0.3
τ=0.4
0.5 1 1.5 2 2.5 3
0
0.2
0.4
0.6
0.8
1
1.2
x 10
-17
RADIUS
FLUX
E=0.02
E=0.021
E=0.023
Fig.2 Variation of flux with radius for different yield stress, when
stress is considered for fixed values of α=0.1,n=0.5,E =0.02.
Fig.3 Variation of flux with radius for different Young’s modulus,
when stress is considered for fixed values of α=0.1,n=0.5,τ =0.3.

16
0.5 1 1.5 2 2.5 3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
x 10
-16
RADIUS
FLUX
p(0)=0.3
p(0)=0.32
p(0)=0.35
0.5 1 1.5 2 2.5 3
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x 10
-16
RADIUS
FLUX
p(1)=0.6
p(1)=0.62
p(1)=0.65
Fig.4 Variation of flux with radius for different Inlet Pressure values,
Fig.5 Variation of flux with radius for different Outlet Pressure values,
V. CONCLUSION
The impact of flexibility on Herschel-Bulkley liquid in a tube is researched. The issue is unraveled
scientifically for two unique sorts: one flux is computed taking the worry of the flexible tube into thought
and the other flux is acquired by considering the weight range relationship. Speed of the inelastic tube is
additionally considered. The impact of various parameters on flux and speed are talked about through
charts. The outcomes acquired for the stream attributes uncover many intriguing practices that warrant
additionally contemplate on the non-Newtonian liquid stream wonders, particularly the shear-diminishing
marvels. Shear diminishing decreases the divider shear stretch..
REFERENCES
[1] Rubinow. S.I. And J. B. Keller: Flow of a viscous fluid through an elastic tube with applications to blood flow, J. Theor. Biology., 1972, 35, 299-
313.
[2] Fung. Y.C., Biodynamics, Spriger-VErlag, New York Berlin Heidelberd Tokyo, 1984, Chap.3, Sec, 3.4.
[3] Vajravelu, K, Sreenadh. S, Devaki. P, and Prasad. K. V. Mathematical model for a Herschel-Bulkley fluid flow in an elastic tube, Central European
Journal of Physics, 2011, 9, 1357-1365.
[4] Sochi T., The flow of Newtonian and power law fluids in elastic tubes, Int. J. Non-Lin. Mech., 2014,67, 245–250.
[5] Vajravelu, K, Sreenadh.S, Devaki.P ,and Prasad. K. V. Peristaltic transport of Herschel Bulkley fluid in an elastic tube, Heat transfer Asian
Research , 2015, 44(7), 585-598.
[6] Krishnan J.M., Rajagopal K.R., Review of the uses and modeling of bitumen from ancient to modern times Appl. Mech. Rev., 2003, 56, 149.
[7] Vajravelu,K.,Sreenadh,S., and Ramesh Babu,V, Peristaltic pumping of a Herschel-Bulkley fluid in a Channel ,Applied Mathematics and
Computation,2005a, 169, 726.
[8] Vajravelu,K.,Sreenadh,S., and Ramesh Babu,V., Peristaltic Transport of a Herschel – Bulkley fluid in an inclined tube, Int.J.of Non-Linear Mech,
2005b, 40,83-90.
[9] Christo Ananth, M.A.Fathima, M.Gnana Soundarya, M.L.Jothi Alphonsa Sundari, B.Gayathri, Praghash.K, "Fully Automatic Vehicle for
Multipurpose Applications", International Journal Of Advanced Research in Biology, Engineering, Science and Technology (IJARBEST), Volume
1,Special Issue 2 - November 2015, pp.8-12.
[10] Sankad G.C., Nagathan P.S., Asha Patil, Dhange M.Y., Peristaltic Transport of a Herschel -Bulkley Fluid in a Non-Uniform Channel with Wall
Effects, Int. J. Eng. Sci. Inn. Tech. 2014, 3(3) 669-678.
[11] Santhosh Nallapu, Radhakrishnamacharya G., Chamkha Ali J., Effect of slip on Herschel–Bulkley fluid flow through narrow tubes, Alex. Eng. J.,
2015, 54(4), 889–896.
[12] Sankad G.C. and Patil Asha.B., Peristaltic Flow of Herschel Bulkley Fluid in a Non-Uniform Channel with Porous Lining , Proc. Eng. , 2015, 127,
686–693.
[13] Bird, R. B., G. C. Dai and B. J. Yarusso, The Rheology and Flow of Viscoplastic Materials, Rev. Chem. Eng. , 1983, 1, 1–70.
[14] S. Abdul Gaffar, O. Anwar Beg, E. Keshava Reddy, V. Ramachandra Prasad, "Mixed Convection Magnetohydrodynamic Boundary Flow And
Thermal Convectionof Non-Newtonian Tangenthyperbolic Fluid From Non-Iosthermal Wedgewith Biot Number Effects.", International Journal of
Advanced Research in Innovative Discoveries in Engineering and Applications[IJARIDEA], Volume 1,Issue 1,October 2016,pp:7-11

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