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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

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 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. HerschelBulkley 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.

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

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, θ , Z ) to study the problem. The fluid enters the tube at the pressure p1 and leaves it at the pressure p2 ,

(< p1 ) . while the pressure outside the tube is p0 . Let z denote the distance along the tube from the inlet end. The pressure of the fluid in the tube decreases from p(0) = p1 to p( L) = p2 . The pressure difference at z inside the tube is denoted by p (0) − p0 . 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 ∂ ∂p (1) ( rτ rz ) = − r ∂r ∂z where τ rz is the shear stress and is given by n

 ∂u  τ rz = µ  −  + τ 0  ∂r  Here τ 0 represents the yield stress of the tube, n is the power – law index and µ is the viscosity.

(2)

A. Non-dimensionalization of The Flow Quantities

The following non-dimensionalized quantities are introduced to make the basic equations and the boundary conditions dimensionless: r z u q r = , z = ,u = ,q = , a0 L U π a02U Q=

τ0 =

a0n +1 q a = = , a , P P, π a02U a0 L µU n

τ0 µ (U a0 )

,τ rz =

τ rz µ (U a0 )

r Ta = , r0 = 0 n a0 µ (U a0 )

(3)

where a0 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 ∂ (4) ( rτ rz ) = P r ∂r where n

 ∂u  τ rz =  −  + τ 0 and  ∂r 

(5)

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

∂p ∂z The non-dimensional boundary conditions are τ rz is finite at r = 0 u = 0 at r = a P=−

(6) (7) (8)

B. Exact Solution Of The Problem

Solving equations (4) and (5) subjected to conditions we obtain the velocity field as k +1 k +1  P 2  P   u=  a − τ 0  −  r − τ 0   . P ( k + 1)  2  2   1 where k = Using the boundary condition n ∂u = 0 at r = r0 the upper limit of the plug flow region is obtained as ∂r 2τ r0 = 0 . P Also by using the condition τ yx = τ a at r = a (Bird et. al., [13]) we obtain 2τ a a Hence r0 τ 0 = =τ, a τa

(9)

0 < τ < 1.

(10)

Using relation (10) and taking r = r0 in equation (9), we get the plug flow velocity as k

1+ k

1+ k P a up =   (1 − τ ) for 0 ≤ r ≤ r0 .  2  1+ k The volume flux Q through any cross-section is given by

Q = ∫ u p r dr + ∫ u r dr = Fa k +3 P k

(11)

(12)

where k +1

(1 − τ ) 1 − 2 (1 − τ )(τ + k + 2 )    2k +1 ( k + 1)  ( k + 2 )( k + 3) 

(13)

From equation (2.12), we find that, 1k

dp Q  1  = −    k +3  ` 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

Q p( z ) = p(0) −   F

n z

∫ ( a( z ) )

3n +1

(15)

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

The integration constant is p (0) , the pressure at z = 0 . The exit pressure is given by equation (2.15) with z = 1 . 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 a0 . 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 ) (Fung, [2]) (16) σ θθ = h 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 (17) eθθ = (σ θθ − vσ zz − vσ rr ) E where υ is the Poisson’s ratio and E is the Young’s modules of the wall material. But σ zz is assumed to be zero and σ rr is, in general, much smaller than σ θθ for thin-walled tubes. Hence equation (17) reduces to σ (18) eθθ = θθ E The strain eθθ is equal to the change of radius divided by the original radius, a0 : a ( z ) − a0 a ( z ) (19) eθθ = = −1 a0 a0 Combining (16), (18) and (19) we obtain: −1

 a  (20) a ( z ) = a0 1 − 0 p ( z )  Eh   Substituting (20) into (14), we may write the result as − (3 n +1) n a0   Q 1 (21) 1 − p ( z ) dp = −   3n +1 dz  Eh   F  a0   Recognizing the boundary conditions p = p (0) when z = 0 and p = p (1) when z = 1 and integrating Equation (20) from p (0) to p (1) on the left and 0 to 1 on the right, we obtain the pressure-flow relationship: (22) Eh   a   a    Q  1 1− p (1) − 1− p (0) = −3 n

 3na10  

 

−3 n

 

 

   3n +1   F  a0

which shows that the flow is not a linear function of pressure drop p (0) − p (1) . where F is given by equation (13) Newtonian fluid : τ = 0, n = 1 In this case equation (21) becomes −3 −3 Eh   a0 1   a   p (1)  − 1 − 0 p (0)   = 12Q 4  1 − 3a10   Eh a0   Eh  

(23) When n = 1 and τ = 0 eq (22) reduces to the corresponding results of Fung [2] for the flow of Newtonian fluid in an elastic tube. n =1 Bingham Fluid: In this case equation (2.21) becomes

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 −3 −3   a0   a0   Q p (1)  − 1 − p(0)   = 4  1 −   Eh   Fa0   Eh where 1 6(1 − τ ) 2 − 4(1 − τ )3 + (1 − τ ) 4  F= 48  Power-Law Fluid: τ = 0 In this case equation (2.21) become

Eh 3a10

Eh 3na10

−3 n −3n n   a0   a0   (12Q) − − − 1 p (1) 1 p (0)   = 3n +1    a0   Eh     Eh

(24) (25)

(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: a = a0 + ap 2

(27)

Hence a0 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 dp dp da 2 da = = dz da dz α dz

(28)

On substituting Eq (28) into Eq(14) and arranging terms, we obtain 3 n +1

[ a( z )]

da 1 da 3n + 2 α Q = =−   2F dz 3n + 2 dz

(29)

Since the right hand side term is a constant independent of z, we obtain at once the integrated result 3n + 2

[ a( z )]

=−

α 2

( 3n + 2 ) 

Q  +C F

(30)

The integration constant can be determined by the boundary condition that when z = 0, a ( z ) = a (0) . Hence 3n + 2 a(0) ] [ C= 3n + 2 Then by putting z = 1 , we obtain the following result from Eq.(30) 1n

 α (3n + 2)    2   = [ a (0) ] 

3n + 2

Q F 3n + 2 1 n

− [ a (1) ]

 

(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

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 3n + 2

a(1)] ( z = 0 ) minus the exit section ( z = 1 ). If the ratio a ( z ) a(0) is 1/2, then [ is only about of 3n + 2 [ a(0)] , and is negligible by comparison. Hence when a(1) is one-half of a(0) 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 : τ = 0, n = 1 5

30α Q = [ a (0) ] − [ a (1) ] (ii)

Bingham Fluid: n = 1

5α Q 5 5 = [ a (0) ] − [ a (1) ] 2 F F=

where (iii)

1  6(1 − τ ) 2 − 4(1 − τ )3 + (1 − τ ) 4  48 

Power-Law Fluid: τ = 0 1n

 α (3n + 2)  12Q   2  

(

3n + 2

= [ a (0) ]

3n +2 1 n

− [ a (1) ]

)

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

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

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

-15

3.5

-17

x 10

1.2

3 τ=0.2 τ=0.3 τ=0.4

0.8 FLUX

FLUX

E=0.02 E=0.021 E=0.023

2.5

1.5

0.6 0.4

0.2

0.5 0 0.5

x 10

1.5 2 RADIUS

2.5

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.

0 0.5

1.5 2 RADIUS

2.5

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.

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 -16

1.4

-16

x 10

2 1.8

1.2 p(0)=0.3 p(0)=0.32 p(0)=0.35

1 0.8

p(1)=0.6 p(1)=0.62 p(1)=0.65

1.6 1.4 FLUX

FLUX

x 10

0.6

1.2 1

0.4

0.8

0.2

0.6

0 0.5

1.5 2 RADIUS

2.5

0.4 0.5

Fig.4 Variation of flux with radius for different Inlet Pressure values, when stress is considered for fixed values of α=0.1,n=0.5,τ =0.3.

1.5 2 RADIUS

2.5

Fig.5 Variation of flux with radius for different Outlet Pressure values, when stress is considered for fixed values of α=0.1,n=0.5,τ =0.3.

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] [2] [3] [4] [5] [6] [7] [8] [9]

[10] [11] [12] [13] [14]

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