Bingham Visco-Elastic Fluid Model
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Blood Flow Behaviour Studied through Bingham Visco-Elastic Fluid Model
This paper is devoted to the survey of blood flow behaviour through Bingham visco-elastic fluid theoretical account. Flow rates for different parts have been obtained. The effectual viscousnesss for one and two stage flows obeying Bingham plastic constituent equation have been obtained. Bingham flow theoretical account of visco-elastic fluid flow in narrow spread with rheological consequence has been analysed. Flow rate of the fluid rises by increasing the PPL thickness at K_{1}= 0, S_{Nitrogen}& A ; gt ; 1 ; K_{1}i‚?0, S_{Nitrogen}& A ; gt ; 1 ; K_{1}= 0, S_{Nitrogen}& A ; lt ; 1 and K_{1}= 0, M_{Phosphorus}= 0, which falls after a peculiar scope of PPL thickness in first two instances merely. While effectual viscousness with PPL thickness at K_{1}i‚?0, S_{Nitrogen}& A ; gt ; 1 ; K_{1}= 0, S_{Nitrogen}& A ; gt ; 1 ; K_{1}= 0, S_{Nitrogen}& A ; lt ; 1 and K_{1}= 0, M_{Phosphorus}= 0 shows opposite tendencies of fluctuations.
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Introduction:
Many mathematical theoretical accounts proposed for the survey of fluctuation of blood viscousness for flows is narrow spreads. Gupta and Seshadri [ 14 ] studied the flow of ruddy blood cell suspensions through narrow tubings. The extent of stopper flow seen to increase with addition in haematocrit. Batra and Jena [ 2 ] studied the being of cell free plasma bed near the wall and a nucleus with changing ruddy cell concentration have been observed by Mishraet Al. [ 17 ] and Bugliarello and Sevilla [ 4 ] . In add-on, several workers like Chaturani and Biswas [ 9 ] , Chaturani [ 6 ] , Bugliarelloet Al. [ 3 ] , Das and Seshadri [ 11 ] , Haynes [ 15 ] , Arimanet Al. [ 1 ] Chaturani [ 7 ] ; Chaturani [ 8 ] , Chaturaniet Al. [ 10 ] , Yamada [ 19 ] , Frigaard [ 13 ] and Ramkissonet Al. [ 18 ] have worked in the same way.
Blood flow in capillaries of internal diameter equal to that of ruddy cell is rather different from that in the big vass. Since the size of RBC is non negligible as compared to vessel diameter, it is necessary to see the flow of blood in capillaries as two stage not homogeneous flow.
The blood consists of two beds: one is plasma of Newtonian fluid and other is cardinal core part of non-Newtonian fluid. The thickness of plasma bed assumed changeless and independent of clip and location.
The evident viscousness of blood depends on several factors such as plasma viscousness, haematocrit, size of the vas, shear rate, rate of flow, rigidness, deformability etc. In add-on, the evident viscousness in capillaries is much lower than in big vass e.g. Dintenfass [ 12 ] . In add-on, it is good established that evident viscousness of blood lessenings as the tubing radius lessenings ( Fahraeus – Lindquist consequence ) . Gupta and Seshadri [ 14 ] , Iida and Murata [ 16 ] , Charmet Al. [ 5 ] have considered the two fluid theoretical accounts in which both beds ( PPL and nucleus haematocrit beds ) are of Newtonian fluids with different viscousnesss.
In the present chapter speeds in the plasma and in the core-regions, flow rate and effectual viscousness have been obtained. Besides, fluctuations of flow rate and effectual viscousness with regard to PPL thickness have been discussed with the aid of Tables and Graphs.
Formulation and Solution:
See a two superimposed Bingham flow theoretical account of blood in a stiff handbill tubing. R is the radius of the tubing and i?¤ is the thickness of PPL.
Figure: Two Layered Bingham Flow Model
The regulating equations of gesture are –
In the cell free peripheral part R – i?¤ i‚? R i‚? R
( 2.1 )
and in the core-region 0 i‚? R i‚? R – i?¤ for unstable stage plasma
( 2.2 )
For particulate stage cells –
( 2.3 )
where i??_{degree Fahrenheit}is unstable viscousness, i??_{s}the suspension viscousness, i?¦ the volume occupied by the ruddy cells per unit volume of the blood, u_{fp}and U_{fc}are the speeds of plasma in the peripheral bed and nucleus parts severally. u_{personal computer}the speed of ruddy cells in the nucleus, P the force per unit area, i?_{0}the magnetic permeableness of the blood and m_{Phosphorus}the magnitude of magnetic strength. Bingham unstable theoretical account is used for nucleus fluid.
The boundary conditions are:
is finite at R = 0 ( 2.4 )
( 2.5 )
i?? = 0 at R = 0
The Bingham constituent equations may be given by
( 2.6 )
where i?? is the shear emphasis, i??_{0}the output emphasis,the strain rate and i?? is the coefficient of viscousness.
Equation ( 2.1 ) can be written as
Integrating above equation we get
( 2.7 )
where C is changeless of integrating
or
On integrating we get
Using boundary status at R = R,
We get
( 2.8 )
From equations ( 2.2 ) and ( 2.3 ) , we get
( 2.9 )
In position of equation ( 2.2 ) , equation ( 2.2 ) becomes
or
Integrating above equation we get
Using boundary conditions i?? = 0 at R = 0 gives C_{1}= 0
Therefore
Now utilizing Bingham constituent equation in above equation we get
or( 2.10 )
where K_{1}= i??_{0}.
Now utilizing boundary status
, R = R – i?¤
We get from equations ( 2.7 ) and ( 2.10 )
Puting the value of C in equation ( 2.8 ) we get
( 2.11 )
On incorporating equation ( 2.10 ) and utilizing boundary statuswe get, at R = R – i?¤
Then
( 2.12 )
Volume flow rate Q is given by
( 2.13 )
where,
,
( 2.14 )
where
or
( 2.15 )
or
( 2.16 )
From equations ( 2.13 ) , ( 2.14 ) , ( 2.15 ) and ( 2.16 ) we get
( 2.17 )
where S_{Nitrogen}is a non-dimensional figure and a ratio of magnetic force to coerce force. Effective viscousness can be derived by utilizing the expression
For non-magnetic instance ( m_{Phosphorus}= 0 ) or when volume fraction i?¦ is zero the look for entire flow rate reduces to
( 2.18 )
When output emphasis is zero i.e. K_{1}= 0,
equation ( 2.17 ) reduces to
( 2.19 )
Effective viscousness
( 2.20 )
For non-magnetic instance ( m_{Phosphorus}= 0 ) and zero volume fraction, the look for entire flow rate reduces to
( 2.21 )
and the effectual viscousness is given by
( 2.22 )
Table – 2.1
Variation of flow rate with PPL thickness when K_{1}i‚? 0, S_{Nitrogen}& A ; gt ; 1, R = 10, i??_{degree Fahrenheit}= 1.25, i??_{s}= 2.5, i?¦ = 0.02, degree Fahrenheit_{vitamin D}= 0.1, S_{Nitrogen}= 10 and
i?¤/R |
Q |
0.01 |
121522 |
0.02 |
129893 |
0.03 |
138670 |
0.04 |
150544 |
0.05 |
145642 |
Table – 2.2
Variation of effectual viscousness with PPL thickness when K_{1}i‚? 0 and S_{Nitrogen}& A ; gt ; 1
i?¤/R |
i??_{vitamin E} |
0.01 |
0.2133 |
0.02 |
0.1973 |
0.03 |
0.1848 |
0.04 |
0.1702 |
0.05 |
0.1760 |
Table – 2.3
Variation of flow rate with PPL thickness when K_{1}= 0 and S_{Nitrogen}& A ; gt ; 1
i?¤/R |
Q |
0.01 |
125419 |
0.02 |
130303 |
0.03 |
138773 |
0.04 |
151548 |
0.05 |
145991 |
Table – 2.4
Variation of effectual viscousness with PPL thickness when K_{1}= 0 and S_{Nitrogen}& A ; gt ; 1
i?¤/R |
i??_{vitamin E} |
0.01 |
0.2126 |
0.02 |
0.1967 |
0.03 |
0.1847 |
0.04 |
0.1691 |
0.05 |
0.1755 |
Table – 2.5
Variation of flow rate with PPL thickness when K_{1}= 0 and S_{Nitrogen}= 0.1
i?¤/R |
Q |
0.01 |
1173 |
0.02 |
1222 |
0.03 |
1274 |
0.04 |
1317 |
0.05 |
1388 |
Table – 2.6
Variation of effectual viscousness with PPL thickness when K_{1}= 0 and S_{Nitrogen}& A ; lt ; 1
( S_{Nitrogen}= 0.1 )
i?¤/R |
i??_{vitamin E} |
0.01 |
2.185 |
0.02 |
2.097 |
0.03 |
2.012 |
0.04 |
1.947 |
0.05 |
1.888 |
Table – 2.7
Variation of flow rate with PPL thickness when m_{Phosphorus}= 0 and K_{1}= 0
i?¤/R |
Q |
0.01 |
10643 |
0.02 |
11051 |
0.03 |
11431 |
0.04 |
11799 |
0.05 |
12157 |
Table – 2.8
Variation of effectual viscousness with PPL thickness when m_{Phosphorus}= 0 and K_{1}= 0
i?¤/R |
i??_{vitamin E} |
0.01 |
2.408 |
0.02 |
2.319 |
0.03 |
2.242 |
0.04 |
2.172 |
0.05 |
2.108 |
Graph – 2.1:Variation of flow rate with PPL thickness when K_{1}i‚?0, S_{Nitrogen}& A ; gt ; 1.
Graph – 2.2:Variation of effectual viscousness with PPL thickness when K_{1}i‚?0, S_{Nitrogen}& A ; gt ; 1.
Graph – 2.3:Variation of flow rate with PPL thickness when K_{1}= 0, S_{Nitrogen}& A ; gt ; 1.
Graph – 2.4:Variation of effectual viscousness with PPL thickness when K_{1}= 0, S_{Nitrogen}& A ; gt ; 1.
Graph – 2.5:Variation of flow rate with PPL thickness when K_{1}= 0, S_{Nitrogen}& A ; lt ; 1.
Graph – 2.6:Variation of effectual viscousness with PPL thickness when K_{1}= 0, S_{Nitrogen}& A ; lt ; 1.
Graph – 2.7:Variation of flow rate with PPL thickness when K_{1}= 0, M_{Phosphorus}= 0.
Graph – 2.8:Variation of effectual viscousness with PPL thickness when K_{1}= 0, M_{Phosphorus}= 0.
Result and Discussion:
- It is shown from Tables – 2.1, that flow rate quickly increases with increasing PPL thickness at changeless value of S_{Nitrogen}but it is maximal at i?¤/R = 0.04 and so decreases with increasing PPL thickness.
- Table – 2.2 shows that the effectual viscousness decreases with increasing PPL thickness but it is maximal at i?¤/R = 0.04 and so increases with increasing PPL thickness.
- From Table – 2.3, the fluctuation of flow rate and PPL thickness are same as in the Table 2.1.
- From Table – 2.4, the fluctuation in flow rate and PPL thickness are shown likewise as Table – 2.2.
- From Table – 2.5, the flow rate additions with increasing PPL thickness for a changeless value of S_{Nitrogen}.
- From Table – 2.6, the effectual viscousness decreases with increasing PPL thickness for a changeless value of S_{Nitrogen}less than 1.
- From Table – 2.7, in instance of non-magnetic field, flow rate varies with PPL thickness.
- In instance of non-magnetic field, the effectual viscousness decreases with increasing PPL thickness as shown in Table – 2.8.
Mentions
[ 1 ] .Ariman, T. ; Turk, M. A. and Sylvester, N. D. ( 1974 ) : “On steady and pulsatile flow of blood” . J. App. Mech. , Vol. 41, p 1.
[ 2 ] .Batra, R. L. and Jena, B. ( 1991 ) : “Flow of a Casson fluid in a somewhat curved tube” . Int. J. Engng. Sci. , Vol. 29, P 1245.
[ 3 ] .Bugliarello, G. ; Kapur, C. and Hsiao, G. ( 1964 ) : “The profile viscousness and other features of blood flow in a non-uniform shear field” . Proc. of Fourth International Congress on Rheology. , P 351, Inter scientific discipline, New York.
[ 4 ] .Bugliarello, G. and Sevilla, J. ( 1970 ) : “Velocity distribution and other features of steady and pulsatile blood flows in all right glass tubes” . Biorheology, Vol. 7, pp 85 – 107.
[ 5 ] .Charm, S. E. ; Kurland, G. S. and Brown, S. L. ( 1968 ) : “The influence of radial distribution and fringy plasma bed on the flow of ruddy cell suspension” . Biorheology. , Vol. 5, pp 15 – 47.
[ 6 ] .Chaturani, P. ( 1976 ) : “Some remarks on Poiseuille flow of fluid with couple emphasis with application to blood flow” . Biorheology, Vol. 13, p 133.
[ 7 ] .Chaturani, P. ( 1978 ) : “Viscosity of Poiseuille flow of a twosome emphasis fluid with application to blood flow” . Biorheology, Vol. 15, p 119.
[ 8 ] .Chaturani, P. ( 1979 ) : “Fluid mechanics of suspensions and its technology applications” . Fluid Mech. Susp. and Engg. Appl. , Ed. pp 1 – 4.
[ 9 ] .Chaturani, P. and Biswas, D. ( 1984 ) : “A comparative survey of Poiseuille flow of a polar fluid under assorted boundary conditions with applications to blood flow” . Rheologica Acta. , Vol. 23, p 435.
[ 10 ] .Chaturani, P. ; Kelker, I. P. ; Ranganath and Seema Maheshwari ( 1987 ) : “A comparative survey of viscousness of blood” . Proc. 15^{Thursday}National Conference of fluid Mechanicss and Fluid Power, Vol. 1, p 337.
[ 11 ] .Das, R. N. and Seshadri, V. ( 1975 ) : “A semi empirical theoretical account for flow of blood and other particulate suspensions through narrow tubes” . Bull. Math. Biol. , Vol. 37, p 459.
[ 12 ] .Dintenfass, L. ( 1965 ) : Exp. Mol. Patnol. , Vol. 4, p 597.
[ 13 ] .Frigaard ( 2003 ) : On 3-dimensional additive stableness of poiseuille flow of Bingham fluids. Phys. Fluids, Vol. 15 ( 10 ) , pp 2843 – 2851.
[ 14 ] .Gupta, B. B. and Seshadri, V. ( 1977 ) : “The flow of blood cell suspensions through narrow tubes” . Biorheology, Vol. 14, pp 133 – 143.
[ 15 ] .Haynes, R. H. ( 1960 ) : “Physical footing of the dependance of blood viscousness on tubing radius” . Am. J. Physiol. , Vol. 198, P 1193.
[ 16 ] .Iida and Murata, T. ( 1980 ) : Biorheology, Vol. 17, p 377.
[ 17 ] .Mishra, J. ; Kumar, J. and Singh, N. L. ( 2004 ) : Acta Ciencia Indica, Vol. 30 M, No. 1, pp 149 – 154.
[ 18 ] .Ramkisson, Harold, Rahaman and Karim ( 2003 ) : “Wall consequence with slip” . ZAMMZ Angew. Math. Mech. , Vol. 83 ( 1 ) , pp 773 – 778.
[ 19 ] .Yamada ( 2002 ) : Equilibrium analysis of a fluxing two unstable plasma. Phys. Plasmas, Vol. 9 ( 11 ) , pp 4605–4014.
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