Abstract— A serious pathological condition is encountered when some blood constituents deposited on the blood vessels get detached from the wall, join the blood stream again and form a clot. The pulsatile flow for peristaltic transport of a fluid with suspended particles may be considered as a mathematical model for the blood flow. We study this model in a annular region with a clot inside it, under low Reynolds number and long wavelength approximation. We model a small artery as a tube having a sinusoidal wave travelling down its wall with a constant velocity c and a clot model inside it. Closed form solutions are obtained for the fluid/particle velocity , as well as the stream function, and the pressure gradient. These solutions contain new additional parameters, namely, δ, the height of the clot, β, the pulsating number and C, the suspension parameter, and the wave amplitude b. The pressure rise and friction force on the outer tube have been discussed for various values of the physical parameters of interest. Finally, the trapping phenomenon is illustrated.
Index Terms— pulsatile flow; peristaltic pumping; Annulus flow ; Clot blood model
eristaltic pumping is a form of fluid transport that occurs when a progressive wave of area contraction or expansion propagates along the length of a distensible duct. Peristalsis is
an inherent property of many biological systems having smooth muscle tubes that transports biofluids by its propulsive move-ment and is found in the transport of urine from kidney to the bladder, the movement of chyme in the gastrointestinal tract, in-tra-uterine fluid motion, vasomotion of the small blood vessels and in many other glandular ducts. The mechanism of peristaltic transport has been exploited for industrial applications such as sanitary fluid transport, blood pumps in heart–lung machine and transport of corrosive fluids where the contact of the fluid with the machinery parts is prohibited[1-12]. Blood being a suspension of corpuscles, at low shear rates be-haves like a non-Newtonian fluid in small arteries . Besides, the theoretical analysis of Haynes  and experimental observations of Cokelet  indicate that blood can not be treated as a single-phase homogeneous viscous fluid in narrow arteries (of diameter ≤ 1000 mm). The individuality of red cells (of diameter 8 mm) is important even in such large vessels with diameter up to 100 cells' diameter . Moreover, an accurate description of flow requires consideration of red cells as discrete particles. It is to note that the average hematocrit in human blood under normal health condi-tions lies between 40-45 percent. In addition, certain observed phenomena in blood including the Fahraeus_Lindqvist effect, non-Newtonian behaviour, etc. can not be explained fully by treating blood as a single-phase fluid. The individuality of eryth-rocytes can not therefore be ignored while dealing with the prob-
lem of microcirculation also. It seems to be therefore necessary to treat the whole blood as a particle-fluid (erythrocyte_plasma) system while flowing through narrow arteries. The interaction of purely periodic mean flow with a peristaltic induced flow is investigated within the framework of a two-dimensional analogue has been studied by N.A.S.Afifi and N.S.Gad . Eytan and Elad  have developed a mathemati-cal modal of wall-induced peristaltic fluid flow in two-dimensional channel with wave trains having a phase difference moving independently on the upper and lower walls to simulate intra-uterine fluid motion in a sagittal cross-section of the uterus. They have used the lubrication theory to obtain a time dependent flow solution in a fixed frame. The results obtained by Eytan and Elad  have been used to evaluate the fluid flow pattern in a non-pregnant uterus. The rheological studies of steady flow of blood are useful in providing reference information on the rheological characteristics of blood, for clinical purpose, in viscometers. On the other hand, in reality, blood flow in arterial system is pulsatile, with time var-ying characteristics, which even extends into the capillarity bed. Ariman et al.  have studied pulsatile flow of blood assuming different modals, they have studied the steady and pulsatile flow of micropolar fluid and have obtained the exact solution for veloc-ity and cell rotation velocity in the form of Bessel-Fourier series. With the above discussion , we are interested to investigate the effect of the pulsatile flow on peristaltic motion of an incompress-ible particle-fluid suspension through a region with flexible walls and a clot inside it.
2 . Formulation and analysis of the physical problem Consider peristaltic transport of an incompressible Newtonian fluid in the region between two coaxial cylinders .
Figure 1. Problem geometry .
The geometries of the cylinders wall surfaces are (see Figure 1).
))t,Z(fa(RR 00 ′′+′=′ , λ≤′≤ Z0 (1)
)),tcZ(2sin(bRh 0 ′−′
00Ra is the radius of the inner cylinder and this keeps the clot
model in position inside the cylinder with 1a 0 << , )t,Z(f ′′
represent an arbitrary shape along the axial direction, 0R is the
radius of the outer cylinder at any axial distance Z′ from the
inlet, the wavelength and the wave amplitude are λ and b
respectively, the propagation velocity of the wave is c and t is
the time. A wave frame )z,r( ′′ is introduced and moving with
velocity c away from the fixed frame ).Z,R( ′′
The governing equations for the flow problem , with no external
forces, are given by,
For a fluid Phase:
)UU( s CRU
2 2(1 ) (1 ) ( )
W W W Wf f f fC U Wf f ft Z R R
W Wf fp Z R
C C CsR W Wf f
R R R
C s W Wp f
′ ′ ′ ′∂ ∂ ∂′ ′− + + − =
′ ′ ′ ′∂ ∂ ∂
′ ′∂ ∂+ +′ ′′∂ ∂ ∂
− − + − +′∂ ′ ′∂
− ′′ ′∂
For a particle phase”
)WW( s C
where )U,W( ff ′′ and )U,W( pp ′′ are the velocity compo-
nents of the fluid and the particle phase in R and Z -directions, respectively, P′ is the pressure, C is the constant (Srivastava
and Saxena , volume fraction density of the particles, )C(sµ is
the mixture viscosity (effective or apparent viscosity of suspen-sion) and S is the drag coefficient of interaction for the force exerted by one phase on the other. The expression for the drag coefficient of interaction, S and the empirical relation for the viscosity of the suspension, sµ for the present problem is select-
• The pressure rise and flow rate relation is a linear one for a steady frame of reference and this relation become a non-linear for the unsteady case ( pulsating flow).
• The pressure rise is higher for an annulus with a clot than that for a tube.
• For a suspended particles with a fluid the pressure rise bcome smaller as C increases.
• The shift displacement clot occurs a sleight effect on the pressure rise.
• The stream lines moves towards the outer wall cylinder as the clot height increase
• The trapping phenomena is obvious for the pulsatin flow than that for the steady one.
• Trapping bolus appeared in the peak wave of the stream lines as the flow rate increases.
THIS PAPER WAS FUNDED BY THE DEANSHIP OF SCIEN-TIFIC RESAEARCH (DSR), TAIF UNIVERSITY(TU) UNDER THE GRANT NUMBER (1-435-3068).THE AUTHORS ACKNOWLEDGETECHNICAL AND FINANCIAL SUPPORT OF TAIF UNIVERSITY. THE SUPPORT IS IN THE FORM OF A PROJECT FOR ACADEMIC RESEARCH AT TU.
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