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In this paper, the pulsatile flow of blood through an inclined catheterized stenosed artery is analyzed. Perturbation method is used to solve the implicit system of partial differential equations with suitable boundary conditions. Various analytical expressions axial velocity, flow rate, wall shear stress and effective viscosity have been derived with the help of MATLAB for understanding the fluid flow phenomena. The combined effect of catheterization, body acceleration, slip and inclination has been seen by plotting the graph and observed that axial velocity and flow rate increases with the increase in body acceleration, inclination angle and slip velocity while axial velocity diminishes on increasing the catheter radius. Wall shear stress increases with the increase in catheter radius and body acceleration but presence of slip velocity reduces the wall shear stress. Effective viscosity diminishes on increasing body acceleration and inclination angle, whereas slightly augmented in non-inclined stenosed artery.

In recent years, the study of blood flow through obstructed arteries has received much attention, due to its ample importance in human cardiovascular system. The theoretical analysis on blood flow is very useful as it plays a significant role to diagnose and understand many cardiovascular diseases such as coronary thrombosis, angina, pectoris, strokes etc. The reason behind the malfunction of cardio-vascular system is the presence of fats, cholesterol and lipoproteins at the sites of atherosclerotic lesion in the artery. By the development of atherosclerotic plaques that protrude into the lumen, arteries get narrowed and stenosed arteries are formed. Presence of atherosclerosis (stenosis), increased the resistance and therefore blood flow reduced inside the artery and also the remarkable change occurred in pressure distribution and wall shear stress. Bennett [

Mathematical models for blood flow through stenosed arterial segment, by taking a velocity slip condition at the constricted wall were developed [

Catheter is a thin hollow flexible tube made from medical grade materials and is used in a broad range of functions. Srinivasacharya and Srikanth [

Cardiac catheterization is a procedure used to diagnose and treat cardiovascular conditions in surgical procedures. During cardiac catheterization, coronary angiography is done by inserting a catheter in an artery or vein in your groin, neck or arm and threaded through your blood vessels to your heart. Back and Denton [

All above mentioned studies are based on the horizontal blood carrying vessels. Although it is well-known that many ducts in physiological systems are no horizontal, few of them have some inclination to the axis. Therefore, a gravitational force has been accounted there due to inclination. Chaturani and Upadhyay [

Consider the pulsatile flow of blood through an inclined axially symmetric ca

theterized obstructed artery in the presence of slip and external body acceleration at the arterial wall. The geometry of the stenosis is described in

The geometry of the stenosis which is assumed to be manifested in the arterial segment is described as [

R ¯ ( z ¯ ) = { R ¯ 0 − δ ¯ s 2 [ 1 + cos 2 π L ¯ s ( z ¯ − d ¯ − L ¯ s 2 ) ] , d ¯ ≤ z ¯ ≤ d ¯ + L ¯ s , t h e s t e n o s e d r e g i o n R ¯ 0 , i n t h e n o r m a l a r t e r y r e g i o n (1)

where R ¯ ( z ¯ ) and R ¯ 0 is the radius of the artery with and without stenosis respectively, d ¯ is the location of the stenosis, L ¯ s is the length of the stenosis and δ ¯ s denotes the maximum height of the stenosis in to the lumen such that δ ¯ s / R ¯ 0 ≪ 1 .

We have used cylindrical polar co-ordinates ( r ¯ , ϕ ¯ , z ¯ ) , whose origin is located on the vessel (stenosed artery) axis and r ¯ , z ¯ denote the radial and axial co- ordinates respectively. It can be shown that the magnitude of radial velocity is negligibly small and can be neglected for a low mean Reynold number flow problems in case of mild stenosis.

The Navier-Stokes equations governing the fluid flow is given by, [

ρ ¯ ( ∂ v ¯ / ∂ t ¯ ) = − ( ∂ p ¯ / ∂ z ¯ ) − ( 1 / r ¯ ) ∂ ∂ r ¯ ( r ¯ τ ¯ ) + B ¯ ( t ¯ ) + ρ ¯ g sin β (2)

∂ p ¯ / ∂ r ¯ = 0 (3)

where v ¯ represents the axial velocity along z ¯ direction, t ¯ is the time, ρ ¯ is the density, p ¯ is the pressure, τ ¯ is the shear stress, β is the inclination angle and B ¯ ( t ¯ ) is the body acceleration.

Newtonian fluid can be represented by the equation

τ ¯ = − μ ¯ ( ∂ v ¯ / ∂ r ¯ ) (4)

where μ ¯ denotes the coefficient of viscosity of blood.

The boundary conditions are

v ¯ = v ¯ s at r ¯ = R ¯ ( z ¯ ) (5)

v ¯ = 0 at r ¯ = R ¯ c (6)

where v ¯ s is the axial slip velocity at the stenotic wall and R ¯ c is the radius of catheter.

Since the pressure gradient is the function of z ¯ and t ¯ , therefore can be represented as ( ≪ R ¯ 0 ).

− ∂ p ¯ ∂ z ¯ ( z ¯ , t ¯ ) = A 0 + A 1 cos ( ω ¯ p t ¯ ) , t ≥ 0 (7)

where A 0 is the steady state pressure gradient, A 1 is the amplitude of the fluctuating component and both A 0 , A 1 are function of z ¯ . It can be seen that the radial velocity is very small in magnitude so that it may be neglected for problem with mild stenosis. The frequency of oscillation of the pulsatile flow is denoted by ω ¯ p and defined as ω ¯ p = 2 π f ¯ p , where f ¯ p is the pulse rate frequency.

The periodic body acceleration B ¯ ( t ¯ ) in the axial direction is given by

B ¯ ( t ¯ ) = a 0 cos ( ω ¯ b t ¯ + ϕ ) (8)

where the amplitude of body acceleration is a 0 and ϕ is the phase angle of body acceleration with respect to the pressure gradient. ω ¯ b = 2 π f ¯ b ; f ¯ b is its frequency in Hz. The frequency of the body acceleration f ¯ b is assumed to be small so that wave effect can be neglected.

Let us introduce the following non-dimensional variables

v = v ¯ A 0 R ¯ 0 2 / 4 μ ¯ , z = z ¯ / R ¯ 0 , R ( z ) = R ¯ ( z ¯ ) / R ¯ 0 , r = r ¯ / R ¯ 0 , d = d ¯ / R ¯ 0 , L s = L ¯ s / R ¯ 0 , t = t ¯ ω ¯ p , R c = R ¯ c / R ¯ 0 , δ s = δ ¯ s / R ¯ 0 , F = A 0 / 4 ρ ¯ g , ω = ω ¯ b / ω ¯ p , v s = v ¯ s A s R ¯ 0 2 / 4 μ ¯ , τ = τ ¯ A 0 R ¯ 0 / 2 , α 2 = R ¯ 0 2 ω ¯ p ρ ¯ μ ¯ , e = A 1 / A 0 , B = a 0 / A 0 . } (9)

where α is the pulsatile Reynold’s number or generalized Womersley frequency parameter.

Using non-dimensional variables, Equation (2) becomes

α 2 ( ∂ v ∂ t ) = 4 ( 1 + e cos t ) + 4 B cos ( ω t + ϕ ) − ( 2 r ) ∂ ∂ r ( r τ ) + sin β F (10)

Equation (4) becomes

τ = − ( ∂ v / 2 ∂ r ) (11)

On substituting the value of τ in Equation (10), we have

α 2 ( ∂ v ∂ t ) = 4 ( 1 + e cos t ) + 4 B cos ( ω t + ϕ ) + 1 r ∂ ∂ r ( r ∂ v ∂ r ) + sin β F (12)

The boundary conditions (5) and (6) reduces to

v = v s at r = R ( z ) (13)

v = 0 at r = R c (14)

The geometry of an arterial stenosis in non-dimensional form is given by

R ( z ) = { 1 − δ s 2 [ 1 + cos 2 π L s ( z − d − L s 2 ) ] , d ≤ z ≤ d + L s , t h e s t e n o s e d r e g i o n 1 , i n t h e n o r m a l a r t e r y r e g i o n (15)

The non-dimensional volumetric flow rate is defined by

Q ( z , t ) = 4 ∫ 0 R ( z ) r v ( z , r , t ) d r (16)

where Q ( z , t ) = Q ¯ ( z ¯ , t ¯ ) π A 0 ( R ¯ 0 ) 4 / 8 μ ¯ 0 ; Q ¯ ( z ¯ , t ¯ ) is the volumetric flow rate.

Effective viscosity μ ¯ e defined as

μ ¯ e = π ( − ∂ p ¯ ∂ z ¯ ) ( R ¯ ( z ¯ ) ) 4 / Q ¯ ( z ¯ , t ¯ )

Can be expressed in the non dimensional form as

μ e = R 4 ( 1 + e cos t ) / Q ( z , t ) (17)

In this paper perturbation method is used with a small parameteric value of pulsatile Reynolds number α of the series expansion to solve this system of non linear equations. Since non-dimentionalized Equation (10), Equation (11) has α 2 term which is dependent on time, therefore expanding Equation (10), Equation (11) about α 2 . The axial velocity v, shear stress τ are expressed as follows in terms of α 2 (where α 2 ≪ 0 ).

v ( z , r , t ) = v 0 ( z , r , t ) + α 2 v 1 ( z , r , t ) + ⋯ (18)

τ ( z , r , t ) = τ 0 ( z , r , t ) + α 2 τ 1 ( z , r , t ) + ⋯ (19)

Substituting Equation (18) in Equation (12), we get

1 r ∂ ∂ r ( r ∂ v 0 ∂ r ) = − 4 r h ( t ) (20)

∂ v 0 ∂ t = 1 r ∂ ∂ r ( r ∂ v 1 ∂ r ) (21)

where h ( t ) = [ ( 1 + e cos t ) + B cos ( ω t + ϕ ) + sin β 4 F ]

Substituting Equation (18) in Equation ((12) and (13)), we have

v 0 = v s , v 1 = 0 at r = R ( z ) (22)

v 0 = 0 , v 1 = 0 at r = R c (23)

Integrating Equations ((20) and (21)),with the help of Equations ((22) and (23)), the expression for v_{0} and v_{1} are obtained as

v 0 = v s ( 1 + log e ( r / R ) log e ( R / R c ) ) + h ( t ) { ( R 2 − r 2 ) + ( R 2 − R c 2 ) log e ( r / R ) log e ( R / R c ) } (24)

v 1 = H ( t ) [ 2 r 2 { ( R 2 − r 2 4 ) + ( R 2 − R c 2 ) log e ( r / R ) log e ( R / R c ) } − ( R 2 − R c 2 ) log e ( R / R c ) ( R 2 + r 2 ) − 3 R 4 2 + log e ( r / R ) log e ( R / R c ) { ( R 4 + R c 4 ) log e ( R / R c ) − 2 R 2 R c 2 log e ( R / R c ) − 2 ( R 4 − R c 4 ) } ] (25)

where H ( t ) = h ′ ( t ) / 8

With the help of Equations ((18), (24) and (25)), the expression for the axial velocity v can be obtained as

v = v s ( 1 + log e ( r / R ) log e ( R / R c ) ) + h ( t ) { ( R 2 − r 2 ) + ( R 2 − R c 2 ) log e ( r / R ) log e ( R / R c ) } + α 2 H ( t ) [ 2 r 2 { ( R 2 − r 2 4 ) + ( R 2 − R c 2 ) log e ( r / R ) log e ( R / R c ) } − ( R 2 − R c 2 ) log e ( R / R c ) ( R 2 + r 2 ) − 3 R 4 2 + log e ( r / R ) log e ( R / R c ) { ( R 4 + R c 4 ) log e ( R / R c ) − 2 R 2 R c 2 log e ( R / R c ) − 2 ( R 4 − R c 4 ) } ] (26)

With the help of Equations ((11) and (19)), the wall shear stress τ w can be obtained as

τ w = − 1 2 { ( ∂ v 0 ∂ r ) + α 2 ( ∂ v 1 ∂ r ) } r = R ( z ) (27)

τ w = { R h ( t ) − ( h ( t ) ( R 2 − R c 2 ) + v s ) 2 R log e ( R / R c ) } − α 2 H ( t ) ⋅ [ R 3 − 1 R log e ( R / R c ) { ( R 4 − R c 4 ) + R c 2 R 2 log e ( R / R c ) − ( R 4 + R c 4 ) 2 log e ( R / R c ) } ] (28)

From Equations ((16) and (26)) the expression for volumetric flow rate Q ( z , t ) can be obtained as

Q ( z , t ) = h ( t ) { R 4 − R 2 ( R 2 − R c 2 ) 4 log e ( R / R c ) } + v s { 2 R 2 − R 2 log e ( R / R c ) } + α 2 H ( t ) 2 . [ R 4 ( R 2 − R c 2 ) log e ( R / R c ) − 2 R 6 − R 2 log e ( R / R c ) { 2 ( R 4 + R c 4 ) log e ( R / R c ) − 4 ( R 4 − R c 4 ) − 4 R 2 ⋅ R c 2 log e ( R / R c ) } ] (29)

The expression for effective viscosity μ e can be obtained from Equationd ((17) and (26)) as

μ e = R 4 ( 1 + e cos t ) ⋅ [ h ( t ) { R 4 − R 2 ( R 2 − R c 2 ) 4 log e ( R / R c ) } + v s { 2 R 2 − R 2 log e ( R / R c ) } + α 2 H ( t ) 2 ⋅ [ R 4 ( R 2 − R c 2 ) log e ( R / R c ) − 2 R 6 − R 2 log e ( R / R c ) ⋅ { 2 ( R 4 + R c 4 ) log e ( R / R c ) − 4 ( R 4 − R c 4 ) − 4 R 2 ⋅ R c 2 log e ( R / R c ) } ] ] − 1 (30)

In the present investigation our objective is, to carry out the combined result of applied body acceleration and slip velocity in an inclined catheterized stenosed artery. The expression of axial velocity, flow rate, wall shear stress and effective viscosity are obtained and computed for the fixed values of F = 0.2, e = 1, ϕ = 0.2, α = 0.5, ω = 1 through the MATLAB. Figures 2-5 reveals the variation of axial velocity with radial distance.

observed in

for different values of body acceleration has been seen in

In the present mathematical model, pulsatile blood flow through an inclined stenosed catheterized artery with periodic body acceleration and axial slip velocity at the constricted wall has been considered. The flowing fluid is represented by a Newtonian fluid. Analytic expressions for flow variables and their variations with different flow parameters have been obtained and are represented graphically. The results based on the mathematical analysis and the subsequent numerical evaluation of the flow quantities show that the axial velocity and flow rate increases with the increase in body acceleration, inclination angle and slip velocity while axial velocity diminishes on increasing the catheter radius. Wall shear stress increases with the increase in catheter radius and body acceleration but presence of slip velocity reduces the wall shear stress. Effective viscosity diminishes on increasing body acceleration and inclination angle, whereas slightly augmented in normal (non-inclined) stenosed artery. This model concludes that slip velocity plays a very eminent role in blood flow models through an inclined stenosed artery with catheter. In future, this study may be helpful for the purpose of simulation and validation of different models in different conditions of stenosis and to investigate that which parameter has the most dominating role.

Corresponding author (Chhama Awasthi) is thankful to the TEQIP (SPFU, World Bank) at HBTU, Kanpur, India for awarding Fellow-ship for this research work.

Siddiqui, S.U. and Awasthi, C. (2017) Mathematical Analysis on Pulsatile Flow through a Catheterized Stenosed Artery. Journal of Applied Mathematics and Physics, 5, 1874-1886. https://doi.org/10.4236/jamp.2017.59157