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Catheter is commonly used by the surgeons for various reasons in the treatment of a patient suffering with cardiovascular diseases. Catheterization increases the mean flow resistance in the arterial blood flow and many other complications are associated with the presence of catheter in the artery. Effects of catheter in stenosed artery can be estimated non-invasively by means of hemo-dynamic indicator-WSS, WSSG, volume flow rate and impedance. The effect of slip at the arterial wall, inclination of the artery and magnetic field on the hemodynamic indicators and flow profiles are computed, presented and discussed through graphs.

A significant change in blood flow, pressure distribution, wall shear stress and resistance to flow has universally observed when an impediment has developed in the arterial lumen. Generally, in the artery, the impediment developments are resulted from the lipoproteins and fatty acids deposition at the sites of atherosclerosis lesion. Consequently, a stenosed artery has been formed. In the stenosed section the velocity gradient near the wall region is steeper due to the increased core velocity resulting in relatively large shear stress on the wall even for a mild stenosis. Several researchers Fung [

The use of catheters is of immense importance and has become a standard tool for diagnosis and treatment in modern medicine. When a catheter is inserted into the stenosed artery, the further increased impedance or frictional resistance to flow will alter the velocity distribution. Kanai et al. [

The study of flow of an electrically conducting fluid through a stenosed artery with permeable walls not only possesses a theoretical importance, but also is useful for many biological and engineering problems such as magnetohydrodynamics (MHD) generators, blood flow problems, plasma studies. In the technical fields, the specification of MHD studies can be found in Moreau [

These researches motivated for the present study of blood flow in catheterized stenosed artery subject to a velocity slip at the stenosed arterial wall under the influence of transverse magnetic field will be quite rational for theoretical study of blood flow and explanation of disease linked with flow dysfunction.

The problem considered here is to study pulsatile blood flow through an inclined axially symmetric catheterized stenosed artery with slip velocity at the arterial wall. The blood vessel geometry is determined by the radius R_{0} of the inlet and outlet unconstricted segment, whereas the radius of the smooth axisymmetric constricted segment is given by

where 2L is the length of stenosis and δ is maximum height of the stenosis.

In the cylindrical coordinate system, the axis of the vessel coincides with the -axis and the origin corresponds to the peak point of the stenosis. The diameter of the artery is assumed to be greater than 1 mm so that Fahreus-Lindquist effect is not significant. The flow through the artery is in the influence of external magnetic field, an electromagnetic force will be produced due to the interaction of current with magnetic field when electrically conducting fluid like blood is flowing in the magnetic field.

The electromotive force is proportional to the speed of motion and the magnetic flux intensity B (Tashtoush and Magableh, [

where, is the electric field intensity, is the magnetic flux intensity is the electric permeability and is the current density. If is the electrical conductivity, Then generalized Ohm’s law is

The induced electromagnetic force is defined as

Following Cowling [

the unit vector in axial direction. Invoking these assumptions the governing equations of the motion of blood as Newtonian incompressible fluid with axisymmetric condition is given by

where, axial velocity, the pressure, time, viscosity of the blood, ρ density of the blood, gravitational acceleration, the aspect ratio of catheter radius to radius of artery, catheter speed, the slip velocity, inclination of the artery as shown in

Introducing the following non-dimensional parameters:

where, H the Hartmann number, Re the catheter speed based Reynold’s number.

On putting these parameters in the Equations (8) and (9), the equation of motion and boundary conditions in dimensionless form reduces to

The corresponding boundary conditions are

Let us assume the pressure gradient in the dimensionless form as

where, f is constant pressure gradient. The governing equation of motion is nonlinear coupled partial differential equation. For its solution, let us consider

where ω is the frequency of the oscillations of pulsatile blood flow.

On plugging Equation (14) into Equation (11) and comparing coefficient of like powers of ε, we have zero-order and first order equations.

The corresponding boundary conditions are

Equation (15) is the modified Bessel’s differential equation whose solution is given by

where

Similarly the solution of the Bessel’s Equation (16) for the transient flow is known and given by

where, and

Wall shear stress is important physical indicator for describing arterial disease due to disturbed flow. High Wall Shear Stress not only damage the vessel wall and cause intimal thickening, but also activate platelets, resulting platelet aggregation and thus formation of thrombus. Wall Shear Stress at the surface of stenosis is given by

where,

Hemodynamic indicator describes the regions of disturbed flow which corresponds with high WSSG. Experimental results of Meng et al. [

The volumetric flow rate Q of blood in the stenotic region is given by

The resistive impedance is physiological important hemodynamic indicator used in the study of resistance to flow of blood in artery. It is defined as

The flow profiles are derived for mild stenoses of thickness 20% of radius of artery. The catheter motion is taken in the positive z-direction.

Re. The increase in the angle of inclination of the artery (

The WSS is reduces with the increase in the catheter speed based Reynolds number as observed in

The volumetric flow rate (Q) diminishes with the increase of H is depicted from

The wall shear stress gradient (WSSG) increases with the increase in H and Re as observed in Figures 14 and 15 respectively.

Figures 17-19 demonstrate that the impedance on the flow is significantly affected by H, Re and k

throughout in the stenosis region of artery.

At the maximum height of stenosis is appreciably diminishes with the increase of H, is in good agreement with the results of Mekheimer and Kot [

Wall shear stress increases with increasing transverse magnetic field.

WSSG high at the apex of the stenosis indicates more disturbed flow at this location.

The impedance at the apex of the stenosis reduces increasing transverse magnetic field strength.

The impedance on the flow in annulus has augmented with the increase of Re and k.