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Steady flow of a couple-stress fluid in constricted tapered artery has been studied under the effects of transverse magnetic field, moving catheter, and slip velocity. With the help of Bessel’s functions, analytic expressions for axial velocity, flow rate, impedance, and wall shear stress have been obtained. It is of interest to note that these solutions can be used for different types of fluid flow in tubes and not only the case of blood. The effects of various geometric parameters, the parameters arising out of the fluid considered and the magnetic field, are discussed by considering the slip velocity, the catheter velocity, and tapering angle. The study of the above model is very important as it has direct applications in the treatment of cardiovascular diseases.

Catheters are semirigid, thin tubes made from medical grade materials serving a broad range of functions and can be inserted in the body to detect and identify diseases or perform a surgical procedure inside the heart, brain, arms, legs, or lungs [

Each year, heart disease is at the top of the list of the country’s most serious health problems. Statistics show that cardiovascular disease in some countries, like America, is the first health problem and the leading cause of death (recent statistics released by the American Heart Association). Catheters are often used to treat stenosis (partial occlusion of the blood vessel), aneurysm (dilation of the blood vessel resulting in stretching of the vessel wall), and embolism (complete occlusion of a blood vessel by a blood clot or some other particle), Figure

Catheters are used to treat stenosis (partial occlusion of the blood vessel), aneurysm (dilation of the blood vessel resulting in stretching of the vessel wall), and embolism (complete occlusion of a blood vessel by a blood clot or some other particle) (a reproduction from Delft Outlook 2004).

The study of blood flow through different types of arteries is of considerable importance in many cardiovascular diseases; one of them is atherosclerosis. The blood flow through an artery has drawn the attention of researchers for a long time up to now, due to its great importance in medical sciences. Under normal conditions, blood flow in the human circulatory system depends upon the pumping action of the heart and this produces a pressure gradient throughout the arterial network.

Many researchers have studied blood flow in the artery by considering blood as either Newtonian or non-Newtonian fluids; since blood is a suspension of red cells in plasma, it behaves as a non-Newtonian fluid at low shear rate. In the stenosed condition, substantial reduction in the lumen of an artery results in size effects (ratio of haematocrit to vessel diameter), which influences flow characteristics significantly. To study the size effect in the fluid flow, Stokes [

The application of Magnetohydrodynamics in physiological problems is of growing interest and it is very important from both theoretical and practical points of view. The application of Magnetohydrodynamics in physiological problems is of growing interest. The flow of blood can be controlled by applying appropriate quantity of magnetic field. Kollin [

There are many treatments available for diagnosing and treating constricted vessels. Catheterization (thin, flexible tube) is one of them, in which balloon angioplasty is a specialized form of catheterization. These procedures are widely used in the medical field for treating the atherosclerosis. Insertion of the catheter in a tube creates an annular region between inner wall of the artery and outer wall of the catheter which influences the flow field such as pressure distribution and shear stress at the wall. In view of its immense importance, the effect of the catheter on physiological parameters was discussed by the researchers [

The shapes of the stenosis in the above aforesaid studies have been considered to be radially symmetric or asymmetric. But while stenosis is maturing, it may grow up in series manner, overlapping with each other, and it would appear like x-shape. Riahi et al. [

The presence of red cell slip at the vessel wall was recommended theoretically by Vand [

In the present analysis a mathematical model for the steady blood flow through tapered stenosed artery under the influence of a moving catheter, slip velocity, and a magnetic field is presented by considering blood as a couple-stress fluid in a circular tube. It is assumed that the magnetic field along the radius of the pipe is present, no external electric field is imposed, and magnetic Reynolds number is very small. The motivation for studying this problem is to understand the blood flow in an artery under the effect of magnetic field alongside with the catheter inserted into the blood vessel also when the fatty plaques of cholesterol and artery clogging blood clots are formed in the lumen of the artery.

The main aim of this work is to study these phenomena, obtain analytic expressions for axial velocity and shear stress, and also study the effect of magnetic field (Hartmann number

Let us consider a two-dimensional steady flow of blood through a rigid tapered stenosed tube by considering blood as an electrically conducting, incompressible, couple-stress fluid. The magnetic field is acting along the radius of the tube. The magnetic Reynolds number of the flow is assumed to be sufficiently small that the induced magnetic and electric fields can be neglected [

Mathematical model of blood flow through a tapered stenosed arterial segment in the presence of a moving catheter and a magnetic field is to be built to study the impact of various geometric, Hartman, and fluid parameters on physiological parameters. The geometry of the tapered stenosed artery is shown in Figures

2D view of a catheterized stenosed artery.

2D view of a tapered stenosed artery.

The current density

The conservation equations which govern the couple-stress fluid flow including a Lorentz force can be written in the following form:

As the flow is steady and incompressible, in the absence of body force and body couple moment (

The flow is considered to take place under the influence of externally applied magnetic field in axial direction

Equation of continuity is

Equation of radial momentum is

Equation of axial momentum is

Also, (

Under the assumption of mild stenosis that is

Equation (

It can be seen that the pressure variation depends only on the axial variable. The pressure gradient

The corresponding nondimensional boundary conditions are as shown below:

As a solution of (

Let

The solution of the above equation is obtained as

Volumetric flow rate

The resistance to the flow (impedance) is obtained from

The dimensionless form of (

The shear stress

Hence, the dimensionless shear stress for the artery is given by

Thus, the shear stress at the wall can be computed from (

The study of blood flow through catheterized stenosed tapered artery with the presence of a transverse magnetic field involves the integration of various geometric and fluid variables, which influences the physiological parameters such as the fluid velocity, rate flow, and wall shear stress. Closed form solutions are obtained in terms of modified Bessel’s functions. The physiological dimensionless quantities such as the fluid velocity in the stenosis region and the wall shear stress at the maximum height of the stenosis are computed numerically for various values of the fluid and geometric parameters using the program

Figures

Variation of axial velocity

Variation of axial velocity

Variation of axial velocity

In general, from Figure

Variation of axial velocity

Variation of axial velocity

Variation of axial velocity

The axial velocity of the blood is high in case of high slip velocity

Variation of axial velocity

Variation of axial velocity

The shear stress at the wall is a significant physiological parameter to be considered in the blood flow study. Precise predictions of the distribution of the shear stress at the wall are particularly useful in assimilating the effect of blood flow in arteries in general. The shear stress at the wall is calculated at the maximum height of the stenosis. When we increase the height of the stenosis

The effect of the Hartmann number

Variation of wall shear stress

Variation of wall shear stress

The slip velocity at the wall of the stenosed artery and the moving catheter significantly influence the shear stress at the wall, which is noticed from Figures

Variation of wall shear stress

Variation of wall shear stress

Variation of wall shear stress

Variation of wall shear stress

Variation of wall shear stress

Variation of wall shear stress

A mathematical model has been built to discuss the flow of blood through a catheterized asymmetric tapered stenosed artery with slip velocity at the stenosed wall and a moving catheter. Closed form solution is obtained and the effects of various geometric, fluid parameters and magnetic field on the axial velocity of the blood and the shear stress at the wall are studied.

There is special importance of couple-stress fluids compared to the Newtonian fluids because of their wide existence such as oil, blood, and polymeric solutions. In view of what is mentioned above, an analytic approach was followed to solve the mathematical model of blood flow through stenosed tapered artery under the assumption of mild stenosis. The resultant observations are summarized as follows:

As the height and the stenosis length are increasing, the obstruction to the flow of blood is increasing.

Converging tapered artery has more shear stress at the wall than the nontapered and the diverging tapered artery.

Diverging tapered artery has least shear stress at the wall.

The increment in the Hartmann number enhances the blood velocity and wall shear stress.

The axial velocity is decreasing while the couple-stress fluid parameters

Three different values of the slip velocity and the velocity of the catheter at the arterial boundary and the catheter wall are considered; they are showing a significant influence on the shear stress at the wall and the axial velocity.

The modeling and simulation of the above phenomena are very realistic and are expected to be very useful in predicting the behavior of physiological parameters in the diagnosis of various arterial diseases.

Magnetic flux intensity

External transverse magnetic field

Constants of integration

Location of the stenosis

Defined constants

Electric field intensity

Body force

Electromagnetic force

Velocity of the moving catheter

Hartmann number

Modified Bessel function of the first kind and order zero

Current density

Modified Bessel function of the second kind and order zero

Body couple moment

Stenosis length

Couple-stress tensor

Defined constant

Defined constant

Pressure

Volumetric flow rate

Radius of the nonconstricted region

Reynolds number

Radius of the tube

Radial velocity

Velocity vector

Slip velocity

Axial velocity

Typical axial velocity

Couple-stress fluid parameter

Laplacian operator

Maximum height of the stenosis

Couple-stress coefficients

Defined constant

Defined constant

Resistance to the flow

Viscosity coefficients

Parameter associated with the couple-stress fluid

Tapering angle

Density of the fluid

Electrical conductivity

Shear stress tensor

Shear stress

Skew-symmetric part of shear stress

Symmetric part of shear stress

Defined constant

Tapering parameter.

The authors declare that there is no conflict of interests regarding the publication of this paper.