Study of Jeffrey Fluid Flow in an Inclined tube with Overlapping Stenosis

C. Uma Devi¹*, R. Bhuvana Vijaya²

¹Assistant Professor, Dept. of Mathematics,TKR College of Engineering and Technology, Hyderabad, India.

²Associate Professor, Dept. of Mathematics, JNTUA College of Engineering, Anantapur, A.P., India.

*Corresponding Author E-mail: uma140276@gmail.com

ABSTRACT:

This problem deals with the theoretical study of Jeffrey fluid flow through an inclined tube with overlapping stenosis. The nonlinear equations are simplified by considering mild stenosis. The exact solutions are obtained for velocity, pressure drop, flow rate, resistance to the flow and wall shear stress. Effects of different physical parameters like Jeffrey fluid parameter and angle of inclination on resistance to the flow and wall shear stress are studied. The effects of various emerging parameters are discussed through graphs for different values of interest.

KEYWORDS: Overlapping stenosis, resistance to the flow, shear stress, stenosis throat, Jeffrey fluid parameter.

INTRODUCTION:

In many cardiac related problems, the effected arteries get hardened due to the accumulation of fatty substances inside the lumen or because of formation of plaques. This accumulation of substances is known as stenosis. This accumulation is caused due to the deposits of cholestral, fatty substances, cellular waste products, calcium and fabrin. As the disease progresses, the arteries get constricted, which leads to serious consequences like cerebral strokes, myocardial infarction etc. by reducing the blood supply. The flow behavior in the stenosed artery is quite different from the normal artery. As a result, there is a significant change in pressure distribution, wall shear stress and flow resistance.

Therefore to understand the mechanics of the circulation of blood, it is essential to have a clear idea of the basic mechanics of fluids.

Some of the basic investigations with different models of Newtonian and non-Newtonian fluid are given by Naz et al. [1], Shukla and Rahman [2]. Several investigators have highlighted the different aspects of blood flow through arteries under different conditions (Rashid Ali et al. [3], Majhi and Nair [4], Randhir Roy et al. [5]). The blood flow through an artery with permeable walls in the presence of composite stenosis was investigated by Misra al.[6]. Nadeem et al. [7] developed a mathematical model to study the blood flow through tapered stenosed artery by considering blood as a power law fluid. Blood flow through an irregular stenosis in the presence of magnetic field was analyzed by Abdullah [8].

Jeffrey fluid is a class of non-Newtonian fluid. It is noticed that Jeffrey fluid as a generalization of Newtonian fluid. Santosh and Radhakrishnamacharya [9] investigated a two fluid model by considering Jeffrey fluid in the core region and Newtonian fluid in the peripheral region. It is noticed that the effective viscosity and hematocrit decreases with the increase of Jeffrey fluid parameter. The flow of Jeffrey fluid with variable viscosity through a tapered stenosed artery was presented by Akbar and Nadeem [10].In all the above models single stenosis is considered. But in general, stenosis may be formed in series or overlapping. Srivastava et al., [11] developed a mathematical model in the presence of overlapping stenosis by treating blood as a Casson fluid. Chakravarthy and Mandal [12] studied the behavior of blood flow through an arterial segment with overlapping stenosis subject to body accleration. Blood flow through an anisotropically tapered elastic artery with overlapping stenosis under the influence of magnetic field was discussed by Mekheimer et al. [13].But, it is known that many ducts in physiological systems are not horizontal but have some inclination with the axis. Maruthi prasad and Radha krishnamacharya [14] studied the inclination effect on multiple stenoses in a non-uniform cross section. Recently Bhuvna Vijaya et al.[14] developed a mathematical model on Herschel-Bulkley fluid flow in an inclined tube with overlapping stenosis. However, the study of Jeffrey fluid flow through an inclined tube with overlapping stenosis has not been studied.

The main objective of the present paper is to study the inclination effects on Jeffrey fluid thorugh an overlapping stenosis. The analytical solutions of velocity, pressure drop, flow resistance and wall shear stress are obtained. The effects of various parameters on these flow variables are anlyzed through the graphs.

FORMULATION OF THE PROBLEM:

Consider the axi-symmetric flow if steady, incompressible Jeffrey fluid flow in an inclined uniform circular tube of radius in the presence of overlapping stenosis. Cylindrical polar coordinates are taken into consideration. The axis of the tube is taken about the z-axis. It is assumed that the tube is inclined at an angle to the horizontal axis as shown in Fig-1.

Fig- 1: Geometry of the overlapping stenosis in an inclined tube.

The radius of the tube is taken as,

, otherwise (1)

Where is the radius of the tube in the presence of stenosis, is the length of the stenosis and indicates stenosis location, the maximum height of the stenosis into the lumen, which appears at two locations , respectively. The critical height of the stenosis is located at , from the origin.

The constitutive equations for an incompressible Jeffrey fluid are

(2)

(3)

Where and are Caushy stress tensor and extra stress tensor respectively, is the pressure, is the identity tensor, is the ratio of the relaxation to retardation times, is the retardation time, is the dynamic viscosity and is the shear rate.

The governing equations for the present problem are as follows

(4)

(5)

(6)

With the extra tensor components

Where and are the velocities in and -directions respectively.

(7)

After substituting the above non-dimensional variables in Eq.(4)-(6), and assuming mild stenosis, then these equations are reduced to the following form

(8)

(9)

Since the flow is assumed axially symmetric, hence the velocity component in - direction is zero.

The non-dimensional boundary conditions are

at (10)

at (11)

SOLUTION:

With the help of the boundary conditions Eqs. (10) & (11), the solution of equation Eq. (9) is

(12)

The flow rate is defined by

(13)

Integrating Eq. (13),

(14)

(15)

When then the equations coincide with the results of Young[16] results.

The flow resistance, is given by

Where is the pressure drop in the stenosed artery and is given by

(16)

The wall shear stress is given by

(17)

RESULTS AND DISCUSSIONS:

In this part, the influence of the pertinent parameters on resistance to the flow , pressure drop ()and wall shear stress are examined through the graphs [Figs. 2-10]. Variation of flow resistance for different values of height of stenosis, length of the stenosis, Jeffrey parameter and angle of inclination are shown in Figs. 2-5. It is observed that, the resistance to the flow increases with the height and length of the stenosis and also with the length of the tube but decreases with the Jeffrey fluid parameter and with the angle of inclination.

The variation of pressure drop with axial distance for different values of is explored in Figs. 6 and7 respectively. It is noticed that the pressure drop increases with the height of stenosis but it decreases with Jeffrey fluid parameter. The pressure drop is maximum at two stenosis throats.

Figs.8-10illustrates the wall shear stress distribution for different values of height of stenosis, Jeffrey fluid parameter and angle of inclination. It is noticed that the wall shear stress rapidly increases from its approached value at to its peak value in the upstream of the stenosis first throat at , it then decreases steeply in the downstream of the first throat to its magnitude at the critical height of the stenosis at , the wall shear stress further increases steeply in the upstream of the second throat at , from its value at the critical height of stenosis to the same peak value as the first throat and then decreases rapidly in the downstream of the second throat when it attains its approached value (i.e at ) at the end point of the constriction. These results have good agreement with the results of Srivastava et al. [11]. The wall shear stress increases with the increase in height of stenosis but decreases with Jeffrey fluid parameter and angle of inclination.

CONCLUSIONS:

This present study examines the effect of inclinationon Jeffrey fluid flow through an overlapping stenosis. Closed form solutions are obtained and the effects of various geometric and fluid parameters on physiological parameters such as resistance to the flow and shear stress at the wall are studied. It is analyzed that the resistance to the flow increases with the height and length of the stenosis, but decreases with Jeffrey fluid parameter and angle of inclination. The pressure drop increases with stenosis height but decreases with Jeffrey fluid parameter. The wall shear stress increases as the height of stenosis increases, while it decreases with Jeffrey fluid parameter and angle of inclination.

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