Electrically conducting fluid flow with Nanoparticles in an inclined tapering Stenoses Artery through porous medium

Objectives: A Mathematical Model is built in an inclined tapered artery having permeable walls for a blood flow with nanoparticles through porous media. Methods/Statistical analysis: The Nanoparticle phenomena and Temperature profiles are determined using Homotopy Perturbation Method (HPM). Findings: Analysis on resistance (or) Impedance to the flow and shear stress distribution in the stenotic area with regard to different flow parameters with stenosis height has been estimated by deriving the flow characteristic expressions and the solutions obtained. For various flow parameters, the variations of flow resistance as well as shear stress with stenosis height are illustrated graphically. For study of the fluid flow properties, streamline patterns are also drawn. It is remarkable to take note that, in converging (ξ < 0), non-tapered (ξ = 0) and diverging regions (ξ > 0), the flow patterns are significantly impacted by magnetic field existence.


Introduction
The investigation of fluid flow through tubes with a permeable wall have many applications in biological as well as engineering systems. The role of human cardiovascular system is to deliver blood to nerves with optimal pressure to move materials across blood vessels. In cardiovascular related issues, the effected arteries get solidified because of accumulation of fatty materials. The substances stored in the arteries is called stenosis. The well-known valvular heart diseases in the world's developed and developing nations is Stenosis. Vascular fluid dynamics play a substantial job in enhancing vascular stenosis, which is among the most problematic diseases in humans that leads to cardiovascular system failure. The movement of the fluid gets interrupted, depending on the severity of the stenosis. Hence, the mathematical modelling of these type of flows may help to better understand and prevent arterial diseases.
Many researchers have done their work by considering the state of vessel's no-slip boundary at walls. Yet, physiological systems, the walls are in general permeable. Many investigators previously stated that the blood stream is Newtonian. Argument of this Newtonian blood conduct is reasonable for high viscosity rate stream. Blood has Non-Newtonian properties in certain cases (1)(2)(3)(4) .
The study of nanofluid particles is attracted by many researchers. (5) , firstly introduced the concept of nanofluid. The non-Newtonian fluids with nano particles have attained much interest by many researchers (6) .
Number of research studies theoretically investigated the flow of blood across permeable walls. (7) studied in detail the movement of fluid into an artery with permeable wall. The nano fluid movement in tapering stenosis arteries having permeable surfaces has been explored by (8) .
Many researchers have surveyed different scientific models to examine fluid behaviour in the magnetic field existence. Investigators such as (9) studied fluid flow through porous media. (10) analysed the flow of nanofluid and heat transfer into porous media.
In (11) investigated the magneto-hydrodynamic impacts on blood flows into porous media. (12) evaluated the peristaltic transport of porous bounding magneto fluid. (13) delivered an abnormal study of the non-Newtonian fluid flow into tapering arteries, having stenosis. It introduces electric and magnetic fields whenever we impose magnetic field to a traveling electrically conductive fluid. For fluid supply in arteries, disorders such as stenosis, the role of magnetic field is used as a fluid pump in cardiac treatments. Likewise, impact of vessels formed by stenosis on the character of the flow seems incredibly significant. The impact of magnetic field on supply of blood across arteries was studied by (14) . (15) has researched slip impacts on unstable MHD pulsatile fluid movement across porous media in the artery, over body acceleration effect.
In this study, the influence of nanofluid flow characteristics under magnetic effect in an inclined tapering porous media has been examined. Pressure drop, Flow resistance and the Shear stress expressions are obtained. Impact of individual fluid parameters on fluid flow variables were discussed through the graphs. Fluid streamline patterns were discussed.

Mathematical Formulation
A polar coordinate system (r, θ , z) is considered and z-axis is along tube axis. An incompressible stream of nanofluid into an inclined tapering artery having stenosis with fluid viscosity µ and density ρ is considered. The radial and circumferential direction be r, θ respectively. Assume that, stenoses form in an axially symmetric direction. The cylindrical tubular radius is taken as Here, f (z) and R 0 are respectively radius of tapering arterial segment in stenotic area and non-tapering artery in non-stenotic area, where f (z) = R 0 + ξ z , ξ is tapering parameter, n(≥ 2) is shape parameter that defines stenosis shape, b is stenosis length.
is a parameter, δ being the maximum height of stenosis at z = a + b n ( 1 n−1 ) . Governing equations are given by (8) where τ = (ρC) P (ρC) f is ratio to nano particle's effective heat capacity and the fluid's heat capacity. Here, being thermophoretic and Brownian diffusion coefficient respectively.
Introducing the Non-dimensional quantities The equations Eq. (2) to (8) becomes Here, w is velocity with radius R 0 . And, N b , N t are respectively Brownian motion parameter and Thermophoresis parameter. G r , B r being local temperature Grashof number and local nanoparticle Grashof number. θ t , σ denotes temperature profile and nanoparticle phenomena respectively. Also, k is porous medium permeability, µ is viscosity of fluid and M = σ B 0 2 is magnetic parameter.
Non-dimensional boundary conditions are

Solution
The coupled Equations (13) and (14) solutions are obtained by using (HPM), given below Here q t is embedding parameter, lies between 0 and 1.
is Linear operator. The initial guesses θ 10 and σ 10 are The series (19) and (20) in many cases are convergent. This convergent is based on the expression's Non-linear part. For q t = 1, https://www.indjst.org/

Substituting equations (21) and (22) in (12) and by taking boundary conditions, the solution of the velocity is
T he dimension less f lux is q = The flux obtained by substituting (23) in (24) is The flow resistance (or) flow impedance λ is given as λ = △p In absence of stenosis h = 1, pressure drop is represented as △p n and is found from Equation (27) as The flow impedance in normal artery is λ n = △p n q https://www.indjst.org/ T he normalized f low impedance is The wall shear stress is

Results and Discussion
The influence of various fluid stream parameters on flow impedance , wall shear stress (τ h ) for three types of arterial forms (converging, non-tapered and diverging tapering artery) are studied.
The physical characteristics of the flow parameters on flow impedance are displayed in Figures 2,3,4,5,6,7,8 and 9 . Impedance to the flow is noted to increase with the increase of local nanoparticle Grashof number (B r ), Inclination (α) and Magnetic parameter (M), but decreases with Brownian motion parameter (N b ) and shape parameter (n).
It is interesting to note that, with the increase of local temperature Grashof number (G r ) and Thermophoresis parameter (N t ), impedance to the flow is also increasing. But there is no much significance upto δ = 0.04 and observed a significant variation in the remaining part of the domain. Similarly, it is concluded that, with the increase of permeability constant (k), impedance to the flow is decreasing, and this impedance is very small in the region 0 < δ < 0.06. This is true for diverging tapering, non-tapered artery and converging tapering.
Effects of various flow parameters on shear stress (τ h ) are shown in Figures 10,11,12,13,14,15,16,17,18 and 19 . The observation noted that, the wall shear stress enhances with the rise of B r , G r , N t and k, but decreases with N b , n, M and viscosity of the fluid.
Figures 20 and 21 displays the temperature profile variation with N t and N b respectively. An increase in temperature profile is observed by enhancing N t , while decreases by enhancing N b . Figures 22 and 23 shows the variation of nano particle phenomena with N t and N b respectively. It is observed that, nano particle phenomena increase with N t , but decreases with N b .
To discuss trapping phenomenon, streamlines are drawn to examine flow pattern in presence of various parameters. Figures 24 and 25 displays stream lines for different values of k and M. Figure 24 displays the streamlines for the permeability constant (k). It reveals that, as we rise (k), bolus area is decreasing and the amount of boluses rises. Figure 25 reveals the stream lines behaviour for Magnetic parameter, which shows that less boluses are found with the increased magnetic parameter and bolus area is also increasing. https://www.indjst.org/

Conclusion
Under the magnetic effect, the influence of various parameters on blood flow having nano fluid particles in an inclined tapering artery across permeable walls is studied.
The conclusions of this model are