Concentric Catheterization to the Fractional non-Newtonian Hybrid Nano Blood Flow through a Stenosed Aneurysmal Artery


 The main theme of this paper is to analyze the effects of concentric catheterization to the diseased arterial segment having both stenosis and aneurysm along its boundary. Fractional second grade hybrid nanofluid model is under consideration. Governing equations are formulated and further linearized for both cases of mild stenosis and aneurysm. Precise articulations for various important flow characteristics heat transfer, hemodynamic velocity, wall shear stress and resistance impedance are attained. Graphical portrayals for the impact of the significant parameters on the flow attributes have been devised and talked about. The worldwide conduct of blood has been examined using an instantaneous streamlines pattern. The present concept plans to be of use in medical regime for the drug conveyance system and biomedicines.


Introduction
Arterial stenosis and arterial aneurysm have attained very widespread attention because of their recurrent occurrence in both young grownup and pediatric patients. A normal blood circulation within human body is vital for the provision of nutrients, oxygen and functions like waste dismissal. The excess of certain nutrients like cholesterol and fat can result in blockage or expansion of the arteries that deliver blood from the heart to different parts of the body. Many people die as a result of it. The constriction or limitation of an artery or heart valve disturbing normal blood stream is known as stenosis and the associated disease is known as arteriosclerosis. Arteriosclerosis occurs as the arteries turn out to be thick and solid causing coronary artery infections, myocardial infarction, strokes, angina, and cardiac arrests. On the other hand, aneurysm is the expansion of an artery brought about by frailty in the arterial wall. Blood flow through veins turns out to be more confounded because of improvement of aneurysm. A cracked aneurysm can prompt lethal intricacies like high blood pressure, atherosclerosis, trauma and abnormal blood flow. In this way, it's important to look at the effects of dilatations and stenosis on blood flow in these narrow arteries. Numerous experts have studied the various blood flow streams through regular, aneurysmal and stenotic arteries, as shown by the references [1][2][3][4][5][6][7][8][9].
Catheterization is now the most standard medical method for diagnosing and treating arterial diseases. Catheter is a dainty, empty cylinder that is injected into the vein. It tends to be utilized to maximize the supply of blood to indispensable organs. Besides, it can every now and again measure the degrees of gases carbon dioxide and oxygen in the circulation system. When a blood vessel is narrowed or blocked, a catheter with a balloon may be used to expand the vein and increase blood flow. The study of blood circulation via stenotic and aneurysmal arteries is gaining prominence, owing to the ever-increasing requirements of science and medicine. Numerous hypothetical examinations have been acquainted with represent the impact of addition of catheter in the presence of stenosis and on the blood stream, as demonstrated by the citations [10][11][12][13][14][15][16] Nanotechnology centers around miniature items and the creation of issue. The nanoparticle organized measurement is 100 nanometers. The innate capacity of nanotechnology permits the transport of prescriptions to various segments of the human body empowering a proficient conveyance of cargo inside tissues and cells. Nanotechnology is acclaimed as having the capacity to build the proficiency of energy utilization and tackle significant medical conditions. Result of nanotechnology is more modest, more practical and entail less energy. Nanotechnology has helped open a scholarly field of science alongside its applications. By intently noticing explicit materials and their designs, new attributes may likewise be found in that capacity materials are changed into nanoparticles, for example, gold and copper nanoparticles. As an outcome, these most recent advances have a critical clinical notoriety in medication. Another realistic form of nanofluids is hybrid nanofluid, which is made of two or more nanomaterials. Adjoining nanoparticles to base liquid upgrade heat transmission qualities of base liquid. A hybrid nanoparticle is an extraordinary compound that blends the compound and actual properties of different segments together and has been endlessly utilized in the production of anticancer medications. Hypothetical examinations that talked about the nanofluid and hybrid nanofluid accessible can be found in the writing [17][18][19][20][21][22][23][24][25][26][27][28][29].
For those blood vessels of which widths are under 0.5 mm, blood shows shear-subordinate thickness, limited yield pressure before the stream can start, partition into two stages and a cellrich focus and a fringe plasma district. These qualities split speculations that are significant for the plan of the overseeing conditions of Newtonian liquid on the grounds that the coronary courses have a distance across under 0.5 mm. appropriately, a non-Newtonian liquid should be utilized for the portrayal of blood development through these arteries. Accordingly, the viscoelastic partial model is picked to depict the blood development through the coronary veins. All things considered, the last model is gotten from the perceived customary worldview by displacing the ordinary timederivative differential conditions to fractional time-derivative differential equation. In a wide scope of conditions, fractional math has been used to manage distinctive rheological problems. Of the couple of models proposed for physiological liquids, a fractional second-grade liquid model is viewed as significant despite the way that this model limits the fractional time-inferred boundary to a second-grade work ( = 0). Further, a set up Naiver Stokes model can be closed from this as an exceptional case by putting the second grade material constant 1 0  = [30][31][32][33][34][35][36].
The main aim of this research is to look at the combined effects of hybrid fractional secondgrade fluid model and concentric catheterization in diseased artery having stenosis and aneurysm which has never been done before to the author's knowledge. For mild stenosis and mild aneurysm situations, the constructed problem is simplified to calculate the exact expressions of temperature, hemodynamic velocity, wall shear stress and resistance impedance for the flow. By plotting graphs and stream lines, the effects of evolving flow parameters are also debated. Finally, in the finishing part, the chief findings of the outcomes are summarized.

Problem Formulation
We consider the fractional second-grade hybrid nano blood flow through an annular region bounded by two coaxial tubes. Outer tube contains an axially symmetric mild stenosis and aneurysm while the inner tube represents the catheter. Balloon catheterization is accomplished in the stenotic segment. We deal with the cylindrical coordinate system ( , , ) rz  while modelling the flow in such a way that z-axis is taken as the axis of the artery and r-axis is in the radial direction. Geometries of the outer and the inner walls are defined by where length of the jth irregular stenosed section emanating from the origin is For fractional second-order derivative we use Caputo's definition where mN  ,  is the order of the derivative. And h is the initial guess of f . For the derivatives we will use Caputo's derivative condition The physical configuration of the blood flow under consideration is given in figure 1.
where r represents radial direction and z -axis is taken along the axis of the artery. u and w are radial and axial velocity components, Q is constant of heat generation or heat absorption, Thermophysical properties of nano blood flow are described as [9] ( ) Thermophysical properties of Hybrid nano blood flow are described as [9] (

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. A distinction between hybrid nanofluid and nanofluid is made. Table 1 provides thermophysical quantitative data for the particular thermal expansion, thermal conductivity and density of blood containing Au and 23 Al O nanoparticles. Table 2 shows the shapes of nanoparticles and their related shape factors.

Table 1
Thermophysical properties of hybrid nano particles Table 2 Nano particles shapes factors

Mild Disease Approximations
We define following variables for the non-dimensionalization .
where U is defined as the velocity averaged over the section of the tube having radius b, temperature is expressed through  and r G is the Grashof number. Under the assumptions of mild stenosis and mild aneurysm we apply the following conditions Equations (7) to (10) in linearized forms after using dimensionless quantities take the forms 0, The dimensionless boundary conditions are 0 at ( ) and 0 at ( ), 0 at ( ) and 0 at ( ).
The dimensionless forms of arterial and catheter walls

Solution of the Problem
The closed form solutions for temperature and hemodynamic velocity are obtained by solving equations (17) to (19) after applying boundary conditions specified in equation (20)

Significant Flow Characteristics
The suitable articulation for volume flow rate in dimensionless form is defines as The volume flow rate () Fz can also be written in the form of linear combination of 1 S and 2 S where 1 S and 2 S are given below Articulations for pressure rise and resistance impedance for non-dimensional arterial segment under consideration is given by 1 0 dp p dz dz Expressions for wall shear stress on arterial wall segment and catheter wall segment in dimensionless forms are given as Stream function  is calculated in the following structure

Graphical Outcomes and Analysis
The determined articulation for temperature and hemodynamic velocity have been acquired for hybrid nano blood stream model. The dynamical features of the wall shear stress, resistance impedance and pressure rise are characterized. Some significant qualities have been explored for different parameters emerging in the solution articulation on the arterial wall segment containing both stenosis and aneurysm in the presence of catheter. Specifically, the effects of the fractional parameter  , the relaxation time 1  , Grashof number r G , heat source Q , nanoparticles shape factor n , nanoparticles volume frictions 1  and 2  , catheter's maximum height parameter  , catheter radius parameter e and maximum height a of stenosis and aneurysm. A graphical examination is set to analyze between the stenosis section (a > 0) and aneurysm section (a < 0). To calculate the numerical closed-form expression for resistance impedance and pressure rise we utilized the numerical integration.

Temperature
The temperature distribution against the radial coordinate r has been plotted for different values of heat source parameter Q , nanoparticles shape factor n, catheter's radius parameter e and catheter's maximum height parameter  for stenosis and aneurysm region and for nanofluid and hybrid nanofluid in Fig. 2. In Fig. 2(a) an examination among hybrid nanofluid and nanofluid is likewise made. It is seen that the heat transfer for nanofluid is lower than that of hybrid nanofluid. It is found in Figs. 2(b) and 2(c) that the temperature drops for spherical-shaped hybrid nanoparticles and by increasing e . And temperature in stenosis segment is much lower than that of dilatation segment. From Fig. 2(d) we can see that the temperature decreases by increasing  in stenosis segment and remains unchanged in dilatation segment. Moreover, it is observed that the temperature in stenosis segment is much lower than that of dilatation segment.  , fractional parameter  , Grashof number parameter r G , catheter's radius parameter e , catheter's maximum height parameter  and nanoparticles shape factor n. In Fig. 3 a comparison is made up between hybrid nanofluid and nanofluid, and between stenosis and aneurysm. In Fig. 3(a) it is discovered that the hemodynamic velocity for nanofluid is much lower than that of hybrid nanofluid. Also by increasing 1  hemodynamic velocity decreases. It is observed in Figs. 3(b) and 3(c) that the hemodynamic velocity increases by increasing  and r G . Also the hemodynamic velocity in stenosis segment is much lower than that of dilatation segment. Figs. 3(d) and 3(f) show that the velocity decreases by decreasing nanoparticles shape factor n and by increasing e . Moreover it is observed that hemodynamic velocity in stenosis segment is much lower than that of dilatation segment. Fig. 3(e) shows that the hemodynamic velocity decreases by increasing  in stenosis segment while remains unchanged in dilatation segment. And hemodynamic velocity is higher in dilatation segment as compared to stenosis segment.

Resistance Impedance
In Fig. 6 the resistance impedance  against the maximum height a for stenosis and aneurysm in the presence of catheter is plotted. Resistance impedance  for different values of fractional parameter  , relaxation time parameter 1  , hybrid nanoparticles fractional parameters 12 and  , catheter's radius parameter e and catheter's maximum height parameter  is plotted. For both stenosis and aneurysm segments, it is discovered that resistance impedance  is inversely proportional to a . It's also worth noting that for maximal a , the resistance impedance  for stenosis is higher than for aneurysm. In Figs. 6(a) and 6(b) it is observed that by increasing fractional parameter  resistance impedance  decreases. And resistance impedance for stenosis segment is higher than that of dilatation segment. Figs. 6(c)-6(h) show that the resistance impedance  increases by increasing relaxation time parameter 1  , hybrid nanoparticles fractional parameters 1 2 and  and catheter's radius parameter e . Moreover, resistance impedance for stenosis segment is higher as compared to dilatation segment. It is seen in Figs. 6(i) and 6(j) that by increasing catheter's maximum height parameter  resistance impedance  increases in stenosis segment and remains unchanged in dilatation segment. It is also observed that resistance impedance for stenosis segment is higher as compared to dilatation segment.

Trapping
The trapping is a significant in connection to the hydrodynamic characteristics of aneurysm and stenosis segments. Drawing contours is the best way to imagine the trapping process. The trapping mechanism is distributed through the revolving bolus internally. The streamlines are enclosed by the scale of the flowing bolus in the fluid. The trapping mechanism for maximum height of stenosis and aneurysm a , catheter's radius e and catheter's maximum height  are discussed for both stenosis and aneurysm segment. It is observed in Fig.7 that the rotated bolus sized increased by increasing a , e and  in stenosis segment. It is found in Fig. 8 that the circulating bolus size increases by increasing a in dilatation segment. Moreover it is also seen that size of the circulating bolus decreases by increasing e in dilatation segment. It is also observed that size of circulation bolus remains unchanged by increasing  in dilatation segment.

Conclusion
In this paper hybrid fractional second-grade nano blood flow through concentrically catheterized stenosed and aneurysmal arterial segment is studied. Exact solutions are obtained by solving governing equations, and the effects of the relevant parameters are examined. Comparisons are established for aneurysm and stenosis, and for hybrid nanofluid and nanofluid. The principle discoveries from the graphical portrayals can be summed up as follows: • As spherical-shaped nanoparticles are compared to other shapes of nanoparticles, the temperature drops. • Temperature for dilatation segment is higher than that of stenosis segment.
• Hybrid nanofluid heat transfer is higher than nanofluid heat transfer.
• As compared to nanofluid, the hemodynamic velocity of hybrid nanofluid is higher.
• As compared to the other nanoparticles' forms, the velocity diminishes for spherical-shaped nanoparticles. • Hemodynamic velocity for dilatation segment is higher than that of stenosis segment.
• The arterial wall shear stress in the stenosis segment increases quickly to its greatest value and then begins to decrease steeply to the end of the stenosis segment, whereas the opposite trend of arterial wall shear stress is detected in the aneurysmal segment when compared to the stenosis segment. • The catheter wall shear stress in the aneurysmal segment increases quickly to its greatest value and then begins to decrease steeply to the end of the aneurysmal segment, whereas the opposite trend of catheter wall shear stress is detected in the stenotic segment when compared to the aneurysmal segment. • Contrary behavior of wall shear stress is observed on arterial and catheter wall segment.
• Wall shear stress on catheter wall segment is more than that of wall shear stress on arterial wall segment. • For the stenosis, the resistance impedance achieves a higher value than for the aneurysm segment. • The trapping mechanism depicts the formation of a rotating bolus in the stenotic and aneurysmal segments with varying parameters of concentration.