Thermo-diffusion effects on mhd casson blood flow in an inclined muti-stenosed artery

Abstract

Keywords: Casson fluid; Fractional derivative; Magnetic particles; MHD; Multi-Stenosed artery; Thermal radiation

Introduction

In contrast to the linear dependency shown by Newtonian fluids, the stress in non-Newtonian fluids can show a nonlinear rate of deformation. According to the literature, from the 1940s and 1950s, there has been an increase in interest in non-Newtonian fluids. NonNewtonian fluids include things like paint, shampoo, soap slurries, tomato paste, greases, food sauces, chocolates, toothpaste, polymer solutions, custard, and blood. NonNewtonian fluid models have been the subject of much study by numerous academics because of its applicability

in numerous industries. Casson [1] developed the Casson fluid model in 1959 to determine how pigment oil suspensions would flow. Human red blood cells, human blood can also be considered as Casson fluid. Because blood is such a vital fluid, its viscosity and other characteristics can be used to detect a wide range of cardiovascular disorders. A great number of theoretical investigations [2-5] have been discussed blood flow circulation in the arteries. It is generally recognized that in sick conditions, an aberrant and unnatural growth arises in the lumen at numerous places throughout the circulatory system. Arteriosclerosis, often known as stenosis, is a common condition. Chaturani and

*Corresponding author. *E-mail address: harshadpatel2@gmail.com This paper was recommended for publication in revised form by Editor-in-Chief Ahmet Selim Dalkilic Published by Yıldız Technical University Press, İstanbul, Turkey Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

Ponnalagar Samy [6] have studied how blood moves via a stenosed artery. Many researchers [7-9] have investigated to impact of stenosis in the lumen of an artery. Casson’s equation is obeyed by blood only for modest shear rate flows, according to Scott Blair et al. [10]. Shukla et al. [11] analysed the effects of stenosis on blood flow which is treated as a power-law and Casson’s fluid. Important applications of magnetic fields include the targeting of cancer drugs, the use of heat to kill tumors, magnetic resonance imaging (MRI), and magneto-therapy. In all these treatments effect of magnetic field plays crucial role. It is currently believed that magnetohydrodynamics (MHD) influences blood flows, or ionic flows, in a manner consistent with the bloodstream. In 1970, Hannes Alfvén [7] was awarded the Nobel Prize in physics for pioneering the field of MHD. The study of blood flow under the effect of magnetic field comes under the title Bio-magnetic fluid dynamics (BFD). Researchers [8-9] are curious about bio-magnetic fluid because Hyperthermia cell death, magnetic drugs targeting, magnetic endoscopy, magnetic devices for cell separation, therapy of cancer tumors, regulation of blood flow during surgery, and many more applications can be found in bioengineering. In the past few decades, the radiation therapy is used in human life where heat is transmitted below the skin’s surface into the tissues and muscles [10-11]. Mekheiner and Kot [12] theoretically addressed the issue of blood flow through a catheterized artery under pathological conditions. To better understand the dynamics of blood flow via stenotic arteries, Majee and Shit [13] conducted a computer study of heat transfer in an unsteady blood flow. Both Rao and Vardynyan [14] demonstrated mathematically that blood can flow through an artery. Very few studies have looked at the effects of different parameter on blood flow and even fewer have taken into account fractional-order derivatives. However, the finite Hankel and Laplace transforms can be used to quickly find exact solutions to this type of problem. Caputo fractional derivatives and Hankel Transform methods are discussed to find the solution by researchers [15-17]. Due to its significance in physiopathology, problems with peristalsis (blood flow) have garnered a lot of attention. Mekheiner [18] determined that the pressure acts as an increasing function, but the behaviour of the pair stress fluid parameter appears to be the complete reverse. In his study, Akbar [19] examined the Prandtl fluid model in a tapering stenosed artery. The pumping features of peristaltic flow were discussed by Sreenadh et al. [20]. The slip effects for the flows investigated by Ramesh and Devakar [21], who then derived analytical solutions. Nanofluid flow onto a stretched, curved surface was studied by Mehdi Mahboobtosi et al. [22] As a unique breakthrough, Moghimi et al. [23] analysed nanofluid flow in a channel and added magnetic field power to momentum and energy equations and evaluated free convection heat transmission. In a non-premixed configuration and in non-adiabatic conditions, Akbari et al. [24] conducted

a theoretical analysis of a counter-flow combustion system that is powered by porous biomass particles. Using the response surface method, Navid et al. [25] optimized a wavy trapezoidal porous cavity that contained a mixture of hybrid nanofluids. Using computational methods, Zangooee et al. [26] examined the mixed nanofluid (NF) flow between two overlapping cylinders. Hosseinzadeh et al. [27] examined the thermal performance of a ferrofluid-wetted hybrid nanofluid with varying cross-sections in a moving porous fin subjected to a magnetic field. Hosseinzadeh et al. [28] studied the flow of a TiO2-ethylene glycol nanofluid across a porous stretched sheet under convective boundary conditions and the presence of heat that is not uniformly generated or absorbed. Muddasar Gulzar et al. [29] presented a nonlinear mathematical analysis of magneto-hyperbolic-tangent liquids that involve the three interrelated concepts of a magnetic field, a heat source, and thermal stratification. As a non-Newtonian fluid moving inside an axisymmetric tube subjected to non-uniform surface heat flux, Shahin Faghiri et al. [30] studied the GraetzNusselt problem for blood. Tashtoush et al. [31] presented a mathematical model of in-arterial multi-stenosis. The effects of a magnetic field on the heat and fluid flow properties of blood flowing through arteries with multiple stenosis are taken into account. In light of what has been said previously, the objective of this investigation is to look into the impact that heat and mass transfer with thermal radiation and thermos-diffusion effects on blood flow in multi-stenosed artery. A thorough search of the relevant literature has witnessed the fact that the existing literature did not present the exact solution of MHD blood flow model in the context of Caputo Fabrizio fractional derivative for inclined multi-stenosed artery. The exact solutions are then calculated by means of significant transformations like Laplace and finite Hankel transforms. For numerical computations, we take the zeros of the Bessel functions to generate the graphical findings by using Matlab for different values of fractional parameters as well as some important physical parameters. The present work furnishes a robust benchmark for magneto-hemodynamics, biomedical engineering, and physiology.

Mathematical Formulation

The focus of the recent research works is discussed on unsteady Casson blood flow, whose physical dimension are as shown in Figure 1. In Figure 1, z-axis consider as axial direction while r-axis indicates radial direction. Blood is treated as non-Newtonian Casson fluid flow with oscillating pressure gradient. The uniform external magnetic fields B0 is applied which is shown in Figure 1. At t = 0, the velocity of blood and magnetic particle are treated as stationary. Blood flow is modelled using Navier-stokes equation. The effects of magnetic fields by Maxwell’s relation whereas, magnetic particle velocity is governed using Newton’s second law.

The unsteady blood flow in an axisymmetric cylindrical tube of radius R0 under the influence of uniform transverse magnetic field and pressure gradient of the form [31].

blood is proportional to the relative velocity. Using Newton’s second law governs the movement of magnetic particles: (4)

(1) Where, a0- systolic pressure gradient, a1-diastolic pressure gradient. Geometrically, the expression of multi-stenosis in the artery can be written mathematically [32] (2)

where m is the average mass of the magnetic particles. The governing energy equation in the cylindrical form (5) The concentration equation in the cylindrical form as, (6)

Where, Rz and R0 represent the constricted region and normal artery radius. x is the stenosed length and α1 stenosis degree. The momentum equation for fluid flow in an inclined stenosed artery [15-33] can be written as

With initial and boundary conditions are as, (7) The dimensionless parameters can be express as,

is the force due to the relative The term motion between fluid and magnetic particles. It is assumed that the Reynolds number of the relative velocity is small. As such, the force between the magnetic particles and the

The governing dimensionless form of governing equation subject to equations (7) after dropping notation and

consider the time-fractional model of governing equations 3-6 can be written as,

Similarly, we process for Concentration equation (11), we get

The FHT (Finite Hankel transformation) is applied in equations (17) with B.C. (12),

Solution of the Problem The Laplace transform technique is applied in equations (10), we obtain

(13) The FHT (Finite Hankel transformation) is applied in equations (13) with B.C. (12), (14)

(28) Now, the Laplace transform of equations (8) and (9) can be express as,

(30) (42) (31) Input the equation (31) in (29), the following equation can be obtained,

The exact expression of blood velocity, Temperature and Concentration profiles are obtained by taking the Inverse Hankel transformation of equations (40) - (42), we get

The FHT (Finite Hankel transformation) is applied in equations (32) with B.C. (12),

(36) (48) Where, represents the finite Hankel transformation of the velocity and temperature function

(37) And are the positive roots of an equation . The I.L.T of the image function can be written as, (38)

(39) The ILT (Inverse Laplace transform) is applied in (16), (28) and (36) are (40) (41)

Results And Discussion

The effects of different physical parameter on velocity, temperature and concentration profiles were studied via graphs which is represented in Fig. 2 to 14. In every case, the fractional parameter that corresponds to 1 is used for comparison, and there are very few cases in which the fractional parameter is strictly less than 1. For the numerical calculations, the following parameters are fixed. a0 = 0.5, a1 = 0.1, ω = π/4, Pe = 0.5, G = 0.5, Ha = 1 and Re = 1. Figure 2-5 shows the effects of systolic and diastolic pressure gradient on both velocity profiles. From the figures, it is concluded that the blood velocity improved with both parameter values increases. Figure 6-7 shows the effects of Casson fluid parameter effects on blood as well as magnetic particle velocity. The Casson parameter is related to the non-Newtonian nature of the blood. Higher Casson parameter is attributed to the Newtonian nature. With an increase

in the Casson fluid parameter, the fluid velocity increases. Casson nature is more significant for small arteries where red blood cells (RBCs) can accumulate due to rotation near the axis of the artery, creating a region depicted in the cells. This statement is in perfect agreement with Jamil et al., [34] for a horizontal cylinder. It is hypothesized that the yield stress declines as γ increases and the thickness of the boundary layer decreases. The magnetic field is used for regulating the blood flow within the human circulatory system. Due to increasing the values of Casson fluid parameter values, blood become thin, so the motion of flow is improved. The blood velocity at different inclination angles ∅ are plotted in Figure 8. From the figure, inclination angles tend to improve the velocity profiles. Figure 9 show the temperature profiles increase with increasing the values Peclet number. Figure 10-11 show the effects of metabolic heat source and heat absorption parameter on temperature profiles. From both figures, it is illustrated that the heat transfer process improves with both parameters. Physically when we increase the heat source parameter, fluid become thinner, due to this effects heat transfer process become faster. The particle mass parameter G is defined as the size of the particles. Figure 12 shows the effects of mass parameter on magnetic particle velocity. From the figure, it is seen that the magnetic particle velocity reduces with increasing the values of said parameter. Figure 13-14 displayed the Reynolds numbers effects on both velocity profiles. Physically, lower viscosity (increased velocity) will increase. So, from the both figures, it is concluded that the blood as well as magnetic fields velocity improved with increasing the values of Reynolds number.

should be noted that the particle has the same tendency as the blood; however, it moves slower. The magnetic particle velocity is slow compared to the blood velocity. These findings will be beneficial for atherosclerosis therapy. The blood as well as magnetic particle velocity improved with Reynold number and Casson fluid parameter. The numerical findings reveal that the inclination angle has a considerable influence on blood and magnetic particle velocities. The development might help in identifying and treatment for specific illnesses. The systolic and diastolic pressure gradients tend to raise blood flow and magnetic particle velocity. Because of these consequences, blood flow in the stenosis artery may be normal. The Peclet number tends to improve heat transfer process.

Acknowledgement

Authors are thankful to Ganpat University- Centre for Advanced Research Studies (CARS) (F. No. 273/GUNI/ CARS/1309/2022, Date: 21/10/2022) for providing financial support to carry out research work.

Conclusion

The following are the most significant findings: The blood flow and magnetic particles distributions are highly influenced by the fractional order parameter. It

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

Statement On The Use Of Artificial Intelligence

Artificial intelligence was not used in the preparation of the article.

Nomenclature

B0 Magnetic field parameter γ Casson parameter ω “Pulsatile frequency” R Particle concentration parameter” Ha Magnetic field parameter ∅ Phase angle F Inclination angle P Oscillating pressure gradient Re Reynolds number Pe Peclet Number Qm Heat Source θm Heat absorption G Particle mass parameter ρ Fluid density r Radial coordinator α Fractional parameter σ Electrical conductivity u(r, t) velocity of the Fluid v(r, t) velocity of the Particle θ Dimensionless Temperature N Magnetic particles number K Stokes constant υ Kinematic viscosity Sc Schmidt number Sr Soret parameter

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