Inclined magnetized infinite shear rate viscosity of non-Newtonian tetra hybrid nanofluid in stenosed artery with non-uniform heat sink/source

Wael Al-Kouz; Wahib Owhaib; Basma Souayeh; Zulqurnain Sabir

doi:10.1515/phys-2024-0040

Article Open Access

Inclined magnetized infinite shear rate viscosity of non-Newtonian tetra hybrid nanofluid in stenosed artery with non-uniform heat sink/source

Wael Al-Kouz , Wahib Owhaib , Basma Souayeh and Zulqurnain Sabir

Published/Copyright: June 3, 2024

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Open Physics Volume 22 Issue 1

Abstract

Many scholars performed the analysis by using the non-Newtonian fluids based on the nano and hybrid nano particles in blood arteries to investigate the heat transport for cure in several diseases. These performances are presented to investigate the blood flow behaviour with extended form of the novel tetra hybrid Das and Tiwari nanofluid system attached by the Carreau fluid. The assessment of energy transport has been achieved based on the thermal radiation, heat source/sink, Joule heating, and viscous dissipation. The obtained partial differential equation from physical problem is transformed into ordinary differential equations (ODEs) by using the similarity variables. Furthermore, system of nonlinear ODEs attached with boundary conditions are transported into the system of first-order ODEs with initial conditions. For the numerical solution of obtained ODEs, the numerical solutions have been performed based on the RK method. The numerical results are plotted through figures, tables, and statistical graphs. Magnetic forces and inclined magnetic effects are caused to reduce velocity of blood. Temperature of blood within the arteries is increased by increasing the parameter of thermal radiation.

Keywords: Carreau fluid; tetra hybrid; Joule heating; viscous dissipation; Tiwari and Das model

1 Introduction

Influence of magnetic field on nanofluid is fully applied in human activities, like engineering devices, electronics, study of plasma in fusion reactors, geophysical fluid dynamics, chemical engineering, medicine, astrophysics, etc. Magnetohydrodynamics (MHD) carries electrically conductive magnetized fluids and creates the interaction between flow of the fluid and magnetic field. Many scholars [1,2,3] made attempts in MHD flow with different nanofluid like hybrid, bi-hybrid, and trihybrid. Shoaib et al. [4] gave results of numerical attempt related to MHD hybrid rotated nanofluid with geometry of a stretching sheet. Localized heat source/sink with MHD effect in hybrid nanofluid on square cavity geometry is discussed by Chamkha et al. [5]. In this study it is concluded that magnetic parameter reduced the velocity of hybrid nanofluid. Zainal et al. [6] studied MHD flow of hybrid nanofluid and heat transport considering permeable moving surface. Furthermore, effect of thermal radiation is also taken and concluded that MHD effect produced higher temperature. Remarkable attempt regarding ternary-hybrid nanofluid with Lorentz force effect was made by Oke et al. [7]. In this attempt they took three-dimensional flow of water including alumina nanoparticles, graphene, and carbon nanotubes. Mahmood et al. [8] investigated about stagnation point flow and influence of suction and heat source attached with ternary hybrid nanofluid.

The need for the Carreau fluid model arises because many real-world fluids exhibit complex rheological behaviour that cannot be described by simple models like the Newtonian model. For example, some fluids exhibit shear-thinning behaviour, meaning that their viscosity decreases as the shear rate increases. Other fluids exhibit shear-thickening behaviour, meaning that their viscosity increases as the shear rate increases. The Carreau fluid model can accommodate both of these behaviours and more. Many authors made their research related to Carreau fluid [9,10,11,12,13] attaching simple and nanoparticles involved fluid. Bibi and Xu [14] worked on hybrid nanofluid using Carreau fluid model. In this study, concentration of nanoparticles is judged by homogeneous/heterogeneous reactions. Chalavadi et al. [15] discussed the vital impact of heat transport via generation/absorption associated with MHD effect. In this article, two mathematical models Casson/Carreau hybrid nano liquids were taken and compared. Application side of different hybrid nanofluids and heat transport via convection was unwrapped by Kumar et al. [16]. Hall and slip impact on Carreau–Yasuda hybrid nanofluid was investigated by Rana et al. [17].

Mixed fluid with nanoparticles is called hybrid nanofluid and when two nanoparticles are involved, it is referred to as bi-hybrid nanofluid and fluid containing three nanoparticles is called ternary hybrid nanofluid. Ternary hybrid nanofluid possess good thermal performance than bi-hybrid, and bi-hybrid has good performance as compared to simple hybrid nanofluid. Another base fluid containing four types of nanoparticles is classified as tetra hybrid nanofluid. A tetra hybrid nanofluid is a type of nanofluid that is made by suspending nanoparticles of different materials in a base fluid. Many researches investigated hybrid, bi-hybrid and ternary hybrid nanofluid with localized magnetic fields with tri-hybrid nanofluids, inclined magnetic aspect of infinite shear rate viscosity with Carreau hybrid nanofluid, cross fluid flow containing nanoparticles and motile gyrotactic microorganisms, ternary hybrid cross bio-nanofluid expanding/contracting cylinder, scrutiny of nanoscale heat transport with ion-slip and radiative effect on hybrid ferroparticles over a porous sheet. Recently Alqudah et al. [18] investigated the ternary nanoparticles to optimize the nanoscale thermal transport of Carreau nanofluid over 3D wedge. Impact of inclined magnetic field in fuzzy hybrid nanofluid is made by Ayub et al. [19]. Al-Kouz et al. [20] explored the thermal proficiency of cross-ternary hybrid nanofluid flow through vertical cylinder. In this study, they took magnetized and radiative environment for thermal transport.

Stenosis is a medical term that refers to the narrowing of a blood vessel, including an artery. In the context of an artery, stenosis can occur when the diameter of the artery is reduced, leading to a decrease in blood flow. Many scholars investigated the fluid dynamics over the geometry of stenosis, artery etc. Shahzad et al. [21] pointed out the interaction of Casson fluid in a bifurcated channel having stenosis with elastic walls. Nowak et al. [22] established mathematical link for the artificial aortic and progressive valve stenosis. Simulation of biomedical nanoparticles to investigate the drug delivery to blood was made by Mekheimer et al. [23]. Haowei et al. [24] employed mathematical model of Sisko fluid to find the characteristics and thermal attitude of blood in stenosis artery. In this study, further several effects like different severities of stenosis, heat flux, and different radii of the artery are taken. Latest study regarding aortic stenosis and its evaluation with mathematics of Navier–Stokes relation was inspected by Gill et al. [25].

1.1 Applications of heat transfer in blood flow

Heat transfer is a very key aspect in engineering point of view. In medical field, heat transfer manages the blood flow within the human body and influence on various medical interventions and physiological processes. One significant application is thermal therapy, where localized cooling or heating is utilized to treat diseases like cancer. Hyperthermia employs heat transfer principles to elevate tissue temperatures and it enhances the efficacy of radiation treatments or chemotherapy. Heat transfer concepts are pivotal for blood flow dynamics within vessels like hypertension and atherosclerosis.

1.2 Motivation

Several scholars analysed the influence of diverse fluids mixed with various nanoparticles like gold, silver, zinc, iron, titanium, etc., in blood though arteries attached several effects to remove the plaque and toxic material from the blood, detection of tumour. Inclusion of hybridity and nanoparticles in traditional fluid is attentive topic nowadays. Based on the motivation received due to the existence of such vital role of nanoparticles in base fluid, current investigation is launched to detect the impact of inclined magnetized radiative Carreau tetra hybrid nanofluid in stenose artery and its effects.

1.3 Novelty

Work on hybrid, bi-hybrid, and ternary hybrid nanofluids is made separately with attachment of several models by scholars, but in this attempt, extended hybrid nanofluid named as tetra hybrid nanoparticles are associated with Carreau fluid model with nonlinearly thermal radiative, heat source (sinking), viscous dissipative flow, Joules heating, and tetra hybrid nanoparticles.

2 Mathematical formula

Figure 1 represents the conduct of blood flow supply in artery. It is assumed that blood is flowing through horizontal artery whose length is L o 2 . Blood is flowing in x-axis and radial axis r. Inclined magnetic field is imposed on blood flow in artery. Investigation on the temperature analysis of blood flow in artery is carried out by incorporating the facts of non-uniform heat sink/source, thermal radiation, and Joule heating. An extension of nanofluid named as prototype which is invention of Tiwari and Das has been included in blood to investigate the impact of four kinds of nanoparticles in blood flow passing through artery. Radius of artery is denoted by R ( x ) ⌢ and λ is maximum height. Area of stenosis is given by R ( x ) ⌢ = R o ⌢ − λ 2 1 + cos 4 π x L o , where x ∈ − L o 4 , L o 4 = R 0 . It means value of x will be between − L o 4 , L o 4 , which is equal to R 0 . So, now R ( x ) ⌢ = R o ⌢ − λ 2 1 + cos 4 π R 0 L o . Furthermore, four nanoparticles are added and their thermophysical properties are presented in Table 1.

Figure 1

Geometry of blood flow via arteries.

Table 1

Thermophysical characteristics [28,29] of TiO 2 , Au , Al 2 O 3 & Ag on blood

Properties	Blood	Gold (Au)	TiO 2	Silver (Ag)	Al 2 O 3
ρ	1050.00	19300.00	4250.00	10500.00	3970.00
C p	3617.00	129.00	690.00	235.00	765.00
K	0.5200	310.00	8.95300	429.00	40.00
σ	0.66700	4.1 × 10 6	2.4 × 10 6	6.3 × 10 7	3.5 × 10 7

Utilizing the boundary layer analysis on obtained system of partial differential equation (PDEs) from Navier Stokes equations, we get equation of momentum, and from Fick’s law heat equation is generated. Therefore, equation of continuity, momentum and heat is given [26,27] as follows:

(1) ∂ ( r u ˜ ) ∂ x + ∂ ( r v ˜ ) ∂ r = 0 ,

(2) ρ tethnf u ˜ ∂ u ˜ ∂ x + v ˜ ∂ v ˜ ∂ r = μ tethnf r ∂ ∂ r ∂ u ˜ ∂ r β ∗ + ( 1 − β ⁎ ) Γ ∂ u ˜ ∂ r 2 ( n − 1 ) 2 − σ tethnf B o 2 sin 2 ( ϖ ) u ,

(3) ( ρ C p ) tethnf u ˜ ∂ T ∂ x + v ˜ ∂ T ∂ x = k tethnf r ∂ ∂ r r ∂ T ∂ x + ∂ ∂ r 16 σ ⁎ ⁎ T ∞ 3 ∂ T 3 κ ⁎ ⁎ ∂ r + k tethnf u ˜ w x v ˜ tethnf [ U ⁎ ( T w − T ∞ ) f ′ ˜ + W ⁎ ( T − T ∞ ) ] ,

where the boundary conditions (BCs) are

(4) r = R : u ˜ = u ˜ o , v ˜ = 0 , T = T w , r → ∞ : u ˜ → 0 , T → T ∞ . .

Portrayals of concerning thermo-physical characteristics for tetra hybridity nanofluid are

(5) μ tethnf = μ f ( 1 − φ 1 ) 2.5 ( 1 − φ 2 ) 2.5 ( 1 − φ 3 ) 2.5 ( 1 − φ 4 ) 2.5 ,

(6) ρ tethnf = ( 1 − φ 1 ) ( 1 − φ 2 ) ( 1 − φ 3 ) ( 1 − φ 4 ) φ f + ρ 4 φ 4 + ρ 3 φ 3 + ρ 2 φ 2 + ( 1 − φ 4 ) φ f + ρ 4 φ 4 + ρ 3 φ 3 + ρ 1 φ 1 ,

(7) ( ρ C p ) tethnf = ( 1 − φ 1 ) ( 1 − φ 2 ) ( 1 − φ 3 ) [ ( 1 − φ 4 ) ( ρ C p ) f + ( ρ C p ) S 4 φ 4 ] + ( ρ C p ) S 4 φ 3 + ( ρ C p ) S 2 φ 2 + ( ρ C p ) S 1 φ 1 ,

(8) k tethnf k hnf = k 1 + 2 k nf − 2 φ 1 ( k nf − k 1 ) k 1 + 2 k nf + φ 1 ( k nf − k 1 ) , k tethnf k hnf = k 2 + 2 k nf − 2 φ 2 ( k nf − k 2 ) k 2 + 2 k nf + φ 2 ( k nf − k 2 ) ,

(9) k tethnf k hnf = k 3 + 2 k nf − 2 φ 3 ( k nf − k 3 ) k 3 + 2 k nf + φ 3 ( k nf − k 3 ) , k tethnf k hnf = k 4 + 2 k nf − 2 φ 4 ( k nf − k 4 ) k 4 + 2 k nf + φ 4 ( k nf − k 4 ) .

Electrical conductivity

(10) σ tethnf σ hnf = ( 1 + 2 φ 4 ) σ 4 + ( 1 − 2 φ 4 ) σ tethnf ( 1 − φ 4 ) σ 4 + ( 1 + φ 4 ) σ tethnf , σ tethnf σ hnf = ( 1 + 2 φ 3 ) σ 3 + ( 1 − 2 φ 3 ) σ hnf ( 1 − φ 3 ) σ 3 + ( 1 + φ 3 ) σ hnf , σ hnf σ nf = ( 1 + 2 φ 2 ) σ 2 + ( 1 − 2 φ 2 ) σ nf ( 1 − φ 2 ) σ 3 + ( 1 + φ 2 ) σ nf , σ nf σ f = ( 1 + 2 φ 1 ) σ 1 + ( 1 − 2 φ 1 ) σ f ( 1 − φ 1 ) σ 1 + ( 1 + φ 1 ) σ f ,

3 Solution procedure

Following similarities are introduced for conversion of PDEs into ordinary differential equations (ODEs).

(11) η = r 2 − R 2 2 R u ˜ o v ˜ L o , u ˜ = u ˜ o x L o f ′ ˜ ( η ) , v ˜ = R r u ˜ o v ˜ L o f ˜ ( η ) , θ ( η ) = T − T ∞ T w − T ∞ ,

By substituting Eq. (11) in Eqs. (2) and (3), following results appear in the form of ODEs.

(12) ( 1 + 2 γ η ) ( 1 + n ( We ( f ″ ˜ ) 2 ) ) ( β ∗ + ( 1 − β ⁎ ) ( We ( f ″ ˜ ) 2 ) ) n − 3 2 f ‴ ˜ + 2 γ f ″ ˜ A 1 A 2 1 + 1 − n 2 ( We f ″ ˜ ) n + A 1 A 2 [ f ˜ f ″ ˜ + ( f ′ ˜ ) 2 − A 3 M f ″ ˜ ] = 0 .

(13) A 5 ( 1 + R d ˜ ) ( 1 + 2 γ η ) θ ″ + 2 γ θ ′ ( 1 + ( θ w − 1 ) θ ) 2 + { 3 ( θ ′ ) 2 ( θ w − 1 ) ( 1 + 2 γ η ) + 2 γ θ ′ ( 1 + ( θ w − 1 ) θ ) + θ ″ ( 1 + 2 γ η ) ( 1 + ( θ w − 1 ) θ ) } + Pr A 4 f ˜ θ ′ + ( U f ′ ˜ + W θ ) = 0 ,

Parallel BCs for ODEs are given as follows:

(14) η = 0 : f ˜ ( η ) = 0 , f ′ ˜ ( η ) = 1 , θ = 1 , η → ∞ : f ′ ˜ ( η ) → 0 , θ ( η ) → 0 . .

Physical quantities are

(15) C f ˜ Re x 1 2 = 1 A 1 f ″ ˜ [ β ∗ + ( 1 − β ∗ ) ( We f ″ ˜ ) 2 ] n − 1 2 ,

(16) Nu x Re x − 1 2 = − { A 5 + R d ˜ [ 1 + ( θ w − 1 ) θ ] 3 } θ ′ ,

where constants A 1 , A 2 , A 3 , A 4 & A 5 are given by

(17) A 1 = 1 ( 1 − φ 1 ) 2.5 ( 1 − φ 2 ) 2.5 ( 1 − φ 3 ) 2.5 ( 1 − φ 4 ) 2.5 , A 3 = σ tethnf k f , A 5 = k tethnf k f ˜ ,

(18) A 2 = ( 1 − φ 1 ) ( 1 − φ 2 ) ( 1 − φ 3 ) ( 1 − φ 4 ) + φ 4 ρ 4 φ f + φ 3 ρ 3 φ f + φ 2 ρ 2 φ f + φ 1 ρ 1 φ f ,

(19) A 4 = ( 1 − φ 1 ) ( 1 − φ 2 ) ( 1 − φ 3 ) ( 1 − φ 4 ) + ( ρ C p ) S 4 ( ρ C p ) f ˜ φ 4 + ( ρ C p ) S 3 ( ρ C p ) f ˜ φ 3 + ( ρ C p ) S 2 ( ρ C p ) f ˜ φ 2 + ( ρ C p ) S 1 ( ρ C p ) f ˜ φ 1 .

4 Numerical scheme

The bvp4c method is a numerical method used to solve ODEs. This method has high accuracy and stability and is particularly useful for solving stiff ODEs where conventional methods may not perform well. It is also more efficient than traditional shooting methods as it eliminates the need for trial-and-error procedures to estimate the initial conditions. For the current issue, the resilience is 10 − 6 and the figuring time frame is [0, 7] instead of [0, ∞]. The complete analytical and Matlab procedure of the recommended mathematical framework is given below. Furthermore, Matlab procedure is depicted by Figure 2.

f ˜ = y 1 , f ′ ˜ = y 2 , f ″ ˜ = y 3 ,

A 1 A 2 [ A 3 M y 2 − y 1 y 3 − y 2 2 ] − 2 γ y 3 A 1 A 2 1 + 1 − n 2 ( We y 3 ) n ( 1 + 2 γ η ) ( 1 + n ( We ( y 3 ) 2 ) ) ( β ∗ + ( 1 − β ⁎ ) ( We ( y 3 ) 2 ) ) n − 3 2 ,

θ = y 4 , θ ′ = y 5 ,

A 5 ( ( 1 + R d ˜ ) ( 1 + 2 γ η ) y 5 2 + 2 γ y 5 ) + R d ˜ ( 1 + ( θ w − 1 ) y 4 ) + P r A 4 y 1 y 5 + ( U y 3 + W y 4 ) = 0 .

Figure 2

Matlab code for bvp4c scheme.

Involved BCs are

η = 0 : y 1 ( η ) = 0 , y 2 ( η ) = 1 , y 4 = 1 , y 2 ( η ) → 0 , y 4 ( η ) → 0 .

5 Validity

Table 2 shows the similarity of literature and current study. Table 2 shows smooth agreement with old literature [30].

Table 2

Agreement shown of current study with old literature

Γ	Ф	Sarwar and Hussain [30]	Present
0.10	0.01	0.9399680	0.9399775
0.12	0.01	0.9247940	0.9247865
0.14	0.01	0.9113110	0.9113542
0.10	0.05	1.3295520	1.3289593

6 Results and discussion

The purpose of this section is to investigate the inclined magnetized radiative Carreau tetra hybrid nanofluid flow in artery. This study describes the flow behaviour through stenosis in an artery with a non-uniform heat sink/source. The data show that the temperature and velocity profiles are affected by the magnetic field, radiative heat transfer, and the nanofluid properties.

The findings also indicate that the inclusion of these physical phenomena results in improved thermal management and increased blood flow through the stenosed artery. In this section, solution of ODEs, which are influenced by physical parameters, is obtained. Numerical domain of physical parameters is taken as

R d ∈ [ 0.5 , 2 , 19 ] , Pr ∈ [ 19, 22 ] , θ ∈ [ 0.1 , 1.5 ] , M ∈ [ 0.5 , 2 ] , We ∈ [ 0.1 , 4 ] , n ∈ [ 0.1 , 1.5 , ] , U ∈ [ 0.5 , 2 ] , W ∈ [ 0.5 , 2 ] , E c ∈ [ 0.1 , 2.1 ] , and γ ∈ [ 0.1 , 2 ]

6.1 Numerical consequences of influential physical parameters attached with M, We, n, and γ on the velocity field of blood

Geometry of blood flow velocity with attached parameter of Weissenberg number We is described by Figure 3a and b for both inclined/non inclined magnetic field by categorizing bi-hybrid/tri-hybrid/tetra-hybrid nanofluid. Velocity of blood fluid is boosted throughout the motion for greater numerical value of “We.” This parameter interlinks between viscous forces and flexible forces and time relaxation constant as well. Viscous forces are weaker than elastic forces. Due to all these factors, motion of blood is more prominent in dihybrid nanoparticles in comparison to trihybrid nanomaterials. Increment in We lowers blood density, hence its advantages can be seen in body, which helps in preventing of heart stokes/attacks. Figure 4 shows the Control of ω on velocity of blood in inclined/non inclined magnetic variable by categorizing bi-hybrid/tri-hybrid/tetra-hybrid nanofluid. Increase in numerical value inclined magnetic parameter causes increase in Lorentz force and this Lorentz force causes reduction in the flow strength and hence, velocity of blood slows down. The velocity of blood in an artery decreases as the power law index (n) increases because a higher value of n indicates a shear thickening and due to this thickening, blood velocity decreases. Moreover, due to greater values of said parameter, thickness of the blood rises. This fact decreases the diameter of a vessel and causes atherosclerosis, a stroke or heart attack. Physically, it can be seen as the dominancy of di-hybrid nanoparticles over tetra-hybrid nanoparticles. Pictorial representation of this fact is displayed in Figure 5a and b. Figure 6a and b demonstrates the impact of magneto variable parameter M on velocity profile. When magnetic field is imposed on nanoparticles fluid, nanoparticles are electrically charged and hence Lorentz force is produced. Lorentz force opposes the fluid flow and as a result velocity of blood becomes slow. One main reason of reduction in velocity of blood is that when Lorentz force is produced, the red cells in blood stick with plasma and this fact leads to slow velocity. The purpose of including nanoparticles in blood is to check the density of fluid. It is known by existing literature that suspension of agglomerative nano molecules in usual fluids increases velocity, and temperature boundary layer thickness seems more dominant in tetra hybrid nano molecules as compared to di-hybrid and tri-hybrid nano molecules.

Figure 3

(a) and (b) Effect of We on velocity of blood in inclined/non inclined magnetic variable.

$Figure 4 Effect of ω \omega on velocity of blood in inclined/non inclined magnetic variable.$

Figure 4

Effect of ω on velocity of blood in inclined/non inclined magnetic variable.

Figure 5

(a) and (b) Effect of n on velocity of blood in inclined/non inclined magnetic variable.

Figure 6

(a) and (b) Effect of M on velocity of blood in inclined/non inclined magnetic variable.

Linkage between curvature parameter γ and the f′(η) is seen in Figure 7a and b. Growth in curvature parameter γ, the f′(η) file of blood increases. Blood flow and cylinder surface is reduced for greater value of curvature parameter γ, hence velocity increases. For further information, it is obvious that blood travels through capillaries, and sudden cardiac death risk is decreased. Momentum boundary layer increases in growing direction and it is seen more dominantly in tetra agglomerative nanomolecular as compared to ternary hybrid nanoparticles.

$Figure 7 (a) and (b) Effect of γ \gamma on velocity of blood in inclined/non inclined magnetic variable.$

Figure 7

(a) and (b) Effect of γ on velocity of blood in inclined/non inclined magnetic variable.

6.2 Numerical consequences of influential physical dimensionless parameters W, M, E _c, U, on temperature field of blood.

Figures 8a, b and 9a, b are designed to present the effect of parameter of heating source U and heat generation “W” on temperature profile of blood for both inclined/non inclined magnetic field by categorizing bi-hybrid/tri-hybrid/tetra-hybrid nanofluid. Basic reaction on which parameters work is endothermic/exothermic reaction and best example of that absorption is photosynthesis. In photosynthesis, plant absorb heat for food preparation. Heat generation occurs in nuclear fission reaction. Figure 10a and b describes the control of Pr on temperature of blood in inclined/non inclined magnetic variable by categorizing bi-hybrid/trihybrid/tetra-hybrid nanofluid. Prandtl number describes the linkage between boundary layer thickness of temperature and boundary layer thickness of velocity. Greater value of Pr gives lower temperature because when value of Pr is high, rate of heat diffusion becomes slow as compared to rate of momentum diffusion. Figures 11, 12a and b show the pictorial representation of ω and M on temperature of blood in inclined/non inclined magnetic variable. Both ω and M create Lorentz force by increasing their numerical values. Due to Lorentz force by imposing magnetic field velocity of blood is decreases and temperature increases. Lower velocity gives higher temperature. Figure 13a and b shows the control of γ on temperature of blood in inclined/non inclined magnetic variable.

Figure 8

(a) and (b) Effect of W on temperature of blood in inclined/non inclined magnetic variable.

Figure 9

(a) and (b): Effect of U on temperature of blood in inclined/non inclined magnetic variable.

$Figure 10 (a) and (b) Effect of Pr \Pr on temperature of blood in inclined/non inclined magnetic variable.$

Figure 10

(a) and (b) Effect of Pr on temperature of blood in inclined/non inclined magnetic variable.

$Figure 11 Effect of ω \omega on temperature of blood in inclined/non inclined magnetic variable.$

Figure 11

Effect of ω on temperature of blood in inclined/non inclined magnetic variable.

Figure 12

(a) and (b) Effect of M on temperature of blood in inclined/non inclined magnetic variable.

$Figure 13 (a) and (b) Effect of γ \gamma on temperature of blood in inclined/non inclined magnetic variable.$

Figure 13

(a) and (b) Effect of γ on temperature of blood in inclined/non inclined magnetic variable.

For increasing values of curvature parameter there is rotation in nanoparticles of blood and linear velocity becomes slow, and hence temperature is increased. Control of E c on temperature of blood in inclined/non inclined magnetic variable is described by Figure 14a and b. Numerical increase in E c gives higher temperature. Amplification is proportional to frictional heating production. Physically, E c represents difference in kinetic energy and the change in enthalpy that exists between both the wall and the fluid. Greater values of E _c turns kinetic energy into stored energy and fluid goes in viscous strains.

$Figure 14 (a) and (b) Effect of E c {E}_{\text{c}} on temperature of blood in inclined/non inclined magnetic variable.$

Figure 14

(a) and (b) Effect of E c on temperature of blood in inclined/non inclined magnetic variable.

Pictorial representation of physical quantities is given by Figures 15a–c and 16a, b. Control of R d , U , W on local Nusselt number by categorizing bi-hybrid/tri-hybrid/tetra-hybrid nanofluid is presented by Figure 15a–c. Greater value of Rd, U, and W, the local Nusselt number increasing for bi-hybrid/tri-hybrid/tetra-hybrid nanofluid. It is shown there is increment rate of heat transport in bi-hybrid/tri-hybrid/tetra-hybrid nanofluid as compared to bi-hybrid/tri-hybrid nano fluid. This shows that tetra-hybrid nanofluid produce rapid heat transport. Furthermore, Figure 16a and b shows the control of n , We on skin fraction by categorizing bi-hybrid/tri-hybrid/tetra-hybrid nanofluid.

$Figure 15 Effect of R d , U , W {R}_{\text{d}},U,W on local Nusselt number by categorizing bi-hybrid/tri-hybrid/tetra-hybrid nanofluid.$

Figure 15

Effect of R d , U , W on local Nusselt number by categorizing bi-hybrid/tri-hybrid/tetra-hybrid nanofluid.

$Figure 16 Effect of n , We n,\text{We} on skin fraction categorizing of bi-hybrid/tri-hybrid/tetra-hybrid nanofluid.$

Figure 16

Effect of n , We on skin fraction categorizing of bi-hybrid/tri-hybrid/tetra-hybrid nanofluid.

7 Conclusion

This study represents a significant contribution to the field of nanofluid research. By combining various theoretical models and physical properties, this study offers a comprehensive understanding of the heat transfer behaviour of tetra hybrid nanofluids in a stenosed artery with a nonuniform heat sink/source. The results show that the use of inclined magnetized radiative nanofluids can significantly improve the thermal performance and stability of the system. This research can have practical applications in various industries, including biomedical engineering and energy systems. Point wise conclusion is given as follows:

Velocity of blood fluid is decreased throughout the motion for greater numerical value of “We” and “n” for bi-hybrid/tri-hybrid/tetra-hybrid nanofluid for orthogonal and non-orthogonal magnetic field.
When Lorentz force is produced, the red cells in blood stick with plasma and this fact gives slow velocity.
Numerical rise in E _c gives higher temperature due to frictional heating production.
Rate of heat transport in tetra hybrid nanofluid is quick as compared to bi-hybrid and ternary hybrid nanofluid.

basma.souayeh@gmail.com

Acknowledgments

This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (GrantA338).

Funding information: This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (GrantA338).
Author contributions: Wael Al-Kouz – original draft preparation and review; Wahib Owhaib – investigation and visualization; Basma Souayeh – data curation and formal analysis; Zulqurnain Sabir – numerical treatment and methodology and editing. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

References

[1] Bilal M, Ullah I, Alam MM, Weera W, Galal AM. Numerical simulations through PCM for the dynamics of thermal enhancement in ternary MHD hybrid nanofluid flow over plane sheet, cone, and wedge. Symmetry. 2022;14(11):2419.10.3390/sym14112419Search in Google Scholar

[2] Ayub A, Shah SZH, Asjad MI, Almusawa MY, Eldin SM, El-Rahman MA. Biological interactions between micro swimmers and cross fluid with inclined MHD effects in a complex wavy canal. Sci Rep. 2023;13(1):4712.10.1038/s41598-023-31853-9Search in Google Scholar PubMed PubMed Central

[3] Shamshuddin MD, Akkurt N, Saeed A, Kumam P. Radiation mechanism on dissipative ternary hybrid nanoliquid flow through rotating disk encountered by Hall currents: HAM solution. Alex Eng J. 2023;65:543–59.10.1016/j.aej.2022.10.021Search in Google Scholar

[4] Shoaib M, Raja MAZ, Sabir MT, Islam S, Shah Z, Kumam P, et al. Numerical investigation for rotating flow of MHD hybrid nanofluid with thermal radiation over a stretching sheet. Sci Rep. 2020;10(1):18533.10.1038/s41598-020-75254-8Search in Google Scholar PubMed PubMed Central

[5] Chamkha AJ, Yassen R, Ismael MA, Rashad AM, Salah T, Nabwey HA. MHD free convection of localized heat source/sink in hybrid nanofluid-filled square cavity. J Nanofluids. 2020;9(1):1–12.10.1166/jon.2020.1726Search in Google Scholar

[6] Zainal NA, Nazar R, Naganthran K, Pop I. MHD flow and heat transfer of hybrid nanofluid over a permeable moving surface in the presence of thermal radiation. Int J Numer Methods Heat Fluid Flow. 2021;31(3):858–79.10.1108/HFF-03-2020-0126Search in Google Scholar

[7] Oke AS, Fatunmbi EO, Animasaun IL, Juma BA. Exploration of ternary-hybrid nanofluid experiencing Coriolis and Lorentz forces: case of three-dimensional flow of water conveying carbon nanotubes, graphene, and alumina nanoparticles. Waves Random Complex Media. 2022;1–20.10.1080/17455030.2022.2123114Search in Google Scholar

[8] Mahmood Z, Iqbal Z, Alyami MA, Alqahtani B, Yassen MF, Khan U. Influence of suction and heat source on MHD stagnation point flow of ternary hybrid nanofluid over convectively heated stretching/shrinking cylinder. Adv Mech Eng. 2022;14(9):16878132221126278.10.1177/16878132221126278Search in Google Scholar

[9] Ayub A, Sabir Z, Shah SZH, Mahmoud SR, Algarni A, Sadat R, et al. Aspects of infinite shear rate viscosity and heat transport of magnetized Carreau nanofluid. Eur Phys J Plus. 2022;137(2):1–17.10.1140/epjp/s13360-022-02410-6Search in Google Scholar

[10] Wang F, Sajid T, Ayub A, Sabir Z, Bhatti S, Shah NA, et al. Melting and entropy generation of infinite shear rate viscosity Carreau model over Riga plate with erratic thickness: a numerical Keller Box approach. Waves Random Complex Media. 2022;1–25.10.1080/17455030.2022.2063991Search in Google Scholar

[11] El Din SM, Darvesh A, Ayub A, Sajid T, Jamshed W, Eid MR, et al. Quadratic multiple regression model and spectral relaxation approach for Carreau nanofluid inclined magnetized dipole along stagnation point geometry. Sci Rep. 2022;12(1):1–18.10.1038/s41598-022-22308-8Search in Google Scholar PubMed PubMed Central

[12] Ayub A, Sajid T, Jamshed W, Zamora WRM, More LAV, Talledo LMG, et al. Activation energy and inclination magnetic dipole influences on Carreau nanofluid flowing via cylindrical channel with an infinite shearing rate. Appl Sci. 2022;12(17):8779.10.3390/app12178779Search in Google Scholar

[13] Shah SZH, Fathurrochman I, Ayub A, Altamirano GC, Rizwan A, Núñez RA, et al. Inclined magnetized and energy transportation aspect of infinite shear rate viscosity model of Carreau nanofluid with multiple features over wedge geometry. Heat Transf. 2021:1–27. 10.1002/htj.22367.Search in Google Scholar

[14] Bibi A, Xu H. Peristaltic channel flow and heat transfer of Carreau magneto hybrid nanofluid in the presence of homogeneous/heterogeneous reactions. Sci Rep. 2020;10(1):11499.10.1038/s41598-020-68409-0Search in Google Scholar PubMed PubMed Central

[15] Chalavadi S, Madde P, Naramgari S, Gangadhar Poojari A. Effect of variable heat generation/absorption on magnetohydrodynamic Sakiadis flow of Casson/Carreau hybrid nanoliquid due to a persistently moving needle. Heat Transf. 2021;50(8):8354–77.10.1002/htj.22280Search in Google Scholar

[16] Kumar KG, Reddy MG, Aldalbahi A, Rahimi-Gorji M, Rahaman M. Application of different hybrid nanofluids in convective heat transport of Carreau fluid. Chaos, Solitons Fractals. 2020;141:110350.10.1016/j.chaos.2020.110350Search in Google Scholar

[17] Rana S, Nawaz M, Alaoui MK. Three dimensional heat transfer in the Carreau-Yasuda hybrid nanofluid with Hall and ion slip effects. Phys Scr. 2021;96(12):125215.10.1088/1402-4896/ac2379Search in Google Scholar

[18] Alqudah M, Shah SZH, Riaz MB, Khalifa HAEW, Akgül A, Ayub A. Neural network architecture to optimize the nanoscale thermal transport of ternary magnetized Carreau nanofluid over 3D wedge. Results Phys. 2024;107616.10.1016/j.rinp.2024.107616Search in Google Scholar

[19] Ayub A, Shah SZH, Sabir Z, Rashid A, Ali MR. A novel study of the Cross nanofluid with the effects of inclined magnetic field in fuzzy environment. Int J Thermofluids. 2024;100636.10.1016/j.ijft.2024.100636Search in Google Scholar

[20] Al-Kouz W, Owhaib W, Ayub A, Souayeh B, Hader M, Homod RZ, et al. Thermal proficiency of magnetized and radiative cross-ternary hybrid nanofluid flow induced by a vertical cylinder. Open Phys. 2024;22(1):20230197.10.1515/phys-2023-0197Search in Google Scholar

[21] Shahzad H, Wang X, Ghaffari A, Iqbal K, Hafeez MB, Krawczuk M, et al. Fluid structure interaction study of non-Newtonian Casson fluid in a bifurcated channel having stenosis with elastic walls. Sci Rep. 2022;12(1):12219.10.1038/s41598-022-16213-3Search in Google Scholar PubMed PubMed Central

[22] Nowak M, Divo E, Adamczyk WP. Fluid–structure interaction methods for the progressive anatomical and artificial aortic valve stenosis. Int J Mech Sci. 2022;227:107410.10.1016/j.ijmecsci.2022.107410Search in Google Scholar

[23] Mekheimer KS, Abo-Elkhair RE, Abdelsalam SI, Ali KK, Moawad AMA. Biomedical simulations of nanoparticles drug delivery to blood hemodynamics in diseased organs: synovitis problem. Int Commun Heat Mass Transf. 2022;130:105756.10.1016/j.icheatmasstransfer.2021.105756Search in Google Scholar

[24] Haowei MA, Hussein UAR, Al-Qaim ZH, Altalbawy FM, Fadhil AA, Al-Taee MM, et al. Employing Sisko non-Newtonian model to investigate the thermal behavior of blood flow in a stenosis artery: Effects of heat flux, different severities of stenosis, and different radii of the artery. Alex Eng J. 2023;68:291–300.10.1016/j.aej.2022.12.048Search in Google Scholar

[25] Gill H, Fernandes J, Chehab O, Prendergast B, Redwood S, Chiribiri A, et al. Evaluation of aortic stenosis: from Bernoulli and Doppler to Navier-Stokes. Trends Cardiovasc Med. 2023;33(1):32–43.10.1016/j.tcm.2021.12.003Search in Google Scholar PubMed

[26] Jena S, Mishra SR, Dash GC. Chemical reaction effect on MHD Jeffery fluid flow over a stretching sheet through porous media with heat generation/absorption. Int J Appl Comput Mathematics. 2017;3:1225–38.10.1007/s40819-016-0173-8Search in Google Scholar

[27] Mohanty B, Mishra SR, Pattanayak HB. Numerical investigation on heat and mass transfer effect of micropolar fluid over a stretching sheet through porous media. Alex Eng J. 2015;54(2):223–32.10.1016/j.aej.2015.03.010Search in Google Scholar

[28] Thumma T, Mishra SR. Effect of viscous dissipation and Joule heating on magnetohydrodynamic Jeffery nanofluid flow with and without multi slip boundary conditions. J Nanofluids. 2018;7(3):516–26.10.1166/jon.2018.1469Search in Google Scholar

[29] Pattnaik PK, Mishra SR, Barik AK, Mishra AK. Influence of chemical reaction on magnetohydrodynamic flow over an exponential stretching sheet: a numerical study. Int J Fluid Mech Res. 2020;47(3):217–28.10.1615/InterJFluidMechRes.2020028543Search in Google Scholar

[30] Sarwar L, Hussain A. Flow characteristics of Au-blood nanofluid in stenotic artery. Int Commun Heat Mass Transf. 2021;127:105486.10.1016/j.icheatmasstransfer.2021.105486Search in Google Scholar

Received: 2024-02-03

Revised: 2024-04-24

Accepted: 2024-05-13

Published Online: 2024-06-03

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2024-0040

Keywords for this article

Carreau fluid; tetra hybrid; Joule heating; viscous dissipation; Tiwari and Das model

Creative Commons

BY 4.0

Inclined magnetized infinite shear rate viscosity of non-Newtonian tetra hybrid nanofluid in stenosed artery with non-uniform heat sink/source

Article

Abstract

1 Introduction

1.1 Applications of heat transfer in blood flow

1.2 Motivation

1.3 Novelty

2 Mathematical formula

3 Solution procedure

4 Numerical scheme

5 Validity

6 Results and discussion

6.1 Numerical consequences of influential physical parameters attached with M, We, n, and γ on the velocity field of blood

6.2 Numerical consequences of influential physical dimensionless parameters W, M, E c, U, on temperature field of blood.

7 Conclusion

Acknowledgments

References

Articles in the same Issue

Articles in the same Issue

Articles in the same Issue

6.2 Numerical consequences of influential physical dimensionless parameters W, M, E _c, U, on temperature field of blood.